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Qqq, then the spacecraft will be in eclipse when the Sun is in a position to shine on the face. In this case F will be reduced to 0. For conditions in between these two extremes, F will be between 0 and its non eclipse value. Specific values will need to be evaluated numerically using Eqs. (1 l-7c) and ( 11 - 8 ). The heat input from both reflected and emitted radiation from the Earth increases the effective value of F. It is significantly more complex to compute than the effect of eclipses because of the extended size of the disk of the Earth and the variability in the intensity of reflected radiation. However, reasonable upper limits for radiation from the Earth are: • 475 W/m 2 for reflected solar radiation (albedo) • 230 W/m 2 for emitted IR radiation (thermal radiation) The radiation balance for the Earth at different times of the year is given in Table 11-2 in Sec. 11.5. For further details, see for example, Gilmore and Bello [1994].
11.3 Transits and Occultations Transit and occultation refer to the relative orientations of a planet, a satellite, and an observer, as illustrated in Fig. 11-8. Transit occurs when a satellite passes in front of the disk of a planet as seen by the observer. Occultation occurs when the satellite passes behind the disk of the planet, and is occulted, or blocked, from the observer’s view. Whether transits or Occultations occur is only a function of the angular separa tion between the bodies involved, so that all of the results of the relative motion discussion in Chap. 10 can be applied directly.
Observer
Fig. 11-8. Transits and Occultations. Transit occurs when the target moves in front of an ex tended object such as the Earth or Sun and occultation occurs when it moves behind it.
11.3
Transits and Occultations
559
Occultation, when applied to satellites, is simply a formal terminology for the fact that the satellites are not visible to each other. This is a topic which has already been fully discussed in Chap. 10. Whenever the angular separation between a central body and another satellite, as seen by either satellite, becomes less than the angular diameter of the central body, and they are on opposite sides of the central body, an occultation will occur and the two satellites will be invisible to each other. The satellites will be able to see each other if and only if they can both see at least one point in common on the surface of the central body. If this does not occur, i.e., the two horizon circles of the two spacecraft do not overlap, then each will be occulted with respect to the other. Transit means that the lower satellite will be in front of the disk of the central body as seen by the observer. Depending on the conditions under which observations are being made, this may make the satellite more or less visible. Figure 11-9 shows a tran sit of one of the moons of Jupiter as seen from the Earth, as well as the shadow cone of the moon falling on Jupiter’s cloud tops.
Fig. 11-9. Transit of one of Jupiter’s Moons as Seen from Earth. Note the shadow of the moon on Jupiter’s surface. For an observer on Jupiter under the shadow, this would be a total eclipse of the Sun. (Photo courtesy of JP L .)
Of course, transit cannot occur for co-altitude satellites. For non-co-altitude satel lites, transit, or occultation will occur whenever the angular separation between the lower satellite and the center of the Earth is less than the angular radius of the Earth. We can easily compute the conditions for transit and occultation in terms of the geom etry as seen from either the higher satellite, the lower satellite, or the center of the Earth. Specifically, we define p h as the angular radius of the Earth as seen from the higher satellite and p/ as the angular radius of the Earth as seen from the lower satellite. We then define A,, the Earth central angle for transits, as:
560
Viewing and Lighting Conditions
K ^ P i-P h and Xa, the Earth central angle for Occultations, as: X0 = 1 8 0 d e g - p /- p ^
113
( n - io ) (11-11)
The necessary and sufficient conditions for transit to occur can then be written in any of three ways: T}h < ph (as seen from higher spacecraft) ( 11 - 12 a) rjl > 180 deg - p (
(as seen from the lower spacecraft)
( 11 - 12 b)
X < Xt
(as seen from Earth center)
(1 1- 12c)
Here rjh is the angle from nadir to the lower satellite as seen from the higher satellite, and ?7/ is the angle from nadir to the higher satellite as seen from the lower satellite* Similarly, the necessary and sufficient conditions for occultation can be written from the same points of view as:
r)h < Ph a°d D >D0
(as seen from higher spacecraft)
(11-13a)
r\{< pi
(as seen from the lower spacecraft)
(11 - 13b)
X > X0
(as seen from Earth center)
( 1 1-13c)
where D() is the distance from the higher satellite to the horizon, = H cos p!v where H is the distance from the higher spacecraft to the center of the Earth. If neither the transit nor occultation condition is satisfied, then the two spacecraft can see each other against the background of space. These results are illustrated for two satellites at altitudes of 1 ,0 0 0 km and 5,000 km in Fig. 11-10. The same geometry is shown on the Earth-centered celestial sphere in Fig. 11-11 and on the spacecraft-centered celestial spheres for both the low orbiting satellite and the high orbiting satellite in Figure 11-12. Note that for the lower satellite, the higher one can appear anywhere in the spacecraft sky. From the higher satellite, the lower satellite is confined by a region about nadir of radius P /0 as given by sin PlQ = f y t Rh (H-14) where R( is the radius of the lower orbit and Rh is the radius of the upper orbit.
Fig. 11-10.
Transit and Occultation Conditions. For drawing convenience, the observer is assumed to be in the orbit plane, but this is not necessary for the computations.
11.3
Transits and Occultations
Fig. 11-11.
561
Conditions for Transit and Occultation as Viewed on the Earth-centered Celes tial Sphere. Th e two orbits need not be coplanar.
When Upper Satellite is Here, Lower Satellite be in Transit
Limit of Possible Positions of Lower Safe •Transit Region for Lower Satellite (When in Front of Earth Disk)
Oocuttation Hegi Satellites in Occultation
(When Lower Satellite
(cannot see each other)
is Behind Earth Disk)
(A ) From Lower Satellite F ig . 11 -12.
(B ) From Upper Satellite
Conditions for Transit and Occultation as Viewed on the Spacecraft-Centered Celestial Sphere for Both the Higher and Lower Satellite. Coplanar orbits not assumed.
In the example shown in the figures, the satellites are at altitudes of 1,000 km and 5,000 km. Therefore, Ri = 7,378 km and Rh = 11,378 km. The angular radius of the Earth from the two satellites is pi = 59.82 deg and ph = 34.09 deg. From Eqs. (11-10) and (11-11), we have Xt = 25.73 deg and ^ = 86.09 deg. Wherever the Earth central angle between the two satellites is less than 25.73 deg, then the angle at the higher satellite from nadir to the lower satellite will be less than 34.09 deg, the angle at the lower satellite from nadir to the higher satellite will be greater than 180 - 59.82 = 120.18 deg, and there will be a transit of the lower satellite as seen from the higher one. Irrespective of their relative inclinations or orbit positions, the lower satellite will always be within p ;0 = 40.42 deg of nadir as seen from the higher satellite.
562
11.3
Viewing and Lighting Conditions
For circular orbits viewed obliquely, we can determine the transit and occultation conditions by making use of the computations in Sec. 6.3.4 for the shape of a circular orbit viewed from nearby. (See Fig. 11-13.) Specifically, if R is the radius of a circular orbit being viewed at distance D from the center of the Earth and at an angle / mea sured at the center of the Earth between the orbit plane and the observer, then the angular height a of the apparent orbit above the Earth’s center (on the far side of the orbit) and the angular distance (3 below the Earth’s center (for a point on the orbit nearest the observer) are given by: R s in l tana = ——------ D+R cos /
(11-15)
R sin I ta n ^
= ~n—
D - Rd c------------o s It
0
1' 1 6 )
Occultation
Fig. 11-13.
G eom etry for Calculating Tra n sit and Occultation Limits far Elliptical Orbits. See text for discussion and formulas.
If p is the angular radius of the Earth seen by the observer then the conditions for eclipse and occultation are p > tan a
(occultation will occur)
(11- 17a)
p> t& nb
(transit will occur)
(11- 17b)
Since the orbit has a larger angular extent on the near side, as illustrated in Fig. 6-23 in Sec. 6.3.4, if transit occurs during the orbit, then occultation must occur. However, occupations can occur without transit occurring on the near side, as illustrated in the figure. As we would expect, the situation is more complex for elliptical orbits. In this case, either transit or occultation can occur without the other. Nonetheless, we can put at least some broad limits on viewing conditions by applying the rules above to apogee.
11.4
Spacecraft Eclipses
563
Thus, if R in Eqs. (11-15) and (11-16) is replaced by the radius of apogee, Ra, then the conditions in Eq. (11-17) will continue to hold, but will be less constraining. In gen eral, we can always return to the conditions of Figs. 11-10 and 11-11 to determine whether transit or occultation occur at any specific time or in any specific orbit.
11.4 Spacecraft Eclipses Eclipses are the phenomena of transits and Occultations relative to the Sun. An eclipse o f the Sun is a transit of an object in front of the sun, blocking all or a significant part of the Sun’s light from the observer. An eclipse of any other object (e.g., an eclipse of the Moon) is an occultation of that object by another object relative to the Sun. Be cause the Sun is the largest object in the solar system, the shadows of all the planets and natural satellites are shaped as shown in Fig. 11-14. As discussed in Sec. 11.1, the umbra, or shadow cone, is the conical region opposite the direction of the Sun in which the disk of the Sun is completely blocked from view by the disk of the planet. Outside the umbra is the penumbra, where a portion of the disk of the Sun is blocked form view and, therefore, where the illumination on objects is reduced.
Fig. 11-14.
Definition of Satellite Eclipse Conditions. Sun, planet, and orbit are all in the plane of the paper.
Unfortunately, the terminology of eclipses depends on whether the observer is thought of as being on the object which is entering the shadow or viewing the event from elsewhere. If the observer enters the shadow, the Sun is partially or wholly blocked from view and the event is referred to as an eclipse o f the Sun or solar eclipse as shown in Fig. 11-15. A total solar eclipse, which is frequently shortened to just eclipse in spaceflight applications, occurs when the observer enters the umbra. If the observer is farther from the planet than the length of the shadow cone and enters the cone formed by the extension of the shadow cone through its apex, the observer will see an annular eclipse in which an annulus or ring of the bright solar disk is visible surrounding the disk of the planet. If the observer is within the penumbra, but outside the umbra, the observer will see a portion of the Sun’s disk blocked by the planet and a partial eclipse o f the Sun occurs. If the observer is viewing the event from somewhere other than on the satellite being eclipsed, the event is called an eclipse o f the satellite. A total eclipse of the sat ellite occurs when the entire satellite enters the umbra and a partial eclipse occurs when part of an extended satellite (such as the Moon) enters the umbra. If the satellite
564
Viewing and Lighting Conditions
(A ) Partial Solar Eclipse Fig. 11-15.
(B ) Total Solar Eclipse
H.4
(C ) Annular Solar Eclipse
Solar Eclipse Geometry. Shaded circle is the planet. Unshaded circle is the Sun.
enters the penumbra only, the event is called a penumbral eclipse. Thus, a total eclipse of the Moon to an observer on the Earth is a total solar eclipse to a lunar observer and a penumbral eclipse of a satellite to an observer on the Earth is a partial eclipse of the Sun to an observer on the satellite. To determine conditions under which eclipses occur, we can apply all of the geom etry of the previous section. From the point of view of the spacecraft, an eclipse occurs whenever all or part of the disk of the Sun is occulted by the central body. If T} is the angle from the center of the planet to the center of the Sun, a , is the angular radius of the planet, and ps is the angular radius of the Sun (= 0.25 deg at the distance of the Earth), thus the following conditions hold: V> Pp + Ps
(Full sunlight)
(11-18)
Pp + Ps> *l> IP p -P sI
(Partial eclipse)
(11-19)
lP p - P s l >? 7 an d pp > p s
(Total eclipse)
( 11-20 )
(Angular eclipse)
( 11- 21 )
Here all of the angles are measured from the spacecraft, i.e., on the spacecraftcentered celestial sphere. Alternatively, we can determine the three-dimensional conditions under which eclipses will occur by determining the size of the shadow cone being produced by the planet. We first determine the length, C, and angular radius, Pc, of the shadow cone for any of the planets or natural satellites (see Fig. 11-16). Let S be the distance from the planet to the Sun, Rp be the radius of the planet, Rs be the radius of the photosphere (i.e., the visible surface) of the Sun, and C be measured from the center of the planet to the apex of the shadow cone. Then, C = ----- ^----( * s -R P )
and
(11-22 )
Spacecraft Eclipses
11.4
565
iex
Sun
Fig. 11 -16.
Shadow Cone Geometry.
For the Earth, the size of the shadow cone for its mean distance from the Sun is C — 1.385 x IO6 km and p c - 0.264 deg. For the Moon, the mean size is C = 3.75 x IO5 km andp c = 0.266 deg. The length of the shadow cone for the Moon is just less than the semimajor axis of the Moon’s orbit of 3.84 x IO5 km. Therefore, eclipses of the Sun seen on the Earth are frequently annular eclipses, and when they are total eclipses they are seen over a very narrow band on the Earth because the maximum radius of the Moon’s shadow cone at the distance of the Earth’s surface is 135 km. The presence of an atmosphere on some planets and the non-negligible radius of the natural satellites may be taken into account by adjusting the radius of the planet, as will be discussed later. Initially, we will assume that there are no atmospheric effects and that we are concerned with eclipses seen by objects of negligible size, such a space craft. The conditions for the satellite to see a total eclipse of the Sun are exactly those for a transit of the satellite as viewed from the apex of the shadow cone. Similarly, the conditions for spacecraft to see a partial eclipse are nearly the same as those for occul tation of the spacecraft viewed from a point in the direction of the Sun equidistant from the planet as the apex of the shadow cone, To develop specific eclipse conditions, let Ds be the vector from the spacecraft to the Sun and let Dp be the vector from the spacecraft to the center of the planet. Three quantities of interest are the angular radius of the Sun, p s, the angular radius of the planet, pp, and the angular separation, 7], between the Sun and planet as viewed by the spacecraft, as shown in Fig. 11-17. These are given by: ps - arc sin (Rs/D s) (11-24) pp = arc sin (Rp /D p)
(11-25)
r) = arc cos (Ds ■D^)
( 11- 26 )
Sun
Spacecraft
Fig. 11-17.
Variables for Eclipse Geometry.
566
Viewing and Lighting Conditions
11.4
The necessary and sufficient eclipse conditions are 1. Partial Eclipse; Ds > S
and
pp + ps > 77 > \pp - ps \
(11-27)
2. Total Eclipse: S < D s < S+ C
and
Pp -P s > 'n
(11*28)
3. Annular Eclipse: S+ C < D s
and
Pp ~ P s >7l
(11-29)
These eclipses are illustrated in Fig. 11-15. The surface brightness of the Sun is nearly uniform over the surface of the disk. Therefore, the intensity, /, of the illumination on the spacecraft during a partial or annular eclipse is directly proportional to the area of the solar disk which can be seen by the spacecraft. These relations may be obtained directly from App. A as: 1, Partial Eclipse COSjC^ “ cos p<* COST]
7o - / =
Jt - c o s p 5 arccos 7C(1 - COS p s )
sin ps sin 77
cos ps - cos p cos 77 - cos Pp arccos
sin Pp sin rj
COST] - COS ps COS
- arccos
pp (1 1 -3 0 )
sin ps sin pp
2. Annular Eclipse 1 = 1,
^1 —cos pp 1 - COS
Ps
(1 1 -3 1 )
where / q is the fully illuminated intensity, and the inverse trigonometric functions in Eq. (11-30) are expressed in radians. The effect of a planetary atmosphere is difficult to compute analytically because the atmosphere absorbs light, scatters it in all directions, and refracts it into the shadow cone. Close to the surface of the Earth, only a small fraction of the incident light is transmitted entirely through the atmosphere. Thus, the major effects are an increase in the size of the shadow and a general lightening of the entire umbra due to scattering. The scattering becomes very apparent in some eclipses of the Moon, as seen from Earth when the Moon takes on a dull copper color due to refracted and scattered light. The darkness of individual lunar eclipses is noticeably affected by cloud patterns and weather conditions along the boundary of the Earth where light is being scattered into the umbra. The atmosphere of the Earth increases the size of the Earth’s shadow by about 2% at the distance of the Moon over the size the shadow would be expected to have from purely geometrical considerations. (See Seidelmann [1992].) Some ambiguity exists in such measurements because the boundary of the umbra is diffuse
11.5
Lighting Conditions Looking at Earth from Space
567
rather than sharp. If the entire 2% at the Moon’s distance is attributed to an increase in the effective linear radius of the Earth, this increase corresponds to about 90 km. In considering the general appearance of the solar system as seen by a spacecraft we may be interested in eclipses of the natural satellites as well as eclipses of space craft. In the case of natural satellites, the large diameter of the satellite will have a con siderable effect on the occurrence of eclipses. This may be taken into account easily by changing the effective linear diameter of the planet. Let R' be the radius of the planet, Rm be the radius of the natural satellite, and define the effective planetary radii Re\ = Rp + Rm and Re2 = Rp ~ Rm- Then, when the center of the satellite is within the shadow formed by an object of radius Re\, at least part of the real satellite is within the real shadow cone. Similarly, when the center of the satellite is within the shadow cone defined by an object of radius Re2, then all of the real satellite is within the shadow cone; this is referred to as a total eclipse of the satellite when seen form another loca tion. This procedure of using effective radii ignores a correction term comparable to the angular radius of the satellite at the distance of the Sun. We may use Eqs. (11-24) through (11-28) to determine the conditions on a satellite orbit such that eclipses will always occur or never occur. Let Dp be the perifocal distance, DA be the apofocal distance, i be the angle between the vector to the Sun and the satellite orbit plane, and C and pc be defined by Eqs. (11-22) and (11-23). We define y and S by:
tan / =
tan 5 =
q, sin; C -L fr cos i Da
C - D a cos
sin/
(11-32)
(11-33)
An eclipse will not occur in any orbit for which / > pc. An eclipse will always occur in an orbit for which 5 < p c.
11.5 Lighting Conditions Looking at Earth from Space A key issue for many space missions is not only eclipses and lighting on the space craft, but also the lighting on the Earth or other central body that the spacecraft is look ing at. This determines both viewing and thermal conditions since the radiation reflected from the Earth is a major source of thermal input for the spacecraft. Viewing conditions are important principally for observation satellites, but can also be used for both orbit and attitude sensing as described in Chaps. 2 and 3. The surface of the Earth is in thermodynamic equilibrium with its surroundings in that the total energy received from the Sun approximately equals the total energy which the Earth radiates into space.* If this were not the case, the Earth would either * They are not exactly equal because some energy goes into chemical bonds and some additional energy is supplied by radioactivity and by thermal cooling of the Earth’s interior. For the Earth, the heat flow from the interior is approximately 0.004% of the energy received from the Sun.
568
11.5
Viewing and Lighting Conditions
heat up or cool down until the radiated energy balanced the energy input. Table 11-2 shows the global average radiation for the Earth. For further discussion, see for exam ple, Gilmore and Bello [1994]. T A B L E 11-2.
Radiation B alance of the Earth -A tm o sph ere System . Data from Lyle, et al., [1971]. Global A ve ra g e D e c .Feb.
M a rMay
Ju n e Aug.
SepN ov.
Annual A ve ra g e
Incident Solar Radiation (W/m2)
356
349
342
349
349
Absorbed Solar Radiation (W/m2)
244
244
258
251
249
Reflected Solar Radiation (W/m2)
112
105
84
98
100
Planetary Albedo
0.31
0.30
0.25
0.28
0.29
230
237
230
Radiation
Emitted Infrared Radiation (W/m2)
223
230
The albedo of an object is the fraction of the incident energy that is reflected back into space. (The word is also used for the reflected radiation itself.) The Earth’s albedo is approximately 0.30, although it fluctuates considerably because clouds and ice reflect more light than the land or water surface. The spectral characteristics of the re flected radiation are approximately the same as the incident radiation. Thus, the Earth’s albedo is most intense in the visual region of the spectrum, i.e., the region to which the human eye is sensitive, from about 0.4 to 0.7 ,um wavelength. Sensors operating in the visible region are called albedo sensors, or visible light sensors. The principal advantage of this spectral region is that the intensity is greatest here. For attitude sensing, however, a significant disadvantage is the strong variation in albedo— from 0.05 for some soil- and vegetation-covered surfaces to over 0.80 for some types of snow and ice or clouds [Lyle, et al., 1971]. The incident energy which is not reflected from the Earth is transformed into heat and reradiated back into space with a blackbody spectrum characteristic of the tem perature. The Earth’s mean surface temperature of approximately 290 K corresponds to a peak intensity of emitted radiation of about 10 /an in the infrared region of the spectrum. This emitted , or thermal, radiation is typically used for attitude sensing because the intensity is much more uniformly distributed over the disk of the Earth than for visible light. Fig. 11-18 shows a photo of the Earth in the visible region of the spectrum. The lit horizon is the illuminated edge of the disk of the Earth. The fuzzy boundary where the edge fades away to darkness, corresponding to sunrise or sunset on the Earth, is called the term inator. The center of the picture is, of course, nadir and the intersection of a vector in the nadir direction with the surface of the Earth is called the subsatellite point, SSP. As we might expect, the most important element in determining lighting conditions on the surface of the Earth is direction to the Sun. This is most easily eval uated with respect to the subsolar point, which is location on the surface of the Earth for which the Sun is straight overhead. This can be determined at any time from the Earth's ephemeris or, except during eclipse, by observing the position of the Sun from the spacecraft. In Earth orbit, the position of the Sun in the spacecraft sky is the same as its position as seen from the subsatellite point or as seen from the center of the Earth
11.5
Lighting Conditions Looking at Earth from Space
569
to within less than 0.01 deg. Thus, the Sun angle, f3, between zenith and the Sun is the same for calculating lighting conditions whether measured from the subsatellite point on the Earth or from the spacecraft.
Fig. 11-18.
Photograph of the Nearly-Full Earth as Seen from Apollo 8. Th e Baja Peninsula in Mexico and most of Florida are visible. Note the bright spot in the ocean that is the specular reflection of the Sun on the ocean surface. (Photo courtesy of N A S A .)
Fig. 11-18 illustrates 3 key features of the Earth as seen from space in the visible region of the spectrum: • The terminator is a very poorly defined boundary, with its apparent location often depending on the amount of cloud cover in the region. (Terminator mod eling is discussed in Sec. 11.5.2) • Near the center of the photo is bright spot about 1 cm in diameter. This is the location on the Earth midway between the subsatellite point and the subsolar point. Here specular reflection will occur off the surface of the ocean with the angle of incidence equal to the angle of reflection. (The Sun angle at various locations on the Earth is discussed in Sec. 11,5.1) • The overall brightness of locations on the Earth varies dramatically, depending primarily on whether there’s cloud cover. The albedo can vary from about 0.05 for some types of soil and vegetation to over 0.80 for snow, ice, and clouds. As the spacecraft goes around the Earth, the Earth goes through a series of phases similar to the Moon’s phases. A schematic diagram of how these come about is shown
Viewing and Lighting Conditions
570
11.5
in Fig. 11-19 and the shape of the illuminated Earth in each of the marked locations is shown in Fig. 11-20. The effect of the lighting conditions on viewing at each location is described in Table 11-1 at the front of the chapter. As an example, a photo of the crescent Earth is shown in Fig. 11-21. As on the gibbous Earth seen in Fig. 11-18, the terminator is poorly defined as the light from the Sun fades out at sunrise and sunset. The cusps of the illuminated disk are the points where the terminator meets the lit horizon. The perpendicular bisector of a line joining the cusps will point toward the Sun with the Sun being in the direction of the illuminated Earth. While the light level is low near the terminator, the shadows are long and the contrast is high. Consequently, more detail is typically visible along the terminator than when the Sun is at the zenith and the shadows have disappeared.
A = Full B = Gibbous C = Quarter D = Crescent E = Eclipse
Fig. 11-19.
As the Spacecraft Goes Around the Orbit, the Lighting on the Earth Changes Depending on the Relationship to the Sun.
Full
G ib bou s
Q u a rte r
C rescent
Eclipse
A
B
C
D
E
Fig. 11-20.
Approximate Appearance of the Earth at Each of the Locations Marked Fig. 11-19.
The sequence of events on any given orbit can most easily be seen in Fig. 11-22.
Because both the terminator and the orbit ground trace are approximately great circles, they will intersect in 2 locations, except when the ground trace lies along the termi nator. If the orbit gets close enough to the subsolar point, the Sun will be high in the spacecraft sky and the Earth beneath will be fully illuminated. At some point the terminator will just become visible along the edge of the disk of the Earth and the Earth
11.5
Fig. 11 -21.
Lighting Conditions Looking at Earth from Space
571
photograph of a Crescent Earth as Seen from Apollo 11. Note that the terminator itself is poorly defined. However, near the terminator objects stand out on the surface because of the long shadows. A portion of East Africa is visible near the terminator. (Photo courtesy of N A S A .)
will take on a gibbous phase. The Earth will become continuously less illuminated until the spacecraft crosses the terminator which will bisect the visible disk at the quarter Earth.* (The width of the band on the Earth representing the gibbous or cres cent phases is just the maximum Earth central angle, A.) The Earth now enters the cres cent phase and continues to become less illuminated. At the same time, the Sun gets closer to the horizon until the spacecraft just enters eclipse with the Sun on the horizon and a very slim crescent Earth visible as sunlight is refracted through the atmosphere. Both the spacecraft and the Earth below will then be in darkness until the process re peats in reverse order on the other side of the orbit. The spacecraft orbit remains approximately fixed in inertial space as the Earth rotates underneath it. Thus, each successive low Earth orbit crosses a new region of the Earth. However, the Sun also remains approximately fixed in inertial space and the illumination cycle remains very nearly the same on successive orbits. Fig. 11-22 is best thought of as being in Earth-centered inertial coordinates such that the ground trace and illumination regions are fixed on the figure as the Earth rotates underneath. Thus, successive “first quarter” terminator crossings will occur with the same geometry at the same point in the orbit, but the land underneath will have changed.
* By analogy with the Moon’s phases, thefirst quarter Earth occurs when the spacecraft crosses the terminator from the dark side toward the bright side and the third quarter or last quarter Earth occurs when the spacecraft is going from the bright side toward the dark side.
572
Viewing and Lighting Conditions
11.5
11.5.1 Sun Angles on the Earth as Seen from Space The Sun angle, is the angle between zenith and the Sun. As discussed above, the Sun angle will be nearly the same at the subsatellite point as it is at the spacecraft. Thus, a very accurate measurement of the Sun angle directly beneath the spacecraft can be made by simply measuring the angle from the Sun to nadir and subtracting from 180 deg. Similarly, it can be done analytically from the ephemerides of the spacecraft and the Sun (which is also the Earth’s ephemeris) and taking the arc cosine of the dot product between the unit vectors to the spacecraft and to the Sun. From low-Earth orbit, the area on the Earth that can be seen is typically between 20 and 30 deg in radius. Observations are typically done near the subsatellite point due to foreshortening and absorption near the edge of the disk. Consequently, in most cases the Sun angle at the subsatellite point is a reasonable estimate of the Sun angle throughout the scene. The direction of the shadows on the surface of the Earth will be just 180 deg away from the direction to the subsolar point. As discussed in Sec. 9.1* azimuth angles about nadir are the same on the spacecraft-centered celestial sphere as they are projected onto the Earth’s surface. Consequently, this measurement can be made either on the surface of the Earth or as seen from the spacecraft, depending on which is more convenient for the problem at hand. A more accurate computation of the Sun angle and shadow direction can be done either on the surface of the Earth, as shown in Fig. 11-23A or on the spacecraftcentered celestial sphere as shown in Fig. 11-23B. On the Earth, the assumed known quantities are the locations of the subsatellite point, SSP, the subsolar point, S, and the
11.5
Lighting Conditions Looking at Earth from Space
(A ) On the Earth
Fig. 11-23.
573
(B ) On the Spacecraft-Centered Celestial Sphere
Geometry for Determining the Sun Angle, jBt , and Shadow Angle, 9'. See text for equations.
target, T, at which the Sun angle and shadow direction are to be determined. Given the coordinates of these points on the Earth, we can immediately determine the arc lengths between. (See App. A.) Thus, the known quantities become the Sun angle, /5, the angle from S to T, p T, and the Earth central angle from SSP to T, X. From the previous discussion, (5j is equal to the Sun angle at the target. From these quantities, we deter mine the angle, % at the target between S and SSP: cos/i~cosA cos/?r ^
sin^sinjSy.
(1 1-34)
The shadow direction on the Earth, 9, relative to the normal to the line to the subsatel lite point is then given by 0 = |9 O d e g - y |
(11-35)
This can be transformed into the shadow direction relative to horizontal as seen on the spacecraft-centered celestial sphere, 6', by using Eq. (9-25) in Sec. 9.1.3. (See Fig. 9-7.) Thus, tan 0 ' = tan#sin£
(11-36)
Here the elevation angle, e, of the spacecraft as seen from the target and the nadir angle, Tj, at the spacecraft from nadir to the target are given by Eqs. (9-3) and (9-4) as _ ^
sin p sin X 1 -s in p c o s A
£ = 90 deg - X - r \
(11-37) (11-38)
Viewing and Lighting Conditions
574
11.5
where p is the angular radius of the Earth a$ seen from the spacecraft as given in Eq. (9-2). While not needed for the shadow computation, it is also convenient to compute the azimuth on the Earth, a, of the target relative to the direction to the Sun: cos fiT - cos X cos S(Z
sinX sin/?
(11-39)
If the measurements are done on the spacecraft rather than the Earth, as shown in Fig. 11-23B, then the known quantities will be a, and t}. A s above, the Sun angle at nadir, /3, will be the same measured on the spacecraft or on the Earth. With the defini tions above, we first compute the elevation angle, e, and Earth central angle, X, from Eqs. (9-4) and (9-6) as: cos £ = sin 77/ sin p
(11-40)
A = 90 deg - r j - e
(11-41)
Next, the Sun angle at the target, /?T, and angle at the target, % from S to SSP, are com puted by: cos Pj- = cos X cos /? + sin X sin j3 cos a cos C0Sr=
^ |_4 2 )
-CO S X cos {$J
sinX sin/3r
(U-43)
Finally, as above, the shadow direction on the Earth, 6, and as seen by the spacecraft, 9', are: = |9 0 d e g - y | tan#' = tan 9 sine
(11_4 4) (11-45)
It is important to understand the meaning of the shadow direction in this context. Specifically, 9 \ is the angle measured from the spacecraft between the horizontal and the target's shadow. Thus, if we photograph the target from the spacecraft and draw a line through the target parallel to the horizon (or perpendicular to the direction to nadir), then 0 'will the angle on the photograph between the shadow and the horizontal line.
11.5.2 The Terminator As we have seen in Figs. 11-18 and 11-21, the terminator between the lit and dark parts of the Earth is very poorly defined. This is due to the three effects: (1) the gradual decrease in overall illumination with increasing Sun angle, (2) the extreme albedo variations between clouds and the planetary surface, and (3) the finite angular diameter of the Sun. We define the dark angle, as the angle at the center of a planet from the antisolar point to the point at which a ray from the upper limb of the Sun is tangent to the surface of the planet, as shown in Fig. 11-24. The dark angle, differs from 90 deg by three small correction terms: £ = 90deg + p E ~ P s
(11-46)
11.5
Fig. 11-24.
Lighting Conditions Looking at Earth from Space
575
Dark Angle of a Planet, £ Showing the Effects of Atmospheric Refraction and Finite Size of the Disk of the Sun.
where pE is the angular radius of the planet as seen from the Sun (i.e., the displacement of the center of the Sun as seen by an observer at the terminator, relative to an observer at the center of the planet), ps is the angular radius of the Sun as seen from the planet, and
(lI_47)
For the Moon, pM is negligible, ps has the same average value as for the Earth, and (7=0. Therefore, = 90deg- 0.267 deg ~ 89.73deg
(11-48)
The fraction,/y, of the area of a planet from which at least some portion of the Sun can be seen is given by: f s =0.5(1 + cos £) = 0.507
for the Earth
= 0.503
for the Moon
^j j _4 9 ^
At best, the expressions for £ are average values. The correction terms are normally dominated by local effects such as terrain (e.g., valleys where sunset is early and
576
11.5
Viewing and Lighting Conditions
mountains where it is late) and albedo variations depending on both the nature of the surface and the local weather. Fig. 11-25 shows the geometry for computing terminator visibility, taking into account the dark angle, as discussed above. As usual, f3 is the Sun angle measured from zenith to the Sun and will be the same at the satellite, the subsatellite point, and the center of the Earth, p is the angular radius of the Earth as seen from the satellite and Aq is the maximum Earth central angle. For convenience, we define the angle at the spacecraft from the Sun to nadir, ft" = 180 deg - (3 and the angle at the center of the Earth from the Sun to the terminator, = 180 deg - £ The resulting conditions for the various Earth phases as seen from the spacecraft, the surface of the Earth, and the center of the Earth are given in Table 11-3.
To Sun
Fig. 11-25.
Geometry for Computing Terminator Visibility.
T A B L E 11-3.
Conditions of Terminator Visibility. Formulas ignore a term equal to the parallax of the orbit as seen from the Sun of approximately 0.005 deg.
% Phase
Illumination
Condition (from Spacecraft)
Condition (from surface)
P < 90° - £" + p P
Full
100
Not Visible
jS" > 90° +£ - p
Gibbous
>50
Visible
90 °+%-p>
Crescent Eclipse
50
<50 0
Through Sub-Satellite Point Visible Not Visible
£>j3"> 90° - £ + p p" < 90° - £
?> P> r+ p
?> 0 >
P=ZU
i3 - r
90* -
0">z. Quarter
Condition (from Earth Center)
Terminator
180° - 4 “- Xq
90° + £ " - p >
> j3> 90°
+p
/8> 90° +%"-p
16 >
+/*q
11.5
Lighting Conditions Looking at Earth from Space
577
If the terminator is visible, then the great circle going through the Sun and nadir will be the perpendicular bisector of the terminator. The angular width on the Earth, Wbright> of the illuminated portion of the area visible to the satellite can be calculated directly as shown in Fig. 11-26: w brlghl =A o+ -T - /3
(11-50)
The angular width of the dark portion, W ^ /,, will be
Wdar k = 2 ^ - 'W Mgh,
( 11-51)
We can then transform these into the same parameters as seen from the spacecraft as: W,w
sinpsin(£"-/3)
„,
(11-52)
and W dark
=2p-
W bright’
(11-53)
Finally, let (j>be the azimuthal range of the terminator measured about the sub satellite point, i.e., the angle measured at the subsatellite point between the two cusps. Ignoring the small deviation of the terminator from being a great circle, we have cos ((p/2) - tan(£" - ($) / tan
( 11-54)
Because azimuth angles do not change in going from the surface of the Earth to the spacecraft-centered celestial sphere, (f>will also be the angle between the cusps mea sured about nadir as seen from the spacecraft.
578
11.6
Viewing and Lighting Conditions
11.6 Brightness of Distant Spacecraft and Planets We would like to determine general expressions for the brightness of planets or spacecraft as seen from any location in the solar system and at an arbitrary distance. However, the brightness depends not only on the relevant distances and angles, as shown in Fig. 11-27, but also on the surface characteristics and attitude of the object we are looking at. A flat, reflective surface that catches the sunlight can be many times brighter than the rest of the spacecraft. Thus, tumbling spacecraft will often appear to twinkle or vary in brightness as different surfaces are presented to the observer.
)
r
Sun
Fig. 11 -27.
T h e Magnitude of a Distant Planet o r Spacecraft. Th e magnitude depends on the distances to the Sun and the observer and the angle between them (known as the Phase Angle), as well as on the surface properties of the object being viewed.
While exact calculations are difficult, some approximate limits and relative rela tionships are available. Astronomers measure the brightness of any observed object in terms of magnitude, m, which is a logarithmic measure of its brightness or flux density, F, defined by m = m0 - 2.5 log F, where m0 is a scale constant.* Two objects of magnitude difference Am differ in intensity by a factor of ( %j100 )Am = 2.51 A"1 with smaller numbers corresponding to brighter objects; e.g., a star of magnitude -1 is 100 times brighter than a star of magnitude +4. The magnitude of an object depends on the spectral region over which the intensity is measured. In this section, we are con cerned only with visual magnitude, V, which has its peak sensitivity at about 0.55 fxm. On a very dark night, the stars visible to the naked eye from the surface of the Earth have visual magnitudes ranging from approximately 0 to +6. In addition to measuring brightness and magnitudes, it is frequently convenient to measure distances in astronomical units, AU, where 1 AU = 1.496 x IO8 km is the mean distance of the Earth from the Sun. Consequently, for spacecraft observers in the * Like many astronomical inventions, the magnitude scale came about before precise measure ments of star magnitudes were possible. Thus, a difference of 1 magnitude represented approximately the difference that could be distinguished by an observer. This was later made more precise by defining 5 magnitudes as exactly 100 times brighter or fainter and, therefore, 1 magnitude as 1000-2 - 2.51.
11.6
Brightness of Distant Spacecraft and Planets
579
general vicinity of the Earth or the Moon, the spacecraft can be thought of as 1 AU from the Sun, which makes logarithmic computations particularly easy. Since the computations are done primarily with logarithms, note that: log D (AU) = log D (km) - 8.17 = log D ( m ) - 11.17
(11-55)
To compute the visual magnitude of an object, let S be the distance of the object from the Sun in AU, r be the distance of the object from the observer in AU, £ be the phase angle at the object between the Sun and the observer, and P(£) be the ratio of the brightness of the object at phase angle £ to its brightness at zero phase (i.e., fully illu minated). Because the brightness falls off as S~2 and r~2, the visual magnitude as a function of £ ,r, and S is given by: V(rS4)= V(1,0) + 5 log (rS) - 2.5 log P(%) = V(1,0) + 5 log r + 5 log S - 2.5 log P(£)
(11-56)
where V(1,Q) is the visual magnitude at opposition relative to the observer* (i.e., £ = 0) and at a distance such that rS~ 1. If we continue to express S in AU but express r in kilometers, Eq. (11-56) becomes: V{r,S& = V(1,0) + 5 log r + 5 log S - 2.5 log F (f) - 40.85
(11-57)
Note that P(£) is independent of distance only as long as the observer is sufficiently far from the object that he is seeing nearly half of the object at any one time; for example, for a low-Earth orbit satellite, the illuminated fraction of the Earth seen by the satellite depends on both the phase and the satellite altitude as discussed previously inSec. 11.5. If the mean visual magnitude, Vq, at opposition'*' to the Earth is the known quantity, then: 7(1,0) = V0 - 5 log [D(D - 1)]
(11-58)
where D is the mean distance of the object from the Sun in AU. Values of Vq and V(1,0) for the Moon and the planets are tabulated in Table 11-4. For planets or other objects for which Vq or 7(1,0) is known, the major difficulty in determining the magnitude is determining the phase law, P(£). Unfortunately, there is no theoretical model available to predict P ( |) accurately for various phases of the planets, or, even more difficult, spacecraft. Thus, the best phase law information is empirically determined and for planets outside the Earth’s orbit only a limited range of phases | = 0 are observed from the Earth. Basically, it is very difficult to determine what the phase law would be for a “first quarter Jupiter” since we have never seen Jupiter from that angle. Although no method is completely satisfactory, the three most convenient methods for estimating the phase law for an object are: (1) assume that the intensity is proportional to the observed illuminated area, that is,P(£)= 0.5(1 + cos | ); (2) for objects similar in structure to the Moon, assume that the Moon’s phase law, which is tabulated numerically in Table 11-5, holds; or (3) for the planets, assume that * Equation (11-56) holds only for objects which shine by reflected sunlight. Additional terms are needed if lighting is generated internally or by planetary reflections. t An object is said to be in opposition to an observer if it is opposite the direction to the Sun, i.e., as seen from the object, the Sun and the observer are in the same direction.
11.6
Viewing and Lighting Conditions
580 T A B L E 11-4.
Values of the A p p a re n t Visual M agnitude for the M oon and Planets at O p p o sitio n to the Earth. See text for definitions. Data from Seidelmann [1992], Visual M agnitude Sm all A n g le Phase Coefficient, a1
V (1 ,0 )
V0
Moon
+ 0 .2 1
-1 2 .7 4
—
Mercury
-0 .4 2
—
0.038
Venus
-4 .4 0
—
0.009
Earth
-3 .8 6
—
—
Planet
Mars
-1 .5 2
-2 .01
0.016
Jupiter
-9 .4 0
-2 .7 0
0.005
Saturn
-8 .8 8
+0.67
—-
Uranus
-7 .1 9
+5.52
0 .0 0 2 8
Neptune
-6 .8 7
4-7.84
—
Pluto
-0 .81
+15.12
0.37
the phase dependence on the magnitude for small £ is of the form V0 + where the empirical coefficients a x are given in Table 11-4. For Saturn, the visual magnitude depends strongly on the orientation of the observer relative to the ring system. Because the ring system is inclined to the ecliptic, the orientation of the rings relative to the Earth changes cyclically with a period equal to the period of revolution of Saturn, or about 30 years. Additional information on visual magnitudes of planets are given by Cox [2000], T A B L E 11 -5.
5 (deg )
Phase Law for the M oon. Note the strongly non-linear character. W hen the Moon is half illuminated (first or last quarter, £ = 9 0 deg), the intensity is only 10% of that of the full Moon,
P<«)
V {r ^ )-V (r ,0 )
0
1.000
0.00
1 (d e g )
m
V (r,4 )-V (r,0 )
80
0.1155
2.34
5
0.929
0.08
90
0.0802
2.74
10
0.773
0.28
100
0.05705
3.11
20
0.5945
0.56
110
0.0391
3.52
30
0.4595
0.84
120
0.0255
3.98
1.13
130
0.01545
4.53
40
0.353
50
0.274
1.41
140
0.0093
5.08
60
0.211
1.69
150
0.0046
5.88
70
0.1585
2.00
160
0.001
7.50
The phase law of the Moon given in Table 11-5 above illustrates the fundamental difficulty with estimating the brightness of an object as a function of phase. A first quarter Moon will have a phase angle of 90 deg and half of the Moon will appear to be illuminated, i.e., the terminator will approximately bisect the disk of the Moon. Although the Moon is half illuminated, the intensity of the Moon is only 10% that of
11.6
Brightness of Distant Spacecraft and Planets
581
the intensity of the full Moon. This is due principally to two effects. First, most of the illuminated area that we are looking at in the first quarter Moon is near the terminator, and therefore, near sunrise on the Moon. Here, the illumination angle is extremely low and sunlight is spread out over a much larger area due to the shallow elevation angle of the Sun. In addition, light tends to be scattered more strongly in a direction back toward the source of illumination, thus, further increasing the brightness of the full Moon relative to what one would expect if it were a purely diffuse sphere. For objects for which no a priori magnitude is known but which shine by reflected sunlight, we may estimate 7(1,0) from the relation: 7(1,0) = V* - 5 log Rp - 2.5 log £
(1 l-59a)
= -26.74 - 5 log Rp - 2.5 log g
(Rp in AU)
(1 l-59b)
= 29.14 - 5 log Rp - 2.5 log g
(Rp in m)
(1 l-59c)
where Vs is the visual magnitude of the Sun at 1 AU, Rp is the radius of the object in AU, and g is the geometric albedo or the ratio of the brightness of the object to that of a perfectly diffusing disk of the same apparent size at £ = 0. For the planets, g ranges from 0.10 for Mercury to 0.57 for Uranus, and is about 0.37 for the Earth, although it is a function of both weather and season. Table 11-4 lists the Bond albedo, A, of the planets, which is the ratio of the total light reflected from an object to the total light incident on it. The Bond and geometric albedos are related by: A = gq
(11-60)
n q = J p (£ )s in £ d f o
(11-61)
where
where P (£ ) is the phase law. The quantity q represents the reflection of the object at different phase angles and has the following values for simple objects: • q= 1.00 for a perfectly diffusing disk • q= 1.50 for a perfectly diffusing (Lambert) sphere • q = 2.00 for an object for which the magnitude is proportional to the illuminated area • q = 4.00 for a metallic reflecting sphere. For the planets, q ranges from 0.58 for Mercury to about 1.6 for Jupiter, Saturn, Uranus, and Neptune. Consider as an example, Fig. 11 -28 which shows the Cassini spacecraft en route to its fly-by of Venus. We assume the spacecraft is about 6 m in diameter, so Rp = 3 m. The geometric albedo is unknown so we will pick an intermediate value of 0.5. Then, from Eq. (1 l-59c), we have 7(1,0) = 29.14 - 5 log (3) -2 .5 log (0.5) = 27.51. We also assume an intermediate phase angle of 45 deg and a phase law proportional to the il luminated area, such that P (|) = 0.5 (1 + cos |) = 0.85. Finally, the distance to the spacecraft, r, is 26,000 km and the distance to the Sun, S, is 1 AU. Consequently, from
582
Viewing and Lighting Conditions
11.7
Eq. (11-57), we have an estimate of the visual magnitude V = 27.51 + 5 log(26,000) + 5 log(l) - 2.5 log(0.85) - 40.85 = 8.91. This is in reasonable agreement with the stars in the photo. Note that the star images buiid up over the 17 sec exposure, while the images of the spacecraft and upper stage are spread out over a trail. In addition, the brightness of the spacecraft fluctuates considerably as sides that are more or less reflective turn toward the observer.
Fig. 11-28.
Cassini Spacecraft and Centaur Upper Stage at a Range of Approximately 26,000 km . A cloud of propellant vented after separation surrounds the Centaur stage. Th e star PPM 204332 is of visual magnitude 8.8. T h e 17 sec exposure had a limiting stellar magnitude of about 15.0. (Photo by Gordon Garradd from New South Wales, Australia.)
11.7 Radar and Laser Illumination of Surfaces Space-based radar and space-based lasers are generally very high-power devices because of the large distances involved in space missions. Nonetheless, they have been used throughout the space program to serve a variety of purposes. There are a number of advantages to the use of both radar and lasers in space. First, we can control both the amount of illumination and the spectral characteristics in order to get as much information as possible. In addition, we can adjust the wavelength to provide cloud penetration as in the case of using radar to map the surface of Venus, as illustrated in Fig. 11 -29. The largest detriment to this approach is that we must carry our own source of radar or laser illumination, which typically requires very high levels of power. It is important to recognize that the dependence of radar power on distance depends on how the radar is used, as shown in Table 11-6. Space-based radar works in a series of dwells, or looks, in which a burst of energy is sent out by the radar antenna and typically received by the same antenna after bouncing off the target. Since the same antenna is used for both transmit and receive, the beam size or angular diameter of the field of view of the radar, will be the same for both transmit and receive. The beam size is inversely proportional to the size of the antenna, thus large antennas correspond to narrow beams and small antennas correspond to large beams, just as it is for para bolic radio antennas.
Radar and Laser Illumination of Surfaces
11.7
Fig . 11-29.
583
R adar M ap of the Surface of Venus. Taken by Magellan spacecraft. (Photo cour tesy of N A S A .)
Ordinarily, we think of the beam size as being much larger than the target we are looking at. For example, in using radar to search for spacecraft, we will use a large beam to search for spacecraft with small angular diameter (row 1 in Table 11-6). Assuming that we hit the target with the radar beam, a single dwell or look is sufficient to see the target. However, the required power per dwell increases as r4, where r is the distance between the radar and the target. As the target gets farther away, the area covered by the radar beam increases as r2, which means we must increase the power as r 2 to provide a given level of illumination at the target. With a given level of illu mination, the surface brightness of the target in terms of the reflected radar energy will be independent of distance, as described in Sec. 11.0. However, the angular diameter of the target as seen from the radar antenna will fall off as 1l r 2 such that the total in tensity will fall off as 1l r 2, This implies a need to further increase the power by r2 to account for the smaller angular size. The net effect is that the power required with a large radar beam and a small target increases a s r 4. Because the distances involved for a space-based radar are typically very large, they are ordinarily very high-power devices. T A B L E 11-6.
Beam Size
P ow er Requirem ents as a Function of R ange for D iffe rin g Beam and Ta rg e t Sizes. See text for discussion. Ta rg e t Size
# of Dwells Required
Power per Dwell
P ow er for W h ole Ta rg e t
Large
Small
1
r4
r*
Small
Large
1//-2
r*
Independent of r
= Target
= Beam
1
Independent of r
Independent of r
584
Viewing and Lighting Conditions
11.7
Next, consider what happens if the target is much larger than the beam size (row 2 in Table 11-6). This would occur, for example, if we want to make a radar map of the state of Texas. As we increase the distance between the target and the radar, die area covered on the ground becomes larger by r2. In order to maintain a fixed level of illu mination per square meter, we must increase the power per dwell proportional to r2. However, we do not have the same problem in the receive antenna. The size of the receive beam is the same as the size of the area being illuminated. The surface bright ness is constant, therefore, the energy return will also be constant. From the point of view of the radar, it is very much like looking at the surface of the Earth illuminated by the Sun. There is a constant illumination level irrespective of distance (due to the increased transmitter power in proportion to r2); therefore, the surface brightness remains the same and the exposure time or received energy also remains the same. Consequently, the total power required per dwell increases only as r2 rather than as r \ What’s more, the area covered by each dwell also increases as r2 and, therefore, the number of dwells required to cover the state of Texas is reduced by 1/r2. Consequent ly, the total power required to make a radar map from space of the state of Texas is independent of the distance so long as the radar beam remains smaller than the size of the state.* Finally, consider the case in which the target and the beam are the same size (row 3 in Table 11-6). Note that for a target of fixed size, this means that the antenna must get bigger as I get farther away in order to make the beam smaller so that it remains the same angular size as the target. In this case, the power per dwell required to provide a given level of illumination is independent of distance. All of the energy transmitted by the beam is falling on the target and none is being wasted. Consequently, the energy per square meter at the target is independent of distance (it is simply the energy in the beam divided by cross-sectional area of the target in m2). As in the case above, I’m looking at a uniformly illuminated target, so once again, the required power per dwell will be independent of distance. Because the target and the beam are the same size, I require only a single look to see it and, consequently, the power required to view the entire target is independent of distance. Of course the practical limitation here is that as I get farther and farther away, I must use larger and larger diameter antennas in or der to make the beam as small as the size of the target. If I do this, however, I can create a radar for which the power requirement is independent of distance. The above results seem strange at first glance. The physical reason that the power required is independent of distance is that I am being efficient in using all of the energy that I am transmitting. None of it is wasted by spilling out into space. This is illustrated conceptually in Fig. 11-30. The same logic applies to laser illumination or the use of a flashlight and a pair of binoculars on the surface of the Earth. For much more extensive discussions of space-based radar and space-based lasers and their application for both military purposes and observation, see, for example Cantafio [1989], Jenn [1995], or Skolnik [1990].
+ Another way to think of this is as follows. I need to illuminate the whole state with so many W/m^ and, therefore, so many watts. (There is a reasonably well-defined number of m2 in Texas, even though that number is large.) I am being efficient in only taking a “picture” of those m2 which are currently being illuminated. Therefore, I only need the prescribed total number of watts, irrespective of how far away the “camera” and the “light source” are.
Jamming and RF Interference
11.8
V
\
585
— -
\ (A ) P ow er Proportional to r 4 F ig . 11-30.
(B ) Pow er Independent of r
P ow er R equirem ents for R adar or Laser Illum ination of Surfaces. (A ) A broad illuminator and a small target. (B) A narrow illuminator and a large target. See text for discussion.
11.8 Jamming and RF Interference Jamming is the process of making an observation useless by overwhelming the signal with noise. It is usually associated with radar or radio signals, but can be done with optical images as well. Typically jamming is intentional to prevent someone else from making an observation or getting a message, such as jamming a radar signal or a radio broadcast.* In contrast RF interference is the unintentional overlap of signals that causes noise, static, or double signals in everything from radio and television reception to cellular telephones and radio telescopes. Interference can be natural, such as when we try to communicate with a satellite too near the Sun. However, as man-made RF signals proliferate, unintentional interference from other broadcasting sources be comes an ever-increasing problem. For example, LEO communications constellations are concerned with RF interference from GEO satellites and may have areas of the world they can not cover because of this. For space applications, signal strength and processing issues associated with jam ming and interference are covered in detail by most references on radar [Cantafio, 1989] and satellite communications (for example, Elbert [1997,1999]). In this section, we are concerned with geometrical and orbit issues associated with when jamming and RF interference will occur and how they can be avoided. Jamming typically occurs when a jammer (i.e., a large RF source operating in the same frequency band as the observation or communications system) is located near a target. Here “target” refers to the object we are trying to see or communicate with, such as a ground station or radar target. As discussed in detail in Sec. 9.1, the ability to see a target depends on both the satellite altitude and the minimum elevation angle. As shown in Fig. 11-31, these parameters define a coverage small circle, called the effective horizon , centered on the target with a radius equal to the maximum Earth cen tral angle. (See Secs. 9.1.1 and 9.5.1.1 for the relevant coverage formulas.) Similarly, * An example of “unintentional jamming” is glare when a star camera looks too close to the Sun.
586
Viewing and Lighting Conditions
ll.S
there is small circle centered on the jammer which represents all of the subsatellites points at which the jammer can see the satellite above the jammer minimum working elevation angle. The minimum elevation angle for the sensor and the jammer need not be the same, but are typically similar. The shaded area in Fig. 11-31 is the region where the satellite can see the target but the jammer can not see the satellite, i.e., the region where observations or communications are possible.
Fig. 11-31.
Ja m m in g Susceptibility. Flying at a low altitude does not reduce jamming suscep tibility. At lower altitudes, both jamming and nominal operations occur closer to the target.
Fig. 11-31 also illustrates a common misconception about space-based radar and jamming systems. It is often assumed that, similar to terrestrial aircraft, there is an advantage to flying low, i.e., to coming in under the field of view of the jammer. This is not the case. The figure illustrates two systems in which the target and the jammer are the same distance apart. The one on the left is for a satellite at a low altitude and the one on right is for a higher altitude. As can be seen, there is no advantage to the lower altitude system and, to the contrary, the higher system has a considerably larger area and range of both latitudes and longitudes from which to work. Of course, the higher altitude system must be designed to work from further away, but that is the case irrespective of jamming considerations. Lower altitude systems will generally require less power (within the constraints discussed in Sec. 11.7), but have no advantage and some disadvantages from a jamming perspective. In contrast to jammers, RF interference usually covers only a segment of the operational area of a system. As illustrated in Fig. 11-32, interference between two sat ellites will typically occur when the line of sight to the two satellites as seen from the target or ground station are within a given keep-out angular radius, q, of each other. For example, we may be able to work with a LEO communications satellite when it is above a minimum elevation angle of 10 deg and further than 20 deg from the line of sight to a GEO satellite operating in the same or very nearby frequency band. In this case, the RF interference region will be a cone of radius 6 centered on the direction to the GEO satellite.
Jamming and RF Interference
11.8
587
ence
Direction to G E O Satellite
(B) Fig.
11 - 32 .
R F interference G eom etry. R F communications can occur when the communica tions satellite is above a minimum working elevation angle and outside a keep-out angular radius of the interference source.
As shown in Fig. 11-33* the percentage of the coverage region that is blocked depends on both the keep out-radius and whether the interference source is near the horizon. However, as can best be seen in Fig. 11-32A, the percentage blocked does not depend on the altitude. The relative angular area between the coverage cone and the interference cone depends only on the two angular radii and not on how far the cones extend into space. Similarly, if the blockage cone is entirely within the com munications cone, then the percentage blocked will not depend on where within the communications cone the interference source lies, In the case of an interference source along the horizon, the percentage is reduced simply because a part of the blockage cone lies outside the effective horizon.
0
10
20
30
40
50
SO
70
80
90
Keep-Out Radius (deg)
Fig. 11 -33.
R F Interference as a Function of the A n g u la r K eep-O ut R adius. T h e percentage blocked is independent of tine altitudes of the satellites. It is also independent of the direction to the interference source so long as all of the interference cone lies inside the communications effective horizon.
588
Viewing and Lighting Conditions
11.8
References Cantafio, Leopold J., ed. 1989. Space-Based Radar Handbook. Boston MA: Artech House. Cox, Arthur N, ed. 2000. A llen ’s Astrophysical Quantities, 4th ed. New York:
Springer-Verlag. Elbert, Bruce E. 1997. The Satellite Communication Applications Handbook. Boston, MA: Artech House. Elbert, Bruce E. 1999. Introduction to Satellite Communication, 2nd ed. Boston, MA: Artech House. Gilmore, David G., and Mel Bello. 1994. Satellite Thermal Control Handbook. El Segundo, California: The Aerospace Corporation. Jenn, David C. 1995. Radar and Laser Cross Section Engineering. Washington D.C.: AIAA Lyle, Robert, James Leach, and Lester Shubin. 1971. Earth Albedo and Emitted Radi ation. NASA SP-8067, July.) Seidelmann, P. Kenneth. 1992. Explanatory Supplement to the Astronomical Almanac. Mill Valley, CA: University Science Books. Skolnik, Merrill. 1990. Radar Handbook, 2nd ed. New York: McGraw Hill. Wertz, James R. and Wiley J. Larson. 1999. Space Mission Analysis and Design (3rd edition). Torrance, CA and Dordrecht, The Netherlands: Microcosm, Inc. and Kluwer Academic Publishers.
PART III
O
r b it a n d c o n s t e l l a t io n d e s ig n
Part III is a practical guide to the process of selecting, designing, and maintaining orbits, including those for single spacecraft (in Earth orbit and beyond), formations, and constellations.
12. Orbit Selection and Design
Chapter 12 Orbit Selection and Design
12.1 The Orbit Design Process 12.2 The AV Budget 12.3 Estimating Launch Cost and Available On-Orbit Mass Launch Options; The Orbit Cost Function; Estimating Launch Cost
12.4 Design of Earth-Referenced Orbits Mission Requirements for Earth-Referenced Orbits; Specialized Earth-Referenced Orbits; Eccentric Orbits for Earth Coverage 12.5 Design of Near-Earth Space-Referenced Orbits 12.6 Design of Transfer and Parking Orbits 12.7 Design of Interplanetary Orbits Overview o f Interplanetary Transfer Trajectories; Navigation, Guidance, and Control; Additional Launch Energy Considerations; Earth Departure Geometry and Constraints; Planetary Arrival Geometry , Orbit Insertions, Orbit Shaping; Other Propulsion Techniques Used in Planetary Mission Design
12.8 Interstellar Exploration
Chapters 2, 9, and 10 introduced the geometry and physics of spacecraft orbits as well as formulas for computing orbit parameters. Those chapters are fundamentally computational in that there is a well defined answer to how the satellite will move in response to a given set of forces or what portion of the Earth I can see with a camera oriented in a specific direction. In contrast, orbit selection and design is a much fuzzier process. The question that we are trying to deal with is not how the spacecraft will move but what orbit should we put it in. This is, of course, a mix of mission objectives, cost, available launch vehicles, and operational requirements to support the mission. There is no precise answer or correct solution.
In most cases, orbit design consists of choosing among relatively similar, simple orbit shapes. However, as illustrated in Fig. 12-1, some orbit design problems can become very complex, particularly when there are multiple mission phases to be sat-
589
Orbit Selection and Design
590
12.1
isfied. In addition, the orbit selection process itself is typically complex, involving trades between many parameters. The orbit normally defines the space mission life time, cost, environment, viewing geometry, and frequently, the payload performance. Nonetheless, the single most common trade is between the velocity required to achieve an orbit as a measure of cost vs. the coverage to be achieved as a measure of perfor mance.
11-23-82
Fig. 12-1.
O rb it Design for the IS E E -C Spacecraft. Most orbit design consists of choosing among similar, simple orbit shapes and sizes. A few, such as IS E E -C shown here, involve complex trajectories in order to meet multiple mission objectives. {Farquhar,
2001].
The orbit selection and design discussion is divided into three chapters: • Chapter 12 introduces the broad topic and lets us choose between high level options, i.e., geosynchronous vs. low-Earth orbit, or Sun-synchronous vs. low inclination. • Chapter 13 addresses specific issues in designing a constellation. • Chapter 14 addresses the details of, having chosen a particular orbit or constel lation, how we select the specific parameters or range of parameters, including all of the orbit elements, launch windows, and intermediate orbit parameters, to get us where we want to go.
12.1 The Orbit Design Process Orbit and constellation selection and design is a process rather than a set of specific computations. Throughout the next three chapters, we will use a series of process tables to summarize the fundamental steps and how to undertake them. These should be treated as guides to the issues involved rather than complete and definitive recipes. There is a wide variety of mission types, each of which will be unique in the orbit selection process. Nonetheless, the tables here should be used as a starting point to create a process appropriate for your particular mission.
12.1
The Orbit Design Process
591
Table 12-1 summarizes the steps in the overall orbit selection and design process. Each step is discussed below and in greater detail in the sections listed in the table. Effective orbit design requires clearly identifying the reasons for orbit selection, reviewing these reasons regularly as mission requirements change, or mission defini tion improves, and continuing to remain open to alternatives. Several different designs may be credible. Thus, communication systems may work effectively through a single large satellite in geosynchronous orbit, or a constellation of small satellites in lowEarth orbit. We may need to keep both options for some time before selecting one for a particular mission. T A B L E 12-1.
T h e O rbit Design P rocess. See text for discussion of each step. This should be treated as a general guide, rather than a definitive process. Step
W h ere D iscusse d
1. Establish orbit types
Table 12-2
2. Determine orbit-related mission requirements
Chaps. 1 ,5
3. Assess applicability of specialized orbits
Secs. 2.5, 12.4, 12.5
4. Evaluate whether a single satellite or a constellation is needed
Chap. 13
5. Do mission orbit design trades Assume circular orbit (if applicable) Conduct altitude/inclination trades
Sec. 12.4,12.5
Evaluate use of eccentric orbits
Sec. 12.4, 12.5
6. Evaluate constellation growth and replenishment (if applicable), or single satellite replacement strategy
Sec. 13.6
7. Assess retrieval or disposal options
Sec. 13.6
8.
Sec. 12.2
Create
Al/ budget
9. Determine launch options and cost Identify launch vehicle options and cost
Sec. 12.3
Identify low cost options (if applicable)
Sec. 12.3
Compute the Orbit Cost Function
Sec. 12.3.1
Estimate launch cost vs. available on-orbit mass
Sec. 12.3.2
10. Document selection criteria, key orbit trades, selected orbit parameters, and allowed ranges
Chap. 14
11. Iterate as needed
Step 1. Establish Orbit Types To design orbits we first divide the space mission into segments and classify each segment by its overall function. In terms of orbit requirements, each segment falls into one of four basic orbit types, listed in Table 12-2. The first two entries are operational orbits in which the spacecraft is intended to spend a large fraction of its operational life, and to do most of the useful work of the mission. In contrast, the bottom two entries are simply means of getting the spacecraft where we want it, when we want it there. Typically, the requirements for the latter two are much less stringent, with the
592
12.1
Orbit Selection and Design
exception that it may be necessary to get the spacecraft where we want it with very precise timing, such as the requirement to attempt to land a probe at a particular loca tion on Mars. T A B L E 12-2.
Principal O rbit Ty p e s. For orbit design it is convenient to divide the mission into a series of orbit types based on the varying requirements as the mission progresses.
Type
Definition
Exam ples
W here D iscusse d
Earth-Referenced Orbit
An operational orbit which provides the necessary coverage of the surface of the Earth or near-Earth space (also applicable to the Moon and planets)
Geosynchronous orbit, low-Earth communications or observation constellations
Sec. 12.4.2, Chap. 13
SpaceReferenced Orbit
An operational orbit whose principal characteristic is be ing or pointing somewhere in space, such that specific orbit parameters may not be criti cal
Hubble Space Telescope orbit, space manufac turing orbit
Secs. 12.4,12.5
Transfer Orbit
An orbit used for getting from place to place
Geosynchronous transfer, inter planetary transfer to Mars
Secs. 12.5, 12.6
Parking Orbit
A temporary orbit providing a convenient location for satellite check-out, storage between activity or at end-oflife, or used to match conditions between phases
After launch, while awaiting proper conditions for orbit transfer; end-of-life 500 km above
Chap. 14
geosynchronous
An example of the changing orbit types throughout the mission would be a geo synchronous communication satellite initially launched along with a transfer stage into a low-Earth orbit. Once ejected from the launch vehicle, the spacecraft stays briefly in a parking orbit to provide test and checkout of the spacecraft and transfer vehicle subsystems. The next mission segment is a transfer orbit which moves the spacecraft from the parking orbit to a geosynchronous equatorial orbit. Frequently, to preserve propellant, the spacecraft is initially put into a drift orbit near GEO such that any errors in the transfer process can be taken out by small adjustments asso ciated with achieving the desired geosynchronous station location. The spacecraft then enters its operational orbit in the geostationary ring where it will spend the rest of its active life. While the spacecraft itself is still functional and before its stationkeeping propellant runs out, we must move it out of the geostationary ring to avoid possible collisions with other satellites and to free the orbital slot for a replacement, as illustrated in Fig, 12-2. Putting the nearly dead spacecraft into a final disposal orbit a few hundred kilometers above the geostationary ring requires a relatively small transfer orbit. Going above the geostationary ring rather than below avoids the potential for collisions with other satellites during subsequent geosynchronous transfers.
12.1
Fig. 12-2.
The Orbit Design Process
593
Ty p ic a l O rbit Phases. At end-of-life, the spacecraft should be either de-orbited or moved into a higher orbit where it will not interfere with other spacecraft.
Step 2. Establish Orbit-Related Mission Requirements For each mission segment defined in Step 1, we define the orbit-related require ments. They may include orbital limits, individual requirements such as the altitude needed for specific observations, or a range of values constraining any of the orbit parameters. Sections 12.4 and 12.5 discuss in detail the requirements for designing operational orbits. Ordinarily, these multiple requirements drive the orbit in different directions. For example, resolution or required aperture tend to drive the orbit to low altitudes, but coverage, lifetime, and survivability drive the spacecraft to higher altitudes. Consequently, the operational orbit trades are typically complex, involving the evaluation of multiple parameters in selecting a reasonable compromise that meets mission objectives at minimum cost and risk. Selecting parking, transfer, and space-referenced orbits is normally conceptually much simpler, although it may be mathematically complex. Here the normal trade is meeting the desired constraints on a mission, such as lifetime, thermal, or radiation environments, at the lowest possible propellant cost. Sections 12.5 through 12.7 dis cuss these types of orbits in more detail.
Step 3. Assess Specialized Orbits In selecting the orbit for any mission phase, we must first determine if a Specialized orbit applies. Specialized orbits are those with unique characteristics, which set them apart from the broad continuum of orbit parameters. (See Table 2-10 in Sec. 2.5 and Table 12-10 in Sec. 12.4.2.) Geosynchronous orbits, Sun-synchronous orbits, M olniya orbits, and Lagrange point orbits are typical examples. We examine each of these specialized orbits to see if its unique characteristics are worth its cost. This examina tion precedes the more detailed trades, because specialized orbits constrain parameters such as altitude or inclination, and, therefore, often lead to very different solutions for a given mission problem. Consequently, we may need to carry more than one orbit into more detailed design trades, such as keeping open a geosynchronous option and a lowEarth orbit option as possibilities for further trade studies.
594
Orbit Selection and Design
12.1
Step 4. Select Single Satellite o r Constellation The principal advantage of a single satellite is that it reduces cost by minimizing the mission overhead. One satellite will have one power system, one attitude control system, one telemetry system, and require only a single launch vehicle. A constellation or multiple spacecraft, on the other hand, may provide better coverage, higher reliabil ity if a satellite is lost, and more survivability. We may also need a constellation to provide the multiple conditions to carry out a mission, such as varying lighting condi tions for observations, varying geometries for navigation, or continuous coverage of all or part of the Earth for communications. To meet budget limits, we often trade a single large satellite with larger and more complex instruments against a constellation of smaller, simpler satellites. This deci sion may depend on the technology available at the time of system design. As discussed in detail by Wertz and Larson [1996, 1999] small satellites have become more capable through miniaturized electronics and onboard processing. Consequently, we may be able to construct constellations of small, low-cost satellites, frequently called LightSats, that were not previously economically feasible. Another issue for large constellations is the operational problem of providing continuous navigation and orbit control. The introduction of low-cost navigation and autonomous orbit control described in Chap. 4 can resolve many of these problems and promote larger constel lations of small satellites in the future.
Step 5. Do Mission Orbit Design Trades The next step is to select the mission orbit by evaluating how orbit parameters affect each of the mission requirements defined in Chap. 5 and Sec. 12.4.1. As shown in Table 12-9 in Sec. 12.4.1, orbit design in Earth orbit depends principally on altitude. The easiest way to begin is by assuming a circular orbit and then conducting altitude and inclination trades. This process establishes a range of potential altitudes and incli nations, from which we can select one or more alternatives. Documenting the results of this key trade is particularly important, so we can revisit the trade from time to time as mission requirements and conditions change. We then evaluate the use of eccentric orbits as discussed in Sec. 12.4.3. If a satellite constellation is one of the alternatives, then re-phasing the satellites within that constellation is a key characteristic, as described in Chap. 13. Note that constellations of satellites are normally at a common altitude and inclina tion because the orbit’s drift characteristics depend largely on these parameters. Satellites at different altitudes or inclinations will drift apart so that their relative orientation will change with time. Thus, satellites at different altitudes or inclinations typically cannot work well together as a constellation for extended times.
Step 6. Evaluate Constellation Growth and Replenishment or Single-Satellite Replacement Strategy An important characteristic of any satellite constellation is growth, replenishment, and graceful degradation. A constellation that becomes operational only after many satellites are in place causes many economic, planning, and checkout problems. Con stellations should be at least partly serviceable with a small number of satellites. Graceful degradation means that if one satellite fails, the remaining satellites provide needed services at a reduced level rather than a total loss of service. Section 13.6
12.1
The Orbit Design Process
595
discusses further the critical question of how we build up a constellation and how to plan for graceful degradation. Constellations are often launched with multiple small satellites on a single large launcher. If this is the case, a key logistics question is how we launch replacement satellites when spacecraft die in orbit.
Step 7, Assess Retrieval or Disposal Options Although given little consideration in many past missions, retrieval and disposal of spacecraft have become critical to mission design. These represent potentially major legal, political, and economic issues. Spacecraft disposal options are discussed in detail in Sec. 2.6.4. Spacecraft that will reenter the atmosphere must either do controlled reentry over the ocean or break up into pieces harmless to the Earth’s surface. If the spacecraft will not reenter the atmosphere in a reasonable time, we must still dispose of it at the end of its useful life so it is not hazardous to other spacecraft. This problem is particularly acute in geosynchronous orbit where missions compete strong ly for orbit slots. (See Cefola [1987] for an excellent analysis of the requirements for removing satellites from geosynchronous orbit.) As pointed out in Sec. 13.5, a colli sion between two spacecraft not only destroys them but also causes a debris cloud dangerous to their entire orbital area. Consequently, this is a major concern for satellite constellations. A third option is satellite retrieval, done either to refurbish and reuse the satellite, or to recover material (such as radioactive products) which would be dangerous if they entered the atmosphere uncontrolled. Currently, the Space Shuttle can retrieve space craft only from low-Earth orbit. In the future, it is likely that we will be able to retrieve satellites as far away as geosynchronous orbit and return them to either the Orbiter or Space Station for refurbishment, repair, disposal, or reuse. Spacecraft beyond geosyn chronous orbit are ordinarily allowed to continue to drift in interplanetary space at the end of their useful life. If a spacecraft is not specifically designed to encounter a plan etary surface, then normal practice calls for preventing collisions with planets or moons in order to avoid the potential of biological contamination. At the end of their normal mission, Pioneers 10 and II became the first spacecraft to begin the explora tion of interstellar space and continued to send back signals for an extended time on the environment which they encountered.
Step 8. Create a AV Budget To numerically evaluate the cost of an orbit, we first create a AV budget for the orbit, as described in Sec. 12.2. This then becomes the major component of the propel lant budget and the Orbit Cost Function as described in Step 9.
Step 9. Assess Launch and Orbit Transfer Cost Section 12.3.1 discusses current satellite launch options, including the options for small systems which cannot afford the cost of a dedicated launch vehicle. The selection of specific launch vehicles is discussed by Wertz and Larson [1999], Chap. 18; Isakowitz [1999] provides detailed numerical summaries of currently avail able launch vehicles worldwide. The launch vehicle contributes strongly to mission cost, and ultimately limits the amount of mass that can be placed in any given orbit. During early mission definition, we must provide enough launch margin to allow for
596
Orbit Selection and Design
12.2
later changes in launch vehicles or spacecraft mass. New designs require more margin than existing ones, with 20% being typical for new missions. Section discusses the orbit cost Junction which is a mechanism for defining the approximate cost of orbit transfer in terms of the mass that must be put in low-Earth orbit to achieve the end mission orbit. Consequently, it provides a cost multiplier to be used in conjunction with launch vehicle cost estimates from other sources.
Steps 10 and 11. Document and Iterate A key component of orbit or constellation design is documenting the mission requirements used to define the orbit, the reasons for selecting the orbit, and the numerical values of the selected orbit parameters. With this documentation, the base line selection can be reevaluated from time to time as mission conditions change. Because mission design nearly always requires many iterations, we must make the iteration process as straightforward as possible and readdress orbit parameters throughout the mission design to ensure that they continue to meet all of the mission objectives and requirements at minimum cost and risk.
12.2 The AV Budget As discussed above, a space mission is a series of different orbits. For example, a satellite may be released in a low-Earth parking orbit, transferred to some mission orbit, go through a series of rephasings or alternative mission orbits, and then move to some final orbit at the end of its useful life. Each of these orbit changes requires ener gy. The AV budget is traditionally used to account for this energy. It is the sum of the velocity changes, or AVs, which must be imparted to the spacecraft throughout mission life. In a broad sense the AVbudget represents the cost for each mission orbit scenario. In designing orbits and constellations, we must balance this cost against the utility achieved. We use the AV budget to create a propellant budget and estimate the propellant mass, mp, weight required for the space mission. (See for example Wertz and Larson [1999] or Humble et al. [1995].) For preliminary design, we estimate the propellant mass, mp, needed for a space mission by using the rocket equation to determine the total required spacecraft plus propellant mass, = m0 + mp, in terms of the dry mass of the spacecraft, mg, the total required AV, and the propellant exhaust velocity, V0: mi = moe
(&y/vn)
0 = m 0e
(A w /„ g )
sp
(12-1)
where the specific impulse, Isp = V0 /g, and g is the acceleration of gravity at the Earth’s surface.* Typical exhaust velocities for chemical propellants are in the range of 2 to 4 km/s and up to 30 km/s for electric propulsion systems. We can see from Eq. (12-1) that a total AV much smaller than the exhaust velocity (i.e., a few hundred meters per second) will require a propellant mass which is a small fraction of the total mass. If the total AV required is equal to the exhaust velocity, then we will need a total propellant mass equal to e - 1 « 1.7 times the mass of the spacecraft. Propulsion systems require additional structure such as tanks, so a AV much greater than the exhaust velocity is * Note that g in the definition of Isp is best thought of as a units conversion factor. It does not depend on the actual acceleration of gravity where the rocket happens to be operating.
12.2
The AV Budget
597
difficult to achieve. It may make the mission effectively impossible, or require some alternative, such as staging or refueling. The fractional mass of the spacecraft that must be devoted to propellant depends directly on the AV that must be supplied during the life of the mission (we will return to this in Sec. 12.3.3). Consequently, calculating AVs is a basic step in assessing the cost of a particular orbit. The process for constructing a AV budget is summarized in Table 12-3. We begin by writing down the basic data needed to compute AVs: the launch vehicle’s initial conditions, the mission orbit or orbits, the mission duration, required orbit maneuvers or maintenance, and the mechanism for spacecraft disposal. We then transform each item into an equivalent AV requirement using the formulas given in Chap. 2 and listed Table 12-3. Each of these formulas is discussed in more detail in the location refer enced. T A B L E 12-3.
C reating a A V Budget. Formulas for the A V required for specific maneuvers are summarized in Chap. 2. Th e right column gives representative values for a 5-year mission in circular orbit at 1000km and 55 deg inclination. W here Discussed
Item
Representative E xam ple
B asic Data Initial Conditions
Sec. 12.3.1
150 km, 45 deg
Mission Orbit(s)
Secs. 12.4, 12.5
1,000 km, 55 deg
Mission Duration (Each Phase)
Sec. 12.1
5yr
Orbit Maintenance Requirements
Sec. 2.7.2
Altitude maintenance
Drag Parameters
Sec. 2.4.4
m/Cd A = 25 kg/m2 p = 3 x 1CM5 kg/m3
Orbit Maneuver Requirements End-of-life Conditions
None Secs. 2 .6.4,14.3
Positive reentry
1st Bum
Eqs. (2-68), (2-71)*
731 m/s**
2nd Burn
Eqs. (2-69), (2-71)*
671 m/s**
Altitude Maintenance (L E O )
Eq. (2-36)
<1 m/s (for 5 years)
North/South Stationkeeping (G E O )
Eqs. (2-46), (2-47)
N/At
East/West Stationkeeping (G E O )
Eq. (2-48)
N/At
Rephasing, Rendezvous
Eq. (2-79)
None
N o d e o r Plane Change
Eq. (2-76)
None
Spacecraft Disposal
Eq. (2-85)
272 m/s
Total A V
Sum of above
1674 m/s
A V B u dg et (m/s) Orbit Transfer
Orbit Maneuvers
Other Considerations A V Savings
Table 2-17, Sec. 12.2
N/A
Margin
Sec. 12.2
Contained in propellant budget
* Sec. 2.6.2 (Eq. [2-76]) if plane change also required.
’ * including 5 deg plane change in both first and second bums. t For GEO missions N/S stationkeeping represents about 50 {m/S)/yr and E/W stationkeeping is about 10% of the N/S amount.
598
Orbit Selection and Design
12.2
What is the AV cost of putting a satellite into orbit at varying altitudes? Figure 12-3 shows representative orbit acquisition and de-orbit AV requirements for a satellite dropped off at 185 km altitude. As can be seen from the figure, a plane change as large as 30 deg tends to dominate all of the other characteristics. Figure 12-4 shows the altitude maintenance A V for spacecraft at varying altitudes and with a ballistic coeffi cient of 100 kg/m2. Recall from Eq. (2-34) that the AV requirement is inversely proportional to the ballistic coefficient such that AVs required for spacecraft with different ballistic coefficients can be easily estimated from the figure.
A ltitude (km )
Fig. 12-3.
Representative Orbit Transfer and De-Orbit A V Requirements. Assumes satellite is dropped oft at 185 km altitude and raised via a 2-burn Hohmann transfer. D e-orbited is assumed to be an elliptical orbit with a perigee altitude of 50 km.
The higher the mission orbit, the more propellant it takes for both orbit acquisi tion and de-orbit, but the less it takes for orbit maintenance. This represents the AV cost which must be compared to the overall utility of the orbit to make an intelligent selection decision. Figure 12-5 shows the total AV for boost, maintenance, and de orbit as a function of altitude for orbits with no plane change and a 30-deg plane change, respectively. Depending on the desired lifetime of the mission and the phase during the solar cycle, there exists an optimal altitude that balances the orbit acquisition process and orbit maintenance. How ever, this may or may not be useful for orbit design because of the dramatically high variability in the atmospheric den sity, depending on the phase in the solar cycle, as discussed in Sec. 2.4.4 and shown in Fig. 12-4. Beyond plane change requirements, the principal factors which will increase the AV cost in low-Earth orbit are a low altitude and long mission lifetime. The atmosphere increases in density exponentially as the altitude decreases, thus
12.2
The AV Budget
599
107
106 105
104
i !
105 IO "1
10” z
100
200
300
400
500
600
700
800
900
1 ,0 0 0
Altitude (km)
Fig. 12-4.
Altitude Maintenance A V for a Ballistic Coefficient of 100 kg/m2. T h e required AV will be inversely proportional to the ballistic coefficient. See Sec. 2.4.4 for a discussion of ballistic coefficients, solar activity, and atmospheric density.
6,000
— A -4 \ i\ V \\ | \
\ \\
' '\
5 ,0 0 0
\
I
/ 1'
J
V
X
Y e a r Life, S o la r M a x im u m A 1 C rtlnr hlin lm u m 1 'J Y e a r Life, S o la r M m im u m 13 V » a r 1 ifo O u o r a Qrtlaf
1
\
\
-------------- ------------------------------ --------------- ---------------3 0 D e g P l a n e C h a n g e 4 ,0 0 0
AV
vT § >
<
/
1 1
*
O 1 1
kJ 1
/ ✓1 Y e a r Life, S o la r M a x im u m
/
3 ,0 0 0
\
A
\ 2, 00 0
/ L
i \ ■\ \
/
X ^ *t ic a i i_ m c , sjwia 1 IVIII HIM
\ /
/
/I
1
1
1
1
> /
V \
1,000
/
\
f
/ -
\ y
\ \
13 Y e a r Life, O v e r a S o la r C y c le
..
X ,
A
N o P la n e c n a n g e -
0 200
300
400
500
600
700
800
900
1 ,0 0 0
1 ,1 00
1 ,2 0 0
1 ,3 0 0
1 ,4 0 0
1 ,5 0 0
Altitude (km ) Fig. 12-5,
Boost, Maintenance, and De-Orbit a V as a Function of Altitude for Low-Earth Orbits with No Plane Change Required.
600
Orbit Selection and Design
12.2
dramatically increasing drag as altitude comes down. In addition, very large solar arrays or other appendages represent a large cross sectional area that will also increase drag. A low mass per unit area, i.e., a large, lightweight satellite will also have higher drag than a small, dense satellite. Note that the shape and surface characteristics typically play only a secondary role. (See Sec. 2.4.4.) They impact the drag coefficient somewhat but usually play only a minor role in overall AV requirements. The AV budget has a strong impact on the propulsion requirements and, therefore, on the final cost and achievability of a space mission. Nonetheless, other factors can vary the propellant requirements relative to a nominal AV budget. For example, al though rocket propulsion usually provides the AV, we can obtain very large AVs from a flyby of the Moon, other planets, or even the Earth itself [Kaufman et al., 1966; Meissinger, 1970)]. As discussed in more detail in Sec. 12.5, a spacecraft in a plane tary flyby leaves the vicinity of some celestial body with the same velocity relative to the body as when it approached, but in a different direction. This phenomenon is like the elastic collision between a baseball and a bat, in which the velocity of the ball rel ative to the bat is nearly the same, but its velocity relative to the surrounding baseball park can change dramatically. We can use flybys to change direction, to provide increased heliocentric energy for solar system exploration, or to reduce the amount of energy the satellite has in inertial space. For example, one of the most energy-efficient ways to send a space probe near the Sun is to use a flyby of Jupiter to reduce the intrinsic heliocentric orbital velocity of the Earth associated with any spacecraft launched from here. A second way to produce a large AV without burning propellant is to use the atmosphere of the Earth or other planets to change the spacecraft’s direction or reduce its energy relative to the planet. The manned flight program has used this method from the beginning to dissipate spacecraft energy for return to the Earth’s surface. It can also be used to produce a major plane change through an aeroassist trajectory [Austin et al., 1982; Mease, 1988]. This was used, for example, on the Mars orbiter mission. The solar sail is a third way to avoid using propellant. A large, lightweight sail uses solar radiation pressure to slowly push a satellite the way the wind pushes a sailboat. Of course, the low-pressure sunlight produces very low acceleration. Hence, very large solar sail areas are required as discussed in Sec. 12.7.6. The aerospace literature discusses many alternatives for providing spaceflight energy because of its importance for a variety of missions. Nonetheless, experimental techniques (i.e., those other than rocket propulsion and atmospheric braking) are risky and costly, so that normal rocket propulsion will ordinarily be used to develop the needed AV, if this is at all feasible. The AV budget described in Table 12-3 measures the energy we must give to the spacecraft’s center of mass to meet mission conditions. When we transform this AV budget into a propellant budget, we must consider other characteristics as well. These include, for example, inefficiencies from thrusters misaligned with the AV direction, and any propulsion diverted from AV to provide attitude control during orbit maneu vers or desaturation of momentum wheels. For most circumstances, the AV budget does not include margin because it results from astrodynamic equations with very little error relative to most other errors in system design. Instead, we maintain the margin in the propellant budget itself, where we can reflect specific elements such as residual propellant. An exception is the use of
12.3
Estimating Launch Cost and Available On-Orbit Mass
601
AV to overcome atmospheric drag. Here the AV depends upon the density of the atmosphere, which is both highly variable and difficult to predict. Consequently, we must either conservatively estimate atmospheric density or incorporate a AV margin for low-Earth satellites to compensate for atmospheric variability.
12.3 Estimating Launch Cost and Available On-Orbit Mass The key characteristic of any orbit is performance vs. cost. What is the performance we can obtain vs. the cost required to put the payload into that orbit? In these terms, GEO is an “expensive” orbit in that approximately 5 kg must be launched into lowEarth orbit for every kilogram that ultimately arrives in geosynchronous orbit. The purpose of this section is to look at alternative launch options and quantify the assess ment of cost as much as possible.
12.3.1 Launch Options A detailed discussion of specific launch vehicles is beyond the scope of this book. Excellent general discussions are provided by Hujsak [1994] and London [1994, 1996]. Isakowitz [1999] has become the standard reference for detailed launch system cost and performance data. This information must, of course, be updated as specific launch vehicles change. The basic goal of the launch vehicle is to expel hot gases at velocities on the order of 2 to 3 km/s in order to achieve a final vehicle velocity of 8 km/s or more, even for low-Earth orbit. This means that the mass of any launch vehicle will be dominated by propellant and that the vehicle itself will need to be designed for minimum excess weight. This in turn implies that launch will be dramatically expensive and will always entail significant risk since each launch vehicle sitting on the pad is more than 95% explosive propellant and virtually nothing to contain the resulting explosion should some error occur. Table 12-4 lists the maximum payload available in low-Earth orbit, the launch cost, and the specific launch cost or cost per kilogram for a representative sample of current launch vehicles. Note that the Saturn V used for the Apollo program was the most efficient launch vehicle ever created by the United States at a bit more than $5,000 per kg. Current vehicles, in part because they are significantly smaller, fall more typically in the range of $10,000 to $20,000 per kg. It is this dramatically high cost per kg that is largely responsible for the very specialized and optimized design of most spacecraft launched today. Because launching spacecraft to geosynchronous orbit costs more than $50,000 per kg, it is worth $40,000 for the spacecraft manufac turer to eliminate a kg of spacecraft in order to provide, for example, more power for additional transponders. The specific launch cost is a valuable figure for comparing launch costs for various launchers and spacecraft. However, it can also be misleading. Launches are almost never sold “by the kilogram” but simply for the entire launch. Thus, once a launch ve hicle has been selected, there is no added cost for “filling it up,” unless doing so moves one into the next larger vehicle, in which case the cost is extremely high. A large number of low-Earth orbit constellations are being developed and deployed. This has the potential for reducing cost due to economies of scale, or increasing them due to limitations in the supply of existing launch vehicles. Unfortu nately, as shown in Fig. 12-6, the cost per kilogram to low-Earth orbit has remained
602
Orbit Selection and Design
T A B L E 12-4.
123
La un ch Vehicle C o sts in FY00$M . T h e data assumes launch from the country's main site. Except where noted, L E O altitude is 185 km (circular orbit) and inclina tion is 28.5 deg (5.2 deg for Ariane). Data from Isakowitz [1999]. See Fig. 12-9 for capability for interplanetary missions.
Maximum Payload-to-Orbit (kg) Launch Vehicles
LEO
G TO
GEO
Unit Cost
Cost per kg to LEO
(FY00SWI)
(FY00$K/kg)
USA Atlas IIIB
10,600
4,510
—
75*
7.1
Atlas V 551
17,420
8,570
3,890
90*
5.2
Athena 1
820
—
—
16.5-17.5
20.7
Athena 2
2,065
590 (Centaur)
—
22.7-26.8
12.0
Athena 3
3,650
31
8.5
Delta 11 (7920, 7925)
5,140
1,870
—
52-62
11.0
Delta III 3940-11
8,290
3,810
1,320
78*
9.4
6,337
1,940
80*
Delta IV 4450-14 Delta IV 4050H-19
12,850
5,950
120*
443
—
—
12.4-15.5
31.4
Saturn V
127,000
—
—
820
6.5
Shuttle-!- (IUS o rT O S )
16.4
Pegasus XL
24,400
5,900
2,360
400
Titan II
1,900
—
—
31—41
19.0
Titan IV
21,680
—
5,760 (Centaur)
361—464
19.0
Taurus
1,380
448
—
18.6-20.6
14.2
ESA Ariane 4 (AR40)
5,000
2,175
—
67-88
15.5
Ariane 4 (AR42P)
6,600
2,890
—
72-93
12.5
Ariane 4 (AR44L)
10,200
4,790
—
103-129
11.4
Ariane 5 (550 km)
18,000
6,800
—
155-186
9.5
—
51.5-72
5.5
1,880
93-101
4.9
—
12.4
8.2 5.9
CHINA Long March (LM-3B)
11,200
5,100
RUSSIA Proton (SL-13)
19,760
Kosmos (C*1)
1,500
4,910
Soyuz
7,000
1,350 (Soyuz/ Fregat)
---
31-52
Tsiklon
4,100
—
—
20.6-25.8
5.7
Zenit 2
13,500
—
—
36-52
3.3
H-2
10,060
3,930
—
170-175
17.2
J-1
850
—
—
44—46
53.3
_
JAPAN
G T O = Geosynchronous Transfer Orbit; G EO = Geostationary Orbit; LEO - Low-Earth Orbit ’ Estimated t Th e re is no official price for a Space Shuttle launch. Following the Challenger loss, only government payloads have been allowed. Th e G A O has assigned a price of $400 million per -flight, but the actual cost depends strongly on the flight rate.
12.3
Estimating Launch Cost and Available On-Orbit Mass
603
approximately constant for 30 years. While many new programs are being initiated to dramatically reduce launch costs, none of these as yet has been strongly successful. An excellent summary of the problem and alternative solutions is given by London [1994, 1996].
Fig. 12-6.
Historical Specific Launch Cost (= Cost per Kilogram) to Low-Earth Orbit. (Data from Koelle [2000].) Costs are in man-years (M Y ) per megagram (M g). See Fig. 12-11 for current values.
Large spacecraft have little choice but to go with one of the dedicated launchers from Table 12-4, unless the program is sufficiently large to justify the nonrecurring development cost of a new vehicle. However, small spacecraft have a number of much lower cost alternatives, as shown in Table 12-5. In particular, small spacecraft should consider non-orbital alternatives, even if it requires multiple flights to achieve mission objectives. Within orbital flights, sharing launch vehicles or being launched as a sec ondary payload on any of the major launchers can dramatically reduce the launch cost for small payloads.
604 T A B L E 12-5.
Orbit Selection and Design
Low-Cost Alternatives to Dedicated Launches. For a more extensive discus sion, see London [1996], from which this table is adapted.
Characteristics
Weight Limits
Balloon Flights
Hours to days at
U p to 70 kg
= 30 km altitude
for low-cost flights
Drop Towers
1 to 10 sec of 0-g with immediate payload recovery
Drop Tubes
1 to 5 sec of 0-g with immediate sample retrieval
Option
123
Principal Constraints
Approximate Cost
Sources
Not in space, not 0-g, weather concerns
$ 5 K to $ 1 5 K
Up to 1,000 kg
Brief “flight,” 5 to 50 g landing acceleration, entire experiment package dropped
= $10K per experiment
ZARM , JA M IC , N A S A LeR C and M S FC , Vanderbilt U.
<0.01 kg
Brief “flight," 20 to 50 g landing acceleration, instrumentation
= $0.02K per experiment
ZARM , JA M IC , N A S A LeR C and M S FC , Vanderbilt U-
U . of Wyoming,
USAFA, NSBF
not dropped
with sample Aircraft Parabolic Flights
Fair 0-g environment, repeated 0-g cycles
Effectively unlimited
Low gravity is only 10-2 g
$6.5K to $9K per hour
N A S A LeR C and JS C , Novespace
Sounding Rockets
Good 0-g environment, altitude to 1,200 km, duration of 4 to 12 min
Up to 600 kg
Much less than orbital velocities
$1M to $2M
NASA G S FC , NR L, E S A , O S C , EER, Bristol Aerosp., Microcosm
GAS Containers
Days of 0-g on board the Shuttle
Up to 90 kg
V e ry limited
$27K for largest container
NASA G SFC
Secondary Payloads
Capacity that is available in excess of primary’s requirements
Up to
Subject to primary’s mission profile
< $10M
Ariane, O S C , M DA, Russia
Shared Launches
Flights with other payloads having similar orbital requirements
Integration challenges
U p to ~ $60 M
Ariane, O S C , Russia
=1,000 kg
Up to = 5,000 kg
external interfaces
Factors other than cost, such as availability, launch site, reliability, and national interests can also have a major impact on launch vehicle selection. Most national programs require that spacecraft be launched on vehicles from that country if there is a national launch vehicle program. The builders of the large commercial commu nications constellations have typically decided that the conservative approach to obtaining launch.is to buy launch services from multiple vendors. The principal rea
12.3
Estimating Launch Cost and Available On-Orbit Mass
605
son for this is to ensure that launch is available in case of problems with any single launch vehicle (A launch vehicle failure normally results in a 4 -1 0 month slip in subsequent launches of that vehicle in order to allow a thorough evaluation and cor rection of whatever problems have occurred). These factors and others which im pact the practical selection of launch vehicle options are discussed in Chap. 18 of Wertz and Larson [1999].
12.3.2 The Orbit Cost Function Given that we can estimate the total launch cost or specific launch cost for putting mass into low-Earth orbit, how can we translate this into the mass available in a mission orbit. Equivalently, what is the ultimate cost per kilogram to get where we want to go and achieve our mission objectives? To address this, we define the Orbit Cost Function, OCF, as the ratio of the mass available in a 185-km circular orbit due East from the launch site to that available in the mission orbit, or at the end of mission life. This can be thought of either as a multiplier on the cost of putting a spacecraft into its mission orbit, or, for a given launch vehicle, an inverse multiplier for the amount of payload that can be put into that mission orbit. Section 12.3.3 will look at the prac tical applications of using the OCF as a means of estimating launch costs and available mass. The OCF depends on how the spacecraft is launched. For example, direct injection into Jupiter transfer orbit has a total OCF on the order of 10. Using a Venus swingby followed by an Earth swingby can reduce this to approximately 6. In this case, we can use the OCF to measure the performance gain associated with the swingbys and com pare that with the time lost to determine whether the swingbys are worthwhile for a particular mission. Thus, the use of a Venus swingby for the Jupiter mission represents an increase in the available mass at Jupiter or a decrease in launch cost by a factor of approximately 10/6 = 1.7. The question for the mission designer is then whether the added complexity and added time associated with these swingbys is worth the savings that are obtained. The orbit cost function is related to the required AV by: AV7K
OCF = (1 + K )e AV!V° - K = ( 1 + K ) e AV/gIsp- K
( 12-2)
where K is the fraction of the propellant mass assigned to tankage and other propellant hardware, typically 10%. lsp is the specific impulse, g is the constant o f gravity at the Earth’s surface, and Vq is the exhaust velocity of the propulsion system. An Isp of 300 sec (typical for a bipropellant system) gives a Vq value of 2.94 km/s. Depending on the purpose of the evaluation, we may choose to compare the Orbit Cost Function for simply getting to the mission orbit, through the useful life, or through de-orbit or disposal. If we calculate the OCF corresponding to the A V for multiple segments, then the cumulative cost function, OCFA+B+£ , is just the product of the individual OCFs. Thus, a series of mission phases or maneuvers A, B, C..., with OCFA corresponding to AV^, and so on, then AVa +b +c „ = A V a + W b + A V c + ...
(12-3 a)
and OCFa +s +c = OCFA x OCFg x OCF q x
(12-3b)
606
Orbit Selection and Design
12.3
Figure 12-7 shows the orbit cost functions for typical Earth orbiting missions as a function of altitude and plane change. For more complex orbits, or interplanetary missions, the OCF is computed by first calculating the required AV, and then account ing for added tank mass and any other additional requirements such as the extra power required for electric propulsion. The OCF is tabulated for various Earth-oriented and interplanetary missions in Table 12-6.
0
5 ,0 0 0
1 0 ,0 0 0
1 5 ,0 0 0
2 0 ,0 0 0
2 5 ,0 0 0
3 0 ,0 0 0
3 5 ,0 0 0
A ltitu d e (k m ) Fig. 12-7.
Orbit Cost Function for Various Earth Orbiting Missions. T h e curves assume starting at a 185 km circular orbit and a circular orbit at the final mission altitude. Th e curves are for orbit acquisition only.
We can also construct the Orbit Cost Function for a launch vehicle alone from the data supplied by the manufacturers. For example, Fig. 12-8A shows the mass as a function of orbit parameters as provided by the launch vehicle manufacturer for the Delta 7925 launch vehicle. The orbit cost function is the ratio of this mass to the mass that can be launched due East to 185 km low-Earth orbit. This is plotted in Fig. 12-8B. As a specific example, the Delta I I 7920 can put 5,140 kg into a 185 km circular orbit at 28.7 deg inclination and 3,220 kg into a Sun synchronous orbit at 800 km and 98.6 deg. Thus, the OCF for the Sun synchronous orbit is 5,140/3,220 = 1.60. Typically, the highest mass to the final orbit can be achieved by using onboard propulsion. The reason for this is that we can then avoid putting the vehicle upper stage into the final spacecraft orbit. For example, if we launch a spacecraft to Jupiter using the traditional approach of a large upper stage, then the upper stage vehicle itself will also be put in the Jupiter transfer orbit. Using onboard propulsion, we will put only the spacecraft and some additional tankage into this very high energy orbit. An example of a high cost approach to orbit transfer is the Hubble Space Telescope, which was launched onboard the Space Shuttle and is re-boosted from time to time by the Orbiter
Estimating Launch Cost and Available On-Orbit Mass
12.3 T A B L E 12-6.
607
Ty p ic a l O rbit C o st Fu nctions for Various E arth -O rb iting and Interplanetary M issions. Interplanetary missions assume a Hohmann transfer. O C F assumes lsp of 300 sec and 10% tank mass. Increm ental A K
Mission
Getting Th e re
M ission Life 12.66
C um ulative O rb it C o s t Function
Disposal
Getting Th e re
M ission Life
0.045
Disposal
1.32
107.01
108.83
200 km @ 90
0.75
4 0 / km @ 51.6
0.39
0.192 0
0.104
1.16
1.24
1.29
500 km @ 2 8 .7
0.24
0.043 2
0.130
1.09
1.11
1.17
500 km Sun-synchronous
1.04
0.043 2
0.130
1.47
1.49
1.57
500 km © 90
0.89
0.043 2
0.130
1.39
1.41
1.48
1,000 km @ 28.7
0.51
0.000 2
0.257
1.21
1.21
1.33
1.66
1.66
1.82
1,000 km Sun-Synchronous
1.38
0.000 2
0.257
G T O (200 x 35,786 km)
2.80
0.411 4
0.024
2.75
3.20
3.23
GEO
4.95
0.521 5
0.018
5.82
7.06
7.11
Lunar F ly -B y
3.14
3.10
Lunar Orbit
3.23
3.20
Lunar Lander
7.07
12.06
Mars F ly-b y
3.61
3.65
Mars Orbit
5.69
7.51
Mars Lander
9.30
25.86
Mission Life A V = AVrequired for 10 years
ijeQ , i* ■
*** deO
Bas elme I
Altitude (km)
(A ) Payload Mass Fig. 12-8.
Altitude (km)
(B ) Orbit Cost Function
Perform ance Data for the Delta 7925 Launch Vehicle (3 m fairing). (A) Mass vs. orbit as provided by the launch vehicle manufacturer. (B) T h e Orbit Cost Function is simply the ratio of the mass that can be launched due East to a 185 km L E O orbit to the mass that can be put in the mission orbit. See text for discussion.
608
Orbit Selection and Design
12.3
at the same time instruments are changed out. The reasons for this approach were to avoid entirely an onboard propulsion system and the potential for contamination from thruster firings as well as to avoid the possibility of propellant slosh providing even a low level of vibration to the telescope. In this case, a higher cost approach to orbit ac quisition and maintenance was used to meet the mission requirements for near-zero disturbances and absolute minimization of potential contamination. In most cases, the dominant reasons for not using onboard propulsion are tradi tion and organizational structure. In a typical mission organization, the responsibil ity for launch is given to some segment of the organization or perhaps an entirely different organization or nation. In this case, the use of onboard propulsion to “fly” the spacecraft from an initial orbit to its final mission orbit constitutes a shifting of responsibility and budget from one organization to another. Consequently, there is a strong tendency to use the more traditional approach of upper stages, even though the available payload will be much less, or alternatively, the cost of launch much higher. In addition, some very simple spacecraft will not have an onboard propulsion system and, therefore, are fully dependent on whatever orbit is provided by the launch vehicle. If there is no inherent need for onboard propulsion, then the funda mental trade becomes the cost of the launch vehicle or the launch vehicle plus upper stage, versus the cost of adding an onboard propulsion system plus the gain in avail able mass at the destination. Table 12-7 compares alternative scenarios for a low-cost Jupiter mission based on (a) the traditional approach of using the launch vehicle upper stage for final injection to the transfer orbit, (b) using a Venus-Earth-Earth swing-by as was done for the Galileo mission described in Sec. 1.1.3, and (c) using on-board propulsion to provide direct transfer to the target planet in a modified launch mode as proposed by Meissinger, et al. [1997, 1998]. The planetary swing-by approach provides 4 to 5 times the payload mass at Jupiter, but at the cost of increasing the transfer time from 2.3 years to over 6 years. The on-board propulsion approach increases the payload available from the traditional approach by 50 to 100%, but with no increase in transfer time. T A B L E 12-7.
Alternative Launch S cenarios for a L o w -C o st Ju p ite r M ission. T h e number below each vehicle is the baseline mass in a due east orbit at 185 km. See text for discussion. In Jupiter Orbit
At Jupiter Arrival Traditional Direct
VenusEarthEarth
MLM Direct* M = 143 kg T = 840 d C =9.8
VenusEarthEarth
MLM Direct*
M = 49 kg T =840 d C = 28.6
M = 249 kg T = 2240 d C =5.6
M = 109 kg T = 840 d C = 12.8
Traditional Direct
Taurus X U $ (1400 kg)
M = 62 kg T = 840 d C =22.6
M =316 kg T = 2240 d C =4.4
Delta II 7925 (5089 kg)
M = 280 kg T = 840 d C = 18.2
M = 1004 kg M = 443 kg T = 2240 d T = 840 d C =5.1 C =11.7
M = 221 kg T - 840 d C = 23.0
M = 792 kg T = 2240 d C =6.4
M = 330 kg T - 840 d C = 15.4
Delta II 7925 H (6107 kg)
M = 336 kg T = 840 d C =18.2
M = 1205 kg M = 520 kg T = 2240 d T - 840 d C = 11.7 C =5.1
M = 265 kg T =840 d C =23.0
M = 950 kg T = 2240 d C = 6 .4
M = 397 kg T =840d C =15.4
M = mass at destination; T = transfer time in days; C = orbit cost function; MLM = modified launch mode ’ U .S . Patent No. 6,059,235, Microcosm, inc.
12.3
Estimating Launch Cost and Available On-Orbit Mass
609
Finally, the selection criteria for launch vehicles for interplanetary missions are similar to those for Earth orbit missions, but use a different set of parameters to char acterize vehicle performance. As discussed further in Sec. 12.7.3, launch requirements are defined in terms of the velocity or energy of the spacecraft at the point at which it has just escaped the gravitational pull of the Earth. The velocity at this time is called the hyperbolic excess velocity, and the energy per unit mass at this time is called the reference launch energy, C 3 = VOT2. Typical values of C 5 are 12 km 2/sec 2 for mis sions to Mars and 75 to 85 km 2/sec 2 for missions to Jupiter, corresponding to values of 3.5 and 8.7 to 9.2 km/sec, respectively. Fig. 12-9 shows the payload mass of representative launch vehicles as a function of the C 3 provided. Because of the strong effect of the upper stage mass and staging process, there is no simple standard form of the rocket equation applicable to all of the vehicles. The data presented is based on de tailed performance analysis by the vehicle manufacturers.
Launch Energy C 3 (km^/sec^)
Fig. 12-9.
Representative Range of U.S. Launch Vehicle Escape Mission Performance. Data should be compared with mass to L E O from Table 12-4 to determine the Orbit Cost Function.
12.3.3 Estimating Launch Cost What does it cost to put a payload in orbit? In general, there are two answers, depending on how we view the problem. On the one hand, having selected a launch vehicle and possibly an upper stage or orbit transfer mechanism, the cost is a fixed dollar value. Changing the mass of the spacecraft will have no impact unless we ultimately are forced to use a larger vehicle, or find that we can launch on a smaller one. The basic message here is that once we have selected a launch vehicle, there is little point in reducing weight beyond what is needed to accommodate that vehicle.
610
Orbit Selection and Design
12.3
The second view is to look at the launch capacity per kilogram, or specific launch cost. Generally, as shown in Fig. 12-10, launching twice as much mass will cost twice as much. From this perspective, reducing the mass of some portion of the spacecraft can allow us to either reduce the launch cost by potentially using a smaller vehicle, or alternatively, to increase the spacecraft mass in some other area, i.e., by flying an additional payload instrument or more propellant to allow a longer mission at the final destination.
J-1: 900kg to LEO ® $63,900/kg
25,000
V
I
--------Pegasus X L --------------------
20,000 Titian I
H-2
Taurus
15,000
Athena 2
+
ARC A 2P*
AR40
A lt A ll A * + Atli
AS
+
Deltail______fll lane 4 iAr44L)
7920,7925)
Athena 3
*
*
Kosmos C'1
Saturn V: 127,000kg to LEO 9 $6,500/kg
Long March
Q.
_raw ♦
CO Tsyklon
Zenit Z
Soyuz
5,000
10,000
Proton SL-13
15,000
20,000
25,000
30,000
Mass to LEO (kg) Fig. 12-10.
S pecific La un ch C osts. Data from Tables 12-4 and 12-5. All = Atlas II. See Fig. 12-7 for historical values.
There are two factors which make the specific launch cost inappropriate or at least a poor measure of effectiveness for launch cost. First is the quantization in launch vehicles. Launchers are not available in an infinite variety o f sizes, so that once we have chosen a vehicle, the cost per kilogram is immaterial. Second, there are econo mies of scale as vehicles get bigger, as shown in Fig. 12-10. In general, launching a spacecraft which is 10 times as massive will cost less than 10 times as much, because of the efficiency associated with larger vehicles. Many of the commercial low-Earth orbit communications constellations have chosen to use a variety of launch vehicles and typically launch several spacecraft on a single large vehicle. This allows them to take advantages of the economies of scale and, at the same time, implies that the cost per kilogram continues to be an important measure. As the mass of an individual spacecraft is reduced, this effect is multiplied by the number of spacecraft per vehicle and can allow either additional propellant on board each of the spacecraft, or poten tially, allow an additional spacecraft to be put on a given launcher. Despite the shortcomings, the specific launch cost is a useful concept in mission design. It allows us to get a rough estimate of the cost of getting our payload into lowEarth orbit at a time before launch vehicle options have been selected. The biggest
12.3
Estimating Launch Cost and Available On-Orbit Mass
611
shortfall of the specific launch cost model is that it only gives the cost of getting into low-Earth orbit. To use this concept effectively, we need to be able to apply it to multiple orbit types—to low-Earth orbit, geosynchronous orbit, Sun-synchronous orbits, interplanetary, and so on, so that we have an effective means of comparing orbit cost. This is the purpose of the Orbit Cost Function defined in Sec. 12.3.2. We can make use of this function to assess the relative costs between, for example, high-inclination posigrade orbits and Sun-synchronous orbits which are retrograde. Table 12-8 summarizes the step-by-step process of estimating the launch cost for a particular mission. Overall, it is an iterative process of selecting launch parameters, spacecraft mass, and launch cost. We need to adjust all of these factors to achieve our mission objectives at minimum cost and risk. T A B L E 12-8.
Estim ating La un ch C ost. Specific launch costs will depend on multiple additional factors, such as the specific vehicle selected and the negotiating leverage of the organization buying the launch. Thus, this process is more accurate at estimating relative cost than absolute cost. (S M A D = Wertz and Larson [1999].) Step
W here D iscu sse d
1. Establish orbit parameters
Secs. 12.1-12.5
2. Establish spacecraft mass
SM AD, Chaps. 10-11
3. Estimate Orbit Cost Function for direct ascent and onboard propulsion, if applicable
Sec. 12.3.2
4. Obtain corresponding L E O mass
Result of (2) X (3)
5. Obtain rough estimate of launch cost from specific launch costs in Table 12-3
Sec. 12.3.1
5A.For dedicated launch, select vehicle or vehicles based on L E O mass or direct injection
Sec. 12.3.1, S M A D Chap. 18, lsakowitz[1999]
6.
See text.
Determine launch cost and available mass margin
We begin the launch cost estimation process by determining the orbit parameters for the mission. We then estimate the spacecraft dry mass consisting of the operational payload and the spacecraft bus exclusive of the tankage required for any propellant used in orbit transfer. Next, we establish the Orbit Cost Function based on the AV bud get as defined in Sec. 12.2. If possible, we would like to do this for both direct ascent using the launch vehicle and for using onboard propulsion. This process allows us to compute a corresponding mass into low-Earth orbit for our particular mission. Given an amount of mass in low-Earth orbit, we can then estimate the launch cost from the specific launch cost given in Fig. 12-10 or using the costs for an appropriate vehicle or range of vehicles listed in Table 12-3, For dedicated launch, we can then do a pre liminary selection of a specific vehicle or range of vehicles based upon either the mass in low-Earth orbit or direct injection. This gives us the first estimate of our launch cost and available mass margin. We then proceed to iterate the process as appropriate. For example, we may be able to reduce our mass estimates somewhat in order to obtain a smaller, lower-cost launch vehicle. Alternatively, if we have large margins, we may choose to increase the system mass in order to provide more payload or greater pro pellant for other mission activities. There are a number potential error sources in estimating launch costs. First, the cost of launch is a negotiated figure based in large part on the law of supply and demand. Thus, a user with a large number of payloads may be able to negotiate a lower price in
612
Orbit Selection and Design
12.4
exchange for a long term contract. However, the capacity to negotiate a lower price depends on the existence of at least some level of competition within the launch vehi cle arena. Thus, if the political decision is made to use only national launch vehicles, then typically the competitive environment is substantially reduced and we can expect to pay significantly more for a given launch. Commercial launches, on the other hand, are usually not constrained to use vehicles from any particular nation or organization and, therefore, are in a better position to negotiate favorable prices based strictly on performance. In addition to price negotiation, there is also the potential for dual payload launches. For example, we may choose to use a somewhat larger vehicle with substantial launch margin, and use that launch margin to include a secondary payload, which can cover a significant fraction of the cost of the launch vehicle. Third, there are overall evolu tionary trends in launch costs that will impact the price which vehicle manufacturers can charge for vehicles of a given size. The net effect of these factors is to imply that the relative costs obtained by the above process will have a higher level of accuracy than the absolute costs. For some missions, it is also useful to run the cost estimation process “backwards.” To do this, we first select a representative launch vehicle, determine the mass available in low-Earth orbit, and use the Orbit Cost Function to estimate the mass available in the final mission orbit. This is a mechanism for sizing the payload to meet the capacity of a launch vehicle that falls within the budget constraints appropriate to the mission. A third way to use the Orbit Cost Function is to use it to determine the orbit ele ments that can be achieved given a launch vehicle and a spacecraft mass. Thus, given a spacecraft mass that I want to put in orbit about Mars, and a launch vehicle that is being provided, I can use the Orbit Cost Function to determine excess mass that can be used as propellant in order to reduce the transfer time and therefore the operations cost of the mission. In general, the Orbit Cost Function provides a much easier mechanism for includ ing launch and orbit transfer costs in system trades than the traditional approach of selecting a vehicle and making the process work within this constraint. We would like to include the mission orbit itself as a part of the system trade process and determine the impact of the mission orbit on the overall mission cost. The Orbit Cost Function is an analytic mechanism doing this.
12.4 Design of Earth-Referenced Orbits Earth-referenced orbits are typically among the most complex in terms of mission design for two reasons. First, there are a variety of specialized orbits that are design options which need to be evaluated and assessed. Second, the orbit parameters impact a large number of requirements in varying ways, such that no one orbit is ideal for all aspects of a mission. For example, a higher orbit typically provides better coverage, but is more expensive to get to in terms of the Orbit Cost Function, has lower reso lution for observations, requires more power for communications, and is in a more adverse radiation environment. Most of the steps in the orbit design process for Earth-referenced missions are the generic ones, defined in Table 12-3. The key issues which are specific to Earthreferenced missions are the orbit-related mission requirements, specialized Earthreferenced orbits, and the potential use of eccentric orbits. These are discussed in the
three subsections below.
12.4
Design of Earth-Referenced Orbits
613
12.4.1 Mission Requirements for Earth-Referenced Orbits As listed previously in Table 12-1, the first step in the orbit selection process af ter dividing the mission into phases is to accumulate the orbit-related mission requirements. The mission requirements which normally have the greatest effect on Earth-referenced orbits are listed in Table 12-9. The principal conclusion here is that a large number of mission requirements are orbit-dependent for Earth-orbiting missions. The second most important conclusion from the table is that the orbit pa rameter which is most important in orbit selection is the altitude, with the secondary parameter being the inclination. Altitude is a primary determining factor in nearly all of the orbit-related mission requirements. The altitude is the primary determi nant of most aspects of coverage, performance, spacecraft environment, launch cost, and even the end-of-life options available to the spacecraft. Consequently, the most common trade here is one of coverage as a measure of performance versus AV or OCF as a measure of cost. The processes for evaluating these numerically were given in Secs. 9.5 and 12.2, respectively. T A B L E 12-9.
Principal M ission Requirem ents that N orm ally Affect Earth-R eferenced O rbit Design. See Sec. 12.1 for a discussion of the orbit design process, (from S M A D = W ertz and Larson [1999].)
M ission R equirem ent C overage Continuity Frequency Duration Field O f View (O r Swath Width) Ground Track Area Coverage Rate Viewing Angles Earth Locations Of Interest Sensitivity o r Perform ance Exposure O r Dwell Tim e Resolution Aperture Environm ent and Survivability Radiation Environment Lighting Conditions Hostile Action Launch Capability Launch Cost On-Orbit Weight Launch Site Limitations G ro u n d C om m u nicatio ns Ground Station Locations U s e O f R elay Satellites
Parameter Affected
W here D iscu sse d Chap. 9
Altitude Inclination Node (only relevant for some orbits)
Altitude
Altitude (Inclination usually secondary) Altitude Inclination
S M A D Chaps. 9 ,1 3 Sec. 11.2,11.3
Chap. 11, S M A D Ch. 8
Sec. 12.3, S M A D Chap. 18
S M A D Chap. 13 Altitude Inclination
Data Timeliness O rbit Lifetime Legal o r Political Constraints Treaties Launch Safety Restrictions International Allocation
Altitude Eccentricity
Sec. 2.4.4 S M A D Sec. 21.1
Altitude Inclination Longitude in G E O
614
12.4
Orbit Selection and Design
Another key factor in altitude selection is the satellite’s radiation environment. As described in Sec. 2.3, the radiation environment undergoes a dramatic change at an altitude of approximately 1,000 km. (See Fig. 12-11.) Below this altitude, the atmo sphere quickly clears out charged particles such that the radiation density is relatively low. Above this altitude are the Van Allen belts, where the high level of trapped radi ation can greatly reduce the lifetime of spacecraft components. Consequently, most Earth-referenced mission orbits separate naturally into low-Earth orbit (LEO), below 1,000 km, and geosynchronous orbit (GEO), which is well above the Van Allen Belts. Mid-range altitudes have coverage characteristics which may make them particularly
Hemispherical Shield Thickness (g/cm2 of Aluminum)
2 ,0 0 0
4 ,0 00
6 ,0 00
8,0 00
10,000
Altitude (km)
Fig. 12-11.
in Practice, L E O is Defined at Below the Inner Edge of the Van Allen Radiation Belts.
valuable for some missions. However, the added shielding or reduced life stemming from this region’s increased radiation environment also makes them more costly. Nonetheless, the increased coverage to be obtained at higher altitudes has tended to push up the minimum altitude defined as LEO. For example, LEO communications constellations are being considered at altitudes of up to 1,500 km. At these altitudes, not only is the radiation environment more harsh, but the potential options for end-oflife are also more difficult. Low-Earth orbit spacecraft are often de-orbited or allowed to reenter at the end of their useful life. Above 1,000 km, this process occurs extremely slowly when driven only by drag and requires a very substantial AV if it is to be done by an onboard propulsion system. Consequently, in the regime above 1,000 km, the end-of-life option often regarded as the most desirable is to raise the spacecraft 50 to 100 km higher in order to remove it from the constellation structure. Note however that this process creates one or more “graveyard” regimes in which a large number of spent non-functioning spacecraft will accumulate. Whether this will ultimately provide a long-term debris hazard remains to be determined.
12.4.2 Specialized Earth-Referenced Orbits After having defined the orbit related mission requirements, the next step in finding an appropriate Earth-referenced orbit is to determine if any of the specialized orbits listed in Table 12-10 are appropriate. The advantages and disadvantages of these alternatives are summarized in Table 12-11 and discussed in more detail below. Chap. 2 provides a detailed discussion of the physical basis of each of these orbit types
12.4
Design of Earth-Referenced Orbits
615
and will not be repeated here. Specific references are listed in Table 12-10. For pur poses of mission design, we need to examine each of these specialized orbits individ ually to see if its characteristics will meet the mission requirements at a reasonable cost. Space missions do not need to be in specialized orbits, but these orbits have come into common use because of their valuable characteristics for many missions. Because they do constrain orbit parameters such as altitude and inclination, we first determine whether or not to use one of the specialized orbits before doing the more detailed de sign trades in Table 12-1. T A B L E 12-10.
Specialized O rbits U sed for Earth-R eferenced M issions. Typically the orbit design process consists of deciding whether any of the specialized orbits should be used followed by a general trade on other orbits." Chapter 2 references pro vide the physical basis for each orbit; Chap. 12 references discuss applications.
Orbit
Characteristic
Applications
Section
Geosynchronous (G E O )
Maintains nearly fixed position over equator
Communications, weather Secs. 2.5.1, 12.4.2.1
Sun Synchronous
Orbit rotates so as to maintain approximately constant orientation with respect to Sun
Earth resources, weather
Secs. 2.5.3, 12.4.2.2
Molniya
Apogee/perigee do not rotate
High latitude communications
Secs. 2.5.4, 12.4.2.3
Frozen Orbit
Minimizes changes in orbit parameters
Any orbit requiring stable conditions
Secs. 2.5.6, 12.4.2.5
Repeating Ground Track
Subsatellite trace repeats
Any orbit where constant viewing angles are desirable
Secs. 2.5.2, 12.4.2.4
’ Specialized orbits tend to divide orbit design into discrete regimes, each of which must be examined separately.
T A B L E 12-11.
Advantages and Disadvantages of Specialized Earth-Referenced Orbits
Orbit
Advantages
Disadvantages
Geosynchronous
Continuous viewing of one region
Poor polar region coverage, very high cost, very long range
Sun-Synchronous
Maintains roughly constant Sun angle
High cost
Molniya
Provides extended coverage of high latitude regions
High cost; widely varying ranges, Earth size, and rates drive up spacecraft cost; high radiation
Frozen Orbit
Maintains stable conditions
None
Repeating Ground Track
Coverage repeats
Orbit perturbations can resonate
It is frequently the existence of specialized orbits which yields very different solu tions for a given space mission problem. Thus, a geosynchronous orbit may provide the best coverage characteristics but may demand too much propellant, instrument
resolution, or power, or have too high an Orbit Cost Function. The trade of value vs. cost can lead to dramatically different solutions depending on mission needs. Thus, a Sun-synchronous orbit is typically 30% more expensive than a low-inclination pro grade orbit. Nonetheless, for many observation satellites such as Earth resources or weather, the advantage of being able to see locations on the ground at the same time of day and under the same lighting conditions on a continuing basis outweighs the additional cost of this type of orbit.
616
12.4
Orbit Selection and Design
12.4.2.1 Geosynchronous Orbits (GEO) Prior to the introduction of low-Earth orbit communications constellations, geosyn chronous orbit represented the single most used orbit in space. It accounted for approx imately 50% of satellite launches during the early 1990s. The physical properties of geosynchronous orbit and orbit perturbations were discussed in Sec. 2.5.1; in addition, Pocha [1987] and Soop [1994] provide very detailed discussions of geosynchronous orbit design and stationkeeping approaches. The principal applications for geosynchronous orbit are communications (both two-way and direct broadcast), weather, and Earth surveillance. However, as shown in Fig. 12-12, GEO is getting full. There are now relatively few orbital slots available over regions of particular interest, such as the mid-Atlantic or mid-Pacific areas used for communications between North America, Europe, and Japan. Similarly, slots over Europe and North America used for internal communications are also becoming full.
100
150
200
250
300
350
300
350
Longitude (deg East) 10
O IU S’ 6 (5 iA 4
° 2L 5 2 £ = 0 100
150
200
250
Longitude (deg East)
Fig. 12-12. Active Satellites in Geosynchronous Orbit as of 1/1/2000.
12.4
Design of Earth-Referenced Orbits
617
Overcrowding and the potential for satellite collisions represent major problems for geosynchronous orbit, as discussed in detail by Reijnen and De Graaff [1989]. In a sense, space is large and the potential for collisions between satellites is small. None theless, the cost of getting spacecraft to GEO is extremely high and there is no current potential for being able to retrieve or actively recover either spent spacecraft or debris. In addition, there are essentially no natural perturbations that will remove debris from geosynchronous orbit. Consequently, any sequence of either spacecraft explosions or inadvertent collisions which produce a significant debris cloud in GEO may make the orbit effectively unusable for the foreseeable future. Because there is currently no realistic mechanism for removing debris from GEO, the only viable mechanism for protecting it is to not allow actions which have the potential for generating debris. This means that spacecraft should be removed from geosynchronous orbit while they are still responding to commands and while there is still propellant available to do so. Thus, in order to preserve this unique resource, we must be willing to turn off operational spacecraft prior to the use of their last few grams of propellant. The “graveyard” for geosynchronous spacecraft is to put them in an orbit approximately 200 km above GEO. At this altitude, natural perturbations will cause the inclination to oscillate as discussed in Sec. 2.5.1. However, these oscillations in or bit elements will not bring the spacecraft back into the geosynchronous ring in any rea sonable amount of time, i.e., within several hundred thousand years. In addition, the AV required to raise the altitude is extremely small. Increasing the altitude by 200 km requires a AV of only 7.4 m/sec. As discussed in Sec. 2.5.1, the principal perturbative forces in GEO are solar/lunar perturbations, which cause a general North/South drift or change in inclination, and the effect of the out-of-roundness of the Earth’s equator, which cause a very slow drift in the East/West direction. In both cases, the effects are extremely predictable and essentially continuous. Consequently, this represents an excellent application for elec tric propulsion for orbit maintenance. A nearly continuous low level of thrust can be used to maintain the spacecraft in geosynchronous orbit. Typically, the North/South stationkeeping is required for operational purposes so that the spacecraft can keep its target regime on Earth in view, and equally important, so that Earth antennas can be pointed at the spacecraft without having to be continuously steered. The East/West sta tionkeeping, while requiring only 10% of the AV of North/South stationkeeping, is perhaps more important in that it prevents either physical or communications interfer ence with satellites in a neighboring orbital slot. Without nearly continuous East/West stationkeeping, geosynchronous satellites would continuously slide through the orbit slots assigned to other spacecraft, thus providing both significant communications in terference and even the potential for satellite collisions. Synchronous orbits are possible around other planets and satellites as well, although, like the Earth, they may be significantly influenced by various perturbations. Both the formulas to determine these orbits and the characteristics of synchronous orbits for a variety of other central bodies are listed in Table 2-11 in Sec. 2.5.1.
12.4.2.2 Sun-Synchronous Orbits As discussed in Sec. 2.5.3, a Sun-synchronous orbit is one in which the perturbation due to the Earth’s oblateness causes the orbit to rotate in inertial space at a rate equal to the average rate of the Earth’s rotation about the Sun. Consequently, in a Sunsynchronous orbit, the position to the Sun relative to the orbit plane remains approxi mately constant. The general motion of the Sun relative to the orbit plane is shown in
Orbit Selection and Design
618
12.4
Fig. 2-27 in Sec. 2.5.1. From an applications perspective, the Sun-synchronous orbit has the principal advantage of maintaining approximately constant Sun angles. This means that for Earth resources satellites, for example, the angle of illumination will be approximately constant as photographs of a given region are taken over time. Conse quently, it is significantly easier to interpret the photos and look for changes over time. Sun-synchronous orbits have also been used by some missions as a means of elim inating a second gimbal in the solar array drive. Thus, rather than requiring a 2-axis gimbal, a single-axis gimbal is sufficient and the cost of the spacecraft can be margin ally reduced. This is an example of an application of the Sun-synchronous orbit which is probably inappropriate in most circumstances. The Sun-synchronous orbit requires an additional cost of approximately 30% to obtain this orbit. This cost is far in excess of the benefit achieved by having only a single gimbal in the solar array drive. In addition, it also adds an additional element of risk to the mission by creating a new failure mode. If the launch vehicle or propulsion system are unable to achieve or to maintain a Sun-synchronous orbit, then a failure of the mission is possible. If the system could operate under non-Sun-synchronous conditions, then there would have been no reason for the orbit in the first place. Consequently, choosing a Sun-synchro nous orbit for purely spacecraft bus functions is an incorrect engineering choice in most circumstances. Sun-synchronous orbits are also possible around other central bodies. Representa tive parameters for these orbits are given in Table 12-12. The advantages of these orbits are similar to their advantages when used on the Earth. That is, a constant level of illumination allows us to compare photographs taken at different times in order to more accurately determine physical changes which may be taking place on the planet or moon that we are orbiting. For example, Sun-synchronous orbits would be appro priate for examining seasonal variations on Mars, or variations as a function of volca nic activity or electromagnetic storms in the vicinity of Jupiter’s moon, Io. Note that in the case of other planets and satellites, Sun-synchronous orbits are significantly less expensive to achieve. Generally, the orbit plane in which the spacecraft arrives can be adjusted by very small AVs applied during the orbit transfer. Consequently, the high cost associated with Sun-synchronous orbits on the Earth is not necessarily a problem for other central bodies. However, the parameters associated with Sun-synchronous orbits for other central bodies, such as the required altitude or inclination may make them inappropriate for other reasons, such as viewing distance or excessive perturbations. T A B L E 12-12.
Representative Sun-Synchronous Orbits for other Central Bodies. A ltitude (km )
Inclination (deg )
Mars
1,000 5,000
94.85 144.52
147.2 388.4
Jupiter
1,000 5,000
90.08 90.09
181.5 196.7
Saturn
1,000 5,000
90.04 90.05
257.8 283.5
Uranus
1,000 5,000
90.01 90.02
183.3 232.4
Neptune
1,000 5,000
90.02 90.04
166.0 206.1
Central B o d y
Period (m in )
12.4
Design of Earth-Referenced Orbits
619
12.4.2.3 Molniya Orbits A Molniya orbit is a highly elliptical orbit inclined at the critical inclination of 63.4 deg (or 180 - 63.4 = 116.6 deg) in order to prevent apogee and perigee from rotating. The fundamental equations and physical properties for Molniya orbits were discussed in Sec. 2.5.4. The basic application of these orbits is for communications or observations at high northern latitudes where geosynchronous spacecraft are ineffec tive or inaccessible. Consequently, Molniya orbits have been used for communications in Russia and the former Soviet Union for an extended period of time. The properties of Molniya orbits for the Earth are summarized in Table 12-13 and the orbit ground track is shown in Fig. 12-13. Note that there are tick marks along the orbit at equal time intervals which clearly illustrates the long period of time that the spacecraft remains over high northern latitudes. Because each of the satellites tends to T A B L E 12-13.
Properties of Representative M olniya O rbits.
Period H o u rs
Eccentricity
2
0.1722
6
0.6020
A p o ge e Height (k m )
Perigee Height (km )
Fraction of O rbit with 9 0 ° < T ru e A n o m a ly < 270°
Fraction of O rbit with 1 2 0 ° < Tru e A n o m a ly < 240°
3,076
300
80.45%
71,72%
20,485
300
92.93%
86.81%
12
0.7493
40,178
300
96.38%
92.46%
24
0.8421
71,440
300
98.16%
95.87%
36
0.8795
97,661
300
98.77%
97.14%
48
0.9005
121,065
300
99.07%
97.81 %
72
0.9241
162,688
300
99.38%
9 8.50%
Fig. 12-13.
A M olniya O rb it with a 12-H our Period. With two satellites in this orbit, one of them will always be within 24 deg of apogee in true anomaly and with three satellites one will always be within 15 deg.
620
Orbit Selection and Design
12.4
“hang” at apogee, it is straightforward to use two or three satellites in orbits of this type to provide good coverage of far northern or far southern regimes. Unfortunately, there are a number of disadvantages of the Molniya orbit, such as the non-stationary ground track with respect to the Earth, and the fact that it travels through the Van Allen radiation belts on a continuing basis, and consequently has a severe radiation environ ment. Nonetheless, the Molniya orbit provides the only realistic mechanism for providing extended coverage in high latitude regions, and therefore, is appropriate for any type of communication, observation, or sampling system that requires coverage of this type. The period and eccentricity of the Molniya orbit depend on the gravitational param eters of the Earth. The fact that it is at the critical inclination of 63.4 deg, however, does not depend on any parameters of the Earth and is a function of the spherical harmonic expansion (i.e., an effect of the relative importance of larger gravitational effects at the equator and lower effects at the pole). Consequently, Molniya orbits about any of the other planets or satellites will always be at an inclination of 63.4 deg relative to the equator of that central body. Thus we might choose a Molniya-style or bit for monitoring the polar regions of Mars or providing communications or observa tions for scientific activities at the poles and the Moon.
12.4.2.4 Repeating Ground Track A repeating ground track is simply an orbit in which the ground trace of the satellite will repeat itself after one or more days. This is extremely convenient for Earth obser vations and communications in that the general characteristics of the flight path with respect to a ground target or observer on the ground is a repeating pattern. Recall from Sec. 2.5.2 that a repeating ground track comes about when we go through an integral number of orbits in a sidereal day, taking into account the orbit rotation due to the oblateness of the Earth. Consequently, the fundamental requirement for a repeating ground track is: (12-4) (12-5) where ( 12-6) (12-7) ( 12-8) where the initial estimate of a for Eqs. (12-6) to (12-8) is just the repeating ground track value of the semimajor axis in the absence of the oblateness correction, i.e., ainitial= ^ l ( / ^ ) -2/3>where j and k are integers. As discussed in Sec. 2.5.2, Q is the node rotation rate, cb is the argument of perigee rotation rate, and M is the mean anomaly rotation rate due to oblateness. As usual, the parameters, a, e, and i are the semimajor axis, eccentricity, and inclination of the repeating ground track orbit and R is the radius of the central body. Values of K\ and K2 for various objects in the
12.4
Design of Earth-Referenced Orbits
621
solar system are given in Table 12-14. (See Sec. 2.5.2 for the formulas for these con stants.) Note that with the constants in the table, the rotation rates are all expressed in deg/sidereal day = deg/day*, where “sidereal day” is the rotation period of the body the satellite is orbiting about relative to the fixed stars. The satellite com pletes j orbits in k sidereal days. T A B L E 12-14. Body
Earth
Repeating Ground Track Parameters for Representative Central Bodies.
R adius, R (k m ) 6,378.136
Day* (s e c )
A* (k m 3/sid d a y2)
8.616 410 04 x 104 2.959 3 1 0 2 0 x 1 0 15
Kz
*1 (k m )
(k m 3-5/sid d a y)
42,164.173
1.029 5 4 9 6 4 8 x 1 0 1 4
Moon
1,738
2.360 591 x I O 6
2.732 032 x 1016
8.023 799 x 1 0 s
7.590 2 7 4 x 1 0 1 °
Mercury
2,439
5.067 0 3 2 x 1 0 6
5.656 674 x 1 0 17
3.666 2 9 2 x 1 0 6
2.013 3 3 8 x 1 0 1 ’
Venus
6,052
2.099 606 x 1 0 7
1.435 7 9 9 x 1 0 2 °
5.988 628 x 1 0 7
8.887 2 9 6 x 1 0 1 2
Mars
3,397.2
8.864 266 x 104
3.382 663 x 1014
2.081 545 x I O 4
3.126 615 x 1011 2.274 883 x 1016
Jupiter
71,492
3.572 986 x IO 4
1.618 757 x 1 0 17
8.884 2 1 4 X 1 0 4
Saturn
60,268
3.780 000 x 104
5.421 019 x 1016
6.405 5 8 9 x 1 0 4
1.043 377 x 101®
Uranus
25,559
5.616 000 x 104
1.827 3 9 6 x 1 0 1 6
5.804 5 2 3 x 1 0 4
7.947 7 9 3 x 1 0 1 4
Neptune
24,764
6.635 5 2 0 x 1 0 ^
2.998 011 x
1016
7.651 2 4 3 x 1 0 4
3.185 5 1 4 x 1 0 1 4
1,151
5 .5 1 8 109 X 105
2.740 457 x 1014
6.566 5 7 3 x 1 0 4
N/A
Pluto Phobos
11.3
2.755 3 8 4 X 1 0 4
5.385 4 1 0 x 1 0 5
1.115318 x 101
N/A
Deimos
6.3
1.090 7 4 9 x 1 0 5
1.889 5 0 3 x 1 0 6
4.241 016 x 101
N/A
Io
1,821.3
1.528 535 x 105
1.392 519 x 1014 2.226 665 x 1 0 4
N/A
Europa
1,565
3.068 220 x 105
3.015 221 x IO 14 4 .5 8 3 899 X 104
N/A
The principal merit of a repeating ground track is that we continue to sample the same region of the planet’s surface on a repeating basis. This is useful, for example, in planet observations where I’m interested in determining changes from one time to the next. This can be particularly important for evaluating changes in vegetation, water levels, pollution parameters, population growth, or deforestation. The principal de merit of a repeating ground track orbit is that it also flies through the variations in the planet’s geopotential in the same fashion on successive orbits. Thus, if I fly just to the left of the Himalayas on one day, I will do so on successive days as well. This allows the potential of resonance effects to build up, comparable to vibrations in a physical structure, which can greatly magnify the impact of the perturbations and cause signif icant larger variations in the motion of the satellite. Consequently, a repeating ground track should be chosen only if there is an operational reason to do so, since it may re quire significant propellant utilization to counter the cumulative perturbative forces. For the Earth, an interesting subgroup of the repeating ground track orbits are those for which the inclination, and therefore node rotation rate, is adjusted so that the repeat occurs on precise multiples of the civil day of 24 hours. For these orbits, the ground track not only repeats but does so at the same time each day, every other day, or on a weekly basis, depending on the orbit parameters chosen. I call an orbit of this type a civil orbit because it is one in which the spacecraft is keeping time with civil functions, rather than requiring the many users on the ground to keep track of a spacecraft ephemeris. Thus, for example, for relaying data to the home office, we could catch the “5:10 to London” on a daily basis such that we could transmit data at the same time each day and have it received at a fixed, later time.
622
Orbit Selection and Design
12.4
12.4.2.5 Frozen Orbits As discussed in Sec. 2.5.6, frozen orbits come about because circular orbits are not inherently stable. On the other hand, stable or frozen orbits are available with very low values of eccentricity, such that they are nearly circular for most practical applications. Frozen orbits can be used with essentially any low-Earth orbit and are largely used as a means of maintaining a more precise orbit and reducing the propellant requirements for orbit maintenance. For applications in which a circular orbit is desirable, a frozen orbit will somewhat reduce the propellant utilization and has no significant drawbacks. For additional details on creating a frozen orbit, see for example Vallado [2001].
12.4.3 Eccentric Orbits for Earth Coverage Eccentric orbits can be thought of in either of 2 ways: On the one hand, they add a level of complexity that increases the design cost of the spacecraft and the mission by requiring the spacecraft to work at multiple altitudes and at differing rotation rates if one side is to remain facing the Earth. On the other hand, eccentricity adds a degree of freedom in the design that can be used to adjust the orbit to best meet specific mission needs. Eccentric orbits can be used to minimize the number of satellites, maximize the coverage with respect to areas of interest, or combinations of these purposes. The use of eccentric orbits has been studied at length by Draim [1997,1998], who has made an extremely strong case for using eccentric orbits to achieve maximum mission perfor mance with a minimum number of satellites [Draim, 1999]. For any specific mission, the question becomes whether this additional level of performance is worth the increased cost and complexity of the spacecraft and mission parameters. As shown in Fig. 12-14, one of the most interesting examples of the use of eccentric orbits is using only 4 spacecraft to provide complete, continuous coverage of the surface of the Earth,* Another interesting example is the proposed, but not built
Fig. 12-14.
Total Coverage of the Earth can be Obtained with O nly 4 Satellite in Eccentric O rbits. Note, however, that the varying altitude may complicate the spacecraft design [Draim, 1 9 8 5 ,1987a, 1987b].
* This orbit has been patented by Draim, 1987a.
12.5
Design of Near-Earth Space-Referenced Orbits
623
Ellipso communications constellation which uses eccentric orbits in combination with a supplemental equatorial orbit in order to optimize a low-Earth orbit communications system with only 18 operational satellites as shown in Fig. 13-8 in Sec. 13.2. As in the case of Molniya orbits, virtually all satellites in elliptical orbits will need to be at the critical inclination of 63.4 deg in order to prevent the line of apsides from rotating. Since this rotation rate can be up to 14 deg per day, it is important to maintain the crit ical inclination with good precision in order to maintain apogee and perigee over an extended mission life. This implies that orbit maintenance will typically be necessary for satellites in long term elliptical orbits. However, the AV requirements to maintain these parameters are very modest.
12.5 Design of Near-Earth Space-Referenced Orbits In a space-referenced orbit, our objective is typically to be in space for manufac turing, low-gravity or high-vacuum experiments, or to observe celestial objects from above the Earth’s atmosphere. These orbits are used by celestial observatories such as Space Telescope, Chandra, or SIRTF. In these orbits we typically have only a minimal concern with our orientation relative to the Earth. Consequently, we se lect such orbits to use minimum energy, while maintaining the orbit altitude and possibly to gain an unobstructed view of whatever celestial objects we may be ob serving. As listed in Table 12-15, the mission requirements that normally affect the design of such orbits tend to be less stringent than for Earth-referenced orbits. We are looking for a reasonable environment, easy accessibility, and good communi cations, all of which can be satisfied by a wide range of orbits. Consequently, the orbit design process is typically much more straightforward than it is for Earth-referenced orbits. Of course, some space-referenced mission orbits are designed spe cifically to sample space, such as magnetic field or solar wind explorers. In these cases, the orbit design is chosen to sample the region of interest on a schedule ap propriate to the science return. T A B L E 12-15.
Principal Requirem ents that N orm ally Affect the Design of S pace -R efer enced O rbits. Typically none of the requirements is as stringent as for Earthreferenced orbits. R equirem ent
W here D iscu sse d
Accessibility (AV'or O C F }
Sec. 12.2
Orbit decay rate and long term stability
Sec. 2.4.4
Ground station communications, especially for maneuvers
Sec. 9.4
Radiation environment
Sec. 12.3
Thermal environment (Sun angle and eclipse constraints)
Secs. 6.3.3,11.2
Accessibility by Shuttle or transfer vehicles
Sec. 12.1
As listed in Table 12-16, there are a few specialized space-referenced orbits that need to be considered in much the same fashion as specialized Earth-referenced orbits. The Sun-synchronous orbit was discussed previously in Sec. 12.4. This type of orbit may be appropriate for maintaining a constant Sun angle with respect to a satellite in strument such as an orbiting solar observatory.
624
Orbit Selection and Design
T A B L E 12-16. Orbit SunSynchronous
12.5
Specialized Space-Referenced Orbits. Characteristic Orbit rotates so as to maintain approximately constant position with respect to S u n
Application Solar observations; Missions concerned
Where Discussed Secs. 2.5.3,12.4.2.2
about Sun
interference or u n ifo rm lighting
Lagrange Point Orbit
Maintains fixed position relative to Earth/Moon system or Earth/Sun system
Interplanetary monitoring; potential space manufacturing
Secs. 2 .5.5,12.5
Statite
Solar radiation pressur© balances gravity
Interplanetary monitoring and communications
[Forward, 1989]
An intriguing space-referenced orbit is the statite, designed and patented by For ward [1989]. The statite is not an orbit in the usual sense, since it uses solar radiation pressure to balance the Earth’s gravitational force to maintain a spacecraft in an ap proximately fixed position relative to the Earth and the Sun. Such a spacecraft would have a rather remarkably large area to mass ratio and would be a substantial distance away from the Earth’s gravitational field. Nonetheless, like nearly all specialized orbits, there are potential applications such as monitoring solar activity. Perhaps the most interesting and most widely used of the space-referenced orbits are the Lagrange point orbits, described in Sec. 2.5.5. These are orbits which use the gravitational attraction of two orbiting bodies such as the Earth and the Moon or the Earth and the Sun to maintain the spacecraft in a constant orientation relative to the two bodies. The locations of Lagrange point orbits are shown in Fig. 12-15 and Fig. 2-29 in Sec. 2.5.5. Because of their long-term stability, these orbits have a wide variety of potential applications as summarized in Table 12-17. Recall from Sec. 2.5.5 that the L4 and L 5 Lagrange points are stable, such that objects orbiting these points will remain there indefinitely. In contrast, Lj, L2, and L 3 are unstable equilibrium points such that an object placed there will ultimately drift away. However, it takes a relatively small amount of propellant to maintain a space craft in a halo orbit about one of the Lagrange points. In any case, the halo orbit is necessary in most cases in order to provide communications. For example, a spacecraft at the L 2 Lagrange point on the far side of the Moon will be useful for communications with any lunar far side stations but would need to be in a halo orbit in order to provide line of sight communications back to the Earth. Similarly, solar monitoring stations have been maintained for extended periods at the Earth-Sun L | Lagrange point in order to provide advance warning for solar storms. Again the halo orbit is required in order to keep the spacecraft away from a direct line with the Sun, which would prevent communications with the spacecraft due to the high level of radio noise from the solar disk. The long term stability of the Earth-Moon L4 and L 5 Lagrange points make these an ideal location for very large space colonies or space manufacturing facilities. Because they are at the distance of the Moon, the gravity-gradient forces will be ex tremely small such that a very large region of extremely low microgravity is possible. In addition, massive facilities are possible there, because no AV is necessary to main tain the orbit and there is no drag to provide orbit decay, as will occur for the Space Station. Material for constructing a large facility at the L4 and L 5 Lagrange points can
Design of Near-Earth Space-Referenced Orbits
12.5
Fig. 12-15.
625
La grange Point O rbits. Th ese orbits are stable with respect to the 2-body system they are part of and are convenient locations for observations, communications, and manufacturing. (See also Fig. 2-29.)
T A B L E 12-17.
Potential Applica tion s of Lagrange Point O rbits. E -S = Earth-Sun, E-M = Earth-Moon, H = Halo orbit Lagrange Point
Application
E -M L4, L5
Space Manufacturing
e - m l 4, l 5
Space Observatory
E -M L4, L5
Lunar Communications
E -M L4i Ls
Large Space Colonies
E -M L2 H
Lunar Far Side Communications
E -S L , H
Monitoring solar activity
E -S L 3 H
Observing events not visible from Earth
be brought at relatively low AV cost from the surface of the Moon. The extensive work by O’Neill* and his colleages in the 1960s and 1970s has clearly established the fea sibility and potential utility of large colonies at these locations. [Heppenheimer, 1977; O’Neill, 1976; O ’Leary, 1982; Faughnam et. al, 1987]. Unfortunately, the continuing high cost of launch has prevented any of these from advancing beyond the mission study phase. * O’Neill originated the concept as part of his undergraduate physics classes at Princeton in 1969. After several years working on technical details, he first published the concept of colo nizing space itself (at L4 or L5), rather than a planetary surface, in the journal Physics Today in Sept., 1974 [O’Neill, 1974]. Substantial public enthusiasm led to the creation of the L-5 So ciety in 1975. In 1987 the L -5 Society merged with the National Space Institute to form the National Space Society.
626
Orbit Selection and Design
12.6
Lagrange point orbits are also possible with respect to other pairs of celestial bodies. For example, spacecraft at the L 4 or L 5 Lagrange point with respect to Ju piter or the Sun could be used for monitoring the asteroid belt, Similarly, a Lagrange point orbit between any of the major planets and their larger moons could be used as a stable orientation with which to observe that planetary system. Thus, long-term stable (or nearly stable) observation platforms could be created at any of several of the Lagrange points within the systems of Jupiter, Saturn, Uranus, or Neptune.
12.6 Design of Transfer and Parking Orbits Typically, the parking orbit, or storage orbit, is a low-Earth orbit high enough to reduce atmospheric drag, but low enough to be easy to reach. We may store sat ellites in these orbits, referred to as on-orbit spares, for later transfer to a higher altitude, or may use them as a place for test and check-out following launch or wait ing for the appropriate transfer conditions to occur. Consequently, the principal is sues for parking orbits are accessibility and matching the orbit conditions on either end to what is needed. For example, a Mars probe may be launched into low-Earth orbit for test and check-out while awaiting the appropriate opportunity to begin the Mars transfer phase. The fundamental purpose of a transfer orbit is to get the spacecraft where it wants to be, when it wants to be there. Ordinarily, the design process is straightforward, and we’ll want to do this with a minimum cost in terms of AV or propellant utilization. The physics of transfer orbits and the equations that govern them were described in Sec. 2.6.1. The principal requirements that normally affect the design of transfer orbits are shown in Table 12-18. As with space-referenced orbits, these are typically less stringent requirements than for Earth-referenced orbits, although there is often a requirement to match the desired end conditions with high precision in order to minimize the propellant cost of any corrective AV. T A B L E 12-18.
Principal Requirements that Normally Affect the Design of Transfer O r bits. Typically none of the requirements is as stringent as for Earth-referenced orbits. Requirement
Transfer A V
Where Discussed Sec. 12.2
Transfer Tim e
Sec. 2.6
Departure and Arrival Conditions
Sec. 2.6.1
Ground Station Communications, Especially for Maneuvers
Sec. 9.4
Radiation Environment
Sec. 12.3
Thermal Environment (Sun Angle and Eclipse Constraints)
Secs. 6.3.3,11.4
Required Navigation and Control Accuracy
Sec. 2.7
A further requirement that can arise in manned flight is for orbit transfer options that are “forgiving.” For example, the Apollo program used a free return trajectory such that if the burn required to go into lunar orbit did not occur on the backside of the Moon, then the spacecraft would return to the Earth with no further AV. This was critical for the successful rescue of Apollo 13, in which the propulsion system in the
Design of Transfer and Parking Orbits
12.6
627
command module was effectively destroyed by an explosion and the free return was necessary in order to bring the astronauts back to the vicinity of the Earth such that the lunar module propulsion system could be used to provide the appropriate deorbit AV for recovery of the astronauts. Similarly, in anticipated manned Mars missions, there is a high premium for reduc ing the transfer time because of the high cost of maintaining people in orbit for extend ed periods. In addition, in any manned planetary mission, we need to plan the transfer time such that a return to Earth can be achieved in an appropriate time. Both of these requirements lead to somewhat higher energy transfers for manned flight than the tra ditional Hohmann transfer. As shown in Table 12-19, there are specialized transfer orbits that are available to the mission designer. The physics of these transfers were discussed in Chap. 2. For each of the specialized transfer orbits, the fundamental objective is to reduce the AV required to achieve a particular set of mission objectives. Lunar or planetary flybys, for example, are used to provide additional energy by providing an effective elastic collision between the spacecraft and some other celestial object. Aeroassist trajecto ries are used principally to provide either a plane change or braking, as in the case of the Mars orbiter program. T A B L E 12-19.
Specialized Transfer Orbits.
Orbit
Characteristic
Application
Section
Lunar or Planetary Fly-By
Sam e relative velocity approaching and leaving flyby body
Used to provide energy change or plane change
2.6.3
Aeroassisst Orbit
Use atmosphere for plane change or braking
Used for major energy savings for plane change or reentry
2.6.2
Reduce total A V fo r orbits with large plane change
2.6.2, 12.6
3-Burn Transfer G o to altitude above final orbit (= bi-ell iptic transfer) to reduce plane change
As was discussed in Sec, 2,6.2, the AV required for simultaneous plane changing and orbit raising can be reduced by combining these maneuvers such that the total AV is the vector sum rather than the linear sum of the individual maneuvers. However, in cases where the plane change is extremely large, this AV can be further reduced by us ing a 3-burn transfer, in which the spacecraft is first raised to an altitude significantly higher than the final orbit altitude. At extremely high altitude, the velocity will be very low and consequently a plane change can be done with very small AV. Thus the plane change is accomplished at the high altitude and the orbit is then returned to the final altitude. Similar to the 3 -bum transfer, Belbruno, et al. [1991], has evaluated and pat ented several trajectories which make use of the Moon to further reduce the AV re quired for plane change maneuvers. The principal demerit of this approach is that it requires a fairly extended time for the maneuver, although substantial AV savings are possible in the case of large plane changes. The principal advantages and disadvantag es of alternative plane change mechanisms are given in Table 12-20. The various alternatives for direct orbit transfers have been described in detail in Sec. 2.6.1 (see specifically Fig. 2-32 and Table 2-16). Table 12-21 lists the relative advantages and disadvantages of using these alternative transfer methods and the prin cipal applications to which they apply. High-energy direct transfers are useful princi pally when the transfer time is critical, as in manned flight or some military missions.
628
Orbit Selection and Design
T A B L E 12-20.
12.6
Alternative Plane C h a n g e M echanism s. See Table 12-17 for physical mecha nisms and computations.
O rbit
Advantages
Typ ica l A pplication
Disadvantages
Do &V at lowest ve locity (highest altitude)
Smallest velocity is easiest to change
Altitude variation may be small
Orbit transfer
Combine AVwith orbit raising
Vector sum is less than sum of components
Works best when large intrack AV' is required
Orbit transfer
3-burn transfer
Plane change can occur at essentially no A!/ cost
Requires going to a higher altitude and then returning
Large plane change with moderate to high altitude final orbit
Use differential node rotation
No propellant cost
Tim e delay
Storage of satellites in a parking orbit
Aero-assist trajectories
No propellant cost
Requires precision ma neuver and aerodynamic surfaces; may be heating
Planetary exploration
Planetary fly-by
No propellant cost
Requires planetary swing-by and typically long delays
Out of ecliptic mission (using Jupiter)
T A B L E 12-21.
Typ e
C haracteristics of the Alternative Direct Tran sfer O rbits S h o w n in Fig. 2-32 in Sec. 2.6.1. See Table 2-16 for transfer characteristics. Ty p ic a l Accel.
O rbit S h ape
Advantages
Disadvantages
Applications
High Energy
10g
Elliptical & hyperbolic
Rapid transfer
Uses more energy than necessary + Hohmann disadvantages
Manned flight, military intercept
Minimum Energy, High Thrust (Hohmann)
1 to 5g
hohmann transfer
Traditional High efficiency Rapid transfer Low radiation exposure
Rough environment Thermal problems Can’t use S/C subsystems
Used for nearly all orbit transfers
Low Thrust Chemical
0.02 to 0.10g
Hohmann transfer segments
High efficiency Low engine weight Low orbit deployment & check-out Better failure recovery Can use spacecraft subsystems
Use when Moderate radiation exposure needed to 3 to 4 day transfer reduce cost, weight, risk, or to G E O disturbances
Electric Propulsion
0.0001 to 0.001 g
Spiral transfer
Can use very high lsp engines = major weight reduction Low orbit deployment & check-out Can have reusable transfer vehicle
2 to 6 month transfer to G E O High radiation exposure Needs autonomous transfer for cost efficiency
Rarely used. Has been proposed for G E O and interplanetary transfer
12.6
Design of Transfer and Parking Orbits
629
Typically, the A V penalty associated with high-energy transfer is small for minor increases in transfer time, but becomes very large for any major reduction in transfer time, such that very short transfers are prohibitively expensive in terms of AK The other alternatives of successive burns using low-thrust chemical or a spiral transfer using electric propulsion both take significantly longer than the Hohmann transfer and are of use principally when transfer time is not a major constraint. The low thrust chemical transfer offers the significant advantage of providing a much more benign environment for the spacecraft, since the principal disturbance forces and maximum loads frequently occur during orbit transfer. In addition, low thrust transfer allows the spacecraft to be fully deployed prior to transfer, such that it can be tested and checked out before being put in a totally inaccessible orbit. Electric propulsion transfer has the merit of extremely high efficiency but can take a very long time. Consequently, elec tric propulsion is an extremely efficient approach for orbit maintenance but has not yet been widely used for orbit transfer. In the design of orbit transfer, it is important to keep in mind the distinction be tween the basic requirements for launch vehicles and the requirements for in-orbit transfer. In launch vehicle design, high thrust is absolutely imperative in order to get to space as quickly as possible and minimize gravity losses which occur when thrust ing is not done at right angles to the central body. In a number of launch attempts in the early space program, the thrust to weight ratio was very close to 1 , such that the launch vehicle lifted a short distance off the pad, burned all of its propellant, held itself in the air, and crashed as soon as the propellant supply was consumed. Thus, a driving characteristic for launch vehicles is the need for a high thrust system. In contrast, once we have gotten into space, high thrust is no longer important and can be a major detriment to an in-orbit transfer system. A high thrust transfer stage frequently imparts the largest loads the spacecraft will ever see. Thus it has a strong potential for damaging the spacecraft. In addition, in any high thrust system, perfor mance is critical and all maneuvers must be done with precision both in terms of the maneuver itself and the timing of the maneuver. Typically, any failures of high thrust transfer will lead directly to mission failure. Similarly, because the forces are extreme ly large, high thrust orbit transfer stages need their own separate control system to control the very large disturbance torques that are generated, Consequently, the trans fer stage is nearly a separate spacecraft with a very large propulsion and control sys tem. However, in the on-orbit environment, much of this performance is either wasted or detrimental. An onboard propulsion system, for example, can provide the AV appropriate to orbit transfer but do so at much lower thrust levels. For most orbit trans fers, the time required for the bum is small relative to the transfer time and is unimportant in successfully completing the maneuver. If the orbit transfer is done by a sequence of low thrust bums, then individual bums become far less critical and an error or misfire on one bum may be able to be corrected on subsequent bums. Conse quently, low thrust orbit transfer tends to be significantly more fail-safe than high thrust transfer. In addition, the disturbance torques imparted by a low thrust system are far less, such that it may be possible to control the transfer using the control system already on board the spacecraft itself. This can significantly reduce the amount of ancillary equipment that needs to be flown, thus reducing both the weight and cost of the mission. As described at the end of Sec. 12.2, the use of onboard propulsion can have a dramatic impact on the mass available in the mission orbit, or, alternatively, on the launch cost because the entire mass of the transfer stage doesn’t need to be sent all the way to the mission orbit.
630
Orbit Selection and Design
12.7
The important point here is to make a clear distinction between the requirements for launch and the requirements for orbit transfer once we have achieved low-Earth orbit. These frequently are regarded as nearly the same process because they are often assigned to a single organization which has the responsibility for putting the spacecraft in its final mission orbit. While this has some possible organizational advantages, it typically provides a system which is less robust, higher risk, more expensive, and low er performing than would be the case using a low thrust system once we have achieved orbit. The critical issue here is to regard in-orbit transfer not as simply an extension of the launch process, but as a separate mission phase. As always, we want to achieve the overall mission objectives at minimum cost and risk. In many cases, we may be able to substantially reduce both cost and risk by making use of straightforward, low-cost, low-thrust technology for in-orbit transfer.
12.7 Design of Interplanetary Orbits Hans F. Meissinger, Microcosm, Inc. The selection and design of ballistic interplanetary mission trajectories have some aspects in common with Earth orbital missions, namely the use of classical Keplerian two-body orbit dynamics around a central gravity field (see Chap. 2), except that suc cessive mission phases involve different central bodies, such as first the Earth, then the Sun, and finally, a target planet such as Jupiter. The alternative of using non-Keplerian interplanetary trajectories will be briefly discussed later in this section. The transition between the motion within essentially one central force field, e.g., the initial geocentric motion, the subsequent heliocentric motion, and finally, the plan etocentric orbit at destination, must be taken into account for a more precise trajectory definition. However, the very short transition phases have only a secondary influence on the overall trajectory design, typically involving few days in a mission that spends many hundreds of days in heliocentric space. A key parameter here is the sphere o f influence of the planets in question which is the region in which the gravitational at traction of the planet is greater than that of the Sun. In the spacecraft-Earth-Sun orbit mechanics applying during the departure phase the radius of the Earth’s sphere-of-influence, of the order of 0.9 million km, as discussed by Wiesel [1998], is the parameter of principal concern. For preliminary trajectory design purposes it often is sufficiently accurate to use the patched conic approximation, as discussed by Bate et al. [1971], Prussing and Conway [1993], Vallado [2001], and others. This technique consists of determining the plane tocentric hyperbolic excess velocities at the departure and arrival planets, in the above example those at Earth and Jupiter, that match the heliocentric departure and arrival velocities, respectively, both in direction and magnitude, while the brief transition phases at departure and arrival are ignored. Computation of precision orbits is a trialand-eiror procedure that involves numerical integration of the complete equations of motion in which all perturbation effects are included. Principal mission objectives lead to the selection of specific trajectory characteris tics, mission durations, preferred launch dates, and the choice of flyby, atmospheric entry, orbiting or landing at the target. Other factors of concern in this process are the past exploration history and known characteristics of the target, the available (or pro jected) technology level required to perform the mission, and any scientific and oper-
12.7
Design of Interplanetary Orbits
631
alional priorities. Budget constraints generally are dominant. Trades between desired and achievable mission objectives generally involve a wide range of alternatives in mission specifications and are an essential part of the mission definition process. Appendix D.4 lists principal orbit parameters and physical characteristics of the planets from Mercury, with a semimajor axis of 0.390 Astronomical Units (AU), to Pluto, with a semimajor axis of 39.5 AU, about one hundred times farther from the Sun than Mercury. (1 AU = 150 million km is the semimajor axis of the Earth’s orbit). Both of these planets have significant orbital eccentricities of 0.206 and 0.249, respectively, and inclinations relative to the ecliptic plane whereas all other planets move in nearly circular orbits, and very nearly in the ecliptic. Planetary exploration, starting in the 1960s with visits to Mars and Venus, has since included flyby or orbiter missions to the other planets as well, except Pluto to which a mission is projected in the early 2000s. Launch energy requirements to reach the plan ets, especially Mercury at its close solar distance, and the outer planets, i.e., Jupiter and beyond, are extremely high (see below). To reduce the required energy planetary grav ity assist is often applied, as will be discussed later in this section (see also Sec. 2.6.3). In the exploration of Jupiter and other outer planets, the first missions were flyby missions, like Pioneer 10 and 11, launched in 1972 and 1973, and Voyager 1 and 2, launched in 1977. The Voyager missions (also known as “Grand Tour Missions”) in cluded successive flybys of Jupiter, Saturn, Uranus and Neptune (see Fig, 12-16). All four of these deep-space probes ultimately left the solar system after passing the orbit of Pluto, traveling at hyperbolic velocities relative to the sun. In this final phase of moving through the heliosphere in four different directions they have been used to gather astrophysical data on particles and fields, and other phenomena. Ultimately, it is of considerable interest to find the heliopause, where galactic influences begin to dominate, a range assumed to be of the order of 100 astronomical units (AU), depend ing on each vehicle’s direction of travel relative to the Sun’s own motion within the galaxy.
Fig. 12-16. Grand Tour Trajectories by Voyager 1 and 2.
632
Orbit Selection and Design
12.7
Having reached distances of more than 70 AU, and still functioning in 1998 after more than 25 years of operation, Pioneer 10 has no longer been actively tracked from Earth (in part for budgetary reasons), since its gradually deteriorating radio-isotope power generator only allows a greatly diminished transmission of scientific data. However, Pioneer 10 was still contacted occasionally for Deep Space Network (DSN) station test purposes. Pioneer 11 ceased to maintain radio contact with the Deep Space Network from its position outside the solar system, in the mid-1990s, i.e., prior to Pi oneer 10, because of its more rapidly diminishing operating power. The Voyager spacecraft, with a system design much more complex than Pioneer and hence more subject to subsystem failure, have had a somewhat shorter (2 0 -year) mission life, al though they achieved a remarkable overall mission success. This section primarily covers ballistic missions with high-thrust departure from Earth, followed by a free-flight heliocentric transfer phase, i.e., missions that have been flown to-date or are projected to be flown in the near future. In addition, the sec tion also briefly discusses “non-ballistic” mission profiles with prolonged low-thrust transfer phases that use solar-electric propulsion or solar-sailing (see Sec. 12.7.6). They offer the principal advantage of large launch cost savings due to either a greatly reduced propellant mass, in the case of electric propulsion, or the use of solar radiation pressure alone for providing propulsion in deep space. Earth-based-laser illumination is being contemplated also as a way to augment solar-sail effectiveness. The first solar-electric propulsion mission in deep space, the Discoverer comet ex ploration mission, DS-1, was launched in 1999 and is performing as intended on its way to close encounters of comets Wilson-Harrington and Borelly. Alternate concepts using nuclear power for electric propulsion also are being considered for deep-space exploration, at distances where solar power is no longer sufficient. However, ade quately dependable, long-life nuclear-electric power sources need considerable further development for this purpose. Table 12-22 gives a summary of some U.S. planetary missions since the launch of Pioneer 10 to Jupiter in early 1972. These missions give evidence of the great progress in space mission sophistication and technology evolution since the 1970s. The trend to ever-greater mission cost has been slowed down, and in some cases has even been reversed, to reflect the current space exploration planning philosophy. In spite of being among the earliest, ambitious deep-space exploration spacecraft, the Pioneer 10 and 11 and Voyager 1 and 2 surviving for many years past the initial mission goals have transmitted astrophysical science observation results from distances up to and more than 70 AU. This was undoubtedly a factor in proceeding with even more demanding planetary mission developments.
12.7.1 Overview of Interplanetary Transfer Trajectories Key parameters in defining interplanetary mission characteristics include the desired flight time to reach the target, the mission departure and arrival dates within a given mission opportunity, i.e., the specific launch and arrival “windows,” as mentioned before, and the departure and arrival velocities that are involved. These parameters affect the type and cost of launch vehicles having the performance capa bility suitable for the desired spacecraft size and mass. They also determine the amount of onboard propellant required for critical deep-space maneuvers, such as orbit insertion at the target planet. Also of critical importance are the velocity vector orien tations in planetocentric and heliocentric coordinates, and various orientation angles like those of the lines-of-sight to the Sun, to the target planet, and to Earth which de-
TABLE 12-22. Representative U.S. Planetary Missions, 1972-2000. M ission N am e
Pioneer 10, Jupiter Flyby
M ission Objectives Jupiter flyby. Safety of crossing asteroid belt determined.
Viking 1, Mars M ars orbital observa Orbiter/Lander tion, surface sample
L a u n ch V ehicle; Date
Ta rg e t E n c o u n te r Dates
Atlas/Centaur/ T E -3 6 4 ; 3/3/72
Jupiter 12/3/73
258
Direct flight to Jupiter. 1stto escape Solar System . = 76 A U from Earth in 2001.
Survival to lO x specified mission life. Discovered gravitational perturbation from a Kuiper Belt object (1995). Pioneer 11 followed Pioneer 10 passing Saturn (9/79) after Jupiter swingby (12/74).
Mars orbit insertion 6/19/76, landing 9/3/76
572
Direct mission to Mars. Lander descends from orbiter for surface obser.
Mars observation from orbit and surface. No signs of life detected. Followed by similar Viking 2 Mission.
722 each
Sw ingby of Jupiter, Saturn (1). Grand To ur of Jupiter, Saturn, Uranus, Neptune (2).
Closeup images of planets/moons. Survived > 20 yrs after launch. V oya ger 2 renamed V oyager Interstellar Mission. Farthest m an-m ade object from Sun.
Used multiple swingby via Venus, Earth (2x), to reach Jupiter.
Observation from orbit of Jupiter environment and moons. Discovered additional moons, Io volcanoes, Jupiter rings. 1 st use of Earth swingby. Reduces launch energy. C 3, from 78 to 13 km2/sec2. Close flyby of asteroid Gaspra.
Used Jupiter sw ingby to achieve required high orbit inclination (79 deg), for solar polar passages.
First look at solar emission in polar directions. Measured particles and fields far from ecliptic plane. Highest injection energy ever used.
Used Venus-V enus-Earth swingbys to reach Saturn.
Extended exploration of Saturn's atmosphere, environment, rings, moons. Surface data of Titan.
Initial excursion to 2.2 A U , return for Earth swingby after 23 mos.
First mission to orbit and land on an asteroid. Flyby, images of asteroid 253 Mathilde (6/27/97). Initial Eros rendezvous aborted; orbit insertion on 2nd approach. Landed 2/01.
Conventional 8-month trajectory to Mars.
N A S A low-cost Discovery mission. First “microrover” on Mars (Sojourner). Left names of 100,000 members of Planetary Society.
analysis. Look for life.
Titan IIIE/ Centaur; 8/20/75
Voyager 1 and 2, Outer PJanet Flyby
Close Jupiter and outer planet observa tions during flyby.
Titan IIIE/ Centaur; 9/5, 8/201977
At Jupiter 2/79 and 7/79, At Saturn 11/80 and 8/81.
Galileo, Jupiter Orbiter and Entry Probe
Detailed exploration of Jupiter and satellites.
Shuttle/1 US; 11/4/89
12/7/95
Shuttle/2 upper stages; 10/6/90
Jupiter swingby 2/9/92 Solar passes 9/13/94, 7/31/95. Again 00/01.
Ulyssesj Solar Observation of solar Polar Mission* polar regions from high latitudes.
(kg)
Orbiter 2223. Probe 339.
370
Cassini, Saturn Orbiter and Huygens Probe
Exploration of Saturn atmosphere, magneto- sphere, satellites, Titan surface.
Titan IV/Centaur. 2 added solid stages; 10/15/97
Saturn orbit insertion, 7/1/04. Huygens Probe at Titan 11/27/04.
Orbiter 2150, Probe, 350
Near-Earth Asteroid Rendezvous (N E A R )t
Rendezvous with and orbit Eros.
Delta 11-7925; 2/17/96
Eros orbit insertion 4/30/00
805
Mars Pathfinder Mission
Mars surface explora tion, use of microrover (Sojourner).
Delta 117925/PAM; 12/4/96
Mars arrival 7/4/97
* Developed b y E S A , launched and m anaged by N A S A .
Tra je c to ry Info
SpcM ass
890 at launch. 360, Lander
iR e n a m e d N E A R -Sho em aker.
Co m m e n ts
634
Orbit Selection and Design
12.7
termine the required spacecraft orientation for uplink and downlink communication, for performing critical maneuvers, for solar power generation, and for other missionspecific objectives. A comprehensive set of reference handbooks for (ballistic) missions to different planetary targets, covering past, present, and future launch and arrival dates have been published by the Jet Propulsion Laboratory starting in the 1980s by Sergeyevsky et al. [1983 and later], showing charts of all relevant parameters for these mission dates. These handbooks cover missions to Venus, Mars, Jupiter, and Saturn for many succes sive launch opportunities. The manuals include detailed definitions of all parameters involved and their geometrical relations. Only a few of these factors will be covered here. Readers who wish to study detailed aspects of planetary missions and their tra jectories are urged to become familiar with this important reference material. A dominant trajectory characteristic, and one of the first to be considered in select ing the mission profile to a planetary target, is the required amount of launch energy, and specifically, the energy required beyond Earth escape. This departure energy, C3,* defined as the square of the hyperbolic excess velocity, V^, is one of the two terms in the equation of the escape orbit vis-viva-energy
V? = V& + v l = v lc (1 + v l l V ^ ) M v}sc (1 + C 3 / V & )
(12.9)
where V} is the path velocity, and Vesc = ^]2jj,frp, the escape velocity at the periapsis (perigee) distance rp. [See Eq. (2-6) in Sec. 2.1.4.] Minimum energy transfers to Mars typically require VU = 3.5 km/s, those to Jupiter require at least 8.5 km/s, or energies C 3 of 12 and 72 km 2 /s2, respectively. The transfer time to the target planet, an impor tant mission parameter, is close to 0.6 years for a minimum energy Mars mission, and 2.4 years for a minimum-energy Jupiter mission corresponding to these C 3 -values. The time interval between successive (minimum energy) launch opportunities to a target planet depends on the synodic period of the Earth’s and that planet’s orbit, as previously discussed in Chap. 2. The inverse of that period is equal to the difference between the inverse of the orbit periods of Earth and target planet. The synodic period is 1.60 years for launch opportunities to Venus, 2.13 years to Mars, 1.09 years to Jupiter, and 1.04 years to Saturn. For the large orbit periods of the outer planets the synodic periods decrease to values that are closer and closer to 1 .0 year the more dis tant that planet is from the Sun. An important launch-time constraint in any planetary mission is the required phase angle at launch of the target planet relative to Earth, i.e., the angular difference in heliocentric space between the orbital positions of Earth and the planet in ques tion. This is illustrated in Fig. 12-17 for the case of Earth-to-Mars transfer. With an assumed transfer time of the Earth-to-Mars trajectory, tT = t2 - tx, the orbital motion of Mars between the launch time tj (at Earth) and the arrival time t 2 (at Mars) is given by 0) t (t2 - 1\), where (Q t is the average Mars orbital rate during that time interval. The phase angle yj is the difference between the spacecraft transfer angle and the angular motion of Mars during the transfer, as shown in the diagram. A launch delay of several days implies a phase angle reduction, and hence will require a shorter transfer time, and, generally, an increase in departure energy. * The subscript of the term C3 derives from an earlier designation in the literature of energy components, such as Cj and C2, that are no longer in use.
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Fig. 12-17.
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635
Phase Angle at Launch, Earth-to-Mars Transfer.
The minimum energy and flight-time to a specific target planet varies between mission opportunities because of the slight eccentricity of the Earth’s and that planet’s orbits and the corresponding slight variations of the perihelion and aphelion distances of the transfer trajectories that are used in different mission years. Thus, the required minimum energy Cg-requirements can vary as much as 5 to 10 km 2/s 2 and more in missions to Jupiter. An additional factor is the generally slight planetary orbit inclina tion relative to the ecliptic plane as will be discussed below. An ideal minimum energy Hohmann transfer, with a 180-deg central angle, involves co-tangential departure and arrival. (See Sec. 2.6.1.) For an actual transfer between planetary orbits the conditions of a strict Hohmann transfer are often not attainable, since the target planet, at the time of arrival, is almost always above or below the ecliptic plane due to its orbit inclination. In a mission to Mars, a planet with an orbit inclination of 1.85 deg relative to the ecliptic plane and a mean orbit radius of 1.52 AU, the maximum distance out-of-plane is only 0.0491 AU or 7.36 million km. Even for conditions with much smaller out-of-plane distances, with the target planet being close to its ascending or descending nodes at the spacecraft arrival time, a coplanar 180-deg transfer will not be possible. For transfer angles near 180 deg the spacecraft’s transfer orbit, in fact, must have an inclination that is considerably larger than that of the target planet and would increase very rapidly the closer the transfer angle approached 180 deg. A very useful illustration of the relationships between departure energy C3, depar ture date and transfer time, and hence arrival date is shown by the chart of departure energy contours, Fig. 12-18A, for a mission to Mars launched in the year 2011 and 2012. This type of chart, often referred to as a pork chop plot because of its distinctive energy contour shapes, shows two separate regions of launch windows, with arrival dates in mid and late 2011 and beyond.* (See Sec. 4.1.). The two nested sets of launch * In the JPL Planetary Mission Handbooks by Sergeyevsky et al. [1983 and later] referred to in the beginning of this section, the C3-contours are plotted in terms of departure and arrival dates, with transfer time indicated by sloping lines. Their appearance, therefore, is somewhat different from those shown here.
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(A )
Departure Time (Days Past Epoch)
(B) Fig. 12-18.
Representative Earth-to-Mars M ission Parameters (“ Pork Chop Plots”) for 2011-2012 Launch Opportunity. Epoch is 1 Jan 2011. (B ) Contours of departure energy C 3(in km2/s2), and (A ) Contours of Mars arrival velocity (in km/sec). Type-I and Type II mission contours are shown, separated by a curve of 180 deg transfer angle.
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energy contours characterize the faster Type-I missions, with heliocentric transfer angles less than 180 deg and the slower Type-II missions, with transfer angles greater than 180 deg. The minimum launch energies fore these mission types are about 9 to 10 knvVsec2, with flight times of about 250 and 320 days, respectively. Fig. 12-18B shows two corresponding nested sets of contours that indicate the hyperbolic arrival velocities, V^, at Mars for the same range of launch dates and flight times as the departure energy, C 3 , contours, for Type-I and Type-II missions. The min imum arrival velocities are about 3 km/sec for Type-I and Type-II transfer trajectories. These minima occur at departure dates and flight times that differ somewhat from the minima of the departure energy contours in Fig. 12-18A. The selection of a preferred launch date and flight time depends on which of the two parameters should best be minimized, the departure energy or the arrival velocity. A compromise date and flight time between the two minima generally will be the preferred choice. The hypothetical Hohmann transfer from Earth to an ideally coplanar Mars orbit would have a flight time of 259 days, between those of the two actual minimumenergy transfer-orbits given in the chart, i.e., in the swath that separates the two mission sets. As indicated by the very steep contour line variation designating this “separating wall” between Type-I and Type-II missions, very much larger launch energies would be required for transfer angles close to 180 degrees. Actually, a purely ballistic transfer would require a departure at a very large, out-of-ecliptic angle, (see Fig. 12-19A) and hence, with an unacceptably high C 3 -value and, therefore, entirely unaffordable launch vehicle performance requirements. In interplanetary mission where no specific target object is to be reached at the required heliocentric distance, i.e., missions with general astrophysical observation objectives in the ecliptic plane, these restrictions do not apply, and the lower-cost Hohmann transfer can be used. (The pork chop contours in Fig. 12-18 correspond to those in the JPL Handbooks referred to above, but differ in terms of the launch, arrival date and transfer time coordinates shown). The geometry constraints that preclude a co-planar departure trajectory can be cir cumvented by performing a midcourse plane change maneuver as illustrated in Fig. 12-19B (from Fimple [1963]). The minimum amount of plane change that will be required for a given out-of-plane position of the target planet occurs at the location 90 deg prior to the target arrival, as indicated by the broken-plane transfer trajectory diagram. However, because of the precise magnitude and direction and the precise lo cation of this plane change being of critical concern, this option is generally not used.
Non-Direct Trajectories with Planetary Gravity Assist A very different class of mission profiles includes one or several encounters with other planets on the way to the target for the purpose of getting an energy boost from the effect of planetary gravity assist obtained at each swingby. This technique can greatly reduce launch energy requirements in some missions, specifically by making the initial target of the mission profile a planet with a much smaller launch energy requirement than a direct mission to the final target (see also discussion in Sec. 2.6.3). The concept of using planetary gravity assist to increase mission energy was first described in 1963 by M. A. Minovitch with reference to a Mars swingby on the way to Jupiter. The first spacecraft that actually used a large Jupiter swingby effect was Pioneer 1 1 , launched in 1973, on its way to Saturn and beyond, as discussed earlier in this Section. Jupiter, Saturn, and their moons were the main objects of observation. To
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®
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EclipticPlane
(A ) Single-Plane Transfer
Optimum Plane Change Point (B ) Broken-Plane Transfer Fig. 12-19.
Broken-Plane Transfer to Target Above Ecliptic Plane.
reach Saturn in 1979 after the Jupiter encounter, the spacecraft followed a trajectory between Jupiter and Saturn with a transfer angle of about 145 deg 7 returning en-route to about 4 AU solar distance. Other missions that used planetary swingbys to reach their targets are the Galileo Jupiter orbiter and entry probe mission, launched in 1989, as shown in Fig. 12-20 [D’Amario et al., 1989], and the Cassini Saturn orbiter mission launched in 1997 [Per alta and Smith, 1993]. With a gross mass of 2,560 kg and 2,470 kg, respectively, these missions could not have been launched directly to their destinations without the ben efit of multiple planet swingbys. Galileo used one Venus and two Earth swingbys, the Cassini mission requires swingbys of Venus, Earth and Jupiter. The Earth departure energy, C3, of about 13 km 2/sec 2 allows a near-Hohmann transfer trajectory to Venus, as the first target, with the Shuttle/Inertial Upper Stage used for launching Galileo, and Titan IV for launching Cassini. This large reduction in departure energy and the associated lowering of launch vehicle performance requirements, however, can only be achieved by accepting the considerably more complex and time-consuming mission profile and added design requirements such as extra thermal protection to withstand the 0.7 AU solar distance at the Venus encounter, as compared with solar distances > 1 AU in direct missions to the outer planets. An extreme case of planetary gravity assist was used by the Ulysses solar-polar mission, launched in Oct. 1990, with Jupiter chosen as the interim target, to allow a close solar approach of 1.3 AU with a departure energy of 128 km 2/s 2 using the Shut tle/Upper Stage as launch vehicle, see Fig. 12-21 [Luthey et al., 1989]. Its departure
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Fig. 12-20.
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639
Galileo 1989 Trajectory to Jupiter via Venus-Earth-Earth Swingby.
energy was by far the highest one used up to that date. The mission included the unique space exploration objectives of passing through interplanetary space at a high (78 deg) inclination relative to the ecliptic plane and of observing phenomena over the Sun’s South and North polar regions that had remained previously unexplored. These obser vations enhanced our understanding of surface phenomena at and near the Sun’s poles, their propagation into the solar system, and especially those affecting particles and fields phenomena near Earth. Without Jupiter’s gravity assist the high out-of-ecliptic orbit inclination of this mission could not have been achieved. It would have required the entirely unrealizable launch energy.
Fig. 12-21. Ulysses Trajectory (1990), Soiar Polar Mission via Jupiter.
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As mentioned before, Sec. 2.6.3 discusses the orbital mechanics of planetary grav ity assist. Of particular concern is the turn angle, y/ (and its supplement (p), which is the angle change of the planetocentric asymptotic velocity during the encounter, and the resulting change in the heliocentric velocity between arrival and departure dur ing the swingby event [Minovitch, 1963], This velocity change can provide a signifi cant increase in magnitude and a direction change either in-plane or out-of-plane, or both. Such changes depend primarily on the direction and magnitude of the hyperbolic velocity and on the closest approach distance, i.e., the periapsis radius, rp. Table 2-19 in Sec.2.6.3 includes a list of turn angles and velocity changes for rep resentative swingbys. For Jupiter and Saturn swingby closest approach distances of 5 to 1 0 planet radii are preferred to avoid unacceptably large, potentially damaging ef fects from planetary radiation belts during the encounter. In the case of Saturn, this also avoids a potential interaction with the planet’s rings. For example, in a Jupiter swingby, at 5.6 km/sec relative velocity, the asymptotic velocity direction changes by 135 deg for a closest-approach distance of 5 planet radii, [see Fig, 12-22], and by 116 deg for 10 planet radii. With a heliocentric arrival velocity of 8 km/sec in this case, the departure velocity would be increased to about 15 km/sec by the trajectory change due to the swingby effect. [Also see Eq. (2-82) to (2-84) in Sec. 2.6.3.] A small delta-V maneuver at the closest approach to the planet produces a powered swingby that can significantly increase the outbound asymptotic velocity at only a minor propellant cost. This mode, therefore, can achieve a much greater gravity-assist effect than the more commonly used passive swingby.
12.7.2 Navigation, Guidance and Control Earth-based tracking of the spacecraft to determine and correct any deviations from the intended flight path has been practiced from the time of the earliest interplanetary missions in the 1960s. More recently, emphasis is being shifted to include autonomous navigation over extended time intervals during the heliocentric phase of the mission and especially during the approach to the target, based on the use of onboard sensing instruments and onboard navigation and guidance computers, with this technology steadily advancing. Optical navigation techniques were used in the Voyager Neptune encounter [Reidel, et al., 1990], the Cassini Saturn orbiter mission [Gray and Hahn, 1995], and the NEAR mission [Miller, et al., 1995]. Serrano et al, [1996] provide a general assessment of various autonomous navigation techniques for interplanetary missions. DS-1, the first in NASA’s New Millennium series, used solar-electric pro pulsion to reach its two cometary targets, and autonomous navigation during the deepspace cruise [Wolff et al., 1998]. Considering the long duration of many planetary missions, a shift to greater navi gation, guidance and control autonomy is a significant cost-saving concern. These operations, if performed entirely by ground-based facilities, contribute significantly to the overall mission cost. Increased autonomy of navigation and guidance serves to reduce this cost element and also helps to mitigate problems that may be caused by un foreseen communication link outages between the ground station and the spacecraft. There generally is a need for several mid-course guidance maneuvers to correct tra jectory errors, followed by approach guidance corrections several weeks or months prior to arriving at the target. Typical guidance maneuvers require a total budget of up to several hundred m/sec in some missions depending on the mission complexity, especially when interim target encounters are included for gravity assist purposes.
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\C (km/sec)
(A)
Fig. 12-22.
S w in g b y A n g le C h a n g e vs. and rP. (A) Angles y and (p change with rp,(B) Trajectory deflection, (C ) Velocity directions.
and
Communication between the spacecraft and ground stations on Earth, i.e., NASA’s Deep Space Network (DSN), is always needed for timely status checks and commands, and for telemetry of data acquired by the science payload instruments. DSN tracking to derive navigation data may be performed at these and other times, but much less frequently, if the spacecraft has autonomous navigation capabilities. The DSN stations involved in tracking and communication are distributed around the globe to provide round-the-clock access as required. The three principal DSN sta tions are located at Goldstone (California), Madrid (Spain), and Canberra (Australia).
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Each of these stations in turn can give up to 10 hr of communication coverage, in accordance with the changing Earth-to-spacecraft viewing geometry. The Space Flight Operations Facility (SFOF) located at JPL controls the global tracking and communi cation activities.
12.7.3 Additional Launch Energy Considerations An overview of the transfer time and the Earth departure velocity required for out bound missions in the ecliptic plane as a function of target distance is shown in Fig. 12-23. This chart (which includes the idealized minimum-energy Hohmann trans fers with 180-deg transfer angles) is useful in providing a preliminary assessment of launch energy requirements and flight times and their variations with target distance. Type I and II trajectory classes are represented in this figure by the segments of the fixed-velocity curves shown here that are below and above the minimum-energy points indicated by a dashed line connecting the points of maximum target distance on each velocity curve. Based on the energy equation (12-9) in Sec. 12.7.1, the relation between the injection velocity, Vt, and K», is readily derived (with Vesc - 10.9 km/s, assumed fixed).
Note: 1. Numbers on curves are injection velocities U, (hyperbolic velocities K J in km/sec 2. Earth Departure from 300 km altitude parking orbit ■ Hohmann Transfer, Assuming Co-Planar ortiits
Fig. 12-23.
Transfer Time to Targets vs. Distance (in AU), for Various Launch Velocities.
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Design of Interplanetary Orbits
643
Figure 12-23 illustrates the important fact that near the minimum-energy Hohmann transfer the flight time varies greatly with only a minor change of the target distance or the departure velocity. This shows, for example, that in a Type-I transfer only little extra launch energy is required to reduce the flight time significantly from that required for the minimum-energy case. Corresponding flight-time and launch-energy variations can also be observed in Fig. 12-18 (see above) between C3-contours near the two minimum-energy points shown. Some interesting relationships between the velocity terms involved in the launch phase— as well as those occurring on arriving at the target planet—can be readily derived from Eq. (12-9) through a geometrical interpretation shown in Fig. 12-24. Figure 12-24A depicts the three velocity terms, and V±, as the two sides and the hypotenuse of a right triangle ABC. (Note: this is a geometrical “construct” which does not reflect any physical velocity directions.) The magnitude of Vesc corresponds to a given perigee radius. A circular arc of radius AB, constructed around point A and intersecting the hypotenuse AC at point P, subdivides the path velocity V) into two seg ments of magnitude Vesc and AVv The latter is the comparatively small velocity incre ment to be added to the escape velocity to achieve the desired amount of which reflects the algebraic definition of the velocity increment = v i ~ v esc =
+ v ~ ~ v esc
(12-10)
as derived from Eq. (12-9). The figure also shows an arc of radius 0.707 Vesc that designates the corresponding circular velocity and indicates the total velocity increment required to reach V,- from a a circular orbit of radius rP. Its magnitude is AVj + 0.293 Vesc.
7 x 103
8 x 1 0 3 104
1.5
rp
(A )
Fig. 12-24.
2
4 6 10s 10s
(km) (B )
Velocity Relationships at Earth Departure for Various Values of Escape Velocity. See text for discussion.
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12.7
A second diagram, Figure 12-24B shows the effect on AV,- of decreasing Vesc asso ciated with increasing the perigee radius rP (see the scales shown below this figure). It is apparent that AV, increases rapidly as Vesc is decreased until, with Vesc approach ing zero for rP « the velocity increment AV, approaches the value of Vm. Figure 12-25 shows the resulting values of AV, as function of rp for several values of , e.g., in a Jupiter mission with V „ = 9 km/s AV,- increases from about 3.5 km/s for low peri gee distances to 7.6 km/s for rp = 20 Earth radii, and to 9 km/s for rp —>
fp (km)
Fig. 12-25.
AV, Variation with
rP and
l ^ . Horizontal axis is on a log scale.
Figure 12-24 also illustrates the fact that a small variation of Vt typically leads to a larger variation of the magnitude of VM. This gain factor, G, can be obtained analyti cally by partial differentiation of in Eq. (12-9) with respect to V,-, viz., G = dVaa/d V i = Vi /V ae
(12-11)
This gain factor is the ratio of the side AC to BC in the triangle shown in Fig. 12-24A. For a Jupiter mission, with = 9 km/s and departure at low perigee, the gain is about 14/9 = 1.56. It decreases to 1.0 as rp approaches infinity. In the interest of obtaining the maximum V^-gain, the departure maneuver should be per formed at a location as close to the planet as possible, i.e., deep inside the planet’s “gravity well.” These characteristics are further illustrated by the curves shown in Fig. 12-25. A key factor in evaluating launch energy requirements concerns the payload per formance capability and cost of the launch vehicles that are under consideration for a specific mission. There is a wide range of launch vehicle candidates available, with injection capabilities ranging from a few hundred to several thousand kg at the reference launch energy C 3 = 0 km 2/s2. Launch vehicle types and performance characteristics, and their relation to mission performance and cost are discussed in Secs. 12.2 and 12.3.
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645
12.7.4 Earth Departure Geometry and Constraints As discussed in Sec. 12.7.1 the launch window selection for a given mission oppor tunity deals with the year, month and calendar dates, based on near-minimum launch energy characteristics, as indicated by the C3-contours given in Fig. 12-18 and also on a trade between minimum launch and arrival velocities. An additional time constraint is imposed by the one or two daily launch windows of typically 1 0 to 2 0 minutes. These windows are related to the orientation, in Earth-centered coordinates, of the de parture trajectory’s asymptotic velocity vector. Coordinates of concern are the decli nation and right ascension of this launch asymptote, designated as DLA and RLA, respectively. Figure 12-26 illustrates the launch geometry constraints imposed by the launch site and the orientation, in geocentric coordinates, of the hyperbolic departure velocity asymptote V^,. Figure 12-26A shows the departure trajectory in two dimen sions, starting at the launch site L, briefly remaining in a circular parking orbit, and then undergoing injection at P into the desired hyperbolic departure orbit. The direc tion of the departure asymptote is the -vector which is reached after going through the transfer angle 7]l . Launch
Fig. 12-26.
To Sun
To Sun
La un ch Constraints. (A) Departure trajectory in 2 dimensions, (B) In 3 dimensions, (C ) ground track, in Earth coordinates.
Figure 12-26B shows the corresponding three-dimensional departure geometry, with the planar motion of Fig. 12-26 A now depicted at some inclination relative to the Earth’s equator. This inclination is dictated by the requirement of the trajectory pass ing through L and V^.
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Orbit Selection and Design
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The geometry constraints are further illustrated in Fig. 12-26C which shows the departure trajectory ground track, passing through L and The departure asymp tote is shown here as a point (K ,) with given longitude and latitude coordinates relative to the launch site coordinates from which the ground track of the departure trajectory originates. The assumed launch site is Cape Canaveral. A due-East launch is assumed in this diagram, which uses a 90-deg launch-azimuth to make best use of the Earth’s axial rotation rate. The first segment of this ground track is the arc between the launch site and the perigee, P, of the hyperbolic departure trajectory, essentially the time interval during which the spacecraft remains in its parking orbit. The second ground track segment extends from P and has the central angle rjL, i.e., the angle which designates the limiting true anomaly of the departure hyperbola. It is important to realize that as a result of launch delay the launch site moves east ward in Fig. 12-26C, which requires a gradual change in launch azimuth. Thus, depar ture ground tracks with different inclinations to the equator and different parking orbit durations will be involved, (This is discussed further below). In any case, as long as the departure trajectory passes through V,*, the three-dimensional heliocentric direc tion of the departure trajectory will be the same, regardless of differences in geocentric orbit inclination. As an important consequence, no plane change maneuver to compen sate for different Earth orbit inclinations, and any associated launch velocity penalty, will be required, although this may appear somewhat paradoxical: Actually the orbit orientation around Earth (i.e. the orbit inclination relative to the equator), is not rele vant as long as the proper departure direction, along the -vector, is reached when leaving Earth. Corresponding conditions also apply to planetary arrival and entry into a planetary orbit of any inclination without requiring a plane change. This will be achieved simply by selecting an appropriate arrival point near the planet, as illustrated in Fig. 12-28, in the next section. Taking the 15 deg/hr eastward shift of the launch site due to Earth rotation into account means that the ground track shown in Fig. 12-26C would miss the target coordinates RLA and DLA even after only a short time interval, unless corrected by launching at a different azimuth angle that increases with time from the preferred 90 deg. This may add a noticeable take-off velocity penalty. However, by restricting the allowable daily launch window duration to about 2 0 min, 1 0 min before and after the optimum takeoff time, only a minor azimuth penalty is incurred. The coordinates DLA and RLA of the target point in Fig. 12-26C are of concern here. Typically DLA is within the range of -3 0 to +30 deg. For a launch that is co planar with Earth’s heliocentric orbit DLA varies seasonally between -23.4 and +23,4 deg, due to the inclination of the celestial equator relative to the ecliptic plane. Zero declination occurs at the times of spring and fall equinox. Ground tracks for a launch from Cape Canaveral can cover a DLA range between +28.4 and -28.4 deg without an azimuth penalty. For departure trajectories that are more high ly inclined relative to the ecliptic the required DLA can exceed this range by at least 10 to 15 deg. This may impose a correspondingly larger azimuth penalty in order to assure reaching the launch asymptote without requiring a significant plane change at point P. Useful references for further information are the series of NASA handbooks entitled Launch Vehicle Estimating Factors [McGolrick, 1971], TRW Flight Operations Handbook [White, 1963], and Orbital Operations Handbook [Wolverton, 1963],
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647
One launch-geometry aspect, readily explained by Fig. 12-26, concerns the number and spacing of daily launch windows. Again assuming a due-East launch, as shown in that diagram, it is apparent that the ground track generally intercepts a given set of target coordinates RLA and DLA twice a day, at times that are at most 1 2 hours apart. The closer the value of DLA is to the upper limit of due-East launch coverage, the smaller is the time difference between the two available daily launch windows. In the limit, i.e., if DLA equals the maximum northern or southern latitudes of a ground track launched due East, the two daily launch windows coincide. A DLA beyond these limits necessitates a launch azimuth that is above or below 90 deg.
12.7.5 Planetary Arrival Geometry, Orbit Insertion, Orbit Shaping In defining the arrival geometry at a planetary target, the transition from helio centric to planetocentric velocity characteristics must be considered. The process is analogous to the conversion from geocentric to heliocentric reference coordinates during Earth departure, but in reverse order. Figure 12-27 shows representative directions of the heliocentric and planetocentric velocities at the time of a rr iv a l. These velocities are the heliocentric velocities of the target, Vr , and of the arriving spacecraft, Generally, they are neither co-tangential nor co-planar with the arrival trajectory. A velocity triangle can be formed that includes these velocities and the relative velocity, V3, shown in Fig. 12-27A, the latter being the planetocen tric hyperbolic excess velocity. Typically, this velocity is oriented at an acute angle, 9, relative to V 7-, with a component pointing in opposite direction. In the case of co ta n g e n tia l arrival, i.e., in a Hohmann transfer, the relative velocity would be anti parallel to both V 2 and \ T. The diagram in Fig. 12-27A defines conditions with an offset or miss distance at arrival. A minimum offset is necessary to avoid a direct impact. Fig. 12-27B shows the hyperbolic approach orbit in planetocentric coordi nates. The offset distance normal to the arrival asymptote is designated as the im pact parameter, b, shown here in a two-dimensional diagram (see Bate et al. [1971]).
(A ) Fig. 12-27.
(B)
Planet Arrival Geometry, Velocity Definitions. (A) Velocity directions and offset miss distance x, (B) Corresponding approach trajectory.
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Orbit Selection and Design
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Figure 12-28 shows the corresponding three-dimensional arrival geometry. Two parallel arrival vectors are shown, as well as a plane normal to them, the so-called Bplane, or impact-plane. The figure shows the impact point B of a specific arrival asymptote, with an offset b from the planet center, i.e., the point in 3-D space that the planetocentric arrival velocity V3 must be guided to, so as to produce a specified en counter trajectory ground track that is indicated by the heavy trace. Other such traces shown in the diagram are various ground tracks, either of a flyby trajectory or a plan etary orbit that are achievable by choosing different impact points on the B-plane. Only a small offset terminal guidance maneuver, to be performed some weeks before arrival, is required to produce any of these ground-tracks, and the corresponding flyby or orbital trajectories. As visualized in this diagram, the set of ground tracks shown form a beach-ball-like pattern that intersect at the vertical impact point of the unde flected arrival trajectory. V„
Fig. 12-28.
Planet Arrival Geometry in 3 Dimensions Showing Impact Plane and Traject ory Ground Tracks, Depending on Choice of Impact Point B.
Table 12-23 lists the various orbital mechanics and geometry relations relevant to the diagrams shown in Figs. 12-27 and 12-28. An important characteristic of the planetary encounter phase is that of the planet impact size, which is defined by the envelope of grazing approach trajectories that separates trajectories that, at the closest approach, clear the planet’s surface from those that impact the planet, see Fig. 12-29 (also see Bate et al. [1971 j). The cross section of that envelope is called the collision cross section, b, having the radius at large distanc es from the planet. As an example, for a Mars mission with Vesc = 5.1 km/s and = 3 km/s, the ratio b/rc is given by b /r c = V1 + (V esc'V ^
)2
= V f + 25.31/9 = 2-93
(12-12)
Here, the collision cross-section is nearly three times the cross section of the planet itself. For a large planet such as Jupiter this ratio is very much larger, i.e., typically 10 = 6 km 2/s2. The collision cross section at Earth, for for Vesc = 60.2 km 2/s 2 and
Design of Interplanetary Orbits
12.7 T A B L E 12-23.
649
Equations U sed for Defining Planetary A rrival C onditions. See Figs. 12-26 and 12-27 for definition of V T, V2, V3. rT is target planet distance from Sun, a T is its orbits’s semimajor axis. See App. D.4 for additional equations.
Equation
Q uantity Heliocentric Arrival Velocity Equation, V2
v 2 = ■yjiVs 1 fr ) ( 2 _ f 7 f ar )
Relationship Orbital velocity at target planet
Relates to Fig. No 12-27A
(//g is Su n’s grav. constant) Heliocentric Flight Path Angle, y
12-27 A / = tan”1[e sin v / (1+e cos v)] (where v is true anomaly at intercept, e is eccentricity)
Rel. Encounter Velocity (= asympt. velocity),
Law of cosines
12-27A
Produced by miss distance along planet orbit
12-27 A
Law of sines
12-27A
= ^ V i + V ? - 2 V 2Vt cosy ( VT is target planet velocity)
Offset Distance (Impact Parameter), b
b = x s \n d
Orientation of Relative Velocity, 9
sin0=siny(l/2 /V 3)
Offset Distance, b, as Related to Closest Approach Distance rp
b = r P VP / V 3
Law of constant 12-27B angular momentum and 12-28
( VP is arrival velocity at rP)
returning planetary spacecraft, is about 2.4 times the Earth diameter, if the arrival velocity is 5 km/sec. It is interesting to note that for meteoroids, or even an asteroid on a collision course, approaching the Earth at a high velocity, e.g., 20 km/sec, the colli sion cross section would only be about 10 percent larger than Earth’s diameter. This means that on Earth we are not receiving as many meteoroid impacts per unit time as we might if the space debris were to arrive at very small velocities. Note that the collision cross section, as shown in Fig. 12-29, at a finite distance from the central body, actually is slightly smaller than its full size that is determined by the corresponding arrival asymptotes. Fig. 12-29 also shows the entry corridor, for planets that are surrounded by an atmosphere, a region of relatively small width adja cent to the collision cross section. Objects arriving inside this corridor enter the atmo sphere and may descend to the surface. The width of the entry corridor is greater than the atmospheric height in the same ratio [Eq. (12-12)] that determines the size of the collision cross section relative to the planet’s diameter. Maneuver requirements for planetary orbit insertion depend on the closest approach distance, the orbit dimensions, the escape velocity at closest approach, and the arrival velocity. Table 12-24 lists the equations to be used in calculating the required retro-maneuver velocity. Figure 12-30 shows the normalized retro-velocity AV/VeP versus the normalized arrival velocity / VeP for various values of the apoapsis-to-periapsis ratio R (where V eP is the escape v e lo c ity at the periapsis o f the a rriv a l tra je c to ry ). T h e s e are tw o
extreme cases:
650
Orbit Selection and Design
Fig. 12-29.
12.7
Collision Cross Section and Entry Corridor. (See also Bate et al. [1971].)
T A B L E 12-24.
Equations Used for Orbit Insertion at Planet Periapsis rp. (See Fig. 12-29).
Q uantity Retro-impulse, A V
Equation
Difference between arrival and orbit velocities at periapsis
A 1/^1/P1 - V P2
(where I/p-,
Escape Velocity at Periapsis, VeP
Relationship
Arrival velocity at periapsis (Vp2 is orbit velocity at periapsis)
- V e2P )
Orbit equation ^eP
/ fp
{fip is planet’s grav. constant) Periapsis Velocity of intended Elliptical Orbit, Vps
V P2 = Ve P ^ jrA I ( rA + rp )
Retro-impulse, AV, Relative to Escape Velocity at Periapsis,
a
Orbit equation (where R = r A / rp)*
v i veP = J v „ i v eP f
+1 -
/ (f l + ■ i)
See also characteristics of large and small planets indicated in Fig. 12-30
VeP T a is apoapsis and rP is periapsis radius of orbit around planet. See also Table 12-23.
(a) For small values of VoaiVeP (typical for large planets) circular orbits, with R = 1, require a very much greater AV than highly eccentric or bits. However, capture is generally less costly than for small planets. (b) For large V^ iVep values (typical for small planets) the normalized retro-velocity AVIVeP is much larger than for case (a), and the AV savings by the choice of a highly eccentric orbit are much smaller than in case (a). An interesting example is the initial choice and the subsequent modifications of the orbit dimensions of the Galileo orbit at Jupiter, shown in Fig. 1-11 in Sec. 1.1.3. The
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651
a. s? s <
Normalized Approach Velocity (VJVep) Fig. 12-30.
Retro-lmpulse Requirements vs. Approach Velocity.
highly eccentric initial orbit has an extremely large apojove-to-perijove ratio calling for a low orbit insertion AV-requirement. Subsequent multiple flybys of Jupiter’s moons allow their close observation and also serve to obtain a gravity assist from each encounter. This is used to gradually shrink the orbit dimensions and reduce the time intervals between subsequent close encounters. The process is known as orbit pump ing, with emphasis on orbit shrinking, pumping down in this case. A different gravity assist effect, known as orbit cranking that would result in successively changing the orbit inclination, is not being emphasized in this mission. The Cassini Saturn Orbiter with its Huygens Titan Probe, was launched on October 15, 1997, to reach Saturn on July 1, 2004 [Peralta and Smith, 1993]. It is one of the largest and most complex interplanetary spacecraft. Its flight included gravity assists by two Venus swingbys, an Earth and a Jupiter swingby in Dec. 2000. After orbit insertion at Saturn it will follow a series of “flower-petal orbits” around Saturn com parable to those flown around Jupiter by the Galileo Orbiter, again using the orbit pumping technique.
12.7.6 Other Propulsion Techniques Used in Planetary Mission Design This section covers mission profiles that employ low-thrust propulsion, rather than high-thrust chemical propulsion to achieve the energy required for transfer to distant planetary and other targets. This includes the use of electric propulsion, at a high specific impulse, and solar sailing which uses large, light-weight, deployed sail sur faces rather than propellant to produce thrust in the desired direction.
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Orbit Selection and Design
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Solar-Electric Propulsion Electric propulsion has been advocated for use in planetary missions since the beginning of space exploration (for example, Stuhlinger [1964]) because of the very large propellant mass savings it offers compared with conventional chemical propul sion. This due to its inherently much greater specific impulse, /^-values which for typ ical ion engines range from 3,000 to 5,000 sec compared with the 300 to 450 sec representative of storable or cryogenic bipropellant rocket engines. Electric propulsion technology and its applications are described in detail in a later volume of this series. Various electrostatic and electrodynamic thruster designs have been developed to date. The most promising type of thruster for primary propulsion in deep-space mis sions is the electrostatic, or ion thruster. Typically, it produces about 40 to 50 millinewtons of thrust per kW of operating power. In applications involving a spacecraft mass of the order of 1,000 kg the corresponding acceleration level is 40 to 50 micro-g. Thus, by using 10 kW of constant operating power to produce an acceleration of 450 micro-g in a mission that requires a velocity increment of 9 .5 km/s, typical for direct low-thrust propulsion transfer to Jupiter after Earth departure, a net thrust duration of about 8 months would be required. Actually, with solar-arrays as a likely power source, the power level decreases with the steadily increasing distance from the sun, and hence, the total thrust duration would be correspondingly longer. This example indicates the nature of mission characteristics based on the use of electric propulsion. Detailed dis cussions of the many factors involved in practical mission design and performance are covered in the extensive literature on the subject (e.g., Sauer [ 1993], and Sauer and Yen, [1994]). Meissinger et al. [1968], XJphoff et al. [1993], and Williams and CoverstoneCarroll [1997]. A fundamental trade must be performed regarding /^-selection between the desired propellant mass savings and the required extra mass of the power source, such as a large solar array. In any solar-electric propulsion application the propel lant mass decreases in inverse proportion with lsp, while the required power and, hence the solar array size and mass increase in proportion with Isp and desired thrust magnitude. An additional factor is the variation of propulsion efficiency with Isp. Without discussing specifics of this trade, the results show that optimal /sp~val ues typically are 2,500 to 3,500 sec for ion engine efficiencies of 60 to 70%, rather than higher Isp-values that would be suggested if propellant saving objectives are considered alone. At the optimum /™-value the propellant mass and the powerdependent mass elements are of equal magnitude, and consequently, the minimum total mass of the propulsion system and power source plus propellant mass is twice the mass of these /^-dependent parts of the system. A thrust level increase shifts the optimum Isp to lower values, whereas an increase of the total impulse require ment shifts it to higher values. Major incentives for the use of solar-electric propulsion in future high-energy missions are the expected cost savings afforded by using smaller launch vehicles for a given net spacecraft mass and, conversely, the much greater net spacecraft mass that can be delivered with a given launch vehicle size and cost. In many application the flight time of the low-thrust trajectory is nearly the same as that of a corresponding trajectory produced with conventional high thrust at Earth departure. This is explained by the cumulative effect of prolonged low-thrust applica tion. Thus, electric propulsion when used in interplanetary missions is not associated with a flight time penalty, a disadvantage that would affect Earth orbital missions with low-thrust primary propulsion.
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653
Advances in ion engine technology as well as in solar arrays and power condition ing subsystems have led to plans for applying primary solar-electric propulsion in the coming decades in high-energy missions to the outer planets, to Mercury, and to comets and asteroids, and especially m missions that include a rendezvous with these targets. Representative propulsion power requirements of such missions have been reduced to the range of 5 to 10 kW at Earth departure, i.e., at 1-AU initial solar dis tance, which typically can be provided by solar array sizes of 50 to 150 m2, depending on the solar cell type, the solar-array design and other design factors. Earlier studies, in the ’60s and ’70s, had projected propulsion power levels that were larger by up to an order of magnitude and therefore offered less of a practical advantage. An important consideration in current solar-electric mission plans is to use the low thrust propulsion only after Earth escape rather than starting with a spiral ascent phase from low Earth orbit. This would add many months of extra flight time and place a much greater demand on continuous thrust time requirements, and hence, propulsion system life. Obviously, thrusting at acceleration levels of 50 to 100 micro-g will be much more efficient after Earth escape at distances where the effective (solar) gravity force is smaller by two to three orders of magnitude. Barber and Meissinger [1969] have shown that optimal departure energies, C3, for Jupiter missions range from to 3 to 5 km 2/s2. This reflects the increased injection-velocity gain, discussed in Sec. 12.7.3 (see Fig. 12-24), for departure velocities V(- that are slightly greater than the Earth es cape velocity Vesc at low altitude. Related studies by JPL and elsewhere [Sauer 1997, and Meissinger, 1970] have shown the performance advantages of including gravity assists from Earth and other planets in the electric-propulsion mission profile. Meissinger and Dawson [2001] show the benefit in outer planet missions of adding an initial out-of-ecliptic mission phase, followed by an Earth swingby, and the advantages of this mission mode relative to one staying entirely in the ecliptic plane, but also using Earth swingby [Sauer, 1997]. Figure 12-31 shows typical trajectory plots for (a) a rendezvous mission to as teroid Ceres, (b) a sample return mission to Comet Temple 1, and (c) a Jupiter orbiter mission with Earth swingby.
Solar Sailing An entirely different technology is involved in missions that would use large solar sails for “passive” low-thrust propulsion as contrasted with active, solar-electric propulsion. Such missions have been under investigation for more than two decades. Solar-sail mission applications have been discussed in the literature by Sauer [1976], Wright and Warmke[1976], Leipold et al. [1996, 1998], Friedman [1978, 1988], Mclnnes [1993, 1999] and others. The principle of using the solar pressure effect is analogous to wind-sailing on water. It permits spiraling inward or outward in the solar system once the sailcraft has left the Earth’s gravitational sphere of influence. As an example, a square sail about 200 m x 200 m in size, made of 7.5 mil Kapton film, and deployed by diagonal carbon-fiber-reinforced plastic booms, has been studied under auspices of NASA/JPL and the German Space Agency DLR [Leipold, 1998]. The flight system includes about 200 kg for the sail system and 115 kg net spacecraft mass, with the sailcraft to be launched by a Taurus rocket. The sail specific-mass is 5.2 g/m2. A potential application is a fast Pluto flyby mission to be launched in the next decade. It would include a “solar-photonic assist” phase with an inbound excursion to 0.45 AU solar distance. Because of the difficulties involved in performing ground-based tests of the solar sailing technique, a preliminary demonstration in Earth orbit is envisioned
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Orbit Selection and Design
(A ) Asteroid Ceres Rendezvous Trajectory
12.8
(B ) Comet Temple 1 Sample Return Mission
Coast
: Jupiter Arrival 2-15-06 iVuo = 6.5 km/s
(C ) SEP Earth Gravity Assist Jupiter Orbiter Trajectory Fig. 12-31.
SEP Transfer Trajectories for Three Mission Types: Ceres, Comet Tem ple 1, and Jupiter. (Sauer [1997].)
[Leipold, 1998]. Materials technology development, system deployment and control processes, and mission design continue to be studied at this time, with further progress expected to lead to greater acceptance of this novel technology by the space explora tion community and government agencies.
12.8 Interstellar Exploration* Pioneer 10 was launched on March 3,1972. In 1991» it became the first spacecraft to go beyond the solar system and enter interstellar space. In a sense, the exploration of the galaxy has begun. However, Pioneer 10, and Pioneer 11 which followed closely * Interstellar travel is not a realistic option in the near-term. Many people believe it to be impos sible, in much the same way that many thoughtful scientists and engineers 100 years ago believed that space travel was impossible. Indeed, 100 years ago, space travel was impossible. And today, many of the ideas put forward for interstellar travel fail the basic test of violating physics. However, many do not. Physicist Robert Forward spent a number of years at the Hughes Research Laboratories studying the problem of interstellar travel and summarized the problem very well: “Interstellar travel is difficult, but it is not impossible” [Forward, 1986]. The purpose of this section is to understand the basic reasons that the problem is so hard, the relativistic equations of motion needed for working in this area and, at a very high level, the nature of a few of the possible solutions.
Interstellar Exploration
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655
behind, were only symbolic in terms of interstellar travel. At the time it left the solar system, Pioneer was traveling at about 13 km/sec. If it were headed in the right direc tion, which it isn’t, Pioneer could get to the nearest star in a bit more than 100,000 years, a somewhat longish time when programs are funded with public money and reviewed on a annual basis. T A B L E 12-25.
Th e 20 Stars Nearest the S un.’ Only 8 of them are visible to the naked eye and 6 are muttiple stars. 8 Eridani and z Ceti are often thought of as a good place to begin exploration because they are reasonably “sun-like".
Star
Distance (Ity rs)
Apparent Magnitude
Absolute Magnitude
Spectral T yp e
-2 6 .7 2
G2V
Centauri C (Proxima)
4.22
11.05
4.85 15.49
Alpha Centauri A
4.39
-0 .01
4.37
M 5V G2V
Alpha Centauri B Barnard's Star G1 4 1 1
4.39 5.94 8.31
1.33 9.54 7.50
5.71 13.22 10.50
K1 V M 5V M2 Ve
Sirius A
8.60 8.60
-1 .4 6
1.42 11.20
A1 V
8.30
9.70
10.45
13.14
M4.5 Ve
10.48 10.72
3.73 7.35
6.14 9.76
10.86 11.36 11.36
13.47 7.56 8.37
11.40
11.10 5.22 6.03 0.37
K2V M2 Ve M 4 .5 V K5 Ve K7 Ve
2.64
F5 IV-V
G1 725 A
11.40 11.45
10.70 8.90
13.00 11.15
wd M4 V
G1 725 B
11.45
9.69
11.94
GXAnd Epsilon Indi
11.64
8.08 4.68
10.39 8.37
M 5V M2 V K5 V
3.50 12.04
5.72
G 8 Vp
14.12
M5.5 Ve
11.94 10.88 8.74 11.87
M3.5 MOV
Sun
Sirius B G1 729 Epsilon Eridani G1 887 Ross 128 61 C ygni A 61 C yg n i B Procyon A Procyon B
Tau Ceti G1 54.1 Luyten Star Kapteyn's Star A X M ic Kruger 60 A Kruger 60 B
11.81 11.89 12.12 12.39 12.78 12.88 13.04 13.04
9.84 8.84 6.66 9.85 11.30
13.30
wd
M0 Ve M2 V wd
* In the table, absolute magnitude is a measure of actual brightness of the star and apparent magnitude is how bright it appears in the sky (i.e., depending on absolute magnitude and distance). Smaller numbers correspond to brighter stars. (See Sec. 13.6.) The spectral type is a classification of stars by their spectra (depending prima rily on temperature, size, age, and composition). The main sequence of stars follow the classification O, B, A, F, G, K, and M with type O being the very hot blue-white stars and types K and M being cool, reddish stars. The classes are further subdivided by tenths and a Roman numeral is added to show the luminosity class which is an indicator o f intrinsic brightness. A lpha Centauri is very sim ilar to the Sun. U nfortunately, its also part o f a triple
star system such that there would be dramatically varying temperatures on any planets in the system. Of our near est neighbors, £ Eridani and z Ceti are the non-multiple stars most like the Sun.
Within the solar system, the basic problem to be overcome is how much energy is required to get there and, therefore, how much fuel must be put into orbit at very high cost to accomplish what wants to be done. For interstellar travel, the problem is not cost, but whether the objective can be achieved at all on time scales consistent with
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human activities. On a human scale, the distances to the stars are enormous and the time scales very long. To get a sense of the magnitude of the problem, Table 12-25 gives the distances to the 20 stars nearest the Sun. Our closest neighbor is Proxima Centauri, one of the 3 components of the Alpha Centauri system, at a distance of 4.22 light years. This is the distance light travels in 4.22 years, which is about 100,000 times the distance from the Earth to Mars when they’re on opposite sides of the Sun, or around 40,000,000,000,000 km. It’s a long way. Of course, a long distance just means I need to use a bigger rocket and go faster. Rockets can provide accelerations of several g’s, but that’s not particularly comfort able for people. If we assume a rocket accelerates continuously at lg for 13 min, this is sufficient to get to low Earth orbit. In only 20 min at lg we will get to the Earth’s escape velocity of around 11 km/sec and will have traveled 7,000 km. If we fire our rocket continuously so as to maintain the lg acceleration for 3 months, we will have gotten to a velocity of about 100,000 km/sec, or 30% of the speed of light (usually written as “0.3 c”). We will have traveled 500 billion km, 3000 times the distance from the Earth to the Sun, or a bit more than 1% of the distance to Proxima Centauri. At this point, we might as well turn off the rocket. Pushing more will increase the velocity only slightly and most of the energy will go into increasing the mass of the rocket. (See Sec. 12.8.1 for an explanation and relevant equations.) Cruising to Proxima Centauri at the speed we’ve gotten to will now take another 14 years. Unfortunately, the scenario is even less appealing from an energy or propellant mass perspective. Using a very efficient hydrogen/oxygen rocket with a specific im pulse of 450 sec, the rocket equation tells us that we can get to low Earth orbit with a propellant mass which is 5 times the mass of the empty rocket plus its payload, ignor ing entirely any loses due to gravity or atmospheric drag. (See, for example, Sackheim and Zafran [1999].) In practice this is a challenging, but workable, problem. However, to keep our rocket going at lg for 3 months requires a propellant mass IO10-000 times the mass of the rocket plus its payload. It is not just a matter of building better, more efficient, or more massive rockets. Getting to even the nearest stars requires a whole new process. The basic problem of interstellar travel is a combination of distance, time, and amount of energy required. This has led Forward [ 1986] to come to four broad conclu sions about the potential for interstellar travel:
1.
We can’t do rapid interstellar travel with simple rocket technology. If standard rockets are used, either the amount of propellant required will be orders of magnitude too large or the maximum velocity will be a very small fraction of the speed of light, such that the trip times will be many centuries. To make the trip within a reasonable human lifetime, we must use something other than normal rockets to avoid carrying all o f the required reaction mass along on the trip.
2.
We can’t do interstellar travel at lg acceleration. As discussed in the next section, a continuous lg acceleration is simply not practical. After having reached some significant fraction of the speed of light, nearly all the propul sion energy goes into making the rocket heavier and harder to push, rather than making it go faster. (Explanations and equations are in Sec. 12.8.1.) This means that a reasonable mission will accelerate to some cruise velocity in the general vicinity of 0 . 2 c to 0 .8 c, and will then coast for the remainder of the trip.
12.8
3.
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We can’t do interstellar travel in round-trip times of less than 10 years. Traveling at 0.8 c to the nearest star and back, with no exploration or turn around time, will take just over 10 years. This implies that interstellar travel and exploration will be a process spread out over many years or decades. However, information can be returned from an interstellar probe at the speed of light. Thus, traveling at 0.5 c means that information can be returned in 3 times the 1-way light travel time. At this speed, we could get information back from Proxima Centauri in 12.5 years and from any of the 20 to 30 nearest stars in less than 40 years.
4.
We can’t do interstellar travel using only resources brought from the surface of the Earth. Interstellar vehicles could certainly be built on the surface of the Earth. However, the energy source or reaction mass must almost certainly come from space. While this is a limitation, it is not necessarily a severe one. There is an enormous amount of mass and energy available in the solar system. There are about 10 million asteroids with a diameter greater than 1 km and a mass of more than IO12 kg [Cox, 2000]. At the distance of the Earth, the Sun is continuously providing 1,368 W/m 2 or 1.4 million kW/km2. Of course, this falls off as 1/r2 as we leave the vicinity of the Sun. Thus, large quantities of energy and mass cannot be brought up from the surface of the Earth and are scarce in interstellar space. However, both are plentiful within the solar system. Energy and mass are not a major limitation, but cannot be brought from the surface of the Earth.
12.8.1 Relativistic Space Travel As we have seen above, traveling to even the nearest stars requires going at speeds that are a substantial fraction of the velocity of light. At these speeds, ordinary New tonian mechanics is no longer a good approximation and we need to use relativistic mechanics to do interstellar travel computations. The physical phenomena associated with both special and general relativity are covered in a variety of texts, such as Ein stein [1936, 1952], Mermin [1968], and Kaufman [1973]. Relativistic time is dis cussed in Sec. 4.1.5 and a detailed discussion is provided by Seidelmann [1992]. The most important relativistic effects for interstellar space travel are as follows;
• Absolute upper limit of the velocity of light No matter how hard or how long we push on a spacecraft, the velocity of light, c, is an upper limit to the spacecraft velocity, v. As we approach the velocity of light, the spacecraft velocity will increase ever more slowly and the mass of the spacecraft will increase. (The rel evant equations are given below.)
• Time dilation. Moving clocks run slow. The clock on the spacecraft runs slower than the clock on the ground by the factor y= (1 - v2 / c 2)0-5. This means that on a long trip less time will have passed for the astronaut than will have passed for someone remaining behind on Earth. (This leads to the famous twin paradox described in the boxed example.) Thus, it is possible that trips of hundreds of years can be completed within the lifetime of a single crew. Time measured by a clock at rest with respect to the observer is called proper time, i.e., time as measured by the astronaut on board the spacecraft. With respect to the proper time, all moving clocks run slow.
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Orbit Selection and Design
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• The Lorentz contraction. Moving objects are shorter. For the astronaut, the distance to the stars has contracted by the same factor of y= (1 -v 2 / c 2) 0 -5 and, therefore, it won't take as long to get there. The various relativistic effects “work together” to produce strange results. Thus, what is a slowing down of the astro naut’s clock as measured by an observer on Earth is a shrinking of the distance to be traveled by the astronaut. The distance in the frame for which the object is at rest is called the proper distance. • The lack of absolute simultaneity. Whether or not two distant events are simul taneous (i.e., happen at the same time), depends on the velocity of the observer. Specifically, if two synchronized clocks are at rest with respect to each other at a proper distance, D, apart, then for an observer moving at velocity v parallel to the line joining them, the clock in the rear will be behind by Dv / c2. It is this feature of special relativity that “resolves” many of the apparent paradoxes in relativity, including the twin paradox. It is also one of the hardest to accept because we want to believe that time has some absolute meaning— either events are simultaneous or they are not. Unfortunately, this is simply not the case when dealing with velocities that aren’t small with respect to c.
• The increase in mass of moving objects. If we continue to push on our space craft, the velocity eventually approaches the velocity of light. As we get close to c, the velocity increases only slightly, but the mass of the spacecraft increases, such that the energy continues to increase as we continue to push.* The ratio of the proper mass, or rest mass, to the moving mass is, once again, y As discussed above, this makes space travel close to the velocity of light very inefficient—we are putting all of our energy into increasing the mass of the spacecraft while hav ing very little effect on the velocity. It is important to recognize that the relativistic concept of a “clock” doesn’t mean just a mechanical or digital watch, but the flow of time however it is measured. A com pelling demonstration of this was done in the 1960s at the Education Development Center for a film on time dilation [Frisch and Smith, 1963]. //-mesons are elementary particles with a short half-life that are produced when cosmic rays hit the upper atmo sphere. In the demonstration the number of //-mesons arriving per hour is first deter mined on a mountain top at an elevation of 2 0 0 0 m. Based on the measured laboratory decay rate only about 5% of the //-mesons should survive until they reach sea level. However, when the experiment is repeated at sea level the count is more than 70% of the count on the mountain top. The //-mesons are traveling very near the velocity of light. Time dilation has caused the //-meson’s internal clock to slow down such that many fewer have decayed. (From the perspective of the //-meson, the Lorentz contrac tion has shortened the distance from the mountain top to sea level.) Thus, even remark ably fundamental clocks, such as those internal to elementary particles, run slow when moving at velocities close to c. * Just as we like to think of time and length as intrinsic properties of the world around us, we tend to think of mass as an intrinsic property of matter. The problem with this is as follows. As we continue to apply force to our rocket, the momentum and energy increase indefinitely as they should. However, as we approach the speed of light, the velocity increases only very slowly. Therefore, we have to either introduce a new “fudge factor” in the definition of momentum and energy or redefine our concept of “mass” to mean the constant of proportionality between velocity and momentum such that the mass is a function of the velocity. While either is possible, physicists have chosen to adopt the latter approach.
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The physical interpretation of relativistic space travel is more complex than the equations. Specifically, assume that a spacecraft of length, L q , and mass, m 0, when at rest is traveling at constant velocity, v. Then the length, L , of the spacecraft when it is moving is given by:
I = l j l - v 2/ c 2 s yIo
(12-13)
where, as usual, c is the velocity of light. As discussed above, Lq is called the proper length or rest length. The Lorentz contraction applies only in the direction of motion. The two axis perpendicular to the direction of motion do not change. Similarly, a time interval, At, measured by a clock on board the spacecraft will be shorter than the same interval, At, measured by a clock which is at rest with respect to the observer on Earth: Az = yA t
(12-14)
where At is called the proper time for the E arth observer. Either an observer on the ground or one on the spacecraft is allowed to consider himself at rest. Thus, for both observers the other clock runs slow. (If this seems strange, see the boxed example on the twin paradox.) If two distant clocks are synchronized (i.e., read the same time) when they are at rest with respect to each other, then in a frame of reference moving at velocity, v, with respect to the clocks, the trailing clock will be ahead by: A T = D v fc 2
(12-15)
where AT is the difference in absolute time* between the two clocks and D is the prop er distance between them. Finally, the increase in mass, m, and energy, E, with increasing velocity are given by: m = niQ i y
(12-16)
and, of course, one of the most famous equations in physics: E - me2 = iuqC2 i J
(12-17)
where, as usual, mg is the proper mass or rest mass, m is the spacecraft mass which now increases with increasing velocity, and E is the total energy consisting of the rest energy (m0 c2) plus the kinetic energy. These and additional equations are derived in nearly all texts on Special Relativity. Particularly good explanations are those of Mermin [1968] and French [1968]. Given the equations above, and some care in interpretation of variables, we can determine the equations for time, distance, and mass for a rocket undergoing a constant acceleration, a, measured by an observer on board the rocket. Since the acceleration must be present for months to reach velocities approaching c, we generally assume that the acceleration will be close to 1 g, but of course this may or may not be true for any specific spacecraft solution. Let rb e time as measured on the rocket and t be time as measured on Earth. Then d r= ydt
(12-18)
* By absolute time we mean, for example, January 13th, 2014, at 1:00 pm UT. This is to distin guish it from a time interval of, for example, 2 hours 30 minutes.
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Orbit Selection and Design
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where, as above, y = ( 1 - v2 / c 2) 0 -5 and v is the velocity of the rocket as measured from Earth or from the spacecraft. (The spacecraft velocity as measured from Earth is the same as the Earth velocity measured from the spacecraft, but in opposite directions, of course.) We also have: d v a=— — dt\_y
(12-19)
from which the velocity as a function of Earth time is ( 12-20) This leads to the equation for distance traveled, x, as measured in the Earth frame of reference: ( 12-21) Finally, we have the interesting equation for time as a function of time: ( 1 2 -2 2 a)
( 1 2 -2 2 b) where again t is measured in the spacecraft frame, t is measured in the Earth frame, and a is the constant acceleration measured in the spacecraft frame. For a more de tailed discussion of these equations see, for example, von Hoerner [1962, 1963], In order to obtain the relativistic equivalent of the rocket equation [Eq. (12-1) in Sec. 12.2)], we need one more element from relativistic kinematics—the addition of velocities. Specifically, if a rocket is moving at velocity v relative to the Earth and shoots a bullet out the front at a velocity u with respect to the rocket, then the velocity of the bullet with respect to the Earth, w, is given by: u+v (12-23) which reduces to the normal addition of velocities if both u and v are small with respect to c. In the case of a small velocity increment, du, provided by the rocket (and mea sured in the rocket frame of reference), we get: dv - y 2 du
(12-24)
This can then be combined with the conservation of momentum to obtain the relativistic rocket equation [von Hoerner, 1963; Oliver, 1990]: dv / dm = - (ue t m )y 2
(12-25)
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Interstellar Exploration
661
The Twin Paradox One of the most famous paradoxes in physics is the twin paradox, introduced by Einstein [1905] in his first paper on relativity. The principle of relativity states that one cannot tell by any experiment the difference between two frames of reference that are moving uniformly with respect to each other, i.e., an observer in either frame is allowed to consider himself at rest and the other frame as moving. Combining this principle with the results of the Michelson-Morley experiment, which showed that the velocity of light was constant in any frame of reference, led Einstein to conclude that time dilation must also occur, i.e., moving clocks run slow. This leads to a paradox in which one twin leaves Earth on a rocket and returns to find that the twin who remained behind is older than the one who traveled out and back. This appears to violate both common sense and the orig inal principle of relativity. In a way, we can resolve the twin paradox by saying that special relativity applies only to frames of reference that are in uniform motion with respect to each other and the twin who has traveled has had to undergo a large change in velocity to return home. Thus, in a formal sense, we can worm our way around having to numerically explain the twin par adox. However, if we do not wish to hide behind general relativity, we can still tackle the problem quantitatively within the framework of special relativity. To make it simple, let’s assume the rocket leaves Earth and accelerates instantly to 0.995 c. It arrives at Alpha Centauri, 4 light years away, and decelerates instantly to 0, It immediately repeats the process, accelerating to 0.995 c towards home and decelerat ing to 0 when it arrives. While traveling, y= 0.1, so the arithmetic is easy, if not the logic. To the observer on Earth, the rocket has gone 8 light years (out and back) at a velocity of 0.995 c, so the trip takes 8.04 years. However, he measures the moving clock running slow by a factor of 10, so when the astronaut returns he is only 0.804 years older. What happens from the perspective of the astronaut on the rocket? He steps into the rocket at time t = 0 at Earth and is immediately transformed into the moving reference frame of the rocket. In this frame the distance to Alpha Centauri is only 0.4 light years, so it will take 0.4 / 0.995 = 0.402 years to get there. At the time he left, the clocks on the Earth and Alpha Centauri were synchronized in the frame of reference in which they weren’t moving. However, our clever astronaut knows about the lack of absolute simultane ity and, therefore, knows that in his current frame the clock on Earth must be Dv Ic2 = 4 x 0.995 = 3.98 years behind. Since he got on board at / = 0, it must now be t = 3.98 years at Alpha Centauri. However, the Centaurian clock (and the Earth clock, of course) are running slow by a factor of l/y= 10. Therefore, he will arrive at Alpha Centauri at t - 0.402 years on his clock and t = 3.98 + 0.402/10 - 4.02 years on the Centaurian clock. Our intrepid traveler immediately transfers to the next rocket back. Having changed his frame of reference again, he knows the Centaurian clock will now be behind the Earth clock by 3.98 years, so it is now / = 4.02 + 3.98 = 8.00 years on Earth. Of course, it takes him 0.402 years to get home by his clock, which, since the Earth clock is still running slow, corresponds to only 0.04 years on Earth. Thus, he arrives home at t = 0.804 years on his clock and t = 8.00 + 0.04 = 8.04 years on the Earth. Both twins come to the same conclusion, but for significantly different reasons. The basic problem we have trouble overcom ing is that not only do moving clocks run slow (which seems a bit strange), but also what time it is at some distant place depends on how fast we’re moving with respect to that distant clock. Another way to look at the twin par adox is to have both twins send out radio pulses to the other at 1 second intervals. We can go through much the same logic as above, counting the pulses that are emitted and received and come to the same conclusion as above. Both approaches to the twin paradox are explained in more detail by Mermin [1968]. French [1968] and Terletskii [1968] also provide excellent discussions of this and similar paradoxes in relativity theory.
662
Orbit Selection and Design
12.8
where m is the rest mass of the rocket and ue is the exhaust velocity relative to the rock et. Two specific cases are of interest for interstellar travel. If the exhaust velocity remains constant with respect to the rocket, then integrating over time yields: m .i
A* = —
rrif
( 1 + v / c ^ c/2m
= \ - -------- —
2i^ / c _ j
c
(
Vl-v/c
^ 2 Uglc
/
12- 26)
x
= tanh — InjU y c J
(12-27)
where the rocket is assumed to start from rest and accelerate to velocity v. Here = mz-/ mf, where mz- is the initial proper mass of the rocket and my is the final proper mass. The second case is a photon rocket for which ue - c. Here Eqs. 12-26 and 12-27 become: 1+ vl c M = J - ----- r \ 1 —v / c
(12-28)
v u2-1 - = c ^ 2+l
(12-29)
Oliver [1990] derives the above equations and provides substantial additional discussion on the performance of interstellar rockets.
12.8.2 Getting to the Stars It is clear from the above discussion that interstellar travel will require not just building bigger and better rockets, but a fundamentally new approach to space travel in at least some respects. The problems of building a spacecraft or, for human missions, creating long-term life support can be solved by the application or extrapo lation of technology that we understand today. The problem of propulsion is more fundamental. We need to build up a velocity at least 1,000 times greater than any cur rent rockets provide and, if we want to stop and explore at the other end, get rid of it again. If we want to return, then we have to start again and stop again. At best, the technology to do this is conceptual and many of the proposed approaches may prove impossible to implement. The systems most often considered are:
• Nuclear Electric Propulsion. Nuclear fission systems that could be launched in the near term could reach the nearest stars in perhaps 10,000 years, cutting 90% off the Pioneer 10 transit time. Solar electric and laser electric are also possible, but even less efficient.
• Nuclear Pulse Propulsion. This approach, originally called “Orion,” uses a series of nuclear fusion bombs to push the spacecraft to its destination. It was originally proposed at Los Alamos National Laboratory and has been worked on by, among others, Freeman Dyson [1968], the developer of the geodesic dome.
• Antiproton Propulsion. This is potentially a remarkably efficient energy source. A 1 ton interstellar probe could be accelerated to 0.1 c by using only 4
Interstellar Exploration
12.8
663
tons of liquid hydrogen and 1 0 kg of antihydrogen as the energy source. Antiprotons are being produced (in remarkably small quantities) and stored for days in current particle physics laboratories.
• Fusion Rockets. Controlled fusion is not yet available. If it does become possi ble, then there is the potential for using it for interstellar rocketry.
• Interstellar Ramjet This concept uses a large magnetic scoop (on the order 1 0 ,0 0 0 km2) to collect interstellar hydrogen and use it for controlled nuclear fu sion such that the vehicle can effectively push against the interstellar medium to reach relativistic velocities. There are several variants on this approach. All remain in the realm of science fiction at the present time.
• Beamed Power Propulsion. In these systems the source of energy remains fixed in the Solar System and power is beamed to the interstellar starship. The three most promising approaches use a stream of small pellets, microwave beams, and lasers. The laser system is particularly interesting in that it can pro vide all four needed phases—acceleration, deceleration, return acceleration, and return deceleration [Forward, 1984]. It requires no new physics, but simply large scale engineering extrapolation of known technologies. An excellent summary of the above approaches is provided by Forward [1986]. All of them represent travel to only the nearest stars. It is the huge distances and the remarkably slow speed limit of the velocity of light that restricts our capacity for far ranging interstellar travel. Table 12-26 summarizes the fundamental problem of travel T A B L E 12-26.
Destination
Tra v e l Data for Representative Destinations. Distances and travel times are all one-way. Travel at the relatively high spacecraft velocity of 25 km/sec is impractical because of the large distances to the stars. Travel at 1 g acceleration is impractical because of the enormous amounts of energy it would require.
Alpha Centauri
Galactic Center
Magellenic Clouds
An dromeda Galaxy
Moon
Sun
Pluto
km
384,000
1.5x108
5.9x109
4 .1 2 x 1 0 ^
2.8 X1017
1.7 X1018
2.3 x1019
AU*
0.00257
1.00
39.5
275,000
1.9x109
1.1 x1010
1.5 x IO 11
4.35
3.0 x104
1.8x105
2.4 x106
Distance
Light Years
4.06 *10-8 1.58 x10-5 6.2 x 1 0 ^
Travel at 25 km/sec Time Ellapsed
4.3 hr
69 days
7.5 yr
52,000 yr
3.6 x108 yr 2.2 x109 yr 2.9 x1010 yr
Travel at 0.5 c Time Ellapsed on Earth Time Ellapsed on Rocket
2.6 sec
16.6 min
11.0 hr
8.7 yr
60,000 yr
3.6 x1 05yr 4.8x106yr
2.2 sec
14.4 min
9.5 hr
7.5 yr
52,000 yr
3.1 x1 05yr 4.2 x 1 0 6yr
1.8 x 1 0 5yr 2 .4x106yr
Travel at 1 g acceleration Time Ellapsed on Earth Maximum Velocity
3.5 hr
2.9 days
18 days
6.0 yr
30,000 yr
0.0002 c
0.0040 c
0.026 c
0.95 c
1.00 c
1.00 c
1.00 c
Time Ellapsed on Rocket
3.5 hr
2.9 days
18 days
3.6 yr
20.0 yr
23.5 yr
28.5 yr
*1 A U (Astronomical unit) = mean distance from the Earth to Sun = 150,000,000 km
664
Orbit Selection and Design
to more distant destinations. One of our nearest extragalactic neighbors is the An dromeda galaxy which contains on the order of 10 billion stars. You can see it with the naked eye on a dark night as a fuzzy patch in the northern sky. (Visually, the galaxy isn’t small, it’s simply faint. The dramatic long exposure photos we see of Andromeda are more than 6 times the size of the full Moon.) But even nearby galaxies are very far away. The light we see from Andromeda left there 2.4 million years ago. Even travel ing at nearly the speed of light, it’s a 5 million year round trip. The literature base on interstellar exploration is enormous. Much of the pro fessional technical literature on this topic appears in the Journal o f the British Interplanetary Society, JBIS, which often runs complete issues on various aspects of interstellar travel Robert Forward published a series of bibliographies, one of which took up an entire issue of JBIS and lists 2,699 articles, books, and reports conveniently sorted by subtopic, including more bibliographies [Mallove, et al., 1980]. Several similarly sorted updates have added a few thousand more entries [Paprotny, et aL, 1983, 1984, 1986], Good summaries of the state of the art in this area, along with more references, are provided by Forward [1986] and Vulpetti [1999]*
References Austin, R. E . 7 M. I. Cruz, and J. R. French. 1982. “System Design Concepts and Requirements for Aeroassisted Orbital Transfer Vehicles.” AIAA Paper 82-1379 presented at the AIAA 9th Atmospheric Flight Mechanics Conference. Barber, T. A. and H. F. Meissinger. 1969. “Simplification of Solar-Electric Propulsion Missions by a New Staging Concept." AIAA Paper 69-251 presented at the AIAA 7th Electric Propulsion Conference, Williamsburg, VA, March 3-5. Bate, R. R., D. D. Mueller, and J. E. White. 1971. Fundamentals o f Astrodynamics. New York: Dover Publications. Belbruno, Edward A.f Rex. W. Ridenoure, and Jaime Fernandez. 1991. “To the Moon from a B-52: Robotic Lunar Exploration Using the Pegasus Winged Rocket and Ballistic Lunar Capture.” Presented at the 5th Annual AIAA/Utah State University Conference on Small Satellites, Aug. 27-29. Cefola, P.J. 1987. “The Long-Term Orbital Motion of the Desynchronized Westar II.” AAS Paper 87-446 presented at the AAS/AIAA Astrodynamics Specialist Confer ence, Aug. 10. Cox, Arthur N, ed. 2000. Allen’s Astrophysical Quantities (4th ed.). New York: Springer-Verlag. D’Amario, L.A., D.V. Byrnes, J. R. Johannesen, and B.G. Nolan. 1989. “Galileo 1989 VEEGA Trajectory Design.” Journal o f the Astronautical Sciences, 37:281—306. Draim, John. 1985. “Three- and Four-Satellite Continuous Coverage Constellations.” Journal o f Guidance, Control, and Dynamics, 6:725-730. -------------. 1987a. “A Common-Period Four-Satellite Continuous Global Coverage Constellation.” Journal o f Guidance, Control, and Dynamics, 10:492-499.
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-------------. 1987b. “A Six-Satellite Continuous Global Double Coverage Constella tion.” AAS Paper 87-497 presented at the AAS/AIAA Astrodynamics Specialist Conference, Aug. 10. -------------. 1997. “Optimization of the ELLIPSO and ELLIPSO 2G Personal Commu nications Systems.” Paper C-4 presented at the IAF International Workshop on Mission Design and Implementation of Satellite Constellations, Nov. 17-19,1997, Toulouse, France. -------------. 1998. “Design Philosophy for the ELLIPSO Satellite System.” Paper pre sented at the 17th AIAA International Communications Satellite Conference. Yokohama, Japan. Draim, J., Helman, G., and Castiel, D. 1997. “The ELLIPSO Mobile Personal Communications System; It’s Development History and Current Status.” Paper IAF 97-M.3.03. Presented at the 48th International Astronautical Congress, Oct. 6 - 8 , Turin, Italy. Dyson, F. J. 1968. “Insterstellar Transport.” Physics Today, October, 41-45. Einstein, Albert. 1905. The Principle o f Relativity. Translated by W. Perrett and G. B. Jeffrey. New York: Dover. -------------. 1936. “Physics and Reality.” J. Franklin Inst, Vol. 221, P. 349-382. Einstein, Albert. 1952, 1961. Relativity: The Special and General Theory. Translated by Robert W. Lawson. New York: Crown Publishers. Farquhar, R. W. 2001. “The Flight of ISEE-3/ICE: Origins, Mission History, and a Legacy.” The Journal o f the Astronautical Sciences, Vol. 49, No. 1, January -March, pp. 23-73. Faughnam, Barbara and Gregg Maryniak. 1987. Space Manufacturing 6 Nonterres trial Resources, Biosciences, and Space Engineering. Proceedings of the Eighth Princeton/AlAA/SSI Conference, May 6-7, 1987. Washington D.C.: AIAA. Fimple, W.R. 1963. “Optimum Midcourse Plane Change for Ballistic Interplanetary trajectories.” AIAA Journal, 1:430-434. Forward, Robert L. 1984a. “Round-Trip Interstellar Travel by Laser-Pushed Lightsails.” J. Spacecraft & Rockets. Vol. 21. -------------. 1986. “Feasibility of Interstellar Travel.” Journal o f the British Interplanetary Society, Vol. 39, No. 9, September. -------------. 1989. “Statite Spacecraft that Utilizes Light Pressure and Method of Use.” U.S. Patent No. 5,183,225. Issued Feb. 2, 1993. French, A. P. 1968. Special Relativity. MIT Introductory Physics Series. NY: W. W. Norton and Company. Friedman, L. 1978. “Solar Sailing—The Concept Made Realistic.” Paper 78-82, AIAA 16th Aerospace Sciences Meeting, Huntsville, Alabama, January.
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-------------. 1988. Starsailing—Solar Sails and Interstellar Travel. New York, NY: J. Wiley and Sons. Frisch, D. H. and J. H. Smith. 1963. Film entitled Time Dilation—An Experiment with fi-Mesons, Education Development Center, Newton, MA. ------— ■— . 1963. American Journal o f Physics, Vol. 31. Gray, D. L. and Y. Hahn. 1995. “Maneuver Analysis of the Cassini Mission.” AIAA Paper 95-3275. AIAA Guidance and Control Conference, Boston, MA, Aug. Heppenheimer, T. A. 1977. Colonies in Space. Harrisburg, PA: Stackpole Books. Hujsak, Edward. 1994. The Future o f U.S. Rocketry. La Jolla, CA: Mina-Helwig Company. Humble, Ronald W., Gary N. Henry, and Wiley J. Larson. 1995. Space Propulsion Analysis and Design. New York: McGraw-Hill. Isakowitz, Steven J. 1999. International Reference Guide to Space Launch Systems, (3rd ed.). Reston, VA: American Institute of Aeronautics and Astronautics. Kaufman, B., C. R. Newman, and F. Chromey. 1966. Gravity Assist Optimization Techniques Applicable to a Variety o f Space Missions. NASA Goddard Space Flight Center. Report No. X-507-66-373. Kaufmann III, William J. 1973. Relativity and Cosmology. NY: Harper and Row. Koelle, Dietrich E. 1991. TRANSCOST, Statistical-Analytical Model fo r Cost Esti mation and Economic Optimization o f Space Transportation Systems. Munich, Germany: MBB Space Communications and Propulsion Systems Division, Deutsche Aerospace. -------------. 2000. Handbook o f Cost Engineering fo r Space Transportation Systems. Germany: TCS—Trans Cost Systems. Leipold, M. 1998. ‘T o the Sun and Pluto with Solar Sails and Micro-Sciencecraft ” ACTA ASTRONAUTICA, Vol. 45, Aug.-Nov. Leipold, Manfred E., Otto Wagner. 1996. “Mercury Sun Synchronous Polar Orbits Using Solar Sail Propulsion.” Journal o f Guidance, Control and Dynamics, Vol. 19, No. 6 , Nov-Dee. Leipold, M., et al. 1999. “ODISEE—A Proposal for Demonstration of a Solar Sail in Earth orbit.” Acta Astronautica, Vol. 45, Aug.-Nov. London, III, JohnR. 1994. LEO on the Cheap—Methods fo r Achieving Drastic Reduc tions in Space Launch Costs. Maxwell Air Force Base, AL: Air University Press. ----------- - 1996. “Reducing Launch Cost.” In Wertz, James R., and Wiley J. Larson (eds.). Reducing Space Mission Cost. Torrance, CA, and Dordrecht, The Nether lands: Microcosm, Inc., and Kluwer Academic.
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Luthey, J. L., F. Peralta, and J. L. Pojraan. 1989. “Ulysses Mission Design After Challenger.” AAS Paper 89-432 presented at the AAS/AIAA Astrodynamics Conference, Stowe, VT, August 7-10. Mallove, Eugene F., Robert L. Forward. 1980. Zbigniew Paprotny, and Jurgen Lehmann. “Interstellar Travel and Communication: A Bibliography.” Journal o f the British Interplanetary Society. Vol. 33, No. 6 , June. McGoIrick, J. E. (ed.). 1971. Launch Vehicle Estimating Factors—For Use in Advanced Space Mission Planning. NASA Headquarters Publication NHB 7100.5, January. Mclnnes, Colin R. 1993. “Solar Sail Trajectories at the Lunar L Sub 2 Lagrange Point.” Journal o f Spacecraft and Rockets, Vol. 30, No. 6 , Nov.-Dee. -------------. 1999. Solar Sailing: Technology, Dynamics and M ission Applications. London: Springer. Mease, K. D. 1988. “Optimization of Aeroassisted Orbital Transfer: Current Status.” The Journal o f the Astronautical Sciences. 36:7-33. Meissinger, H.F., R.A. Park, and H.M. Hunter. 1968. “A 3-kw Solar Electric Space craft for Multiple Interplanetary Missions.” Journal o f Spacecraft and Rockets, Vol. 5, June. Meissinger, Hans F. 1970. “Earth Swingby— A Novel Approach to Interplanetary Missions Using Electric Propulsion.” AIAA Paper No. 70-1117, presented at the AIAA 8 th Electric Propulsion Conference, Stanford, CA, August 31 September 2. Meissinger, H.F., S. Dawson, and J.R. Wertz. 1997. “A Low-Cost Modified Launch Mode for High-C 3 Interplanetary Missions.” AAS Paper No. 97-711, presented at the AAS/AIAA Astrodynamics Specialist Conference, Sun Valley, Idaho, August 4-7. Meissinger, Hans F. and Simon Dawson. 1998. “Reducing Planetary Mission Cost by a Modified Launch Mode.” Presented at the Third IAA International Conference on Low-Cost Planetary Missions, Pasadena, CA, April 27-May 1. Meissinger, H. F, and S. Dawson. 2001. “Solar-Electric Planetary Missions with an Initial Out-of-Ecliptic Thrust Phase.” AIAA Journal Spacecraft and Rockets, Vol. 38, No. 2, March-April. Mermin, David N. 1968. Space and Time in Special Relativity. NY:McGraw-Hill Book Company. Miller, J. K., B. G. Williams, W. E. Bollman, R. P. Davis, C. E. Helfrich, D. J. Scheeres, S. P. Synnott, T. M. Wang, and D. K. Yeomans, “Navigation of the Near Earth Asteroid Rendezvous Mission.” AAS Paper 95 -111 presented at AAS/AIAA Spaceflight Mechanics Meeting, Feb. 13-16.
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Minovitch, M- A. 1963. “Determination and Characteristics of Ballistic Interplanetary Trajectories under the Influence of Multiple Planetary Attractions,” Technical Report No. 32-464, Jet Propulsion Laboratory. O ’Leary, Brian 0 . 1982. Space Industrialization, Vol. I. BocaRotan, FL: CRC Press. Oliver, B. M. 1990. “A Review of Interstellar Rocketry Fundamentals.” Journal o f the British Interplanetary Society, Vol. 43, pp. 259-264. O ’Neill, Gerard K. 1974. “The Colonization of Space.” Physics Today, Vol. 27, No. 9, pp. 32-41. -------------. 1976. The High Frontier: Human Colonies in Space. NY: William Morrow and Company. Paprotny, Zbigniew and Jurgen Lehmann. 1983. “Interstellar Travel and Communica tion Bibliography: 1982 Update.” Journal o f the British Interplanetary Society. Vol. 36, No. 7, pp. 311-329, July. Paprotny, Zbigniew, Jurgen Lehmann, and John Prytz. 1984. “Interstellar Travel and Communication Bibliography: 1984 Update.” Journal o f the British Interplanetary Society. Vol. 37, No. 11, pp. 502-512. --------. 1986. “Interstellar Travel and Communication Bibliography: 1985 Update.” Journal o f the British Interplanetary Society. Vol. 39, No. 3, July, pp. 127-136. Peralta, F. and J. C. Smith, Jr. 1993. “Cassini Trajectory Design Description.” AAS Paper 93-968 presented at AAS/AIAA Astrodynamics Conference, Victoria, B.C., Canada, Aug. 16-19. Pocha, J. J. 1987. An Introduction to Mission Design fo r Geostationary Satellites. Dordrecht, The Netherlands: Kluwer Academic. Prussing, J. E., and B. A. Conway. 1993. Orbital Mechanics. New York: Oxford University Press. Reidel, J., W. Owen, J. Stuwe, S. Synnott, and R. Vaughan. 1990. “Optical Navigation During the Voyager Neptune Encounter.” AIAA Paper 90-2877 presented at AIAA/AAS Astrodynamics Conference, Portland, OR, Aug. 20-22. Reijnen, G. C. M. and W. De Graaff. 1989. The Pollution o f Outer Space, in Particular o f the Geostationary Orbit: Dordrecht, The Netherlands: Kluwer/Martinus Nijhoff Publishers Sackheim, Robert, L. and Sidney Zafran. 1999. “Space Propulsion Systems.” Chap. 17 in Wertz, J.R. and Larson, W J. (eds). Space M ission Analysis and Design (3rd ed.). Torrance, CA, and Dordrecht, The Netherlands: Microcosm Press, and Kluwer Academic. Sauer, C. D. 1976. “Optimum Solar Sail Interplanetary Trajectories.” AIAA/AAS Astrodynamics Conference, San Diego, CA, Aug. 18-20.
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670
Orbit Selection and Design
-------------. 1999. Space Mission Analysis and Design (3rd ed.). Torrance, CA, and Dordrecht, The Netherlands: Microcosm Press, and Kluwer Academic Publishers. White, J. F. 1963. Flight Performance Handbook fo r Powered Flight Operations. NY: John Wiley & Sons. Williams, S. N., and Coverstone-Carroll, V. 1997. “Benefits of Solar Electric Propul sion for the Next Generation of Planetary Mission.” Journal o f the Astronautical Sciences, Vol. 45, No. 2. Wolff, P. J., F. Pinto, B. G. Williams, and R. M. Vaughan. 1998. “Navigation Consid erations for Low-Thrust Planetary Missions.” AAS Paper 98-201 presented at AAS/AIAA Space Flight Mechanics Meeting, Monterey, CA, Feb. 9-12. Wolverton, R. W. ed. 1963. Space Technology Laboratories, Inc. Flight Performance Handbook fo r Orbital Operations. New York: John Wiley & Sons, Inc. Wright, J., and J. Warmke. 1976. “Solar Sail Mission Applications.” Presented at the AIAA/AAS Astrodynamics Conference, San Diego, CA, August 18-20.
Chapter 13 Constellation Design
13.1 13.2 13 .3 13.4
Coverage in Adjacent Planes Constellation Patterns Selection of Constellation Parameters Stationkeeping Controlled Orbits and Absolute vs. Relative Stationkeeping; In-Track Stationkeeping; Cross-Track Stationkeeping 13.5 Collision Avoidance 13.6 Constellation Build-Up, Replenishment, and End-of-Life 13.7 Summary—The Constellation Design Process
A constellation is a set of satellites distributed over space intended to work together to achieve common objectives. If the satellites are close together, such as a flying in terferometer, or spacecraft next to each other exchanging data or material, then it is called a cluster or formation, rather than a constellation. A number of past and pro posed constellations are listed in Table 13-1. Notice that nearly all of the characteristics of the constellations vary dramatically, including size, altitude, and in clination. Because constellations are inherently very expensive, most are used for communications, navigation, or similar functions that take advantage of the global Earth coverage provided by satellites and difficult to achieve by other means. The key background material on Earth coverage, large and small scale relative motion of satellites, and the design process for single satellite orbits has been covered in Chaps. 9, 10, and 12. This chapter provides additional relevant information on sta tionkeeping, collision avoidance, and constellation build-up, replenishment, and endof-life, and makes use of this material to provide specific recommendations on con stellation design. Finally, Sec. 13.7 provides an overview of the constellation design process and how to go about achieving this in a systematic way. A major trend in the evolution of satellite systems is an increase in the number of smaller, lower-cost satellites. This has led to a dramatic increase in the number of Earth-orbiting constellations for observations, communications, navigation, and sci ence. Consequently, constellation design has become a topic of considerable current work and will continue to grow in importance, both as constellations become more achievable and as the potential problems of collision avoidance and debris mitigation become more critical.
671
672
Constellation Design
T A B L E 13-1.
Representative Past and Proposed Satellite Constellations.’
Name Aries
Operator
Constellation Comm Inc.
Purpose
comm (voice, data,
Alt. (km)
Inc.
# of
# of
(deg)
Sats
Planes
planned
1,022
90
48
4
— —
4
—
6/8
— 1
Status
radiolocation)
A$trplink
TRW
comm (broadband)
planned
GEO
Bankir
NPO Larochkine
comm (data)
planned
Celsat
Celsat Inc
comm (voice, video, data, radiolocation)
planned
LEO GEO
Coscon
KB Polyet
comm (data, radiolocation)
demo
Discoverer II
US DoD
Earth obs radar
planned
DM SP
Military
meteorological
operational
Draim 4
N/A
continuous whole
theoretical
0
1/2 +1 spare
1,000
83
32
TBD (LEO)
TBD
TBD
TBD
822 >38,736
99 31.3
2
—
4
4
—
16
1
Earth coverage DSP
Military
comm
deploying
GEO
ECCO
Constellation
comm
planned
Eltipso
MCHI
comm (mobile voice) planned
1,018 7.506 8,050
E-Sat
E-Sat Inc.
comm (message)
planned
1,262
—
6
—
F A IS A T
Final Analysis
comm (message
planned
1,000
66/83
36 (+4), 2
F L TS A TC O M
USN
UHFcomm
operational
35,800
4—5
navigation
planned
24,000
ocean obs
planned
—
57-65 90
4 24
— 1
GalileoSat
48 116 6 0
10 7
2 1
3
16
— — 14
Gander
BNSC
GGS
Japan/ESA
scientific
operational
Globalstar
Loral
planned
Glonass
Russian govt
comm (voice) navigation
GO ES
NASA/NOAA
meteorological
operational
GEO
0-7
Gonetz
comm (voice, data)
planned
1,400
82.6
G P S Block I
NPO Prikadnoy USAF/USN
navigation
operational
20,200
63
24
3
G P S Block II
USAF/USN
navigation
deploying
20,200
63
55
21
G P S III
USAF/USN
navigation
planned
—
—
ic o Teiedesic Global
comm (voice, broadband)
planned
— 10,335
—
New IC O
45
10
3
Inmarsat
Private
comm (mobile voice) operational
—
—
Intelsat
C O M S A T , Intel
com m (voice, data, T V )
operational
GEO
0
— 17
— 1
operational
—
—
2-5
1,386
47,55
—
—
24/48 24 — 24
—
1 4
Iridium
Motorola
comm
bankrupt
805
90
66
6
IN X
iridium LLC
comm (voice)
planned
850
90
96
—
iSky
iSky
comm
planned
—
—
—
—
Leo-One
LEO One USA
comm (message)
planned
950
50
48
8
LEO SAT
Low-Earth Orbit Satellite Corp
comm (data radio-message)
operational
1,000
42
24
Magellan
navigation
planned
—
comm (voice, telex, telecopy)
operational
— 3 GEO, 8 elliptic
—
Marathon
ESA/EC MarathonZemlia, NPO
Molniya
USSR
comm
operational
M -S A T
American Mobile comm {voice, data) Satellite Corp, Telesat Mobile
operational
M S SP
Military
comm
proposed
LEO
Norstar 1
Norris Satellite Communication
comm (voice, data)
operational
GEO
—
0, 63
11
1,000/ 26,600
63.4
many
—
GEO
0
2
1
“
various 200-400 0
1
many 1
Constellation Design T A B L E 13-1.
673
Representative Past and Proposed Satellite Constellations/ (Continued) Operator
Name
Purpose
Status
Odyssey
TRW
comm (voice)
planned
OR B CO M M
Orbital Comm Corp
comm (radio-message)
operational
Earth obs (visual)
in design
Alt. (km)
Inc. (deg)
#of Sats
#of Planes
10,354
50
12 + 3 spare
3
970
40
20
3
—
—
—
—
planned
elliptical
—
14
—
comm (voice, data)
planned
—
—
35/40
—
Kayser-Threde
Earth obs (agricul)
planned
S-80
CNES
planned comm (radio* location & message)
SBIRS
US DoD
Earth obs (IR)
Skybndge
Skybridge Ltd, Alcatel
comm (broadband, data)
Spaceway
Hughes
Stamet
Starsys Inc
TDR S
NASA
TechSat21
USAF AFRL
Virgo
Virtual Geosatellite
Vitasat
Walker 5
Orbview
Orbital
Pentrisd
Denali Telecom
Project 21
Inmarsat
RapidEye
4
Earth obs (agricul)
Resource 21
1,0001,500
45-55
—
—
planned
GEO/ME O/LEO
TBD
TDD
TDD
planned
1,469.3
53
80
comm (broadband)
planned
10,352
—
comm (voice, radiolocation)
planned
1,300
50/60
comm
operational
GEO
0-2
5
1
planned
TBD
TBD
—
constel
—
planned
800 x 27,000
Volunteers in Technical Assistance
comm (data)
planned
LEO
NA
continuous whole Earth coverage
theoretical
Earth obs (SAR)
>38,655
. 20
20
—
—
—
—
15 + 3 spare
43.7
5
—
5
* Status as of January 1, 2000.
One of the most important results of decades of constellation studies has been that no absolute rules exist for constellation design. As described in the boxed example in Sec. 13.1, a constellation of satellites in randomly spaced low-Earth orbits is a possi bility for a survivable communications system. One of the most interesting character istics of the low-Earth orbit communications constellations is that constellation builders have invested billions of dollars and come up with distinctly different solu tions. For example, a higher altitude means fewer satellites, but a much more severe radiation environment, such that the cost of each satellite will be higher, and the life potentially shorter. Both solutions are being used by commercial constellations. Sim ilarly, circular orbits provide a simple, low-cost satellite design. However, elliptical orbits allow an additional degree of freedom, which allows the constellation to be optimized for multiple factors, but requires a more complex satellite operating over a range of altitudes and velocities, and passing through heavy radiation regimes. (See, for example, Draim et al. [1996, 1999].) Again, both solutions are being used by modem constellations. Because the constellations’ size and structure strongly affect system cost and performance, we must carefully assess alternative designs and document the reasons for trades which are made. It is this list of reasons that allows the constellation design process to continue in order to achieve the system objectives at minimum cost and risk.
674
Constellation Design
While there are no absolute rules, there are a number of key issues which dominate constellation design trades, as listed in Table 13-2, These are discussed briefly below, and in more detail later in the chapter. Because virtually any multi-satellite system will be inherently expensive, all constellation design trades are ultimately based on cost vs. performance. Thus, each of the issues in Table 13-2 is a major determinant of system cost, performance, or both. T A B L E 13-2.
Principal Issues which Dominate Constellation Design. Virtually all constella tions are designed on a cost vs. performance basis. See Secs. 13.3 and 13.7 for a detailed discussion of the contellation design process.
Issue
W hy Important
What Determines It
Principal Issues or Options
W here
Discussed
Coverage
Principal performance parameter
Altitude, minimum elevation angle, inclination, constellation pattern
Gap times for discontinuous coverage; Number of satellites simultaneously in view for continuous coverage
Chap. 9, Sec. 13.3
Num ber of Satellites
Principal cost driver
Altitude, minimum elevation angle, inclination, constellation pattern
Altitude, minimum elevation angle, inclination, constellation pattern
Sec. 13.3
Launch Options
Major cost driver
Altitude, inclination, spacecraft mass
Low altitude, low Secs. 12.3, inclination costs less 13.3
Environment
Radiation level and, therefore, lifetime and hardness requirements
Altitude
Options are below, in, or above Van Allen radiation belts
Fig. 2-12, Sec. 12.4
Orbit Causes Perturbations constellations to (Stationkeeping) disassociate over time
Altitude, inclination, eccentricity
Keep all satellites at common altitude and inclination to avoid drifting apart
Secs. 2.4, 13.4
Collision Avoidance
Snowball effect can destroy entire constellation
Constellation pattern, orbit control
No option— must design entire system for collision avoidance
Sec. 13.5
Constellation Build-Up, Replenishment, and End-of-Life
Determines level of service over time and impact of outages
Altitude, constellation pattern, build-up and sparing philosophy
Sparing: on-orbit spares vs. launch on demand; End-of-life: Deorbit vs. raise to higher orbit
Secs. 2 .6 .4,13.6
N um ber of Orbit Planes
Determines performance plateaus
Altitude, inclination
Secs. Fewer planes means more growth 2.6.2, 13.6 plateaus and more graceful degradation
Coverage Coverage is ordinarily the principal performance parameter, (See Chap. 9, Sec. 13.3.) It is normally the reason we are creating a constellation in the first place. If the coverage requirement is not continuous, then we are typically interested in mini
Constellation Design
675
mizing the gap times for coverage of regions of interest. Continuous coverage most often requires that a region of interest, and perhaps the whole world, be seen at all times by at least one satellite in the constellation. However, multiple continuous coverage may also be required in some circumstances. For example, GPS requires continuous coverage of the entire world by a minimum of four non-coplanar satellites.
Number o f Satellites Typically, the number of satellites is seen as the principal cost driver, and the cost is usually assumed to be proportional to this number. (See Sec. 13.3.) Thus, the most common goal in constellation design is to achieve the desired coverage with the minimum number of satellites. Nonetheless, a minimum number of satellites may not represent minimum cost. A smaller constellation may require larger or more complex satellites. Since constellations with fewer satellites are typically higher, they may also have greater launch costs. Thus, a constellation of 20 satellites in low-Earth orbit may or may not be significantly cheaper than a constellation of 3 or 4 satellites in geosyn chronous orbit.
Launch Options After the number of satellites, the launch requirements normally represent the prin cipal cost driver for the system, and may also represent the largest risk, since only about 90% of all launches are successful. (See Secs. 12.3, 13.3.) Typically, a given level of coverage can be achieved with fewer satellites at a higher altitude. However, the higher altitude and increased mass due to increased power and performance requirements tends to drive up the launch costs, which in turn, pushes the altitude down again.
Environment For purposes of constellation design, the vacuum, thermal, and zero-g environment are largely independent of orbit. (See Fig. 2-12 in Secs. 2.3,12.4.) However, the radi ation environment is dramatically dependent on the altitude. Being in or above the Van Allen belts can increase the radiation by several orders of magnitude and therefore put demanding and expensive requirements on the spacecraft and potentially limit the life time of radiation-sensitive components, such as solar arrays or computers.
Orbit Perturbations (Stationkeeping) The dominant perturbations for most Earth satellites are drag, which is a function principally of altitude, and the Earth’s oblateness effects, which are a function of al titude, inclination, and eccentricity. (See Secs. 2.4, 13.4.) Because we cannot realisti cally use propellant to cancel the effects of oblateness, constellations with satellites at different altitudes, inclinations, or eccentricities will drift apart rapidly over time. In order to maintain the constellation structure, nearly all constellations have satellites at a common altitude, inclination, and eccentricity, except for a few special cases, such as the addition of an equatorial orbit for which orbit rotation does not matter. In addi tion, constellations of satellites in eccentric orbits will nearly always be at the critical inclination of 63.4 deg, so that apogee and perigee do not rotate.
Collision Avoidance The single largest threat to constellations in the long term is the problem of satellite collisions as discussed further in Sec. 13.5. Specifically, collisions within a constella tion can have a cascading or snowball effect in which a debris cloud remains within
676
Constellation Design
13.1
the constellation structure, thereby dramatically increasing the probability of subse quent collisions. Thus, while the probability of collision is inherently low because space is large and spacecraft are small, it is nonetheless critical to design the entire system (i.e., the constellation, the spacecraft, and the launch and de-orbit processes) for collision avoidance.
Constellation Build-Up, Replenishment, and End-of-Life Historically, most constellations spend most of their time in a less-than-complete configuration. Consequently, it is critical to consider as a fundamental part of con stellation design the process of how the constellation is to be constructed and what happens when spacecraft fail. In practice, this area is frequently given only minimal attention. However, it may represent a major operations cost driver. Equally critical is the end-of-life problem because dead satellites must be removed from the constella tion pattern or unacceptable collision probabilities will result. (See Secs. 2.6.4 and 13.6.) N umber o f Orbit Planes The key issue in terms of the number of orbit planes is that moving satellites within a plane takes extremely small amounts of propellant, whereas changing the orbit plane takes very large amounts of propellant. Consequently, rephasing within planes is eco nomical, and can be done often, This allows much more flexibility for the in-plane structure, and means that constellations with a smaller number of orbit planes will have more performance plateaus as the system is built up, and more graceful degrada tion if there are spacecraft failures. In general, higher constellations can have fewer orbit planes, and there will be a series of altitude plateaus below which additional planes will be required. (See Secs. 2 .6 . 2 and 13.6.)
13.1 Coverage in Adjacent Planes Before discussing constellation patterns, we need additional background on cover age in adjacent orbit planes of a constellation. Chapter 10 provided a detailed discussion of spacecraft relative motion and of the coverage and viewing angles by sat ellites in adjacent planes. Most constellations involve multiple satellites per plane and a series of planes spaced more or less uniformly along the equator. One way to view this problem is to look at a series of relative motion analemmas, as described in Sec. 10.1. For satellites in the same plane, there will be no relative motion to first order, and the relative motion analemma reduces to simply a dot in the orbit plane, ahead of or behind the reference satellite. The series of satellites in adjacent planes will each move on an analemma for which the half-height is equal to the relative inclination between the two planes. Note that all of the satellites in an adjacent plane will move on relative motion analemmas which are identical in size and shape. (See Fig. 10-9 in Sec. 10.1.1.) Differing phases in the adjacent plane will simply move the relative motion analemma along the reference orbit plane. Thus, choosing any satellite in the constellation as our reference satellite, all other satellites which are not in the same plane will move back and forth across the plane of the reference satellite, reflecting the fact that any two orbit planes either coincide or intersect at two points 180 deg apart. The larger the relative inclination, the larger the relative motion analemma. An alternative approach to the relative motion analemma is discussed here, in which we look at the relationship between coverage circles both within a single orbit
13.1
Coverage in Adjacent Planes
677
plane and between adjacent planes. The simplest coverage relationship between satel lites is, of course, for those in the same orbit plane as shown in Fig. 13-1. As can be seen from the figure, the spacing between satellites in a single orbit plane determines whether coverage in that plane will be continuous, and the width of the continuous coverage region. “Bulge”
Fig. 13-1.
T h e “Street of C ove ra ge ” is a Sw ath Centered on the G ro u n d T ra c k for w hich there is C o n tin u o u s C overage.
Assume that there are N satellites equally spaced at S - 360/N deg apart in a given orbit plane and that X ^ is the maximum Earth central angle, as defined in Sec. 9.1. [See Eqs. (9-2) to (9-7).] There will be intermittent coverage throughout a swath of half-width Xmax. If S > 2 Xmax, the coverage is intermittent throughout the entire swath. If S < 2 Xmax, there is a narrower swath, often called a street o f coverage, centered on the ground trace and of width 2 Xstreet>m which there will be continuous coverage. This width is given by: cos Xstreet = cos X ^ 1 cos (SI2 )
(13-1)
In the coverage diagrams such as Fig. 13-1, the ground traces and coverage circles are projected onto a fictitious, non-rotating Earth (i.e., one fixed in inertial space). If we are interested in continuous coverage, this is not an issue. If the system continu ously covers all of the surface of a non-rotating Earth, then it will also cover all of the surface of the real Earth, and we do not need to take into account the rotation of the Earth to determine coverage. However, if coverage gaps are allowed in a particular constellation, then we need to be concerned about the rotation of the Earth carrying one coverage gap into another, such that the gap is longer than we would otherwise have calculated. A second issue for the figures is how to treat elliptical orbits. In this case, the ground trace in inertial space will still be a great circle. However, the rate of travel along the great circle and the size of the coverage circle will both vary. For moderate eccentricities, we can use the series expansion in Sec. 2.1.8 (see Eq. 2-12). Xmax will then be a function of the altitude and, therefore, of the position along the ground trace. Returning to the coverage patterns themselves, if the satellites in adjacent orbit planes are going in the same direction, then the “bulge” in one orbit can be used to off set the “dip” in the adjacent orbit, as shown in Fig. 13-2. In this case, the maximum
678
Constellation Design
13.1
Orbit 1
Orbit 2
Fig. 13-2.
C o ve ra g e in Adjacent Planes. If the satellites in the planes are moving in the same direction, the coverage pattern can be designed to provide maximum overlap.
perpendicular separation, Dmax, between orbit planes with continuous coverage is Dmaxs - K treet + Knax
(moving in the same direction)
(13-2)
If the satellites are moving in opposite directions, then the bulge and the dip cannot be made to continuously line up and, therefore, DmaxO = 2 ^street
(moving in opposite directions)
(13-3)
Note that the equations above are correct only in terms of the perpendicular dis tance between the orbit planes at their maximum separation. Because the ground traces are great circles, this maximum separation occurs at only two points in both orbits, 90 deg on either side of the point where the two orbits cross. While this is apparent for polar orbits, it is true for any pair of orbits. For most constellations, the inclination of adjacent orbit planes will be the same in order to maintain the same node rotation rate. Consequently, the relevant orbit parameters will be the spacing between the nodes, AN, and the common inclination of the orbits, i. This geometry is shown in Fig. 13-3. Each orbit crosses the equator at inclination i, reaches its maximum latitude 90 deg later, then comes back toward the equator. The relative inclination, discussed in more detail in Sec. 10.1, is the angle at which the two orbits intersect and is also the angular separation between the orbit planes, 90 deg from this intersection. If we draw the great circle arc connecting the two orbits at their greatest separation, this arc will cross the equator midway between the two nodes, forming two right spherical triangles that are convenient for computation. From this triangle, we find: sin {iRt2) = sin {AN12) sin i
(13-4)
Similarly, we define the phase offset, A0, between the two orbits as the difference in path length along the two orbits from the nodes to the point at which the orbits intersect. This would also be the relative phase between two satellites in these orbits which cross the ascending nodes at the same time. Again, from the small spherical
13.1
Fig. 13-3.
Coverage in Adjacent Planes
679
Relative Geometry Between T w o Planes at the Same Inclination. The great circle 90 deg from the intersection of the two planes crosses the equator midway between the two nodes and forms two identical right spherical triangles.
triangles in Fig. 13-3, we have; tan (A0/2) = tan(AM2) cos i
(13-5)
To connect Fig. 13-3 with Figs. 13-2 and 13-1, note that the maximum spacing between the two orbit planes, D, is just equal to the relative inclination, thus: D = iR
(13-6)
We can then use Eq. (13-4) to solve for the node spacing corresponding to two orbits of inclination i and maximum separation D, as follows: sin (AN/2) = sin (D/2) / sin i
(13-7)
Finally, it is convenient to determine the longitude shift Lx, and latitude Hx, of the point at which the two orbits intersect. From Fig. 13-3, we see immediately: Lx = 90 deg - AN/2 tan Hx = cos (AN/2) tan i
(13-8) (13-9)
Surrounding each ground trace in Fig. 13-3 is a swath of half-width Xmax or Xstreet, depending on the application. These swaths are illustrated in Fig. 13-4, in which the ground trace is now the dashed line in the center of each swath. In some respects, Fig. 13-4 is simply a more accurate representation of Fig. 13-2, in which the geometry on the sphere is now taken into account. Note that the swaths between adjacent orbit planes overlap substantially, as shown by the shaded region in Fig. 13-4. This overlap is at a maximum where the two orbit planes cross, and at a minimum at the equator. The shaded area is the region of double coverage in which satellites from either orbit
680
Constellation Design
13.2
plane can be used. Similarly, if the spacing between nodes is sufficiently far apart, there will be a similar diamond-shaped gap in coverage centered on the equator. The size of the overlap region or uncovered region can be easily determined for any particular problem using the geometry of Figs. 13-3 and 13-4. This allows us to do the basic coverage calculations for any two orbit planes in a constellation.
Fig. 13-4.
T w o Sw aths with the G eom etry Defined in Fig. 13-3. Overlapping coverage is shaded. T h e minimum overlap occurs at the equator. Th is provides a more realistic view than the schematic representation in Fig. 13-2.
13.2 Constellation Patterns As discussed briefly in the introduction, one of the most remarkable facets of constellation design is the diversity of both constellation patterns and methods of an alyzing them. Table 13-3 from Mora et al. [1997] provides an excellent summary of this historical process. Conceptually, we like to think of there being a small number of possible constellation configurations, roughly equivalent to the five regular solid geometry figures, and beyond that the number of satellites needed would simply decrease smoothly and uniformly with increasing altitude. Of course the problem is more complex because each of the satellites is in an orbit and therefore is moving with respect to the other satellites in the pattern, unless the others are simply leading or trail ing in the same orbit. In practice, there has been no single universal constellation pattern. Instead, there are multiple issues that tend to drive constellation design, each leading to somewhat different characteristics. For example, we may want complete global coverage but may also want to concentrate a higher fraction of our resources on high northern lati tudes where most of the world’s population resides. In most constellations, we would like to minimize the number of satellites, but would also prefer to stay below the Van Allen radiation belts to reduce the cost and complexity of each satellite. These diverse requirements have led to diverse solutions.
13.2
681
Constellation Patterns
T A B L E 13-3.
Review of Constellation
A u th o r
Year
D e s ig n .t
Orbit type
Design method*
Rem arks
Luders
1961
circ., inclined, symmetrical
S oC
full single coverage
Ullock, Schoen
1963
circ., polar, non-symmetr.
SoC
phasing of co-rotating planes
Walker
1970 1977
star pattern
satellite triad
T/P/F and inc.
1972
circular
symm. group
hard to find
Emara, Leondes
1976
circular, inclined
pt covsim ul.
optimum 4X2 and 3X4
Beste
1977
circ., polar, non-symmetr.
SoC
systematic, mult. cov.
Ballard
1980
rosette (delta pattern)
satellite triad
systematic, mult. cov.
Lang, Hanson
1983
delta pattern
pt cov simul.
minimum revisit time
Rider
1986
delta pattern
SoC
analyt. closed form, mult, cov.
Draim
1986
elliptic, high altitude
tetrahedron
4 s/c continuous cov.
Adams, Rider
1987
polar, circular, non-symm.
SoC
systematic computational approach
Lang
1987
circular, inclined
point cov + s/c triad
2 step approach.
Mozhaev
delta pattern
5 sat. for global cov.
Hanson, Linden
1988
circular, inclined
SoC
B TH double, A T H single
Mainguy et al.
1989
geos., elliptic, inclined
zonal cov.
S Y C O M O R E S , 2 -3 s/c constellation
Rondinelii etal.
1989
G E O (3) + T U N D R A (2 )
zonal cov.
orbit control analysis
Hanson , Higgins
1990
geosynchronous
pt cov simul.
geostationaty, Walker, elliptic, com.
MaraI et at.
1990
L E O circular
zonal cov.
network topology
Baranger et al.
1991
G P S like
pt cov simul.
adaptative random search, PDOP
Hanson et al.
1992
c irc u la r, incline d
cov. timeline meshing
time gap, partial cov, repet. g. track
Lang
1993
circ., polar, non-symmetr.
ptco v simul.
up to 100 s/c
Werner et al.
1995
L E O - IC O
analytical approx.
network topology sim.
Radzik, Mara1
1995
Walker and Beste types
SoC
min. revisit time
Sabol et al.
1996
E LLIP S O
refinement
orbital perturbations
Ma, Hsu
1997
Repeat, ground track
cov. timeline meshing
part, cov, oblate earth
Kelley, Fischer
1997
G P S orbit type
simulated annealing
V D O P optimization
Pablos, Martin
1997
G E O + IG S O
zonal coverage
availability, integrity
Lansard, Palmade
1997
L E O circular
Design to cost
Cost/efficiency, spares, availability
682
Constellation Design
T A B L E 13*3. Author
Review of Constellation Year
D e s ig n .t
Orbit type
13.2
(Continued) Remarks
Design method*
Palmade et al.
1997
L E O circular
Double Walker
G E O interference constraint
Boudier et al.
1997
G E O + M EO
Hybrid
Communication, stationkeeping
Renault
1997
G E O + IG S O
Hybrid
Navigation const., H D O P , VDOP
Micheau, Thiebolt
1997
G E O + LEO
Walker
Accuracy, integrity, continuity
Perrota etal.
1997
L E O circular
Walker
Nav., 75 s/c constel.
Palmerini
1997
Elliptical
Hybrid
Addl. local cov.
Draim
1997
Elliptical + circular
Hybrid systems
Stationkeeping, collision avoidance
Comara et al.
1997
G E O , LEO
Various
Geom. and dynamics s/c interaction
Ulivieri et at.
1997
L E O sun-sync. Circ.
Optimal revisit time
Earth observation
Lang, Adam s
1997
L E O circular
pt cov simul.
Comparative table of opt. Const.
Lucarelli et al.
1998
circular
Hippopede
General analytic approach
* S o C = street of coverage pt cov = point coverage simulation t Adapted from Mora et al. [1997], which also includes a detailed list of references.
An interesting example of this solution diversity is the straightforward question for constellation design, “What is the minimum number of satellites required to provide continuous coverage of the Earth?” In the late 1960s, Easton and Brescia [1969] of the United States Naval Research Laboratory analyzed coverage by satellites in two mutually perpendicular orbit planes and concluded that at least six satellites were needed to provide complete Earth coverage. In the 1970s, John Walker [1971, 1977, 1984] at the British Royal Aircraft Establishment expanded the types of constellations considered to include additional circular orbits at a common altitude and inclination. He concluded that continuous coverage of the Earth would require five satellites. Be cause of his extensive work, Walker constellations are a common set to evaluate for overall coverage as discussed below. Finally, in the 1980s, John Draim [1985, 1987a, 1987b] found and patented a constellation of four satellites in elliptical orbits which would provide continuous Earth coverage. A minimum of four satellites is required at any one instant to provide full coverage of the Earth. Consequently, while the above progression looks promising, the 1990s didn’t yield a three-satellite full Earth cover age constellation and the 2 0 0 0 s are unlikely to provide a two-satellite constellation. In principle, specifying a constellation pattern means specifying the number of satellites and the six orbit elements for each. Thus, in some sense, the process of con stellation design is the process of filling in an n x 6 matrix where n is the number of satellites. In practice, it rarely works that way. Because of the need for symmetry, there are usually far fewer free parameters. At the same time, a great many practical constel lations, such as GPS, are not fully symmetric, such that more than the minimum number of parameters is often needed. In addition, a non-orbit parameter, the swath
13.2
Constellation Patterns
683
width or minimum working elevation angle, is a key element of constellation design and should be specified as a constellation parameter, although this is rarely done. As discussed numerically in Secs. 13.1 and 13.3, the swath width is a function of the ele vation angle and the altitude. In turn, the node spacing for the constellation is a func tion of the swath width and the inclination. We discuss below the most common constellation patterns and then summarize some Other patterns which have been proposed, including a constellation pattern which has no structure at all. (See the boxed example at the end of the section.)
Geosynchronous Constellations The simplest constellation pattern is a set of geosynchronous satellites that work together to perform a common function. For example, TDRS consists of multiple satellites in geosynchronous orbit that cover a large fraction of the Earth as shown in Fig. 13-5. The basic coverage parameters are defined in Eqs. (13-1) to (13-4) above. Geosynchronous constellations have been used extensively for many purposes, includ ing communications, weather, and tracking of low-Earth orbit satellites.
Fig. 13-5.
Geosynchronous Constellation. This is the simplest of the constellations, consist ing of several satellites in geosynchronous orbit working together to achieve a com mon purpose, such as T D R S .
Geosynchronous constellations require very few satellites. However, they are in very high energy orbits and occupy prime real estate in the space arena. These constel lations cover equatorial and mid-latitude regions extremely well but do poorly over high latitude regions. The biggest advantage of geosynchronous constellations is that the individual satellites remain essentially fixed with respect to the surface of the Earth, and therefore provide continuous coverage of one region and antennas on the Earth’s surface can maintain a fixed orientation when pointing at the satellite.
Streets o f Coverage Constellations The concept of streets of coverage, introduced in Sec. 13.1, can be used to define a constellation pattern also known as “streets of coverage, ” in which n satellites in each
684
Constellation Design
13.2
of m approximately polar orbit planes are used to provide continuous global coverage. As illustrated in Fig. 13-6, at any given time satellites over half of the world are going northward and satellites over the other half are going southward. Within both regions, the orbit planes are separated by DmaxS as defined in Eq. (13-2). Between the two halves is a seam in which the satellites are going in opposite directions. Consequently, the spacing between planes on opposite sides of the seam must be reduced to Dm(aQ as defined in Eq. (13-3) in order to maintain continuous coverage. In order to provide global continuous coverage for a streets of coverage pattern using m planes of polar orbits, we must have: (m + 1) ^ treet + (m - 1)
> 180 deg
(13-10)
where hstreet and Xmax are defined in Eqs. (13-1) and (9-2).
Fig. 13-6.
“Streets of C o ve ra g e ” Constellation Pattern. View seen from above the North Pole. Northward portions of each orbit are shown as solid lines; southward portions are dashed.
Notice that it is the swath width and not the altitude which determines the number of orbit planes and ultimately the number of satellites required to provide global coverage. This illustrates clearly one of the most critical characteristics of constella tions— coverage does not vary continuously and smoothly with altitude. There are discrete jumps in coverage which depend primarily on k max which in turn depends upon the minimum elevation angle Emin and the altitude [see Eqs. (9-2) to (9-7)]. If we keep £min fixed and lower the constellation altitude, then we will reach an altitude plateau at which we will need to add another orbit plane and n more satellites to cover the Earth. The Iridium communications constellation was originally intended to have 77 satellites in a streets of coverage pattern. (The element iridium has atomic number 77.) By slightly increasing the altitude and decreasing the minimum elevation angle, the number of orbit planes was reduced by 1 , and the number of satellites required for continuous coverage was reduced to only 6 6 . (Unfortunately, dysprosium is not a good constellation name.)
13.2
Constellation Patterns
685
Figure 13-6 illustrates another key constellation characteristic— what appears to be a major collision problem at the poles. One can almost see the satellites colliding there. Many practical constellations spread the intersections out somewhat over the polar region to avoid this. Intersatellite collisions are indeed a problem, as discussed in Sec. 13.7. However, they are no more of a problem for the streets of coverage pattern than for any other constellation pattern made of circular orbits. Every great circle orbit will intersect every other great circle orbit exactly twice. Whether those intersections are bunched together at the pole or spread out over the globe doesn’t matter. It is the pairwise probability of collision that matters and this depends only on the fact that all pairs of orbits intersect. Thus, there are just as many collision possibilities in any con stellation pattern with a given number of satellites and a given number of orbit planes as there are in any other pattern with the same number of satellites and orbit planes. The streets of coverage pattern is generally very efficient, allowing good overlap between adjacent satellites, and allowing the space between adjacent planes to be increased by taking advantage of the bulge in one plane accommodating the dip in the adjacent plane. There are two principal drawbacks to this pattern. First, the seam represents an asymmetry in coverage and satellite relative motion. Consequently, satellites on either side of the seam must behave differently than those in other parts of the pattern. Second, and perhaps most important, the streets of coverage constella tion has the greatest coverage over the pole where there are few people or facilities needing its services. This overcoverage at the pole leads us to look for ways to reduce the inclination so as to have more coverage at mid-latitudes where there is greater demand.
Walker Constellations The most symmetric of the satellite patterns is the Walker constellation, named after John Walker, who made an extensive study of their properties at the Royal Air Force Establishment from the late 1960s to the early 1980s [Walker, 1971, 1977, 1984]. The most common of these constellations is the Walker Delta Pattern, contain ing a total of T satellites with S satellites evenly distributed in each of P orbit planes. All of the orbit planes are assumed to be at the same inclination, i, relative to the equa tor. (For constellation design purposes, the reference plane need not be the Earth’s equator, however, since orbit perturbations depend on the inclination relative to the equator, this is the most practical reference plane.) Unlike the streets of coverage pat tern, the ascending nodes of the P orbit planes in Walker patterns are uniformly dis tributed around the equator at intervals of 360 degIP. Within each orbital plane, the S satellites are uniformly distributed at intervals of 360 degIS. The only remaining issue is to specify the relative phase between the satellites in adjacent orbit planes. To do this, we define the phase difference, A, in a constellation as the angle in the direction of motion from the ascending node to the nearest satellite at a time when a satellite in the next most westerly plane is at its ascending node. In order for all of the orbit planes to have the same relationship to each other, A0 must be an integral multiple, F, of 360 deg/7*, where F can be any integer from 0 to P - 1. So long as this condition holds, each orbit will bear the same relationship to the next orbit such that there is no starting or ending point in the constellation. The pattern is fully specified by giving the inclination, and the three parameters T, P and F. Usually such a constellation will be written in a shorthand notation of i:T/P/F. For example, Fig. 13-7 illustrates Walker constellations of 65:15/3/2 and 65:15/5/4. Table 13-4 gives the general rules for Walker Delta Pattern parameters.
Constellation Design
686
(A ) 15/3/2 Configuration Fig. 13-7.
13.2
(B ) 15/5/4 Configuration
Typical Walker Constellations for /= 65 deg.
T A B L E 13-4. Characteristics of a Walker Delta Pattern Constellation [Walker, 1984]. T/P/F— Walker Delta Patterns T = Number of satellites P = Number of orbit planes evenly spaced in node F = Relative spacing between satellites in adjacent planes Define S = V P - Num ber of satellites per plane (evenly spaced) Define Pattern Unit, P U = 360 degI T Planes are spaced at intervals of S PU s in node. Satellites are spaced at intervals of P P U s within each plane. If a satellite is at its ascending node, the next most easterly satellite will be F P U s past the node. F is an integer which can take on any value from 0 to ( P - 1). Example: 15/5/4 constellation shown in Fig. 13-7B 15 satellites in 5 planes ( T = 15, P - 5) 3 satellites per plane (S = T / P = 3) P U = 360/7= 360/15 = 24 deg In-plane spacing between satellites =. P U X P = 2 4 X 5 = 120 deg Node spacing = P U x S = 24 x 3 = 72 deg Phase difference between adjacent planes = P U x F - 24 X 4 = 96 deg
Walker constellations have the advantage of having complete symmetry in longi tude. Because both the ascending and descending nodes cover the full equator, they give up some efficiency relative to the polar orbit streets of coverage patterns. How ever, they gain efficiency by allowing the inclination to be reduced to provide the highest level of coverage at mid latitudes, which has by far the largest population concentration. Perhaps the greatest advantage of Walker constellations is that there is a finite number of them and all can be identified and investigated. This has made them popular
Constellation Patterns
13.2
687
with mission designers, since a “complete” analysis can be done for any specific characteristic of interest. This completeness has made them one of the most analyzed constellation designs, but does not necessarily make them better or more successful than other patterns.
Elliptical Orbit Patterns The oldest elliptical orbit constellations are Molniya orbits, used by the former Soviet Union for communications since the opening of the Space Age. (See Sec. 2.5.4.) While geosynchronous satellites are inherently useful as single spacecraft, communications systems using Molniya orbits require multiple satellites to provide continuous coverage of any region. Nonetheless, they can provide coverage of high northern latitudes which cannot be successfully served from geosynchronous orbit. It is this characteristic that caused their use by the Soviet Union which has extensive territory at high latitudes. As discussed in Sec. 12.4.3, elliptical orbits add complexity to both the system and spacecraft design and, at the same time, add an additional degree of freedom to allow the constellation designer to optimize coverage to fit specific needs. Draim, et al. [2 0 0 0 ] have done an extensive analysis of these trades in order to use this additional design freedom to maximize the performance benefits in terms of coverage of appro priate latitudes, longitudes, and time of day. The Ellipso constellation shown in Fig. 13-8 takes full advantage of this freedom to optimize coverage for both the locations of the Earth and the time of day at which peak traffic is anticipated.
Fig. 13-8.
Th e Ellipso Communications Constellation. Elliptical orbits are used to optimize coverage as a function of longitude, latitude, and time of day. As discussed in the text, there will also be a significant cost, (Adapted from Draim [2000].)
The largest advantage of elliptical constellations is that they provide additional free parameters to optimize the constellation. The penalty for this added freedom is a greater level of spacecraft complexity, because it must work at varying altitudes which, in turn, means variations in range, angular size of the Earth’s disk, in-track velocity, and relative position for satellites in the same orbit. Elliptical orbits are effectively restricted to the critical inclination of 63.4 deg, and will be in the Van Allen belts much of the time. Consequently, components for these spacecraft will need extensive radiation hardening.
688
Constellation Design
13.2
Other Constellation Patterns There are many types of non-standard constellation patterns, some of which may be much better suited to specific problems than either Walker or streets of coverage patterns. For example, the Ellipso constellation described above adds an equatorial ring to improve coverage characteristics at low latitudes. Similarly, a polar ring could be added to a constellation for high latitude coverage, in conjunction with an inher ently low latitude or low altitude constellation. Figure 13-9 shows specific examples of non-Walker constellations, i.e., ones which break specific symmetries in order to achieve other objectives.
(A ) 2-plane Polar
(B ) 3 M utually Perpendicular Planes
(C ) 2 P erpendicular N on-pola r Planes
(D ) 5-plane Polar '‘Streets of C o ve ra g e ”
Fig. 13-9.
Exam ples of Ty p ic a l N on-W alker Constellations. All orbits are assumed to be circular. There is also a variety of constellations with non-circular orbits as discussed in the text.
A major problem for constellation design is the fact that it is not a systematic study. Many variations are possible. GPS follows nearly a Walker pattern. Ellipso provides excellent coverage characteristics with eccentric orbits. Iridium and others have demonstrated how the asymmetric streets of coverage constellation can also provide
13.2
Constellation Patterns
689
excellent coverage. One of the most intriguing examples for a constellation which defies nearly all of the rules of constellation design is the Multi-Satellite System Program described in the boxed example. MSSP— A Counterexample for All the Rules of Constellation Design Constellations use symmetry to provide well defined relationships between satellites that persist over time so that the total number of satellites can be minimized. Thus the streets of coverage pattern provides a series of satellites which moved "together” over the surface of the Earth and the Walker patterns provide a regularly recurring coverage mechanism. However, it is this very symmetry which represents a weakness from the perspective of providing continuous coverage. For example, if one or two satellites in the GPS navigation constellation are destroyed, there will be long and regular breaks in coverage that are easily predicted and could in principle be used for a military advantage. One way around this problem of allowing an opponent to take advantage of coverage holes is to create a constellation consisting of satellites in random or pseudorandom orbits, such that the constellation pattern does not repeat but is continuously changing. Although it requires a larger number of satellites than ones having a high degree of symmetry, it has the advantage that the loss of even a significant number of satellites either intentionally or by accident will not significantly disrupt the communications system as a whole, since the system does not depend on a regular pattern to provide communications. This was the basic concept of the Multi-Satellite System Program, which was an inherently survivable com munications constellation illustrated in Fig. 13-10.
Fig. 13-10.
M S S P T h i s c o n ste lla tio n c o n s is ts of a la rg e n u m b e r of s a tellites in ra n d o m orbits. Drawing from Fleeter [1999].
While MSSP requires a larger number of satellites than most other constellation patterns, it has a number of inherent advantages in addition to fundamental survivability. Because the orbits are random, they do not need to be controlled or maintained. This significantly reduces the operations cost and complexity. In addition, insertion into specific orbits is not required so that piggyback launches or other cheap mechanisms of putting spacecraft in orbit can be used because the orbits themselves are not critical. Similarly, any launch dis persion that comes from errors or anomalies in the launch process makes no difference, so long as the satellites have achieved orbit. The idea of deploying MSSP was dropped because of the large number of satellites required. However, it remains an interesting possibility, as microsatellites and nanosatellites become more competent. Certainly one characteristic of this type of constellation is that it violates virtually all of the rules of constellation design.
690
Constellation Design
13.3
13.3 Selection of Constellation Parameters Constellation parameters are frequently thought of as the orbit elements of the satellites in the constellation. Nonetheless, one of the most critical of the constellation parameters is the swath width or maximum Earth central angle, which defines the coverage for each of the satellites in the constellation. The eccentricity of the orbits is also a key parameter although in most cases it is set to zero or near-zero (i.e., a frozen orbit. See Sec. 2.5.6). Non-zero eccentricity orbits are discussed in Sec. 13.2 and the references therein. In this section, we will discuss the basis for selecting the other prin cipal constellation parameters consisting of: • Swath width (maximum Earth central angle) • Altitude • Inclination • Node spacing Formulas for the relevant parameters were defined in Chaps. 2 ,9 and Sec. 13.1. This section defines the logic and trades ordinarily used in the parameter selection process.
Swath Width (Maximum Earth Central Angle) The key coverage parameter for any constellation is the area that can be covered by each individual satellite. For discontinuous coverage, this is most conveniently mea sured by the swath width or maximum Earth central angle. For constellations requiring continuous coverage, the maximum Earth central angle represents the circle of cover age that can be seen by the satellite and the minimum sw ath width or width of the street of coverage represents the required spacing between orbit planes. We tend to regard the altitude as the principal determinant of coverage for satellite systems. However, it is the swath width or the maximum Earth central angle which actually determines the coverage for individual satellites. The maximum Earth central angle is a function of both the altitude and the minimum elevation angle or grazing angle at which the system operates. (See Table 13-5.) Particularly at low elevation angles, the maximum Earth central angle is a very strong function of the minimum elevation angle. As shown in Fig. 13-11, a given swath width can result from a variety of combinations of elevation angle and altitude. Formulas for the maximum swath width as a function of both altitude and minimum elevation angle were provided in Sec. 13.1 and 9.4. Recall that the maximum swath width is simply twice the maximum Earth central angle. As shown in Fig. 13-12, the minimum swath width or width of the street of coverage is determined by both the maximum Earth central angle and the spacing between satellites, as defined in Eq. (13-1). Depending on the application, we want either the coverage circles or the swaths to overlap in order to maintain good satellite coverage. If this does not occur, then there will be gaps in coverage between the orbit planes. Similarly, the swath width and maximum Earth central angle will also determine the reach of constellation coverage between the ground track and the Earth’s poles. Thus, the wider the swath width, the lower the orbit inclination can be, while still maintaining coverage at high latitudes. The minimum elevation angle is fundamentally determined by payload perfor mance. Nearly all payload types will perform better at higher elevation angles because of the angle above the horizon, distance, and atmospheric absorption. Consequently, payload designers would prefer to limit the minimum elevation to relatively high
Selection of Constellation Parameters
13.3 T A B L E 13-5.
691
Swath Width (deg) as a Function of Altitude and Minimum Working Elevation Angle. It is the swath width that is the most basic determinant of coverage. Minimum Working Elevation Angle (deg)
Altitude (km)
0
5
10
15
20
30
100
20.16
12.48
8.32
6.02
4.61
3.00
200
28.33
2001
14.56
11.04
8.68
5.78
300
34.48
25.86
19.71
15.40
12.34
8.39
400
39.56
30.76
24.15
19.29
15.68
10.84
500
43.96
35.03
28.09
22.80
18.76
13.15
600
47.87
38.84
31.65
26.02
21.61
15.34
700
51.39
42.29
34.90
28.99
24.28
17.41
800
54.62
45.46
37.90
31.75
26.78
19.38
900
57.59
48.38
40.68
34.34
29.13
21.26
1,000
60.36
51.10
43.29
36.77
31.35
23.05
1,500
71.89
62.49
54.25
47.11
40.93
30.96
2,000
80.85
71.36
62.87
55.33
48.65
37.51
2,500
88.15
78.60
69.94
62.12
55.08
43.05
3,000
94.30
84.70
75.90
67.87
60.55
47.83
3,500
99.57
89.93
81.03
72.83
65.29
52.00
4,000
104.16
94.50
85.51
77.17
69.45
55.69
4,500
108.21
98.52
89.46
81.01
73.13
58.97
5,000
111.81
102.11
92.99
84.43
76.43
61.91
10,000
134.16
124.35
114.90
105.81
97.07
80.58
15,000
145.28
135.42
125.83
116.50
107.44
90.05
20,000
152.01
142.12
132.45
122.99
113.73
95.83
(A ) Fig. 13-11*
(B )
(C )
Three Systems with Equal Swath Widths but Different Altitudes. The only dif ference is the orbit period and, therefore, the longitude shift per orbit. Swath width = 4,000 km for all three orbits: (A ) altitude = 525 km, minimum elevation angle = 5 deg; (B) altitude = 735 km, m in im u m elevation angle = 10 deg; (C ) altitude = 1,225 km, minimum elevation angle = 20 deg.
692
Fig. 13-12,
Constellation Design
13.3
Characteristic M axim um Sw ath W idths. T h e minimum swath width or width of the street of coverage will depend on the in-plane spacing between satellites, but is typically chosen as approximately 7 0 % of maximum swath width.
values. From a constellation design perspective, however, a high minimum elevation angle implies either a larger number of satellites or a higher altitude, both of which significantly impact cost. Consequently, a key issue in the constellation design process is to find a reasonable compromise between altitude and minimum elevation angle in order to achieve the widest possible swath width for a given constellation. This becomes one of the most important system engineering trades in the design of most constellations. Because of the importance of swath width in determining the number of satellites required and the payload performance, a reasonable alternative is to consider using more than one performance plateau. For example, in observation systems, it might be acceptable to have one level of performance to provide complete Earth coverage on a daily basis if better performance (i.e., a higher elevation angle) can be provided every several days. This would allow frequent coverage at moderate resolutions with less frequent coverage at higher resolution with a smaller number of satellites. Similarly, a communications satellite system might be willing to allow lower minimum elevation angles in the vicinity of the equator where the population density is not as high as in the mid-latitude regions. As satellites move away from the equator, the orbits converge and the spacing between the orbit planes becomes less. Consequently, one could envi sion a satellite system with a 1 0 -deg minimum elevation angle in high latitude regions where the population density is high and an 8 -deg minimum elevation angle over the equator. Such a constellation could potentially be built with a significantly fewer satellites because of the much wider swath width that can be accommodated at the equator and, therefore, the smaller number of orbit planes that would be required.
Altitude Generally, all of the satellites will need to be at the same altitude in order to have the same period. This is important in order to maintain a uniform relationship between the satellites over time. If all of the orbit periods in the constellation are the same, then the pattern will repeat every orbit. Otherwise, it will not and a variety of different
13.3
Selection of Constellation Parameters
693
patterns will need to be accommodated. While this is not impossible, most constella tions tend to work at a single common altitude. A second reason for this is that it also keeps the node rotation rate the same such that the orbit planes continue to maintain their relative orientation. If the minimum elevation angle has been fixed by the payload (typically as a result of extensive trades as discussed above), then the altitude is the key driver for constel lation design. Specifically, higher altitudes will mean a smaller number of satellites and a smaller number of orbit planes. However, the number of satellites and number of planes is not continuously variable with the altitude. Rather, there are a series of altitude plateaus below which an additional plane will be required. For example, for circular polar orbits and a 0 -deg minimum elevation angle, the following plateaus exist: • Above 2,642 km, complete coverage can be provided with two perpendicular planes • Below 2,642 km, at least three planes are required for full Earth coverage • Below 987 km, at least four planes are required for full coverage • Below 526 km, at least five planes are required for full coverage Altitude plateaus are a function of both the minimum working elevation angle and the inclination. As the minimum elevation angle becomes higher, the swath width becomes smaller, and the altitude plateaus for a given number of orbit planes become higher. Conversely, at lower inclinations, the required node spacing at the equator becomes larger and a smaller number of planes can suffice. Table 13-6 provides rep resentative examples of altitude plateaus for various minimum elevation angles and inclinations. In general, systems will need to be at somewhat higher altitudes than those specified to accommodate the stationkeeping box. The values in this table should be taken only as representative limits. The actual values depend not only on the mini mum elevation angle but also on the portions of the Earth to be covered, the overlap area needed for stationkeeping and margin, and the desired coverage pattern. The values in Table 13-6 are representative for streets of coverage constellations in which coverage at both ascending and descending nodes are used. Walker constella tions will need approximately twice as many planes because the Walker patterns are meant to provide complete coverage at both the ascending and descending nodes. Also note that less planes may be required in a lower inclination orbit in order to cover the equator. However, they may leave coverage gaps in some higher latitude regions in the streets of coverage type patterns. In addition, more planes may also be required due to the need for overlap. The altitude plateaus for any particular constellation pattern will be unique to that pattern. The principal point to bear in mind is that for any coverage objective and con stellation pattern a series of altitude plateaus exists. As we go down in altitude, we will get less and less overlap until at some point the overlap will disappear entirely and we will need to add one more plane, or revise the constellation pattern in some other way in order to ensure coverage. The altitude plateaus in Table 13-6 take only coverage into account. However, as described in detail in Chap. 12, a variety of other issues are strongly affected by the altitude. Of particular importance are the launch capability and the radiation environ ment. Above approximately 1,000 km, the radiation environment increases extremely rapidly. This significantly drives up the complexity and cost of individual spacecraft.
694
Constellation Design
T A B L E 13-6.
13.3
Altitude Plateaus (k m ). These plateaus represent the minimum number of planes required for complete equatorial coverage in a streets of coverage pattern. Walker constellations will typically require twice as many planes. Th e actual altitude for a given constellation will typically be higher. See text for limitations and discussion.
M inim um Elevation A n g le = 0 deg
M inim um Elevation A n g le = 5 deg
Inc.
90 deg
75 deg
60 deg
Inc.
90 deg
75 deg
60 deg
Planes
Altitude (k m )
Altitude (km )
Altitude (k m )
Planes
Altitude (k m )
Altitude (k m )
Altitude (k m )
10
80
74
59
10
170
162
137
9
98
92
73
9
200
189
160
8
125
116
93
8
240
227
192
7
164
153
122
7
297
281
235
6
225
209
167
6
384
361
301
5
328
305
241
5
524
493
406
4
526
486
382
4
785
734
596
3
987
906
698
3
1,379
1,275
1,008
2
2,642
2,354
1,690
2
3,507
3,132
2,276
M inim um Elevation A n g le = 10 deg Inc.
90 deg
75 deg
60 deg
Planes
Altitude (km )
Altitude (km )
Altitude (km )
M inim um Elevation A n g le = 20 deg Inc.
90 deg
75 deg
60 deg
Planes
Altitude (km )
Altitude (k m )
Altitude (k m )
10
265
253
218
10
475
454
397
9
306
292
251
9
543
519
451
8
361
344
295
8
633
604
524
7
438
416
355
7
757
721
622
6
552
523
443
6
939
892
764
5
736
695
582
5
1,228
1,163
986
4
1,069
1,004
828
4
1,751
1,648
1,373
3
1,821
1,691
1,353
3
2,946
2,735
2,197
2
4,573
4,078
2,966
2
7,804
6,860
4,857
In addition, launch costs also increase significantly with altitude. For example, the payload available in geosynchronous orbit is only one-fifth of that available in lowEarth orbit for a given launch system. Consequently, higher altitudes will generally imply a smaller number of satellites in a constellation; however, the satellites themselves may be significantly more expensive to build or launch and therefore the total system cost may or may not be less than with a low-altitude constellation. See Table 12-9 in Sec. 12.4.1 for a list of the principal system parameters impacted by the altitude. In constellation design studies, each of these issues should be addressed in terms of their impact on overall system cost and performance as a part of the altitude selection process.
13.3
Selection of Constellation Parameters
695
Inclination One of the most important “rules” of constellation design is driven by the oblate ness of the Earth. As discussed in detail in Sec. 2.5 and 12.4, the oblateness, represented by J 2 in the spherical harmonic expansion of the Earth’s gravitational potential, causes both the node and argument of perigee to rotate rapidly with respect to the lifetime of most constellations. This rotation rate is a function of the altitude, the inclination, and the eccentricity. However, rotation itself is typically not a problem
so long as the whole constellation rotates together. This will occur if the satellites are all at the same altitude, inclination, and eccentricity. For example, if we construct a constellation at 700 km with some satellites at an inclination at 70 deg and some at 30 deg, those at 70 deg will have a node rotation rate of 2.62 deg per day, and those at 30 deg will have a node rotation rate of 6.63 deg per day. This means the planes will move with respect to each other at the rate of 4 deg per day. While it may not be impossible, this certainly makes the construction of a long term constellation challenging. If the orbits are noncircular, than the rotation of the argument of perigee due to J2 also becomes important. This implies that constellations with satellites in eccentric orbits will almost certainly be at the critical inclination of 63.4 deg, so that perigee doesn’t rotate.
(A ) Viewed from Pole Fig. 13-13.
(B ) Viewed from Equator
Ground Swaths as Seen from Equator and from the Pole for Satellites with a Swath Width of 30 deg and Inclination of 0 deg, 45 deg, and 90 deg.
The inclination has a strong impact on both how coverage patterns are formed and on coverage as a function of latitude. Figure 13-13 shows the ground swath as seen from both the pole and the equator for satellites in orbits with inclinations of 0,45, and 90 deg. The coverage as a function of latitude for these ground swaths is shown in Fig. 13-14. Note that it is not simply the coverage as a function of latitude but also the character of the coverage curves that changes significantly between the three cases. For equatorial orbits, coverage is essentially constant as a function of latitude. At lat itudes below the half width of the streets of coverage, the coverage will be 1 0 0 % and
696
Constellation Design
13.3
at latitudes above the maximum Earth central angle there will be no coverage. Of course the width of the equatorial band will depend on both the altitude and the mini mum elevation angle, as discussed above. For the polar orbit, the highest level of cov erage is at the pole itself, which will be covered by every swath in the constellation. Coverage of the equatorial regions will be divided into two symmetric segments at the ascending and descending nodes. The streets of coverage constellation pattern is typ ically formed by multiple near-polar orbits. The moderate inclination orbits provide asymmetric coverage over the mid-latitude regions. Along the equator, there are two bands of coverage similar to the polar orbit. However, coverage is best at mid-latitude regions centered approximately on the inclination, which is also the maximum latitude of the ground track of the satellite. The actual maximum coverage is achieved at a lat itude equal to the inclination minus the swath width. Above a latitude equal to the in clination plus the swath width there will be no coverage. Algebraic formulas for the coverage and breakpoints were provided in Sec. 9.5.1.3. The key issue here is to note the asymmetry of the coverage and how the inclination can be adjusted to provide varying levels of coverage at varying latitudes.
Latitude (deg)
Fig. 13-14.
Single Satellite Coverage vs. Latitude. All three curves are for a satellite with a swath width
of 30
deg. See also Tables 13-14 and 13-15.
An interesting constellation design that provides complete global coverage could be provided by a combination of polar and equatorial orbits. So long as the altitude is above the plateau for which only two planes are required, then an equatorial orbit can be combined with a polar orbit with any node orientation to provide complete global coverage. For these orbits, the orbit rotation due to the oblateness of the Earth will not be important and complete continuous coverage can be maintained over time with a relatively small number of satellites. For example, at an altitude of 8,660 km and a minimum elevation angle of 5 deg, the maximum Earth central angle is 60 deg and the half width of the streets of coverage is 45 deg for 4 satellites. For this constellation, complete continuous global coverage can be provided by four satellites in an equato
13.4
Stationkeeping
697
rial orbit and four satellites in a polar orbit, with the maximum overlapping coverage occurring in the general vicinity of 45 deg latitude.
Node Spacing The actual location of the ascending node for individual orbits in a constellation is generally irrelevant, since the whole constellation will rotate with respect to inertial space and with respect to the Earth’s surface. (For some repeating ground track orbits, the node location may be important in order to provide appropriate coverage at specific longitudes). The key issue is to have all of the nodes rotate at the same rate, which implies a single altitude and inclination for the entire constellation, except for the possibility of adding an equatorial ring to improve coverage there, since node rotation doesn’t matter for equatorial satellites. Four types of node spacing are most commonly used: • Equal node spacing over the complete equator as done for Walker constellations • Equal node spacing over half the equator to avoid coverage gaps between orbits (see Fig. 13-8). Except for polar orbits, this process tends to leave a hole in one region of the Earth • Equal node spacing except for a seam between satellites going up and coming down, as used in streets of coverage patterns (see Fig. 13-9D) • Node spacing adjusted in pairs or triplets to provide “fill-in coverage” for suc cessive satellites. For example, a synthetic aperture radar can not see targets which lie along the ground track. Thus, a system of this type intended to provide nearly complete coverage would need successive satellites having the nodes shifted slightly to fill in the nadir hole for the preceding satellite. While equal node spacing around the equator is the one most often evaluated, it is not necessarily the only or best choice. There may well be clever selections of node and inclination which provide strong coverage characteristics for specific applications. The key issue in node spacing is to look at the coverage patterns to see how best to achieve the specific mission objectives. For example, Fig. 13-9C illustrates a coverage pattern with asymmetric nodes which is achieved by using a two-plane polar constel lation and simply reducing the inclination in order to provide greater launch mass.
13.4 Stationkeeping Stationkeeping is the process of maintaining a satellite within a well defined box or position relative either to inertial space or to other satellites. In formation flying, the position is always maintained relative to another spacecraft, such as maintaining close proximity to the Space Station or building a multi-satellite interferometer. In global constellations, stationkeeping may be done either relative to other satellites or, as in the case of geosynchronous spacecraft, relative to inertial space or the surface of the Earth. The principal reason for stationkeeping is to maintain the overall constellation structure. At the same time, it also provides collision avoidance and allows intersatel lite communications. Within constellations, the objective of stationkeeping is to maintain the relative position between satellites. We want to minimize the propellant utilization and the system cost and complexity needed to do this. The need for stationkeeping arises from two sources:
698
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13.4
• Orbit Perturbations Atmospheric drag causes overall system decay that must be compensated in an long-lived LEO constellation to keep the constellation from ultimately decaying and reentering. In addition, drag and other orbit perturbations impact each satel lite in the constellation differently. These differences can accumulate with time such that small variations in perturbations can lead to large separations over the course of many orbits. • Variations in Initial Conditions Variations in the initial conditions for each individual satellite cause the satel lites to drift in time with respect to each other. If these drifts are oscillatory or accumulate only small differences, we may be able to simply live with them. However, small differences in altitude, for example, cause different periods such that the in-track error grows exponentially with time. Small differences in incli nation cause differences in the node drift rate which also grows with time. There are several different ways to evaluate stationkeeping. Geosynchronous stationkeeping is well understood and will be treated only briefly here. For extended discussions, see for example Pocha [1987] or Soop [1994]. We will concentrate primarily on stationkeeping in low-Earth orbit. Ultimately, the implementation will be divided into in-track stationkeeping, which controls the true anomaly or phase in the orbit, and cross-track stationkeeping, which controls the component perpendicular to the orbit plane, i.e., the inclination and node. The radial component (corresponding to the semimajor axis) is maintained as a part of in-track stationkeeping, since the mean altitude determines the orbit period. Each of the orbit perturbations described in Sec. 2.4 will have an impact on the motion of the spacecraft and the stationkeeping requirements. Generally, each pertur bation can be dealt with separately using one of three possible approaches. 1.
Leave the Perturbation Uncompensated This is the best method for accommodating any perturbation, since it requires no propellant and no control. We simply increase the stationkeeping budget to incorporate variations that occur over time. It is the only realistic method for accommodating short-period oscillations in the positions of the satellites, such as those due to higher order harmonics. There is simply not enough propellant on board any spacecraft to do otherwise for an extended period.
2. Control the Perturbing Disturbance to be the Same fo r AU Satellites
in the Constellation In this case satellites will maintain the same relative position but will not follow a perfectly Keplerian orbit. Providing the same level of perturbation for all of the satellites requires less propellant than negating the perturbation, and is the best approach if the perturbation cannot be ignored. For example, in low-Earth orbit, the node drift rate will be controlled to be the same for all satellites by controlling the inclination. There is insufficient propellant (and no reason) to attempt to stop the node drift rate. 3.
Negate the Perturbing Force This approach maintains the orbit characteristics over time. Consequently, it requires continuous propellant usage and should be used only when necessary. For example, atmospheric drag in low-Earth orbit must be negated in order to
Stationkeeping
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699
maintain the constellation altitude and prevent the system from decaying. In geosynchronous orbit, both the north-south perturbations due to solar-lunar effects and the east-west perturbations due to the non-spherical character of the Earth’s equator are negated in order to maintain the satellite within a well defined box over the equator. T A B L E 13-7.
Recommended Methods for Handling the Principal Perturbations in Low Earth Orbit.
Perturbation
Impact
How Handled
Atmospheric Drag
Secular decay ranging from 1-100 m/day
Negated by altitude maintenance
J2
Secular node rotation of - 6 cos /deg/day
Controlled by inclination
S e c u la r p h a s e rotation of u p to 1 4 d e g / d a y
C o n tro lle d a s pa rt of altitude
m a in te n a n c e
(Oblateness)
maintenance Changes in shape of orbit up to ~5 km variation between adjacent satellites
Uncompensated
Higher Order Harmonics (Zonal)
Eccentricity oscillation of 0.001 about a mean value of 0.0013— a value of -0 .0 0 1 3 can be maintained naturally
e » 0.0013 maintained naturally
Solar/Lunar
Secular drift in inclination and node of up to
Negated by inclination maintenance or reduced by inclination change
3.5 X 1 0 -5 deg/day
Solar Radiation Pressure
Low amplitude oscillations in inclination and node
Uncompensated
Small
Typically uncompensated
Section 10.1.1 provided a detailed discussion of the consequences of each of the perturbations. The basic stationkeeping process is to look at each of these orbit pertur bations within the context of the specific mission and determine how it should be handled. Table 13-7 provides recommendations for handling the principal perturba tions in low-Earth orbit. Table 13-8 provides similar recommendations for geosyn chronous spacecraft. Similar tables can be constructed for other orbits such as interplanetary ones or orbits around a comet or asteroid by examining the effect of the individual perturbations and determining whether it should be left uncompensated, controlled, or negated. T A B L E 13-8. Recommended Methods for Handling the Principal Perturbations in Geosyn chronous Orbit. Perturbation
Impact
H o w Handled
Drag
Negligible
Uncompensated
J 2 and Higher Order Harmonics
Negligible
Uncompensated
Sectorial Harmonics
East-West drift about stable longitudes
Negated by East-West stationkeeping
Solar/Lunar
North-South (inclination) drift of -0 .9 deg/year
Negated by North-South stationkeeping
Solar Radiation Pressure
Small
Secular component handled as part of stationkeeping
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Constellation Design
13.4.1 Controlled O rbits and Absolute
ys.
13.4
Relative Stationkeeping
To begin, it is critical to understand the distinction between alternative approaches to stationkeeping and orbit maintenance. Specifically, absolute stationkeeping main^ tains each satellite in a predefined mathematical box relative to the Earth or inertial space (this is equivalent to geosynchronous stationkeeping, but in low-Earth orbit). Relative stationkeeping maintains only the relative positions of the satellites with respect to each other and not the absolute positions. Thus, relative stationkeeping is what would be done in formation flying. Orbit maintenance is the general process of maintaining the average orbit elements over time except that the in-track phase or true anomaly is not necessarily maintained. In contrast, orbit control maintains the value of all of the orbit parameters including the in-track phase. Thus, in a controlled orbit, we will know in advance the future positions of the spacecraft because they will be controlled to have specific values to within the stationkeeping errors. We will return to the importance of these distinctions shortly. In constellation maintenance, our only goal is to maintain the relative positions of the satellites. Thus, it is natural to ask whether it is possible to save propellant by doing relative stationkeeping rather than absolute stationkeeping. A great deal of effort has been expended, both analytically and in real constellation maintenance, to achieve this. However, in most cases this is not necessary and may be counterproductive in terms of saving propellant. That is, in most practical orbit maintenance applications, absolute stationkeeping uses less AV and less propellant than relative stationkeeping. The answer to the issue of absolute vs. relative stationkeeping depends upon another key question—whether we wish to maintain the system altitude or allow the constellation to slowly “fall” to lower altitudes because of atmospheric drag. Allowing the system to fall saves propellant only in the short term (i.e., several years). Pre sumably replacement satellites would be launched at the lower altitude, drag would continuously increase as the altitude decreases, and coverage holes would begin to appear as the constellation goes lower. Eventually, if left for a long enough period, the constellation would reenter. Maintaining the altitude and negating atmospheric drag is the only way to give the system long-term viability without having performance degradation grow with time. For many individual low-Earth orbit satellites, the alti tude is not maintained. The orbit is allowed to decay and the spacecraft eventually reenters, typically well after the end of its useful life. For most low-Earth orbit con stellations, however, the altitude should be maintained over the long term in order to maintain constellation utility. If the decision is made to never do altitude maintenance, then it is possible to save propellant by doing relative stationkeeping. In this case, we would, for example, allow all the satellites in the constellation to decay at the same rate as the most slowly decay ing one. It would be undesirable to use any other satellite as the basis since that would mean pushing some satellites downward and augmenting rather than compensating for drag. If the decision is made to do altitude maintenance at any time, including re boosting the satellites after they’ve decayed for many years, then there are no substan tial advantages, and quite a few disadvantages, to relative stationkeeping. Ultimately, if we are to maintain the satellite altitude, we have no choice but to put back in the AV which the atmosphere takes out. If we do relative stationkeeping such that we allow the constellation to “decay” for a period of time, then the constellation will be moving further down into the atmosphere and the level of atmospheric drag and rate of decay will increase significantly. At the altitude of most constellations, the atmospheric
Stationkeeping
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701
density decreases exponentially with altitude, with a doubling of the density every 2 0 to 50 km. Thus, allowing the constellation to decay even a few kilometers can signif icantly increase the decay rate and require a greater amount of propellant to raise the satellite back to its original altitude. If the constellation is maintained at its higher altitude, the impact of atmospheric drag will be minimized and the total AV and propellant required will also be minimized. Table 13-9 summarizes the advantages and disadvantages of absolute vs. relative stationkeeping for the circumstance where the altitude of a constellation will be maintained over time. In this case, nearly all of the advantages favor absolute stationkeeping. The absolute stationkeeping process itself is inherently simple in that we con tinuously push on each satellite in order to put back the AV taken out by atmospheric drag. Each satellite is flown by itself and does not require interaction with the rest of the constellation. Because I’m generally using a larger number of smaller bums, the size of the thrusters is also smaller for absolute stationkeeping. With both smaller thrusters and shorter bums, the disturbance torque on the spacecraft is smaller, which reduces the size and speed required of the attitude control system, since thruster firings are normally the largest disturbance the spacecraft sees once it is on orbit. Propellant utilization is also minimized, since I am firing only in one direction to combat atmospheric drag, and am maintaining the constellation at its highest altitude (lowest density and lowest drag) at all times. T A B L E 13-9. Absolute vs. Relative Stationkeeping Trade. See text for discussion. Method
Additional Propellant Cost
Advantages
Disadvantages
May be some Relative Stationkeeping added propellant cost depending on implementation
• Minimizes maneuver frequency
• Stationkeeping depends on interrelationship between all satellites in the system • Complex commanding m ay lead to command errors and greater risk • High operations cost • Different logic for system build-up than operations
Absolute None, assuming Stationkeeping drag makeup is the only in-plane stationkeeping
• Simple commanding • Each satellite maintains itself in the pattern • Position of all other satellites are known without sending data around continuously • Can be fully autonomous • Sam e logic for build-up as for normal operations • Easily monitored from ground • Stationkeeping will not interfere with normal system operations
• More frequent stationkeeping (m ay be an advantage since burns will be smaller and more easily controlled).
The principal advantage to relative stationkeeping is that it minimizes the maneuver frequency and, therefore, minimizes the amount of commanding required of the space craft. Note that the advantages of absolute stationkeeping go away if it is appropriate
702
Constellation Design
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to allow the entire constellation to decay. In this case, we can do relative stationkeep ing only and, in some cases, may even be able to achieve this with differential drag, such that the propulsion system can be eliminated entirely. This approach has been used for the ORBCOMM constellation [Burgess and Gobrieal, 1996]. There is a secondary but potentially substantial benefit to absolute stationkeeping. In this case, we can create a purely deterministic constellation with a deterministic pat tern in which the positions of all of the satellites are known at all future times. Because we are controlling the absolute position of all of the satellites, the long-term error in our ability to project these positions is simply the size of the stationkeeping control box. This has the advantage of dramatically simplifying the planning and scheduling process for both housekeeping and payload operations. Ordinarily, this scheduling process is dominated by multiple iterations as we’re able to more accurately propagate the future position of the spacecraft. Thus we may begin with a preliminary plan for a particular stationkeeping pass or observation activity 2 to 3 weeks in advance, based on an approximate propagation of what we believe the orbit will be. For a critical op eration, this would then be refined, say, 72 hours before the planned observations, and refined again 24 hours before, with a final update a few hours or minutes prior to the actual event. This is how we establish where the satellite will be at various critical events, such as acquisition of the satellite by a ground station, or photographing a ground target. In absolute stationkeeping, this replanning process can be entirely elim inated. Because we know the positions of the satellites in advance, we can do our “final” observations and communications scheduling as far in advance as is conve nient for business purposes, say at the beginning of the month, or on Monday after noons. In most cases, the absolute stationkeeping algorithms are sufficiently simple that the ground station, or even a hand-held receiver can know where the satellites are at future times without having that information conveyed or updated. Essentially, the satellites run on time because they are controlled to run on time. This is what we call a controlled orbit.
The Controlled Orbit Concept The basic distinction between orbit maintenance and orbit control is important in terms of how a constellation functions. In normal orbit maintenance, the orbit elements are maintained only in an average sense. We need an orbit propagator to estimate where the satellite will be at any future time. We use the word “estimate” since the ac tual future positions of the satellite will depend on drag and other variables which are normally poorly known in advance. Orbit propagation is inherently inaccurate in terms of predicting future positions of spacecraft primarily because of the difficulty in being able to predict the levels of drag. In contrast, a controlled orbit is one which uses absolute stationkeeping to maintain all of the elements of the orbit, including the in-track phase. An orbit propagator is not needed to determine future positions, and indeed will not work over an extended period, since small bums are being made on a regular basis to force the satellite to follow a predefined mathematical algorithm. This means that the position of the satellite at any future time is easily determined, as discussed in Sec. 13.4.2. This determination is not done by an orbit propagator, but by a much simpler mathematical projection. A relatively simple orbit propagator can be used to determine the position of the stationkeeping box, if it is desired to know the position of the satellite to a higher level of accuracy than the short-term variations produced by higher order harmonics. Even in this case, the orbit propagator would not be run from the current time to some
13.4
Stationkeeping
703
future time, but only for one orbit corresponding to the time at which the information was needed. In many respects, a controlled orbit is similar to the attitude control system on board most spacecraft. In most communications satellites, for example, we know that 3 years from now, if the spacecraft is functioning, the spacecraft’s 2 -axis will be pointed at nadir, simply because it is a nadir-oriented spacecraft, and a particular axis is con trolled to be in that direction at all times to within the control system accuracy. We can at best crudely estimate the sum of all of the disturbance torques which the spacecraft feels over that extended period of time. However, irrespective of the magnitude or sequence of these torques, we expect the spacecraft to be nadir pointed at any given future date. Similarly, in a controlled orbit, we do not know all of the orbit perturba tions that the satellite will need to overcome. Atmospheric drag will vary in unpre dictable ways; there may be a small leak or an explosive bolt which will change the orbit. Irrespective of these effects, if the system is operating, the spacecraft in a controlled orbit will be at a well defined position at any given future date and time, just as the spacecraft attitude will be.
13.4.2 In-Track Stationkeeping In low-Earth orbit, we can use either relative or absolute in-track stationkeeping. In relative stationkeeping, we will maintain each satellite to follow the most drag-free, i.e., the one which decays the most slowly. Otherwise, we would be using the thrusters to increase the rate of decay for some satellites, thereby wasting system propellant. The most direct approach to achieving this is to determine which satellite is decaying the most slowly and fire thrusters in the other satellites in the system so as to match as closely as possible this decay rate. A particularly clever alternative approach is to make use of differential drag to adjust the decay rates to be the same for all of the satellites, as was done on the ORBCOMM constellation .1,1 For many spacecraft the amount of drag can vary by up to a factor of 1 0 , depending on the orientation of the spacecraft, and particularly the solar arrays, with respect to the velocity vector. If the arrays are turned sidewise to the velocity vector, the drag level and decay rate will be significantly reduced relative to what will occur if the arrays are normal to the velocity vector. Adjusting the angle of the arrays by even small amounts can impact the decay rate of the satellite with only minor impacts on the power output of the arrays. In ad dition, during eclipse periods the arrays can be adjusted to whatever angle is needed to achieve the most appropriate decay rate. The ORBCOMM satellites shown in Fig. 13-15 used this approach to provide relative orbit maintenance without the use of propellants. For most constellations, altitude maintenance is required such that absolute in-track stationkeeping is more appropriate than relative stationkeeping. f (See Fig. 13-16.) In in-track stationkeeping in which the altitude is maintained, the fundamental problem is to put back the AV that drag takes out. This is equivalent to bouncing a pingpong ball with a paddle and maintaining it in the air. Gravity is continuously pulling the pingpong ball downward. I hit the ball upwards such that it follows a small parabolic arc; it returns to the position of the paddle, where it is hit again, and the process repeats. * U.S. Patent No. 5,806,801. + The process described here is covered by patents by Microcosm (U.S. Patent Nos. 5,528,502 and 5,687,084) and Glickman (U.S. Patent No. 5,267,167).
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Constellation Design
13.4
G P S A n te n n a (4)
Battery Subscriber Receiver Battery Charge Regulator. Antenna Stowage Trough A vio nics B o x
Magnetometer VH F/UH F Antenna Deployed (129" Long)
Fig. 13-15.
-Sub scribe r “ Thruster (2) Transmitter
O RBCOM M Satellites use the Rotational Position of the “ Ears” for Differential Drag Control. (See Burgess, [1996].)
Similarly, I maintain the satellite in a stationkeeping box b y letting it drift to one side of the box, “hitting” it with a thruster, letting it drift across the box and back, and then hitting it again, as illustrated in Fig. 13-17.
Satellite 1 Box
Fig. 13-16.
Absolute Stationkeeping Maintains Each Satellite Within a Pre-Defined Mathematical Box. T h e boxes are equivalent to stationkeeping boxes in G E O , except that they are moving with respect to the surface of the Earth.
Stationkeeping
13.4
Rear of box
Fig. 13-17.
705
Front of box
Satellite In-Track Stationkeeping.
The total AV which we must apply to the spacecraft does not depend on how often the stationkeeping maneuvers are made. At each maneuver, I must put back the AV that the atmosphere has taken out since the last maneuver. Consequently, the total AV over any given time period will be the same irrespective of the number of maneuvers used to apply this AV. The size of the stationkeeping box and the timing and frequency of in-track bums is a function of the atmospheric drag, which in turn is a strong function of the altitude and the phase in the solar cycle as described in Sec. 2.4. At low altitudes, where drag is strong, the disturbance is substantial and the control process is straightforward. Thruster bums are made at frequent regular intervals such as once per orbit, and tight control can be maintained. As the altitude increases, the drag perturbing force goes to zero. As this occurs, propellant utilization goes down, but the control process itself becomes more difficult to manage. At the point where there is no effective drag, the in-track control process becomes equivalent to the cross-track stationkeeping dis cussed in Sec. 13,4.3, where bums in both directions must be done in order to maintain the satellite within a well-defined box. As illustrated in Fig. 13-18, the size of the stationkeeping box depends in part on how it is defined and what it will be used for. If the stationkeeping box is to be used for determining or controlling the relative distances between satellites, it must include the differential in-track perturbations discussed in Sec. 10.1. If the only requirement is position knowledge, then these terms are not required. Similarly, if we wish to model the center of the position of the box with a simple circular model, then the size of the box must be increased to take into account the non-circular character of a true space craft orbit. On the other hand, these variations from pure circular motion can be easily computed such that we can also consider a smaller stationkeeping box moving in a more complex path. Generally, an ideal (Keplerian) reference orbit is used to define an overall mission plan or constellation pattern. This reference orbit can be either elliptic or circular. This idealized orbit is perturbed by the complex structure of the Earth’s geopotential field. These higher order perturbations are not controlled because of the enormous amount of propellant that it would require, and the fact that their effect is cyclic, returning to the original values when the spacecraft again passes over
706
Constellation Design
13.4
the same place on the Earth. While this more complex motion cannot be eliminated, it can be known very precisely because the Earth’s geopotential field is extremely well known. This more complex orbit shape defines the center of the orbit control box in which the satellite is maintained. The maximum variation between the motion of the control box and the much smoother Keplerian reference orbit is about ±5 km. This variation can be thought of as the sum of a smooth longitude-independent variation of nearly ±5 km due to the oblateness of the Earth, and a small, much more complex motion due to higher order harmonics of approximately ± 2 km. In-plane stationkeeping should be done by controlling the orbit elements as determined at a fixed time (e.g.), at the node crossing). This results in the following error budget build-up: Control of node crossing times to within specified m easured value s
+ Measurement error in node crossing times results in Absolute box for node crossing times
+ Position variations throughout the orbit due to errors in orbital elem ents
+ Position variations throughout the orbit due to natural perturbations results in Absolute whole orbit stationkeeping box for the constellation This must be less than the stationkeeping requirement box. Typically, stationkeeping box must be 10 km to 50 km long. Fig. 13-18.
Build-up of the Stationkeeping Box. Which terms are included depends on the purpose for which the stationkeeping box is used.
Stationkeeping accuracy depends on both the navigation accuracy and the control mechanization. Generally the repeatable accuracy with respect to an absolute standard is on the order of ±0.15 sec = ±1 km for low-Earth orbit in-track stationkeeping. Typ ical results are shown in Fig. 13-19. Note that there is also a problem with keeping absolute positions relative to the surface of the Earth because the rotation of the Earth itself is non-uniform.* This variation is compensated by leap seconds, which are added or not added to civil time at 6 -month intervals. The leap second causes a 1 sec pertur bation in the stationkeeping problem if we wish to maintain coverage with respect to a civil clock. In geosynchronous orbit, stationkeeping is done both in-track (also called east-west stationkeeping) and cross-track (north-south stationkeeping). While the nature of the perturbing forces is substantially different, the process is essentially the same for both of these as it is for in-track control in low-Earth orbit. The reasons for the geosyn*There is a general slowing of the Earth’s rotation caused primarily by tidal friction with the Moon. In addition, there is an irregular non-uniformity in the rotation caused primarily by freezing and thawing of the polar ice caps. When the ice caps recede, water is transferred from the polar regions to the equator, and the Earth slows down. See Fig. 4.1 on Sec. 4.1.
13.4
Stationkeeping
2000
707
2500
3000
3500
4000
Time (days)
Fig. 13-19.
Simulation Results of In-track Stationkeeping over a 10-Year Period.
chronous orbit perturbations were discussed in Sec. 2.5.1. In both cases, there is a perturbing force pushing the satellite continuously in one direction. Therefore, the stationkeeping box and stationkeeping process are essentially similar to that defined above.
13.4.3 Cross-Track Stationkeeping Cross-track stationkeeping refers to maintaining the orientation of the orbit plane. This is done by maintaining the inclination and the right ascension of the ascending node, or, with respect to the Earth, the longitude of the ascending node. These two parameters are coupled due to the perturbation caused by the Earth’s oblateness. Specifically, the J 2 term representing the Earth’s oblateness causes the ascending node to drift at a rate that is proportional to the cosine of the inclination (see Secs. 2.4 and 2.5). This means that small differences in the inclination of each satellite will cause a cumulative differential drift in node due to the oblateness. This in turn implies that the inclination must be maintained in order to control the node rate. For example, in a polar orbit at 700 km, an 0.1 deg difference in inclination will cause a relative drift rate in the value of the node of 0.01 deg per day - 4.4 deg per year. For a 5-year mission life, this small difference in inclination would cause a 2 2 -deg separation in ascending node which is more than enough to destroy the structure of nearly any constellation. It is neither necessary nor realistic to attempt to drive the node drift rate to zero. AU that needs to be done is to make the node drift rate the same for all of the satellites in the constellation. This in turn means adjusting the inclination as needed to provide a common drift rate. The cross-track control problem is fundamentally different in character than in-track control. In the in-track direction in low-Earth orbit there is a continuous
708
Constellation Design
13.5
perturbing force (atmospheric drag) always acting in the same direction that must be negated. This in turn gives the control system a force to push against in order to main tain tight control. In the cross-track direction, there is no continuously growing pertur bation that must be balanced. The orbit inclination is inherently stable over long periods of time. The problem to be solved is that each satellite will have a slightly different inclination when it is delivered to orbit. These inclination variations must be corrected in order to provide the same level of long-term drift. How this is achieved depends upon the accuracy with which the node is to be maintained and the period of time over which the constellation is to be stable. If we can measure eitheT the node drift rate or inclination with sufficient precision, then we can make fine adjustments in the inclination so as to provide each satellite with nearly the same drift rate, such that the constellation will be stable over the life time of the spacecraft. Typically, this would imply establishing the inclination to better than 0.01 deg. If the inclination cannot be adjusted sufficiently accurately when the satellites are placed in orbit, then it will be necessary to make control maneuvers from time to time through the life of the spacecraft. This is basically a bang-bang control process in which the inclination is pushed back and forth with a very long time con stant to try and achieve a mean value appropriate to the constellation as a whole. Once a nearly correct value has been established at the beginning, it will take very small amounts of propellant to provide continuing adjustments. However, it may be neces sary to burn in die cross-track direction from time to time. Therefore, thrusters need to be mounted in this direction or alternatively, a mechanism needs to be provided such that the spacecraft can be rotated to use the in-track thrusters in the cross-track direc tion on an occasional basis. The typical logic involved in the process would be to sense the inclination error by sensing the difference in drift rate of the ascending node rela tive to the desired value. This drift rate error then leads to an adjustment by correcting the inclination. This type of control approach is appropriate whenever there is no continuing perturbing force that must be balanced. This occurs in the cross-track direction in lowEarth orbit and in both the in-track and cross-track when I get sufficiently far from low-Earth orbit that atmospheric drag is no longer a significant perturbation. However, when the orbit goes all the way to geosynchronous altitudes, then we need the “in-track style” stationkeeping approach for both in-track and cross-track components, due to the continuing perturbations caused by the Sun and the Moon, and by the Earth’s high er order harmonics. Again, the type of control is determined by the nature of the per turbing forces which are being balanced.
13.5 Collision Avoidance On a first look, collisions between spacecraft should not be a problem, and, in gen eral, they are not. Space is remarkably big, spacecraft are small, and there are only small numbers of them. If we had only a few hundred cars scattered over the surface of the world and moving at random, car collisions would be unheard of, even though there are only two dimensions for avoiding collisions, rather than three, as in space. However, if we confine all of the automotive traffic to a small number of well defined high velocity superhighways and if the superhighways intersect each other with no traffic lights, then not only do collisions become more likely, but as the traffic density increases, they become nearly inevitable. This is the fundamental problem for colli
13.5
Collision Avoidance
709
sion avoidance in constellations. For the satellite population in general, it is a relatively minor problem. Historically, collisions between spacecraft or spacecraft and large pieces have been very rare. However, constellations force spacecraft into narrowly defined intersecting orbits. Because constellations frequently have a high level of sym metry, there is often a desire to design the pattern in such a way that in principle, two satellites will end up at the same position at the same time at some point in the orbit. The obvious alternatives of, for example, putting satellites in eccentric orbits so that they pass over and under each other, may serve to make the problem worse, rather than better. Continuing our automotive analogy, if a collision does occur, then we leave a large amount of debris scattered about the roadway which dramatically increases the potential of secondary collisions. As discussed in detail in Sec. 2.5, the increasing problem of artificial space debris has received significant attention. Johnson and McKnight [1991], Reijnen and deG raaff [1989], and Simpson [1994] provide detailed assessments of the artificial debris and collision problem. Akella and Alfriend [2000] discuss alternative computations for debris collisions with the Space Station. Finally Chobotov et al. [1997], Kamprath and Jenkin [1998], and Jenkin [1995,1993a, 1993b] discuss specifically the problem of collision and debris hazards for constellations, including the geostationary ring which, for purposes of collision analysis, can be thought of as a one-orbit constellation in a very unique environmental regime. Thus, while collision avoidance is a workable problem in constellation design, it is one that should be given serious consideration as a part of the fundamental systems engineering of constellations. In order to assess the importance of collision avoidance, we need to estimate two key elements. First, what is the probability of a collision, and second, what are the con sequences of a collision? We will find that the consequences of a collision are poten tially very detrimental and that the number of collision opportunities in constellations is extremely large. This in turn implies that a key issue in constellation design is to drive the probability of collision per opportunity to be extremely small. We define a collision opportunity as an incident in which two satellites pass more or less close to each other. Assume that we have two satellites, A and B, which are in different orbits but at approximately the same altitude. Each satellite passes through the orbit plane of the other satellite twice per orbit. In most cases, the crossing satellite, say A, will pass over, under, in front of, or behind the other satellite, B. However, if the center of mass of the two satellites come within a distance of each other which is less than the sum of the radii of the two satellites, then a collision will occur. This process of one satellite passing through the orbit plane of another is a collision opportunity.
For computations, it is convenient to think of the two satellites as spheres of radii i?1 and R2The collision cross section, <7, is the area surrounding the center of mass of satellite A such that if the center of mass of satellite B falls within that area, a collision will occur. For the case of spherical satellites of radii RA and RB, the cross sectional area will be n(RA + RB)2. Thus, if the two satellites are both 10 m in diameter, the collision cross section will be approximately 300 square meters. Assume that satellite A is confined to a stationkeeping box 3 km long and 100 m high. Therefore, it must be within an area of 300,000 m2. Any time satellite B passes ran domly through that stationkeeping box, there will be a probability of 300 -s- 300,000 = 0.001 that the two satellites will collide.
710
Constellation Design
13.5
How many collision opportunities will there be in a given constellation? Consider as an example a low-Earth orbit constellation of 100 satellites with 10 satellites in each of 10 orbit planes. Assume that we have spread our satellites out within each orbit plane but are unconcerned about collisions so we have decided not to control the sat ellites in different planes with respect to each other. Further, let us assume that each time a satellite crosses one of the other orbit planes, it has a non-zero probability of colliding with only one other satellite, i.e., whichever of the 1 0 in that plane is the clos est. Each satellite crosses 9 other planes twice and, therefore, faces 18 collision oppor tunities per orbit, with approximately 15 orbits per day. Therefore, the constellation as a whole has 100 x 18 x 15 = 27,000 collision opportunities per day = IO7 per year = IO8 in a 10-year constellation life. If we want less than a 1% probability that there will be a collision in 1 0 years of the constellation life, then the probability of a collision in any single opportunity should be less than IO-10, and less than IO- 1 2 would certainly be more comforting, given the extremely adverse consequences of a collision. This is a remarkably small number. The chances of anything man-made working correctly every time in a trillion failure opportunities is not good. Table 13-10 gives the collision opportunities and collision probabilities for repre sentative constellations. Note that these calculations assume that the satellites are ran domly located in a stationkeeping box and that the centers of these boxes are designed to collide in order to provide the proper coverage characteristics. In a realistic constel lation, we will not do this. We will design the stationkeeping boxes such that they do not collide. However, the large number of collision opportunities implies that we must do more than this. We should attempt to maximize the probability that collisions will not occur. In the above discussion, we have implied that the consequences of a collision are extremely deleterious. There are two reasons for this. First, as illustrated in Table 13-11, the energy associated with colliding objects in space is extremely large. This comes about because the energy imparted by the colliding particles is propor tional to the square of the relative velocity. The orbital velocities are extremely high and the velocity with which they collide will be proportional to the linear velocities and the sine of the angle between their velocity vectors. As can be seen from the table, 7 km/s times the sine of almost anything is a large number, such that even particles far too small to be tracked carry dramatically large amounts of kinetic energy. By and large, spacecraft do not do well when hit by high speed bullets, cannonballs, trains, other satellites, or fragments of satellites. The second problem with collisions is the consequence of that collision over time. An explosion of a satellite creates a large number of fragments of various sizes and imparts an additional velocity relative to the previous center of mass. However, the ve locity imparted by the explosion is extremely small relative to the orbital velocity, Consequently, the particles will form a debris cloud consisting of fragments of various sizes and moving fundamentally in the same orbit as the satellite which exploded.* What the explosion has done is transform a spacecraft which is trackable and poten tially controllable into a large cloud of fragments in a nearly identical orbit, many of * The natural analog to this process is comets, which tend to evaporate and break up as they near the Sun. The dust and rock left behind stays in the comet orbit and produces a meteorite swarm. When the Earth crosses the comet’s orbit and encounters (his swarm, it results in a me teor shower, which will be more or less spectacular depending on the density of particles in that particular portion of the comet’s orbit.
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13.5 T A B L E 13-10.
Collision Opportunities and Collision Probabilities for Representative Con stellations. Collision probability represents how often a collision would occur if the satellites were randomly located in a stationkeeping box and the centers of the stationkeeping box occupied the same space at the same time— i.e., if we ignore the potential for collisions and simply use the “space is big, satellites are small” philosophy.*
100 Sats in Intersecting Boxes, Mod. Cntl.5
Sats1
Sats2
No. of sats considered
2
100
100
100
100
No. of orbit planes
2
100
10
10
10
Satellite diameter (m )
5
5
5
5
5
Diam. of keep-out box (m)
10
10
10
10
10
Collision cross-sect, (m 2)
100
100
100
100
100
1
1
1
1
0.1
4,500
10
2
Vertical dispersion (km)
100 Random Co-Alt
100 Sats in intersecting Boxes, Poor Cntl.4
100 Sats In-Plane Control Only3
System Modeled
2 Random Co-Alt.
45,000
In-track dispersion (km)
45,000
45,000
45,000
4,500
10
0.2
Collision prob. peropport.
2.22 X 10-9
2 .2 2 x 1 0 -9
2 .2 2 X 1 0 -8
1.00 X 10-5
5.oo x i c r 4
Collision opport. per orbit
4
19,800
1,800
1,800
1,800 900
Potential impact area (km2)
2
9,900
900
900
100
100
100
100
100
10,519
5.21 x 107
4.73 x 10®
4.73 x 106
4.73 x 106
1.05 x 105
5.21 x 10S
4.73 x 1 0 7
4.73 x 107
4.73 x 107
x 0 5 for joint collisions Orbit period (min) Collision opport. per yr Collision opport. per 10 yrs
2.34 x 10-5
0.12
0.11
47.34
2,367
Mean no. of years between collisions
42,780
8.64
9.51
0.021
0.00042
Mean no. of days between collisions
1.56 x 107
3,157
3,472
7.72
0.15
Mean no. of collisions per yr
* Collision probabilities are exceptionally low under ordinary circumstances. A controlled constellation is dramatically different— without a collision avoidance strategy, collisions are essentially inevitable in the two rightmost columns
1 2 satellites in ra n d o m circular Orbits at the same altitude 2 100 satellites in random circular orbits at the same altitude 3 100 satellites in 10 planes, controlled in phase within each plane, but uncontrolled with respect to ihe other orbit planes 4 100 satellites in 10 km x 1 km stationkeeping boxes with boxes allowed to intersect
5 100 satellites in 2 km x 0.1 km stationkeeping boxes with boxes allowed to intersect
T A B L E 13-11.
Consequences of a Collision Between a Spacecraft and Small Debris Frag ments. Th e fragments are assumed to have a density of 1 g/cc and an impact angle of 30 deg. A 10 cm fragment is approximately the smallest that can be tracked by ground radar systems.
Fragment Diameter
Equivalent Kinetic Energy of Impact
1 mm
marble @ 25 mph
1 cm
baseball @ 500 mph
10 cm
bowling ball @ 2,000 mph
1m
truck @ 4,000 mph
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Constellation Design
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which will be too small to be tracked. As shown in Fig. 13-20, this debris cloud evolves over time, due to both the differential initial velocities from the explosion and differential drag due to the varying properties of the individual fragments. Initially, the cloud grows, following closely the path of the original satellite. Over time the debris will spread throughout the orbit plane of the original satellite. Thus we create a debris ring similar in many respects to components of Saturn’s rings, although, of course, with far fewer particles. This dramatically increases the collision cross section with other spacecraft, since each of the individual fragments now has the potential for a col lision.* While these fragments will eventually decay, the lower the level of drag, the longer that decay process will take.
(A )
(B)
(C) Fig. 13-20.
Evolution of a Collision Debris Cloud. Immediately after the collision (A), there is a dense cloud of pieces traveling in essentially the same orbit as the original satellite with small differential velocities imparted by the explosion. Because of the differen tial velocities and differential drag, the particles spread out (B) and eventually form a ring (C) in the original orbit and then decay due to atmospheric drag.
* In the earlier example, two spacecraft each of 10 m diameter had a collision cross section of 300 m2. Assume one of these has exploded or hit a fragment and been transformed into 10,000 small pieces, each of negligible size but substantial kinetic energy. The collision cross section per fragment is now n x5- = 80 m2, so the collision cross section for the entire set of fragments that used to be a satellite has increased to 800,000 m2. This increases the collision probability by a factor of more than 1,000. As an analogy, your chance of being hit by a shotgun fired in your general direction is far higher than the chance of being hit by a single rifle shot fired in the same direction.
13.5
Collision Avoidance
713
The possible consequences of a collision with another satellite in the constellation, or even with a small fragment are distinctly bad. A 10 cm diameter fragment deposits as much energy as a bowling ball hitting the spacecraft at 3,000 miles per hour, or a large car hitting it 250 miles per hour. In either case, there will not be a lot left of the satellite that is hit, other than another debris cloud. This greatly increases the probabil ity of secondary collisions, which in turn can result in a chain reaction, or snowball effect. A single collision creates a debris cloud that greatly increases the collision probability. This, in turn, creates more collisions and a larger cloud, such that over time the entire constellation may be transformed into a debris cloud. This cloud slowly decays, proceeding to take out large portions of those constellations which are beneath it and, after a large number of years, the International Space Station. On the whole, this would make for a very bad day. Of course, this is a hypothetical result. An assessment of the potential for this occurring for any specific constellation requires a detailed anal ysis and modeling of the constellation, both in terms of the collision probabilities and the consequences of any collisions that do occur. Our principal message is that such an assessment is clearly worth doing before investing billions of dollars in a low-Earth orbit constellation. All of the above implies two specific issues for constellation design. First, collision avoidance analysis should be given significant attention. For your constellation, how important is collision avoidance, in terms of both probability and consequences? Secondly, the constellation design should be such that the probability of collision is extremely low, even for satellites that are no longer functioning. Thus, we want to consider collision avoidance, not only in normal satellite operations, but also in the process of putting satellites in orbit, taking them out of orbit, and working with or around satellites which have died in place. Fortunately, the importance of collision avoidance is straightforward to evaluate. Using the techniques described above, we can assess the collision probabilities for most types of orbits, at least in a first approximation. The leftmost column in Table 13-10 is applicable, for example, for a satellite in a geosynchronous transfer orbit. Here, we are going through virtually all of the low-Earth orbit constellations. Nonetheless, we are doing so only a very small number of times such that the total probability of collision is remarkably small. Clearly it would be worthwhile in such a transfer to avoid large trackable particles. Nonetheless, collisions are not generally a problem for single satellites or spacecraft in transfer orbits. Space is indeed big, and spacecraft are small. The real problem arises with constellations, which are con strained to operate within a narrowly defined space, and which will occupy that space for an extended period. In these cases, we need to proactively design the constellation to avoid collisions.
Designing the Constellation to Avoid Collisions Table 13-12 summarizes the critical issues for designing a constellation for colli sion avoidance. The most basic problem is to design the constellation so that the stationkeeping boxes do not intersect. This in turn means that we must keep track of all of the mutual intersections of all possible pairs of planes. Generally, the larger and more symmetric a constellation, the more difficult this problem will be. This is illustrated in Fig. 13-21, which shows our hypothetical constellation of 100 satellites in 10 orbit planes, 10 satellites per plane. The heavy lines on each orbit represent the stationkeeping box for each satellite. Our problem then is to design the constellation such that none of the 900 interactions between the varying orbits occurs when a second
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stationkeeping box is at the point of intersection. How to do this will be discussed below. T A B L E 13-12.
Key Issues in Designing a Constellation for Collision Avoidance. Approach or Issue
Comment
1. Maximize the spacing between satellites when crossing other orbit planes.
May impact phasing between planes and, therefore, coverage.
2. Remove satellites at end-of-life.
Either deorbit or raise them above the constellation, if still functioning.
3. Determine the motion through the constellation of a satellite that “dies in place."
Constellations at low altitude have an advantage.
4. Remove upper stages from the orbital ring or leave them attached to the satellite.
Do not leave uncontrolled objects in the constellation pattern.
5. Design the approach for rephasing or replacement of satellites with collision avoidance in mind.
All intersatellite motion should address the collision potential.
6. Capture any components which are ejected.
Look for items such as explosive bolts, lens caps, or Marmon clamps.
7. Avoid the potential for self-generated explosions.
Vent propellant tanks of spent spacecraft.
Fig. 13-21.
Constellation with 10 Orbit Planes and 10 Satellites per Plane has 90 Different Locations Where Tw o Planes Intersect Which Results In 900 Potential Collisions per Orbit.
In addition to simply avoiding collisions between the stationkeeping boxes, we want to maintain the largest possible minimum separation between satellites for two reasons. First, we would like to provide the largest possible stationkeeping margin because of the dramatically adverse consequences of a collision. Second, maintaining a large minimum separation provides the most time and greatest safety margin if a satellite fails in place, as is virtually certain to occur many times in any large constel lation.
13.5
Collision Avoidance
715
It is important to understand the motion of a satellite that “dies in place,” i.e., that loses the capacity for control while it is still in the constellation pattern. This would occur, for example, if the satellite power system fails, or if the command link fails for a satellite which is controlled from the ground. Relative to the rest of the constellation pattern, the dead satellite moves both down and forward as illustrated in Fig. 13-22. We would like the satellite to be below the rest of the constellation when it moves
In-track Motion (km) Fig. 13-22.
Consequences of a Satellite “ Dying in Place.” In a high drag environment, the satellite moves down and forward more steeply than in a low drag environment. O n both curves, dots are at an interval of 7 days.
forward enough to intersect the next stationkeeping box crossing its orbital path. If this occurs, then the dead satellite will dive under the constellation and we will have a greater assurance of avoiding collisions which might generate debris within the constellation. This “cleaning” of the constellation occurs best at low altitudes. In a high drag environment, the satellite will move forward rapidly. However, it also moves downward faster than it moves forward, moving the satellite on a steeper slope relative to the constellation. In a very low drag environment, the satellite moves down ward slowly with only a very small amount of downward motion. Therefore, the dead satellite slices the next forward stationkeeping box which crosses its orbit very finely, maximizing the inherent probability of a collision. Spacecraft in a low drag envi ronment will evolve out of their pattern very slowly, and will do so in a way that is potentially very detrimental to the constellation as a whole. At first, it might appear that collision avoidance is best done by the choice of con stellation design pattern. For example, the fundamentally polar orbits in the streets of coverage pattern imply all of the satellites come together at the poles, which appears to maximize the collision potential. However, this is not the case. Any two distinct or bit planes will intersect twice and only twice, irrespective of their relative inclination or positions of the node. Consequently, major changes in the constellation structure,
716
Constellation Design
13.5
such as polar vs. low inclination, will have no fundamental impact on collision oppor tunities. Reducing the number of orbit planes will reduce the collision opportunities simply because satellites will not collide with other satellites in the same plane. Thus, collision avoidance is not done by making major changes in the constellation pattern, but rather by “fine tuning” the locations of the stationkeeping boxes. There are three ways to make these fine adjustments in how the stationkeeping boxes intersect:
1.
Eccentricity. Maintaining the satellites in eccentric orbits such that stationkeeping boxes pass over and under each other seems like an inherently strong solution but has significant problems. Eccentricity gives the constellation pat tern thickness such that it takes far longer for a dead satellite to decay through the constellation. Recall that for a dead satellite in an eccentric orbit, drag ini tially maintains perigee at an approximately constant altitude and lowers apogee until it is the same height as perigee. The satellite then spirals inward (see Fig. 2-18 in Sec. 2.4.4). During the entire time apogee is being reduced, the satellite is continuously within the constellation pattern. Further, because the orbit is eccentric and is drifting in terms of argument of perigee and eccen tricity, it is also sliding in phase throughout the constellation. This provides a very large number of collision opportunities over an extended period, which is the opposite of what we desire.
.
Inclination. Changing the inclination shifts the along-track position relative to the equator at which the orbit planes intersect. Since constellations are typ ically defined by the relative phase when they cross the equator, changing the inclination will shift where the satellites are at the time the orbit planes inter sect. The geometry of this phase shift is shown in Fig. 13-23. Here we assume the two planes are at the same inclination i and have a node spacing AN. We wish to determine the phase shift, A<j>c = A0c2 = -2A0 c2 correspond ing to a small change in inclination, Ai. From the right spherical triangle in the figure we have:
2
tan
(13-1 la)
0c2 =180 deg - <j)cj
(13-1 lb)
A(j>c = A0^2 - M C] = “ 2 A
(13—12a)
from which we find:
= -2 sin2(0c) tan (A N /2) sin i
(13-12b)
For high inclination orbits and AN ~ 30 deg, so that A
In-Plane Phasing Between Adjacent Planes. Shifting the phase between adjacent orbit planes shifts the relative time at which the satellites cross the equator. This shift has the same effect as a change in inclination in that it slides the intersection between the stationkeeping boxes forward and back ward. Note that a shift in in-plane phasing will typically break the symmetry that ordinarily occurs as we go from the last plane to the first in a given constellation.
The inclination and phase shift (options 2 and 3 above) are typically the most practical approaches for designing a constellation for collision avoidance. These can
13.5
Fig. 13-23.
Collision Avoidance
717
Effect of Inclination on the O rbit C ro ssin g Location. Both orbits are assumed to be at the same inclination.
be mixed as needed to maximize the spacing at the time of plane crossings. This problem is most conveniently done numerically by examining the closest approach between the constellation satellites as a function of inclination and phase offset. The results of a typical analysis of this type are shown in Fig. 13-24. The horizontal coor dinates are the inclination and phase offset for a near polar constellation. The vertical coordinate is the closest approach between any two satellites in the constellation, assuming that each is maintained at the center of its defined stationkeeping box. Thus the valleys in the plot represent collisions between stationkeeping boxes. At an ab solute minimum, we must be at a higher “elevation” on the plot than the size of the stationkeeping box. Our real goal for collision avoidance is to live on the highest mountaintop, in order to give ourselves margin, and to minimize the potential problem with dead satellites within the constellation. Within the figure, each of the valleys represents a specific collision. Thus the valley labeled 11 is the potential collision between satellites in plane 1 (the base satellite) and the nearest satellite in plane 1 1 . Valley 11- is a collision between the base satellite and the next satellite behind the closest one in plane 11. Plane 11+ would be collisions with the next satellite in front of the closest satellite. Consequently, with various combinations of orbit planes, there are many “valleys” to be avoided. In the particular pattern illustrated, we can design the constellation to have a minimum separation between satellites of as much as 2.5 deg, or -300 km. Note that these are fine-tuning adj ustments that are imposed on the overall structure of the constellation. While they ordinarily break the perfect symmetry of the constel lation, they are at a small scale, and will typically have minimal impact on system coverage as a whole. For example, assume that we have a baseline streets of coverage constellation with perfectly polar orbits. In this case, satellites in the adjacent plane will be midway between satellites in the base plane in order to maximize the overlap in the coverage areas. However, if each plane has satellites midway between those in
718
Constellation Design
8 7
Fig. 13-24.
13.6
' ^
Closest A p p ro a ch Between Satellites as a Function of Inclination and Phase Offset for a Streets of C o ve ra g e Constellation with 10 Satellites in each of 11 O rb it Planes. Numbers on the “valleys" indicates which plane interfaces with plane 1. See text for explanation.
the adjacent plane, then satellites two planes over will have the same phase as the base plane, and will result in collisions between stationkeeping boxes at the pole. It is this symmetry which needs to be broken by some combination of small adjustments in inclination and phase offset. In addition, an adjustment which is too large will result in potential collisions between stationkeeping boxes further out in the pattern, say, between plane 1 and plane 6 . Thus we need to evaluate the pattern as a whole in terms of collision avoidance probabilities. Finally, we can evaluate the cost of collision avoidance by assessing the impact of lack of symmetry on coverage. Basically, the lack of symmetry that is necessary to meet our collision avoidance criteria will cause nonuniform coverage that must be compensated by going to a smaller elevation angle, higher constellation altitude, or both. However, these numbers are typically small and can ordinarily be accom modated by changing the altitude by only a few kilometers, or the minimum elevation angle by a fraction of a degree.
13.6 Constellation Build-Up, Replenishment, and End-of-Life Most constellations spend a major portion of their lives in less than their full con figuration. It may take several years to launch and test the constellation; spacecraft die on orbit; and new configurations take over for old ones. Consequently, it is important to understand both how to build the constellation in a configuration sense and how to achieve incremental performance. One of the most fundamental goals of building up a constellation is providing performance plateaus, incremental performance, and graceful degradation. We cannot
13.6
Constellation Build-Up, Replenishment, and End-of-Life
719
have a 1 0 0 -satellite constellation in which all of the performance comes only when the last satellite is in place. The funding source would never be willing to build the constellation, and a single satellite failure would be catastrophic. What we need is an incremental improvement in performance with each small group of satellites, both for building up the constellation initially, and to provide as much residual performance as possible when satellites fail. Recall that moving satellites within the orbit plane is easy, while moving between planes is hard. Putting up multiple satellites in one orbit plane is much easier for launch but provides limited utility because the coverage will tend to be very nonuniform. If we put up one satellite in each orbit plane, we get much more uniform coverage. More important, as we add one more satellite to each plane, we can rephase the satellites to maintain consistent spacing, with almost no propellant usage. If we have 4 satellites per plane at an early stage of constellation building, it would be desir able to have them 90 deg apart. When a 5th satellite is added, we can rephase all of the satellites such that they are 72 deg apart with a very small amount of propellant. As we add one more satellite to each orbit plane, we will achieve a performance plateau with overall better performance than we had before. The ease of moving satellites within an orbit plane provides a significant advantage in system utility to having a constellation with a smaller number of orbit planes. With two planes, we will have performance plateaus at 1, 2, 4, 6, 8, 10, 12, 14, and 16
satellites. With eight planes, there will be plateaus at 1, 8 , and 16 satellites. This rein forces the importance of altitude plateaus as discussed in Sec. 13.3. Because of this, there is a significant advantage in constellation design to having a small number of orbit planes in terms of both incremental performance and graceful degradation. We can build up the full constellation in a wide variety of ways. The detailed process of buildup tends to be unique to each constellation and is strongly related to the choice of launch vehicle and the number of satellites that it can place on orbit in a single launch. Nonetheless, there are several general characteristics applicable to most constellations: • Constellations often begin with a test satellite or set of test satellites which usu ally become a part of the subsequent constellation • An attempt is made to build some utility at an early stage in the buildup process (i.e., performance plateaus) • Rapid completion occurs after the initial slow test and evaluation phase These general criteria are driven primarily by the economics of constellation buildup. We need an initial test satellite to verify that the system functions as intended and that all of the components are correctly working before committing to launching the entire constellation. Having shown that the system works, we would like to obtain at least some initial utility so that the system can begin generating useful results, either money for a commercial system or scientific or military results for a government system. Finally, having begun the process, we would like to complete it rapidly so that die en tire constellation can be in place as soon as possible. Constellations are dramatically expensive, and a large amount of money is tied up in nonrecurring and development which will only be recovered after the constellation becomes operational. In addition, the early satellites launched will be going through their operational life. If the buildup process is very slow, the early satellites may die before the constellation is complete.
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Constellation Design
13.6
Spacecraft Outages and Satellite Replacement What happens when a satellite dies? The response will depend upon the nature of the constellation and the importance of having complete coverage. We will consider as an example the case of a constellation intended to provide continuous coverage such as a low-Earth orbit communications constellation.
Fig. 13-25.
Typical C o ve ra g e O utage Due to Lo ss of a Sing le Spacecraft. This assumes a continuous coverage constellation with an approximately polar orbit.
As shown in Fig. 13-25, when a single satellite in a constellation dies, a hole is created in the coverage pattern at the equator. The hole typically becomes smaller, and may disappear entirely at higher latitudes. Depending on the constellation altitude and pattern, points near the equator will experience outages at one or two successive as cending nodes, and then again approximately 1 2 hours later at the descending node of the failed satellite. Consequently, the result of a single satellite failure is typically 1 to 4 outages per day for points on the equator, with durations on the order of 5 to 20 min. As we move away from the equator, the outages will become less frequent and shorter. With a well designed constellation, we may experience either very brief or no outages at high northern latitudes where much of the population is centered. (See Fig. 13-26.) How do we accommodate a satellite outage? The answer depends on the level of need to maintain full coverage. The only instantaneous solution available to the constellation operator is to change the minimum working elevation angle so that neighboring satellites pick up more of the burden, as illustrated in Fig. 13-27. In this case, performance may be degraded within the coverage hole but this may be more de sirable than a performance outage. The principle advantage of changing the minimum elevation angle is that it can be done immediately by simply changing the operating parameters of the system. This reduces the size of the hole at the equator and reduces the latitude at which the hole closes. It will shift a number of areas from “no coverage” to “possibly degraded coverage.” This can have a major impact on coverage interrup tions. The service will, of course, not be as good in regions which have gone to a lower working elevation angle. For marginal users (i.e., those with limited signal strength), it may be that the degraded performance mode will prove unacceptable. However, for
13.6
Constellation Build-Up, Replenishment, and End-of-Life
721
Fig. 13-26.
C o ve ra ge Hole at High Latitude for the Sam e C onstellation S h o w n in Fig. 13-25.
Fig. 13-27.
O utage Reduction du e to U sing Lo w er M inim um Elevation A n g le for N earby
Satellites.
users with significant margin, it may be that the coverage will remain indistinguishable from the previous uninterrupted coverage. Thus, while not ideal, changing the mini mum working elevation angle can provide an excellent short term solution. We can further reduce the outage due to a dead satellite by rephasing the satellites on all sides of the coverage hole while at the same time increasing their coverage region by going to a lower working elevation angle. Recall that this rephasing can be done with relatively little propellant and that the fuel required to rephase is propor tional to the rephasing time as shown in Fig. 13-28. (See Sec. 2.6.2 for the relevant equations.) As we slide neighboring satellites along their orbits toward the coverage
722
Constellation Design
13.6
T im e of F o re -W a rn in g (D a y s )
Fig. 13-28.
& V Required for to Rephase Satellite by 10 deg In-Track. Th e satellite is as sumed to be at 800 km.
hole, we begin to fill in the coverage gap and as shown in Fig. 13-29, and may be able to eliminate the gap entirely by combination of lower minimum elevation angles and rephasing. However, the rephasing process needs to be well established in advance in order to avoid potential problems of intersatellite collisions. Thus, the satellite outage and rephasing plan should be well thought out prior to implementation.
Fig. 13-29.
Elimination of Outage by Shifting to Lower Minimum Elevation Angle Plus Rephasing of Nearby Satellites.
Summary—The Constellation Design Process
13.7
723
For most constellations, we will ultimately choose to replace a satellite which has died on orbit. The two basic approaches for satellite replacement are an on-orbit spare and launch on demand. On-orbit spares are typically stored above or below the base line constellation, since storing them within the constellation poses a potential colli sion hazard. Storage above the constellation requires extra propellant for both raising and lowering but reduces the drag makeup requirements while the satellite is on orbit. Storage below the constellation reduces the demand on raising and lowering propel lant but subjects the satellite to greater atmospheric density and, consequently, higher drag and more propellant utilization during the period of on-orbit storage. Launch on demand, in which a satellite is ready to be launched when needed, reduces the number of spacecraft required on orbit and the on-orbit propellant cost, but places a much heavier (i.e., more expensive) burden on the launch segment. In an extreme circumstance, we would have the spacecraft ready to be launched and a vehicle prepared to go on an as-needed basis. In practice, both on-orbit spares and having spacecraft available for launch on demand are used by most constellations. Thus some spares will be available on orbit and some spare spacecraft will have been built up on the ground such that they are prepared to be launched quickly. Thus, if a satellite fails on orbit, one of the on-orbit spares will be moved into the empty slot as promptly as possible and a new spacecraft will be moved to the launch site and put into the position of the spare which has been used.
End-of-Life Finally, it is important to address the issue of removing satellites at end-of-life. As discussed in detail in Sec. 2.6.4, de-orbiting the satellite is certainly the best solution. If this is impractical due to the altitude, then raising the dead satellites above the con stellation minimizes the probability of future collisions. The AV for this process has been previously calculated and is relatively small. (See Sec. 2.6.1.) The fundamental problem is that raising the satellite or lowering it requires that the satellite be moved while it is still operational. If the satellite dies in orbit, then removing it becomes sig nificantly more difficult. Whatever the method chosen, the fundamental rule for all constellations is the same:
When you’re done with it, take it out of orbit.
13.7 Summary—The Constellation Design Process In most chapters, we begin by setting out a process and then discuss the various steps in that process. Because of its complex character, constellation design is best done in reverse, We began by defining the key elements of constellation design, dis cussing the reasons for them, and how they are selected. We are now in a position to go back and summarize the constellation design process, the principal factors which need to be defined, and the general rules for a good design. The overall process of constellation design is summarized in Table 13-13. The key requirement in this process is to understand the mission objectives and what is needed to fulfill the mission, particularly with respect to coverage. This means understanding both the coverage needs for the mission and the spacecraft needs to provide that cov erage, such as swath width or constraints on lighting, power, or communications. Can
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13.7
the system operate in the Van Allen belts? A second key part of the requirements is the growth and degradation goals. Are outages acceptable if a satellite dies on orbit? How soon is a replacement required? How soon will the constellation as a whole be replaced and upgraded? T A B L E 13-13.
Th e Constellation Design Process. See also Tables 13-14 and 13-15. Step
1. Establish constellation-related mission requirements, particularly -
Latitude-dependent coverage
-
Goals for growth and degradation plateaus
-
Requirements for different modes or sensors
-
Limits on system cost or number of satellites
Where Discussed Chap.5 Secs. 12.1,13.1
2. Do all single satellite orbit trades except coverage
Chap. 12, Sec. 13.3
3. Do trades between swath width (or maximum Earth central angle), coverage, and number of satellites.
Chap. 9 Secs. 1 3 .3 ,1 3 .6 ,1 3 .7
Evaluate candidate constellations for: -
Coverage Figures of Merit vs. Latitude and mission mode
-
Coverage excess
-
Growth and degradation
-
Altitude plateaus
-
End-of-life options
Consider the following orbit types -
Walker Delta pattern
-
Polar constellations with seam
-
Equatorial
-
Equatorial supplement
-
Elliptical
4. Evaluate ground track plots for potential coverage holes or methods to reduce the num b e r of satellites
Sec. 9.5.1.2
5. Adjust inclination and in-plane phasing to maximize the intersatellite distances at plane crossings for collision avoidance
Sec. 13.5
6. Review the rules of constellation design in Tables 13-14 and 13-15.
Sec. 13.7
7. Document reasons for choices and iterate.
The principal factors to be defined during constellation design are listed in Table 13-14, along with the major selection criteria for each. These factors have been discussed in detail earlier in the chapter. A key point to keep in mind here is that constellation design consists of more than just orbit elements. The minimum working elevation angle is perhaps the single most critical parameter in defining constellation coverage. Collision avoidance parameters, including the method for stationkeeping and size of the stationkeeping box are key to insuring the integrity of the constellation over its lifetime. While we often think of a constellation as a single, coherent pattern, it may be that some combination of patterns will satisfy our mission objectives at a lower cost and risk. This approach has been adapted, for example, by Ellipso with a mix of inclined eccentric and circular equatorial orbits.
13.7
Summary—The Constellation Design Process
T A B L E 13-14.
725
Principal Factors to be Defined During Constellation Design.
Effect
Factor
Selection Criteria
Where Discussed
P rin cip a l D e sign Variables: Num ber of Satellites
Principal determinant of cost and coverage
Constellation Pattern Determines coverage vs. latitude, plateaus
Minimize number consistent with meeting other criteria
Chap. 9 Sec. 13.3
Select for best coverage
Sec. 13.1
Minimum Elevation Angle
Principal determinant of single satellite coverage
Minimum value consistent with payload performance and constellation pattern
Secs. 9.1, 13.3
Altitude
Coverage, environment, launch, and transfer cost
System level trade of cost vs. performance
Chap. 9, Sec. 12.4, 13.3
Num ber of Orbit Planes
Determines coverage plateaus, growth and degradation
Minimize consistent with coverage needs
Sec. 13.6
Collision Avoidance Parameters
Key to preventing constellation self-destruction
Maximize the intersatellite distances at plane crossings
Sec. 13.5
S e co n d a ry D e sig n Variables: Inclination
Determines latitude distribution of coverage
Compare latitude coverage vs. launch costs'
Sec. 13.3
Between Plane Phasing
Determines coverage uniformity
Select best coverage among discrete phasing options*
Sec. 13.1
Eccentricity
Mission complexity and coverage vs. cost
Normally zero; non-zero may reduce number of satellites needed
Sec. 13.3
Size of Stationkeeping Box
Coverage overlap needed; cross-track pointing
Minimize consistent with low cost maintenance approach
Sec. 13.4
End-of-Life Strategy
Elimination of orbital debris
Any mechanism that allows Secs. 2.6.4, you to clean up after yourself 13.6
* Fine tune for collision avoidance
Typically the purpose of creating a constellation is to provide Earth coverage, or at times, coverage of near-Earth space. The demerit of a constellation is that you need multiple satellites, and therefore, multiple reaction wheels, communication systems, and so on. A typical goal of constellation design is to provide the needed coverage with a minimum number of satellites. Consequently, the principal system trade is most fre quently coverage as a measure of performance vs. number of satellites as a measure of cost. Nonetheless, it is important to bear in mind that the number of satellites is not necessarily an accurate representation of system cost. For example, raising the altitude of the constellation will normally result in needing fewer satellites. However, as the altitude increases, the launch cost for each satellite increases, and the radiation hard ening requirements increase dramatically as we enter the Van Allen radiation belts. Minimizing the number of satellites may or may not minimize the system cost and it is the cost which is typically of most interest to constellation sponsors, both govern ment and commercial.
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13.7
Coverage and coverage figures of merit were described in detail in Chap. 9. Perhaps the most important coverage issue to remember in terms of constellation design is that compiling statistical data on orbit coverage may not provide useful physical insight into the constellation design problem and can, in many cases, be dramatically misleading. The mission designer must always be aware that in constellation design, statistical data is being used to represent a dramatically non-Gaussian process. A num ber of examples of this problem have been given throughout the book. See, for exam ple, Secs. 9.5.1.2 and 9.5.1.4. There are two fundamentally different types of constellations — those that demand continuous coverage and those that do not. The former group represents the more straightforward design problem. What is the minimum number of satellites (or mini mum total system cost) needed to provide the necessary single or multiple coverage? For example, GPS wants continuous coverage of the entire world by a minimum of four satellites. An interesting figure of merit for continuous coverage constellations is the excess coverage—i.e., the total instantaneous coverage available as a percentage of the total required. For N satellites in circular orbits at a common altitude with coverage defined as being within of the subsatellite point, the total instantaneous coverage, COV, is given by: COV = N ( 1 - cos Xm a) / 2 K
(13-13)
where K is the coverage multiplicity that is needed (i.e., K = 4 for GPS). Our goal is to drive COV toward 1. For the 24-satellite GPS constellation, the satellites are all at half GEO (a = 26,561.75 km) and the assumed minimum elevation angle is 5 deg. From Eqs. (9-4) and (9-5), we find that = 71.2 deg and COV = 2.03. This means that it takes approximately twice as many satellites as would be needed if the coverage were static and could be perfectly distributed. Iridium is a communications constellation with 6 6 satellites at 780 km. Here K = 1 and the minimum working elevation angle is 9.5 deg corresponding to = 19.0 deg and COV ~ 1.80, Although the two con stellations are very different in use and in coverage pattern, the excess coverage in the implemented constellations is nearly the same. For constellations which do not demand continuous coverage, the most representa tive figure of merit is typically the mean and maximum response time as a function of the number of satellites (see Sec. 9.5.1.4). This is a substantially more difficult trade because it now needs to be worked iteratively with the basic objectives of the constel lation and may require extended interaction with the end user to determine how those needs are best met. The complexity of choosing orbit parameters for noncontinuous coverage is illus trated in Fig. 13-30, which illustrates the hypothetical mean response time vs. number of satellites for a satellite system for detecting forest fires. If we assume that the initial goal of the system was to have a mean response time of no more than 5 hours, we see from the plot that a system of 6 satellites can meet this goal, while a 4-satellite system can achieve a mean response time of 6 hours. Is the smaller response time worth the increased number of satellites and the money required to build them? Only the ultimate users of the system can judge. The additional fire warning may be critical to fire containment, and therefore, a key to mission success. However, it is also possible that the original goal was somewhat arbitrary and a response time of approxim ately 5 hours is what is really needed. In this case, firefighting resources could probably be
13.7
Summary—The Constellation Design Process
727
Number of Satellites
F ig . 13-30.
H y p o th e tic a l C o v e r a g e Data fo r F ire D e te c tio n S y s te m . S e e text for definitions
and discussion. As discussed in Sec. 13.6, satellite growth comes in increments or plateaus. These are assumed to be two-satellite increments for the example shown.
used better by flying a 4-satellite system with 6 hours response time and applying the savings to other purposes, such as more ground equipment or a larger number of firefighters. We can only determine how this is best done by working closely with the end users.
The Rules for Constellation Design It is clear from the above discussion, and particularly the example of the MSSP program at the end of Sec. 13.2, that there arc no absolute rules for constellation design. It is not a systematic process, where we can simply take, for example, all possible Walker constellations and enumerate how well they do, and then select an absolute winner for the best constellation. The constellation design process is much fuzzier than that. The key is to look at the fundamental mission objectives and deter mine how these objectives can best be met at minimum cost and risk. Ordinarily, our goal would be to minimize the number of satellites while achieving the appropriate level of coverage. Nonetheless, as discussed above, this may not be the lowest cost approach. For example, we may be able to achieve our coverage objectives with 20 satellites in the Van Allen radiation belts, where 25 would be required below the belts. The key question then becomes whether the lower cost and longer life per satellite for the lower constellation is worth the additional number of satellites. The constellation design process per se cannot answer that question. It can only be answered within the broader context of mission, system, and spacecraft design. While there are not absolute rules, there are broad guidelines which assist in the process of constellation design. These are summarized in Table 13-15. In the end, constellation design is coupled only moderately to astrodynamics, and strongly to the process of meeting mission objectives at minimum cost and risk.
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T A B L E 13-15.
13.7
Rules for Constellation Design. While there are no absolute rules, these broad guidelines are applicable to most constellations.
Rule
Where Discussed
1. T o avoid differential node rotation, all satellites should be at the same inclination, except that an equatorial orbit can be added.
Sec. 13.1
2. T o avoid perigee rotation, all eccentric satellites should be at the critical inclination of 63.4 deg or (180 - 63.4)deg.
Sec. 13.1
3. Collision avoidance is critical, even for dead satellites, and may be a driving characteristic for constellation design.
Sec. 13.5
4. Symmetry is an important, but not critical element of constellation design.
Secs, 13.0,13.1
5. Altitude is typically the most important of the orbit elements, followed by inclination. 0 eccentricity is the most common, although eccentric orbits can improve some coverage and sampling characteristics.
Sec. 13.1
6. Minimum working elevation angle (which determines swath width) is as important as the altitude in determining coverage.
Sec. 9.1, 13.3
7. Tw o satellites can see each other if and only if they are able to see the same point on the ground.
Sec. 10.1.1
8. Principal coverage Figures of Merit for constellations:
Secs. 9.5.1.4, 13.1
• • • • •
Percentage of time coverage goal is met Number of satellites required to achieve the needed coverage Mean and maximum response times (for non-continuous coverage) Excess coverage percent Excess coverage vs. latitude
9. Size of stationkeeping box is determined by the mission objectives, the perturbations selected to be overcome, and the method of control.
Sec. 13,2
10.For long-term constellations, absolute stationkeeping provides significant advantages and no disadvantages compared to relative stationkeeping.
Sec. 13.2
11. Orbit perturbations can be treated in 3 ways:
Sec. 13.2
• Negate the perturbing force (use only when necessary) • Control the perturbing force (best approach if control is required) • Leave perturbation uncompensated (best for cyclic perturbations) 12. Performance plateaus and the number of orbit planes required are a function of the altitude.
Sec. 13.1
13. Changing position within the orbit plane is easy; changing orbit planes is hard; implies that a smaller number of orbit planes is better.
Sec. 13.4
14. Constellation build-up, graceful degradation, filling in for dead satellites, and end-of-life disposal are critical and should be addressed as part of constellation design.
Sec. 13.4
15. Taking satellites out of the constellation at end-of-life is critical for long-term success and risk avoidance. This is done by:
Sec. 13.4
• Deorbiting satellites in L E O • Raising them above the constellation above L E O (including G E O )
References
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References Akella, Maruthi R., and Kyle T. Alfriend. 2000. “Probability of Collision Between Space Objects.” Journal o f Guidance, Control, and Dynamics, Vol. 23, No. 5, pp. 769-772, September-October. Burgess, E. L. and H. S. Gobrieal. 1996. “Integrating Spacecraft Design and Cost-Risk Analysis Using NASA Technology Readiness Levels ” The Aerospace Corpora tion. Presented at the 29th Annual DoD Cost Analysis Symposium, Leesburg, VA. February 21-23. Burgess, Gregg E. 1996. “ORBCOMM.” In Reducing Space Mission Cost. J.R. Wertz and W. J Larson (eds.) Torrance, CA: Microcosm Press and Dordrecht, The Netherlands; Kluwer Academic Publishers. Chobotov, V. A., D. E. Herman, and C.G. Johnson. 1997. “Collision and Debris Haz ard Assessment for a Low-Earth-Orbit Constellation.” Journal o f Spacecraft and Rockets. 34(2): 233-238, March-April. Draim, John. 1985. “Three- and Four-Satellite Continuous Coverage Constellations.” Journal o f Guidance, Control, and Dynamics. 6:725-730. --------- . 1987a. “A Common-Period Four-Satellite Continuous Global Coverage Constellation.” Journal o f Guidance, Control, and Dynamics. 10:492-499. --------- . 1987b. “A Six-Satellite Continuous Global Double Coverage Constellation.” AAS Paper 87-497 presented at the AAS/AIAA Astrodynamics Specialist Conference. --------- . 1998. “Optimization of the ELLIPSO™ and ELLIPSO 2G™ Personal Communications System, Mission Design & Implementation of Satellite Constel lations.” In Space Technology Proceedings, Jozef van der Ha, editor. Dordrecht, The Netherlands: Kluwer Academic Publishers. Draim, J., and D. Castiel. 1996. “Optimization of the Borealis and Concordia SubConstellations of the Ellipso Mobile Communications System.” Paper No. IAF-96A.1.01, presented at the 47th International Astronautical Congress, Beijing, China October 7-11. Draim, John E., Cecile Davidson, and David Castiel. 1999. “Evolution of the ELLIP SO™ and ELLIPSO 2G™ GMPCS Systems.” Paper No. IAF-99-M.4.02, presented at the 50th International Astronautical Congress, Amsterdam, The Neth erlands, October 4-8. Draim, John, Paul J. Cefola, and David Castiel. 2000. “Elliptical Orbit Constella tion— A New Paradigm for Higher Efficiency in Space Systems.” Easton, R. L., and R. Brescia. 1969. Continuously Visible Satellite Constellations. Naval Research Laboratory Report 6896.
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Fleeter, Rick. 1999. “Design of Low-Cost Spacecraft,” Chap. 22 in Space Mission Analysis and Design, 3rd ed., James R. Wertz and Wiley J. Larson, eds. Torrance, CA, and Dordrecht, the Netherlands: Microcosm, Inc. and Kluwer Academic Publishers. Jenkin, A. B. 1993a. “DEBRIS: A Computer Program for Debris Cloud M odeling.”
Paper No. IAA.6.3-93-746, presented at the 44th Congress of the International Astronautical Federation, Graz, Austria, October. --------- . 1993b. “Analysis of the Non-Stationary Debris Cloud Pinch Zone.” Paper No. AAS-93-625, presented at the AAS/AIAA Astrodynamics Conference, Victo ria, BC, Canada, August. --------- . 1995. “Probability of Collision During the Early Evolution of Debris Clouds.” Presented at the 46th Congress of the International Astronautical Federa tion, Oslo, Norway, October. Johnson, Nicholas L. and Darren McKnight. 1991. Artificial Space Debris. Malabar, FL: Orbit Book Company. Kamprath, M. F. and A. B. Jenkin. 1998. “Debris Collision Hazard from Breakups in the Geosynchronous Ring.” In Proceedings o f the SPIE Conference on Character istics and Consequences o f Space Debris and Near Earth Objects, San Diego, CA, 23 July. Mora, Miguel Bello, Jose Prieto Munoz, and Genevieve Dutruel-Lecohier. 1997. “Orion—A Constellation Mission Analysis Tool.” International Workshop on Mission Design and Implementation of Satellite Constellations, International Astronautical Federation, Toulouse, France. Nov. 17-19. Pocha, J. J. 1987. An Introduction to Mission Design fo r Geostationary Satellites. Boston: D. Reidel Publishing Company. Reijnen, G. C. M. and W. de Graaff. 1989. The Pollution o f Outer Space, in Particular o f the Geostationary Orbit. Dordrecht, The Netherlands: Kluwer/Martinus Nijhoff Publishers. Simpson, John A. (ed.). 1994. Preservation o f Near-Earth Space fo r Future Generations. Cambridge, UK: Cambridge University Press Soop, E. M. 1994. Handbook o f Geostationary Orbits. Dordrecht, The Netherlands: Kluwer Academic Publishers. Walker, J. G. 1971. “Some Circular Orbit Patterns Providing Continuous Whole Earth Coverage.” Journal o f the British Interplanetary Society. 24: 369-384. --------- . 1977. Continuous Whole-Earth Coverage by Circular-Orbit Satellite Patterns. Royal Aircraft Establishment Technical Report No. 77044. ----- . 1984. “Satellite Constellations.” Journal o f the British Interplanetary Society. 37:559-572.
Chapter 14 Operations Considerations in Orbit Design —Launch, Orbit Acquisition, and Disposal Lauri Kraft Newman, NASA Goddard Space Flight Center 14.1 Definition of Complete Orbit Parameters Orbit Error Boundaries
14.2 Launch Window Parameters Operational Considerations for Launch Vehicle Targeting
14.3 End-of-Life Disposal Predictability of Impact, Disposal Options', Choosing an Orbit for Uncontrolled Reentry 14.4 Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra Definition of Complete Orbit Parameters; Orbit Error Boundaries 14.5 Example 2: End-of-Life Disposal of CGRO
Chapters 12 and 13 discussed in detail the process of defining orbit parameters based on mission requirements. However, in most cases the fundamental mission requirements will define only a subset of the orbit parameters. Nonetheless, each spacecraft must ultimately be launched into a specific orbit for which all of the param eters are chosen and assigned allowed error limits. Trajectory designers are faced with the prospect of choosing values for those elements for which there have been no spec ifications. Of the many factors which feed into this choice, operations considerations are key. In other words, knowledge of how the spacecraft will be operated, how it will perform on orbit, and how it will interact with the launch vehicle are used to make element selection in the absence of other criteria. Sec. 14.1 addresses some of these secondary criteria that may be used to choose a complete set of orbit parameters, including error boundaries. Once the element values are chosen in the early phase of mission design, the focus shifts to planning for operations. The remaining sections of the chapter discuss the launch, orbit acquisition, and end-of-life disposal phases, and form a link between the definition of the initial elements and the mission operations. Sec. 14.2 provides the process for translating detailed orbit criteria into launch param eters and launch window boundaries. The process of achieving the mission orbit after launch is highly mission dependent. The various elements of this process have been previously discussed in Secs. 2.6, 12.6, and 12.7. Sec. 14.3 provides detailed disposal
731
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14.1
methods which implement the general approach outlined in Sec. 2.6.4. Finally, Sec. 14.4 documents the definition of orbit parameters for the Earth Observing System Ter ra while Sec. 14.5 covers the analysis and execution of an end-of-life disposal scenario for the Compton Gamma Ray Observatory.
14.1 Definition of Complete Orbit Parameters Typically, the orbit designer is faced with a set of mission and or payload require ments that must be met by the final orbit design. However, not all of the classical Keplerian elements are driven by requirements in every case, and those that are not may be selected arbitrarily or used to achieve secondary mission goals. The flexibility in how these elements are driven by primary mission requirements usually depends on the trajectory design complexity and typically differs from mission to mission. Trade studies are performed, costs are evaluated, and the desired values are chosen. Often these decisions are not documented well, if at all, since they occur early in the mission design and are often the result of informal analysis or taken from previous experience. As in all mission elements, documentation of these decisions would be helpful to the spacecraft support team in understanding the rationale behind what eventually become untraceable “requirements” on the orbit elements. The values of various orbit elements have different importance for different classes of orbits. However, the most common requirement is on altitude, whether it be chosen to support an instrument field of view, to yield a particular period (such as for geosyn chronous spacecraft), to maximize time in sunlight, to avoid high drag, or to synchro nize the orbit with a particular ground trace or with another spacecraft. The second most common requirement is on inclination, which is often chosen based on the desire to view certain portions of the Earth or the celestial sphere. Small payloads are often flown “piggyback” on the same launch vehicle as larger missions in order to share the launch cost. These missions often cannot choose any of their orbital elements. Since they are not paying the majority of the launch cost, they must choose to piggyback with a spacecraft that has orbit requirements similar to their own desires and be flexible in how closely they achieve the desired orbit. Many of these spacecraft have no propulsion system with which to change their orbit once they are injected by the launch vehicle. However, the primary payload often has some flex ibility in orbit mission parameters. For instance, the SAC-C mission is in a Sun-synchronous orbit, but is limited in the local mean time it can use because it is flying piggyback with E O -1, which is required to fly in constellation with Landsat-7. Since changing the mean time is costly, SAC-C will not have much flexibility in altering this parameter. The following paragraphs discuss the operational considerations that may affect the selection of mission parameters not defined by the primary mission requirements. Semimajor Axis. If the altitude of the orbit is not defined by mission requirements, other factors which could be used to choose the altitude include avoiding the Van Allen radiation belts, increasing on-orbit mass, determining the length of eclipses, or providing optimal ground station coverage, For interplanetary missions, the semimajor axis is often driven by the time of flight desired, avoiding eclipses, or minimizing fuel use by employing a lunar or planetary swingby. For instance, for the MAP mis sion, a lunar swingby is required for the spacecraft to reach the L2 Earth-Moon libration point within the available fuel budget. The timing o f this swingby is critical, so phasing loops, a series of Earth orbits following injection and prior to the swingby, are
14.1
Definition of Complete Orbit Parameters
733
used to allow the swingby to occur at the appropriate time. The semimajor axis of each loop is determined to achieve the correct timing of the swingby. Eccentricity. If the orbit eccentricity is not specified by mission requirements, it is generally set to 0. However, a frozen orbit could be chosen to minimize altitude vari ations over a given latitude. (See Secs. 2.5.6 and 12.4.3.5.) Often, a near-frozen orbit can be implemented. This is one in which the eccentricity and argument of perigee vary slightly, but not enough to affect the accuracy of instrument measurements. Since an exact frozen point is not being maintained, near-frozen orbits allow more flexibility in maneuver placement, timing, and, potentially, fuel use. In the case of a geosynchro nous mission, eccentricity affects the range of east-west drift over the course of a day. (See Sec. 12.4). Inclination. If the orbit inclination is not pre-determined, several factors can be considered in choosing it. Launch vehicle capability is maximized if the inclination equals the latitude of the launch site, which is the normal default if no specific value is required. Choosing a polar orbit minimizes the amount of time spent in the diumal bulge, therefore decreasing atmospheric drag and lengthening mission lifetime. Choosing a Sun-synchronous inclination, if it is available at the mission altitude, pro vides thermal and power stability. For geosynchronous orbits, a near-zero inclination provides minimal north-south excursions over the course of a day. This reduces dis tortion in image data such as that provided by weather satellites. However, choosing any inclination other than the latitude of the launch site will reduce the available onorbit mass. Right Ascension of the Ascending Node. Regression of the nodes will cause the ascending node to rotate continuously for non-polar missions. If the node is not defined by the mission, it is often left as a free variable for the launch vehicle. How ever, there may be benefits in choosing the initial node because it defines the eclipse cycle. For missions with lifetimes of less than a year or very sensitive power require ments during the checkout, choosing an initial right ascension to maximize sunlight times at launch could be advantageous. Geosynchronous spacecraft normally require an initial right ascension that will maximize the amount of time that the spacecraft remains below the maximum allowable inclination. This optimum node will cause the inclination to approach zero, then return to the maximum desired value before the first north-south stationkeeping maneuver is required. Argument of Perigee- The argument of perigee could be selected along with the eccentricity to freeze the orbit, or to place the perigee above a particular sub-satellite point such as a ground station. For critically-inclined elliptical orbits (Sec. 12.4) the argument of perigee is chosen to maximize the desired coverage, typically by placing apogee over the desired latitude. Anomaly. The anomaly is often not something that can be chosen by the mission because most launch vehicles use this as a free variable for trajectory design. This can pose challenges for spacecraft attempting to rendezvous with another or being placed into a constellation. Often those missions have to carry enough fuel to alter their anom aly after launch by either injecting low and raising the orbit at the correct time, or by performing a pair of maneuvers to raise and lower the orbit at the proper time. How ever, the added fuel requirement is typically small. Once the orbit elements have been broadly determined, we must decide how to compute the specific value of each. Often this decision is based on the development phase of the mission, which drives the accuracy level required. Missions in early phas es of development only require a coarse orbit definition to feed the design of orbit-
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14.1
dependent subsystems, such as power, thermal, propulsion, and altitude control. How ever, missions in final phases of development, especially during integration and test phases, will require that the orbit be defined in much greater detail to ensure, for ex ample, that the correct mass of fuel is loaded. Section 14.5 gives a detailed example of how the orbit elements were chosen for the Terra mission.
14.1.1 Orbit Error Boundaries Each mission parameter also has error bars on how tightly it must be achieved in order to meet mission requirements. These error boundaries drive operational param eters such as m aneuver size, location, and frequency, as well as launch window dura tion. Although the choice of error bounds for a given orbit is very mission-specific, the following paragraphs provide general examples of requirements which could drive the error bounds for some of the orbit elements. Sem im ajor Axis. A simple example would be the choice of altitude bounds for an earth-looking scientific payload. The instrument may require a viewing range of between 400 and 450 km to take accurate images. Therefore, the altitude bounds are set by the instrument requirement, the orbit raising maneuver frequency is determined by how often the orbit decays to the lower edge of the box, and the maneuver size is determined by how much AV7is required to achieve the upper edge of the box. As another example, a spacecraft which controls its ground track to within ±5 km of a reference ground track may have chosen those bounds to keep the image angle or size within science requirements. This ground track accuracy translates directly into bounds on the orbit altitude, since as the altitude changes, so does the ground track error. The initial altitude is chosen such that the error with respect to the reference is +5 km, the eastern edge of the control box. As atmospheric drag causes the altitude to decay, the orbit period decreases, causing the equator crossing to drift westward rela tive to the reference track. Eventually, the altitude will reach the value that yields a -5 km error. At this point, the orbit altitude starts to fall below the nominal value, causing the period to be less than nominal and hence causing the drift to turn around and be come eastward. Once the track has drifted back to the eastern control boundary, a maintenance maneuver is required to raise the orbit altitude and begin the cycle again. The difference in altitude between the value at the 4-5 km edge of the box and the value at the -5 km edge of the box then form the error bounds for the mission. Mission requirements are sometimes specified in terms of derived requirements, such as a ground track accuracy, instead of an actual requirement on the orbit altitude itself, since that value can vary, as in this case in which it would depend on the expected solar flux level during the decay time between maneuvers. However, requirements defined in this manner actually require further analysis. For instance, while a satellite may wish to control its ground track to within ±5 km of a reference, instead of choosing the exact orbit altitudes which yield the edges of this box to the altitude bounds, the actual altitudes chosen are usually within the box, so that the ground track is controlled more tightly than the requirement. This operational consideration prevents uncertainties in predicting the solar flux, and hence the altitude rate of change, from causing an unex pected violation of constraints. Thus, many orbit error boundary definitions depend on the perturbations expected on that orbit element throughout the mission life. Orbit per turbations are discussed in more detail in Sec. 2.4. Inclination. A Sun-synchronous spacecraft may have a requirement to maintain a 10:30 am descending mean local time within ±15 min. This 30 min error boundary could be used by a launch vehicle with variable targeting capabilities to place the
14.2
Launch Window Parameters
735
spacecraft in any orbit plane between 10:15 and 10:45, allowing an approximately 30 min launch window duration. Anomaly. Sometimes the orbit error boundaries do not determine operational parameters to meet the mission requirements, but, conversely, are driven by opera tional considerations. For instance, satellites in a formation or constellation may experience different orbit perturbations due to differences in mass properties and at mospheric drag. Therefore, they may oscillate with respect to each other over a period of weeks or months. If two such spacecraft desire to use the same ground station with out conflicts, they need to be separated by at least 20 min, allowing time for the ground station to take a IO min pass from one spacecraft and then repoint to acquire the next. Normally this means that the two spacecraft should be placed 20 min (-70 deg) apart in mean anomaly. However, if the error boundary on the mean anomaly is 8 deg due to the oscillations of the spacecraft from perturbations, then the spacecraft should be placed 78 deg apart to prevent any conflicts at the station.
14.2 Launch Window Parameters The launch window is the time during which a spacecraft may be launched to meet all mission requirements. The launch period is the set of dates on which there exists a viable launch window. For many low Earth spacecraft, there exists a launch window every day of the year, so launch period is not a factor. However, for many interplane tary missions the launch period becomes critical because of the changing relative position of the planets. Choosing a vehicle and launch site also often plays a key role in launch window determination. The launch site depends on the chosen launch vehicle as well as the orbit to be achieved. Each launch vehicle has facilities in certain geographic locations, some in more than one. Figure 14-1 shows a selected set of launch sites and Table 14-1 lists data about these and other available sites. These sites can change and should be con-
Fig. 14-1.
W orld La u n ch Sites [C h iulli, 1994]. (Reprinted with permission of T h e Aerospace Corporation.)
736
Operations Considerations in Orbit Design
14.2
TABLE 14-1. World Launch Sites. Data from Kraft [1996], Isakowitz [1999], and Chiulli [1994],
Launch Site
Latitude Longitude
Alcantara, Brazil
2.28°S
44.38°W
Andoya, Norway
69.28°N
16.02°E
Baikorur Cosmo drome, Kazakhstan Russian submarine launch
45.6° N
63.4°E
69.3° N (mobile)
35.5°E (mobile)
Cape Canaveral AFB, USA
28,5° N
80.57°W
Allowed Inclinations 2.2°—100°
Vehicles VLS, sounding rockets Sounding rockets, orbital capability planned
50.5°-99° (not Dnepr, Ikar, Molniya/Soyuz, contiguous) Proton, Rokot, Tsyklon, Zenit 79° Shtil 28.5°-57°
Athena, Atlas, Delta, Pegasus, Taurus, Titan
Cape York, Australia
12°S
143°E
Esrange, Sweden
68°N
21°E
Zenit
Kourou, Guiana
5.23°N
52.776W
5.2° -100*
Ariane 4 and 5
Sounding rockets
Hammaguir, Algeria
30.9° N
3.1 DW
34°-40°
Diamant (closed)
Jiuquan Sat. Launch Cen., China
40.7° N
100.0°E
57° -70°
LM-1D, LM-2C, LM-2E, LM-2F, LM2E(A)
Kagoshima Space Center, Japan*
31.2°N
131.1°E
31=_ioo°
M-V
Kapustin Yar, Russia
48.4° N
45.8°E
51°
Kosmos 3M
Kodiak Island, AK, USA
57.5° N
152.2°W
63°—116D
Athena I and II
Marshall Islands
19.5°N
166.1 °W
<10°
Pegasus XL
NASA Kennedy Space 28.5°N Center, USA
81.0°W
28.5° -57°
Space Shuttle (STS)
Nevada Test Site, USA 37.2°N
116.3°W
45-60°, 84-99°
Kistler K-1 (not active)
Poker Rat, AK, USA
65.12°N
147.47°W
Sounding rockets 143°
Shavit 1
Palmachim AFB, Israel 31.9°N
34.7°E
Plesetsk, Russia
62.S°N
40.1 °E
San Marco Platform, Indian Ocean, Italy
2.9°S
40.3°E
2.9-38°
Inactive, planned for Vega
Pacific Ocean
0°N (mobile)
154°W (mobile)
0-100°
Zenit 3SL
Kosmos, Molniya/Soyuz, Tsyklon, Zenit
Sriharikota, India
13.68°N
80.23°E
18-50°, SSO
PSLV, GSLV, sounding rockets
Svobodny, Russia
51.8°N
128.4°E
SSO
Start, Start-1, Strela
Taiyuan Launch Center, China
37.8° N
111.5°E
87°, 96-98°
LM-4, LM-2C/SD
Tanegashima Space 30.4®N Center, Japan* Vandenberg AFB, USA 34.7°N
130.6°E
30-100°
H-ll, H-IIA, J-l
120.6°W
63.4-110°
Athena, Atlas, Delta, Pegasus, Taurus, Titan II and IV
Wallops Island, VA
37.85°N
75.47°W
38-55°
Pegasus XL, sounding rockets
Woomera, Australia
31.1°S
136.6°E
45-60°, 84-99°
Kistler K-1 (not active)
Xichang Launch Center, China
28.2°N
102.0°E
27.5-31.1°
LM-2E, LM-3, LM-3A, LM-3B, LM3C
‘ Seasonal launch periods of 45 days in Jan/Feb & Aug/Sept due Io range safety and the fishing fleet.
14.2
Launch Window Parameters
737
firmed with the launch vehicle provider. The inclination achievable from a site is a function of its latitude. The latitude of a site is the minimum orbit inclination that can be achieved, unless the vehicle performs a plane-change maneuver during ascent or the spacecraft performs one after insertion. If the inclination required is greater than the latitude of the launch site, two launch windows are available, one yielding a descending orbit, the other an ascending orbit. Range safety limits the set of incli nations that can be achieved from a site based on surrounding land masses. Geosyn chronous or other low inclination spacccraft are launched from equatorial or lowlatitude sites, since the inclination of these spacecraft needs to be near zero degrees and the maneuver to adjust the inclination to that value is cheaper from low-inclination sites. Typical performance values for various vehicles are provided in Sec. 12.3.2 and by, for example, Isakowitz [1999]. However* only the launch vehicle manufacturer can provide the most current and most accurate information regarding launch vehicle capabilities. Detailed performance information must be obtained from a users guide which is typically updated every 3 to 4 years. Users guides for the Delta [Boeing, 1996, 1999a, 1999b], Atlas [International Launch Services, 1998], Pegasus [Orbital Sciences Corp., 2000] and Taurus [Orbital Sciences Corp., 1999] are given in the references. Once a site and vehicle have been selected, the launch window may be determined. The optimum launch time is when the desired inertial orbit plane rotates through the launch site. The orbit plane is defined by the orbit inclination and right ascension, while the launch site is defined by the latitude and longitude of the launch pad. Figure 14-2 shows the geometry of a launch site and inclination of the orbit. The latitude of
Fig. 14-2.
Launch Site Geometry. See text for explanation.
738
Operations Considerations in Orbit Design
14.2
the site is given by L, while the longitude difference between the site and the orbit node is given by 5. Az is the launch azimuth, the angle measured clockwise from north to the velocity vector. If the inclination of the desired orbit is less than the latitude of the launch site, the desired orbit cannot be achieved. (Small differences may be accommo dated by the powered flight of the launch vehicle, but large differences are too expen sive to correct.) The launch azimuth required to achieve the desired inclination from the latitude of the launch site is given by Wertz & Larson [1999] as: A z -A z f± 7 ~ A z i
(14-1)
sin Az{ = cos i I cos L
(14-2)
tan 7 = VL cos AZl/(V 0 - V€q cos i) = (VL!Va)cos AZ}
(14-3)
VL = (464.5 m/s) cos L
(14-4)
where
Here VL is the inertial velocity o f the launch site, VeCj —464.5 m/s is the velocity of the
Earth’s rotation at the equator, and V0 ~ 7.8 km/s is the velocity of the satellite imme diately after launch, y is a small correction to account for the velocity contribution caused by the Earth’s rotation, and has a value of 0 deg for a due east launch, and 3 deg for a polar launch. Equation 14-3 is accurate to within 0.1 deg for low Earth orbits. In Eq. 14-1, the minus sign is used for ascending node orbits, and the plus sign is used for descending node orbits. Interplanetary and deep space trajectories have very tightly constrained windows based on the alignment of celestial bodies with the spacecraft trajectory. Often fuel costs for adjusting these trajectories are high, so accuracy of launch time is important to meet mission goals. The boxed example describes the launch window design for Clementine, a military lunar probe. For low-Earth spacecraft, often there are few requirements on either the launch date or time, unless the mission is Sun-synchronous. In that case, the spacecraft must be launched at the time that the desired orbit plane passes through the launch site longitude, which occurs once per day in the appropriate (ascending or descending) direction. The second boxed example shows the computation of launch time for a Sun-synchronous mission. The length of the window may be widened around the exact launch time by making use of the permissible error range on the mean time requirement; however, this error box is often used to eliminate inclination maintenance maneuvers as described in the example in Sec. 14.4. In that case, the launch window must be as short as possible (seconds) or make use of guided targeting for widening the window by altering the powered flight trajectory. Powered flight trajectories are designed by the launch vehicle supplier due to the proprietary nature of the propulsion system models. Another possible restriction on the powered flight trajectory is the desire to have contact with the vehicle or spacecraft during flight. This coverage can be provided by Instrumented Aircraft, which fly to the injection point or other relevant point in the trajectory to provide coverage; by local ground antennas if one is available in an appropriate location; or by the Tracking and Data Relay Satellite System, TDRSS, which are geosynchronous tracking satellites with a continuous view of most of the Earth.
14.2
Launch Window Parameters
Launch Window Determination for Clementine* Clementine, launched on January 25, 1994, was intended to map the lunar surface and then to perform a flyby of the asteroid 1620 Geographos. The nominal mission plan involved spending 7 days in LEO, then injecting into a cislunar transfer orbit con sisting of 2.5 phasing loops about the Earth. The first loop had a 5-day period, the sec ond loop a 10-day period, and the last half-loop had a 5-day period as illustrated in Fig. 14-3. At the end of this half-loop, the spacecraft was inserted into a polar lunar map ping orbit, where it remained for two months. At the end of the mapping phase, Clem entine left the Moon on a planned asteroid-intercept trajectory and was lost during the first post-lunar bum.
Fig. 14-3. Clementine Transfer Phase Trajectory Schematic [Sch iff, 1993].
The Clementine launch window had to accommodate all of the mission require ments, including lunar mapping and the Geographos encounter which was a time-critical rendezvous; therefore, the trajectory was designed assuming a specific date for the Geographos encounter and working backward. The launch window was designed to allow the spacecraft to meet its mapping orbit constraints by controlling which lunar longitude was achieved at lunar injection and to meet its asteroid rendezvous constraints by controlling the time of maneuvers. Phasing orbits were used for the transfer phase to allow flexibility of launch date, sincc changing the sizt and n u m b e r of phasing orbits allowed flexibility on when to perform the fixed AVtransfer insertion maneuver and still reach the moon on the target date. The launch period was deter mined by performing analysis to see how late or early the transfer maneuver could be *Adapted from Richon [1995].
739
740
Operations Considerations in Orbit Design
performed and still have enough fuel to perform the necessary additional maneuvers. Moving the initial maneuver date caused the phasing loops to change size in order to meet the mission constraints, and this change in size drove the AVbudget. Based on available AV, a preferred launch date of January 25,1994 was determined that allowed 7.5 days in the parking orbit before the transfer maneuver. The launch period was divided into two parts: the nominal period, and an extended period. The nominal period, from January 25 to 31, consisted of the launch opportunities when the transfer maneuver occurred on February 2. The stay in the parking orbit decreased throughout the period, from 7.5 days to 1.5 days. The extended launch period, from February 1 to 8, consisted of the launch opportunities during which the transfer maneuver epoch moved one day later for each day into the extended period. In order to maintain lunar arrival on the same date, the time of flight was shortened by decreas ing the period of the second phas-ing loop. This was accomplished by removing AV from the first phasing perigee and applying an essentially equal AV at the second peri gee to ensure the correct lunar arrival conditions. On February 8, the first phasing peri gee maneuver reached zero, and all of the AV was applied at the second perigee. To determine the daily launch window for each day within the launch period, a nom inal trajectory for that day, the “daily nominal,” was designed. This was not necessar ily the center of the window open and close box, but the time when all of the spacecraft AV occurred at the first perigee for the nominal launch window, or where all AV occurred at the second perigee on February 8 for the extended window. A contingency allowance of 10 m/s from the baseline 603 m/s AV requirement was used to calculate the daily window open and close. Figure 14-4 shows the daily launch window as a function of launch date.
Launch Date Fig. 14-4. Clementine Launch Window for 2966 m/s Transfer Burn [Richon, 1995].
14.2
14.2
Launch Window Parameters
741
Launch Time Computation for Sun-synchronous Missions The equation for computing the Greenwich Mean Time (GMT) of launch for a Sunsynchronous orbit is: GMTlo = GMTan - A / l - At2 (14-5) where GMTL0 is the GMT of liftoff, Afj is the nominal time of launch vehicle flight from launch to spacecraft separation, and Ar2 is the nominal time from spacecraft sep aration to the descending node. GMT an is the GMT of the ascending node. If times are expressed in sec, then: GMTan = [LMSTan - LAN x (86,400 sec/360 deg)]mod 8M00 sec (14-6) where LMSTan is the local mean solar time of ascending node, LAN is the East longi tude of ascending node (in deg). It is the angle measured east from the Greenwich meridian to the ascending node at the time of the ascending node crossing. (For more information about time systems and the Earth’s rotation, see Sec. 4.1.) For example, for the Earth Observing System Aqua spacecraft, which has a 1:30 pm ascending node local mean time requirement and a launch date of Dec. 30, 2000, the above equations can be used to compute the launch time as follows. From the trajectory data provided by the Delta launch vehicle, the LAN is 39.74°, At] = 150.0 sec, and At2 = 3114.0 sec. Also, 1:30 pm converted to seconds of the day equals 48600 sec. Then, GMTAN = [48,600 - 39.74 x (86,400 sec/360 deg)]mod 86i400 sec = 39,062.4 sec (14-7) and GMTlo = 39,062.4 - 150 - 3114 = 35,798.4 sec = 9 hours, 56 min, 38.4 see (14-8)
14.2.1 Operational Considerations for Launch Vehicle Targeting For some missions, a direct ascent to the mission orbit is available due to the per formance capabilities of the launch vehicle. For other spacecraft, the vehicle cannot provide enough lift to attain the desired orbit, so the spacecraft must carry sufficient fuel in its own propulsion system to perform ascent maneuvers. Spacecraft with no propulsion systems must use a direct transfer to the desired mission orbit, thus dictat ing the choice of a launch vehicle with sufficient capability. For spacecraft having propulsion systems, various targeting strategies are available for choosing the orbit into which the vehicle places the spacecraft. It is sometimes desirable to plan for the launch vehicle to place the spacecraft into a 3a low injection orbit. This low targeting prevents the need for negative AV maneuvers to achieve the desired mission orbit, even if a 3 cr high injection is achieved. Negative AV maneuvers are those which require the spacecraft to bum in a direction opposite to the velocity direction. These maneuvers waste propellant and often require yawing the spacecraft 180 deg to align the thrusters properly, a potentially risky and typically avoided oper ational practice. For instance, a spacecraft requiring an orbit altitude of 700 km and is targeted for an altitude of 690 km. Then, even a 10 km injection error would not re quire the spacecraft to perform a negative AV maneuver. However, if the launch vehi cle performs flawlessly, the spacecraft will have to raise the orbit from 690 km to 700 km—i.e. the probability of needing to do a maneuver to achieve the mission orbit is increased. Even if the target orbit is not chosen to be 3(7 low, choice of a slightly low target allows the opportunity to perform some small trim maneuvers to place the spacecraft precisely on station. Allowing atmospheric drag to lower the altitude over time is an alternative to performing a negative AV maneuver; however, the amount of
742
Operations Considerations in Orbit Design
14.3
time required to achieve the desired altitude may be undesirably long, preventing the collection of data during the drag period. The risks versus benefits of which targeting scenario to use must be weighed based on the spacecraft's maneuvering capabilities and operational constraints. For spacecraft that cannot obtain a direct transfer, an ascent scenario must involve combinations of transfer orbits. Given the types of transfer orbits discussed in Chap. 12 and the ability to compute the AV required to maneuver from one to the other, a combination of transfer orbits may be determined. Various factors must be taken into account during this planning process, including the drift rates of the orbit elements while in the transfer orbits to ensure meeting all mission requirements once on orbit. Note that if a spacecraft is planned to be in a certain transfer orbit for a given period of time, and this time is based on achieving a certain amount of drift in one of the orbit elements, the element could drift to an undesirable value if the spacecraft is forced to remain in the transfer orbit longer or shorter than planned. One example of ascent planning is the GOES mission, which uses restartable en gines to affect the transfer from injection to mission orbit. Constraints used in planning a geosynchronous ascent include the number and magnitude of maneuvers which must be performed (both in- and out-of-plane), the number of orbit revolutions between ma neuvers, the Sun-spacecraft geometry, ground station coverage, and the longitude drift rate. Other launch and early orbit operational concerns must also be addressed. For instance, each maneuver in the sequence is time critical if the spacecraft is to be placed on station correctly due to the drift rates experienced in each orbit. Also, between each maneuver, orbit determination must be performed to ensure the next maneuver will be accurate. In order to feed the orbit determination process, 4 to 8 hours of tracking data are required from a combination of 2 or 3 ground stations. Finally, the attitude must be determined between each maneuver, usually to within ±2 deg, requiring an amount of time which depends on the subsystem configuration and required accuracy.
14.3 End-of-Life Disposal The growing amount of orbital debris surrounding the Earth poses a potential haz ard to all Earth orbiting spacecraft. Some space agencies have established policies for minimizing orbital debris. One method of doing this includes ensuring proper end-oflife disposal for a spacecraft to reduce the number of spacecraft being left on orbit after the end of their useful lifetimes. NASA Policy Directive 8710.3 states that spacecraft must reenter the Earth’s atmosphere within 25 years of the end of their mission life. Guidelines on performing analysis to ensure compliance with 8710.3 were published to aid compliance [NASA, 1995]. The trade studies performed to arrive at the 25 year requirement were also published [Reynolds, 1995], The most current version of the policy can be obtained from the NASA Code Q website. Spacecraft designers must be aware of the policies limiting orbital debris because they may have an impact on the required fuel load, as well as other design features. Often a great deal of analysis must be performed to design a spacecraft to meet debris limitation policies. These analyses are complicated by the inherent uncertainties in predicting events that will occur many years after the spacecraft is launched. Choosing an analysis method that provides sufficient accuracy while minimizing the necessary effort requires care. This section provides a comparison of three available disposal analysis methods and recommends which reentry analysis method to choose for a given situation.
14.3
End-of-Life Disposal
743
14.3.1 Predictability of Im pact It is impossible to accurately predict the way a spacecraft will behave during decay, the way it will break up, or the altitude and time at which breakup will occur. The spacecraft attitude during decay significantly affects the decay time but is often unpre dictable because most attitude control systems are designed to function only above a minimum threshold altitude. Once the spacecraft passes below this threshold, any deboost maneuvers could not be reliably pointed, and tumbling may cause variations in the frontal area which are hard to model and which affect the impact time. Break up factors depend on the material composition and structural integrity of the spacecraft components [Newman, 1993]. However, it is possible to adequately bound the prob lem using judicious choice of analysis methods. It is important to know the limitations and appropriate applications of the available methods, and to choose the method for performing a given piece of analysis accordingly. The trajectory and lifetime of a few pieces of the spacecraft, such as the largest and smallest, may be computed to deter mine the area over which the debris is expected to impact the Earth. Sometimes the choice of pieces to study is made by the fact that certain materials are known to survive reentry, such as a telescope mirror which is fairly large and made of a solid base metal. If available, a thermal analysis can predict which pieces could survive reentry heat ing. Johnson Space Center has the capability to perform such analyses (see Newman [ 1993]). Then the trajectory of each of the surviving pieces can be propagated from the breakup point. Variations in mass and frontal area will affect the landing site, creating the scatter field. If breakup and thermal models are not available, the whole spacecraft may be assumed to reenter simultaneously and survive to impact.
Date
Fig. 14-5.
Solar Flux Predictions as of April, 2001. See also Fig. 2-19 in Sec. 2.4.
It is also very difficult to predict the solar flux far into the future, and even the best models are updated frequently to account for changes. Yet this factor has the greatest effect on the lifetime calculations. Figure 14-5 shows the solar flux levels as predicted in April, 2001 by Schatten [2001]. Note that predictions are available for mean and ±2 sigma flux levels. In addition, early and late timing of the solar maximum are predicted similarly to the nominal timing values shown here. For historical variations, see Fig. 2-19 in See. 2.4. For further information concerning the historical and future predic tion of solar flux, see Schatten [2001].
744
Operations Considerations in Orbit Design
14.3
14.3.2 Disposal Options During the initial design phase for a spacecraft, programs that follow the NASA guidelines must perform a lifetime analysis to determine whether the spacecraft will be able to meet existing disposal requirements. If it is determined that the spacecraft cannot meet the end-of-life time on-orbit constraint through natural decay, several dis posal options are available. These include: • Uncontrolled Reentry • Controlled Reentry • Place in Disposal Orbit • Retrieval • Reuse Uncontrolled reentry is an option for spacecraft that, due to their ballistic coeffi cients and the level of solar activity at their end of life, are able to reenter the Earth’s atmosphere within 25 years. If initial analysis shows that a spacecraft will not reenter within this time, there are still ways to take advantage of an uncontrolled reentry, which is the cheapest of the disposal options. The spacecraft area to mass ratio may be altered sufficiently to change the decay lifetime of the spacecraft. This could be done before launch by physically changing the mass or area, although this is generally not feasible due to spacecraft design constraints. After the spacecraft has finished its mis sion life it could be put into a high drag attitude to promote rapid decay. However, this alternative assumes an ability to control this attitude throughout the decay period. In addition, the spacecraft orbit could be lowered to one which will decay during the required time frame. This option can be relatively cheap if only a small orbit change is required. If the spacecraft is unable to reenter naturally within an appropriate time, a con trolled reentry may be performed.* “Controlled” means that the ground operators plan maneuvers which drive the spacecraft back to a specific location on Earth. This requires that the spacecraft carry the necessary fuel, and so must be planned before the spacecraft is fully designed and built. There must also be sufficient attitude control at low altitudes to ensure that the final maneuvers are executed in the proper direction. These operations are potentially hazardous, as not all spacecraft completely bum up during reentry. Components that are very large, have materials with high melting points, or have a shape which helps reduce atmospheric heating may survive reentry, posing a threat to those on Earth. Statistics regarding the probability of the reentering components striking a human being are used in determining the size and location of the disposal location needed to perform a controlled reentry of a spacecraft, based on models of how the spacecraft is expected to break up and which parts are expected to reenter (for more information, see Mrozinski [2001]. This option is very expensive from a fuel point of view. Carrying such a large amount of propellant is generally pro hibitive for small, low-budget spacecraft. However, some larger spacecraft may carry the extra fuel to perform a controlled reentry to avoid causing casualties on impact or the perception that damage could occur. Section 14.5 contains a detailed discussion of the reentry analysis and execution for CGRO, a NASA spacecraft designed to be * This option may also be preferred to ensure that the spacecraft doesn’t reenter overpopulated areas.
14.3
End-of-Life Disposal
745
reentered at the end of its useful lifetime due to its large lead components which could potentially injure someone if left to reenter in an uncontrolled fashion. For orbits such as geosynchronous which are too high to use the reentry methods, the spacecraft may be placed in a disposal orbit that is out of the way of other space craft in the vicinity. Geosynchronous spacecraft are usually boosted to a higher orbit at the end of their useful life to keep them out of the path of new spacecraft in geosyn chronous transfer orbits. This option is also available to low Earth spacecraft which do not carry enough fuel to reenter completely, but which can be taken out of crowded orbits. The region allocated by NSS 1740.14 for disposal of LEO spacecraft is between 2.500 km and 18,833 km altitude. Since most LEO spacecraft have orbits well below 2.500 km, placing these spacecraft in such a storage orbit would be more costly than maneuvering to an orbit which will allow decay within 25 years. Some spacecraft may take advantage of the retrieval option in which the spacecraft is captured by the Space Shuttle and brought back to Earth for testing or reuse. How ever, this can only be done for spacecraft within reach of the Shuttle, which typically goes only to a maximum altitude of 300 km and an inclination of 28 deg to 57 deg. Retrieval requires substantial rendezvous planning and special hardware on the space craft to enable grappling by astronauts. The LDEF retrieval is an example of this tech nique. However, the purpose was not to eliminate debris, but to retrieve the information LDEF had collected about the space environment. A final option is to reuse the spacecraft bus for a new mission. This can be accom plished through extensive initial planning to allow various payloads to use the same support systems. The Explorer Platform was designed in this manner, to first carry the EUYE payload, then have that replaced by the XTE payload. However, this capability was not used and XTE was launched aboard a different bus. The Hubble Space Tele scope is also an example of reuse, since its instruments are switched out every few years, potentially extending the lifetime of the telescope indefinitely.
Disposal of the M ir Space Station In 2001, the Russian space agency, Rosaviakosmos, decided to reenter the Mir space station, successfully placing the station in a planned deorbit box in the Pacific Ocean on March 23, 2001. At the end of its useful lifetime, the Russians had allowed Mir to decay naturally from its mission orbit until they were ready to perform the reentry op eration. Mir’s altitude had gradually dropped from 350 km in the end of December to about 250 km before the final reentry sequence began. The station was then reentered using a control sequence powered by fuel contained in a Progress vehicle docked to the station for that purpose. Three reentry maneuvers lasting approximately 20 min each were executed by Russian Mission Control at 6:35 pm EST on March 22,9:01 pm and 12:07 am March 23 [Sietzen, 2001]. Splashdown occurred about 50 min after Bum 3 began. This method of providing extra fuel for the operation was available due to the flexible nature of docking ports on the station. A typical satellite may not be equipped with such a feature that would allow the addition of extra fuel for end-of-life disposal. The center of the Mir debris impact area was approximately 3,700 km south of Tahiti and 4,500 km east of New Zealand in an area completely free of islands, human habi tation, shipping lanes, or air traffic. Controllers expected that most of the 135-ton sta tion would bum up in the atmosphere, but that up to 30 tons of debris, including some pieces the size of an automobile, would reach the surface. Observers near Fiji noted large fireballs streaking through the sky that broke into smaller pieces as they dropped toward the Pacific.
746
Operations Considerations in Orbit Design
14.3
14.3.3 Choosing an Orbit for Uncontrolled Reentry As discussed in Sec. 2.4.1 satellite decay varies dramatically with time relative to the solar cycle. (See Fig. 2-21 in Sec. 2.4.1). Consequently choosing an end-of-life orbit altitude that will meet the NASA guidelines of reentry within 25 years can be analysis-intensive. Three methods are readily available: the method presented in NASA guidelines [NASA, 1995], the modified NASA method developed at GSFC, and the full numerical approach. The NASA guidelines method is easy to use and requires virtually no knowledge of orbit perturbations and decay. The analysis is quick using software available with the guidelines. However, the results obtained are coarse and conservative, as they do not model the varying solar flux levels. The numerical method is most accurate because it accounts for orbit perturbations such as the solar flux cycle and the diurnal bulge. However, it is complicated and time consuming and requires experience in orbit analysis and reentry planning. The GSFC method is a compromise between the other two. It uses solar flux predictions to model the solar cycle and the timing of orbit decay, yet is still simple and easy to use. Differences in fuel calculated between methods could drive mission cost, mass margins, tank sizes, or other spacecraft design concerns. The method used for analysis of the uncontrolled reentry option should be chosen based on the level of development of the spacecraft. The NASA guidelines and GSFC methods are appropriate for space craft undergoing feasibility studies, in which a worst case estimate of the disposal AV" required would be used for tank volume and mass calculations. A more accurate approach may be needed for spacecraft in the design and test phase, when the space craft configuration has been established and disposal planning becomes necessary to determine fuel budgets. The NASA guidelines method uses a series of plots and the spacecraft ballistic properties to determine whether the spacecraft meets the 25 year lower lifetime requirement, or if moving the spacecraft to a disposal orbit or reentering it will be required. From these results, the fuel required for spacecraft lowering or reentry may be computed. Figure 14-6 is one step in the process. The figure shows, for a given spacecraft mass and area, how much AV and hydrazine are required for reentry. The plot was generated assuming a constant solar flux of 130 x IO-22 W/m2/Hz over peri ods of up to 50 years, a physical impossibility given that the solar flux varies between approximately 75 and 250 during a typical 11 year solar cycle. (See Fig. 14-5 or Fig. 2-19 in Sec. 2.4.) Therefore, the lifetimes calculated using the NASA guidelines meth od are erroneously long. While the long lifetime may be viewed as a conservative estimate, it may also cause unnecessary mitigation techniques to ensure the spacecraft will be able to comply with the guidelines, since under true solar conditions the space craft might meet the 25 year requirement more economically. Reynolds [1995] states that the fuel required for reentry is on the order of the amount usually put on board as margin in case of emergency. However, this is not always the case. The percent of the Terra fuel budget required for the uncontrolled reentry option ranges between 16% and 38%, much greater than the 10% typically held in reserve (see Table 14-4). Therefore, each spacecraft should examine its dispos al options to find the one that is most efficient and least costly. The GSFC method involves scaling the NASA guidelines results to be more consistent with a full numerical analysis. By fitting predicted solar flux data with a 9th order polynomial, the NASA guidelines calculations can be scaled to the numerical data. The polynomial is of the form:
14.3
End-of-Life Disposal
747
500
400
_
300
I <1
200
100
0
0
500
1,000
1,500
2,000
Orbit Altitude (km)
Fig. 14-6. NASA Guidelines Method [Reynolds, 1995]. Select the area to mass ratio (A/M) for a spacecraft design and use this chart to determine the required reentry fuel (right axis) or A V (left axis).
flu x = c + C } X +
c 2x
(14-9)
2 + ...+
where x is the time in years from the epoch of the solar flux predictions. The mean value over the appropriate interval should be found by evaluating the polynomial over the desired time interval and dividing by the NASA Guidelines solar flux value of 130 x IO-22 W/m^/Hz to compute the scale factor. The interval to be used should begin at end-of-life and include at least one complete solar cycle. The perigee deter mined from the NASA Guidelines is then multiplied by the scale factor. The full numerical method is most accurate but very time consuming. It applies all perturbations to the propagation over the entire predicted lifetime of the spacecraft to determine when reentry will occur. This method need only be applied once a configu ration is well defined and an exact fuel budget is needed for the reentry trajectory. Table 14-2 compares all three methods for the Terra spacecraft, showing the perigee height to which the 705 km circular mission orbit would have to be lowered for the spacecraft to reenter within 25 years. The higher the perigee height, the less AV required, so the NASA Guidelines approach is the most fuel-expensive while the GSFC and numerical methods provide good agreement for a lower cost solution. TABLE 14-2. Terra End-of-Life Uncontrolled Reentry Orbits Based on Varying Models.
EOL Conditions
NASA Guidelines Perigee Height (km)
GSFC Perigee Height (km)
Numerical Perigee Height (km)
EOL on 6/2003, mean flux
550
571
571
EOL ©solar max, mean flux
550
580
587
EOL @solar max, +2s flux
550
634
638
748
Operations Considerations in Orbit Design
14.4
14.4 Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra In order to illustrate the design principles discussed in this chapter, this section de scribes the selection of orbit elements for the Earth Observing System (EOS) Terra spacecraft (Fig. 14-7) launched Dec. 18,1999. Terra carries five scientific instruments and is designed for a 6-year lifetime as part of a constellation of Earth-observing sat ellites. Originally proposed in the late 1980’s to be a polar orbiting platform as part the Space Station program, the program was called POP-A and carried dozens of instru ments. In the early 1990’s, this design was scaled back to eventually become EOS Terra. Through all of the configuration changes, the orbit design stayed fairly constant. Because of its complexity, Terra is a good example of the process of orbit design.
Fig. 14-7. The Terra Spacecraft.
14.4.1 Definition of Complete Orbit Parameters The Terra mission orbit requirements are: 1. Sun-synchronous 2. Repeating ground track with cycle of 233 orbits in 16 days 3. 705 km mean altitude over the equator 4. Frozen Orbit 5. 10:30 am (±15 min) descending node mean local solar time 6. ±20 km ground track control accuracy with respect to the World Reference System 7. ±5 km radial constraint 8. Fly the same ground track as Landsat-7 within 15 min to 1 hour As discussed in Sec. 14.1, not all of the classical Keplerian orbit elements are explicitly defined by the mission requirements listed above; however, all are defined
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
749
implicitly and some are over-defined. Table 14-3 shows which elements are affected by which requirements. The equations which define these relationships are covered in Chaps. 2 and 12. While argument of perigee and mean anomaly are only affected by one requirement, the semimajor axis and inclination are affected by many. Obviously, Terra has a very strictly defined set of mission requirements with little room for further choices. In fact, the requirements need to be checked to make sure they are not over constraining the orbit element selection. The best place to start is to determine mean values for the elements, beginning with the altitude and inclination. The mean values will account for basic orbital forces such as the geopotential, but will not take into ac count higher order terms such as atmospheric drag or lunar and solar gravitation. Those types of forces affect the maintenance of the orbit elements as opposed to the initial selection. TABLE 14-3. Mapping of Terra Orbit Requirements to Orbit Elements Affected. Requirement
a
e
i
1. Sun-synchronous
X
X
2. Repeating Ground Track
X
X
3.705 km Altitude
X
4. Frozen Orbit
X
5 . 10:30 am Mean Time
X
6. ±20 km Cross Track
X
7. ±5 km Radial 8. Some Ground Track as LandSat-7
£2
(!)
M
X X
X
X X
X
The first step in the mean element selection process is to see if it is possible to choose an inclination that meets both the repeat cycle and Sun-synchronous node rate requirements for the required altitude. Figure 14-8 illustrates combinations of altitude and inclination that result in various ground track repeat cycles and also those which result in a Sun-synchronous node rate. For Terra the repeat cycle of 233 orbits in 16 days equals 14.56 revs/day, and the mean altitude is required to be 705 km. These requirements can both be met at an inclination of 98.2 deg. Since one unique solution was successfully found, mission orbit requirements 1 through 3 have now been met. The second step in the design process is to choose the values of argument of perigee and eccentricity for a frozen orbit. In order for this to occur, the change in eccentricity and change in argument of perigee must be zero. As discussed in Sec. 2.5.6, in order for the change in eccentricity to be zero, the argument of perigee must be either 90 deg or 270 deg, that is, fixed above the North ernmost or Southernmost latitudes, respectively. The chosen argument of perigee is then plugged into Eq. 2-67 in Sec. 2.5.6, along with the altitude and inclination deter mined above. In the case of Terra, a value of 90 deg is chosen for the argument of peri gee based on the science desire to have perigee over the Northernmost latitude for data calibration purposes. The frozen eccentricity is computed to be 0.0012 and require ment number 4 has now been met. The third step is to select the node such that the mean local time meets the 10:30 am descending node mission requirement. The node may be computed using:
750
Operations Considerations in Orbit Design
14.4
Ground TVack Repeat Cycles e = 0.0001
Fig. 14-8. Sun-Synchronous and Repeating Altitude/Inclination Combinations.
Q = (M L T ^ -1 2 )
360 deg 24 hours
SUN
(14-10) mod 360
where MLT^y is the mean local time of the ascending node &sun ->*s the right ascen sion of the Sun on the epoch date, and £2 is right ascension of the ascending node. As suming a launch date for Terra of 6/20/98, the original target launch date, we obtain Q, = 246.19 deg. If the mean local time requirement is specified as a descending node time, as is the case for Terra, simply add 12 hours to get the corresponding ascending node time before applying Eq. 14-10. This process meets requirement number 5. The above analysis fully defines a stand-alone mission for the Terra spacecraft to meet all the original science requirements. However, we have not yet addressed requirement 8, to fly in constellation with Landsat-7, which was levied a few years before launch but well after the design and fabrication of the spacecraft parts. The only orbit element not yet specified is the mean anomaly. Fortunately, Landsat-7 has iden tical orbit requirements to Terra, except that it is in a 10:00 am descending orbit. Therefore, specifying the mean anomaly allows Terra to fly in constellation with Landsat-7 such that the two spacecraft trace the same ground track within 15 to 60 min of each other. While none of the other orbit elements are affected. In order for both spacecraft to see the same point on the ground and still meet their respective mean time requirements, they must be separated in mean anomaly enough so that as the Earth rotates under the two orbit planes such that a given point on the equator is at 10:00 am as Landsat-7 crosses over head, and at 10:30 am when Terra is overhead. This geom etry is shown in Fig. 14-9. The two orbit planes are separated by 30 min in mean local time corresponding to 7.5 deg of right ascension. A separation of 108 deg (360 x 30 min/100 min orbit period
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
751
Terra Landsat
Orbit Plane
Fig. 14-9. Relative Positions of Terra and Landsat-7 fo r 15-min Separation on Same Ground Track [McIntosh, 2000].
= 108 deg) in anomaly will permit the two spacecraft to be separated by 30 min in mean local time. Any value can be assumed for the anomaly pre-launch, with the un derstanding that the anomaly of the final orbit must be constrained. No particular value should be chosen during the pre-launch phase because the anomaly of the first space craft at the time that the second reaches orbit determines the target anomaly for the sec ond spacecraft. In addition, most launch vehicles will not permit a particular value of mean anomaly to be used as a target becausc it is a free variable for the launch vehicle trajectory. Variations in launch time, date, and powered flight trajectory experienced by the second spacecraft prevent the specific value from being known in advance. These steps have yielded a mean element set that meets mission requirements. Now, the elements must be refined to include higher order terms and perturbation effects and be converted from mean to osculating elements for the launch vehicle to target.
14.4.2 Orbit Error Boundaries Requirements 6 and 7 have not yet been addressed. They affect not the initial choice of orbit elements, but the maintenance of these values over time. (Orbit maintenance is discussed in Sec. 2.7.2 and 13.4.) These requirements do not over-constrain the choice of orbit altitude or eccentricity, but define error bounds on them. As discussed in Sec. 14.1, the orbit error bounds for a given mission are often a measure of the flex ibility of the requirements driving a particular orbit element, and thus the selection of those bounds is very mission specific. In order to determine how tightly each element must be achieved, it is necessary to analyze the long-term perturbations on the ele ments. The following paragraphs provide a definition and analysis of the Terra orbit error bounds in an effort to provide a complete case study for the mission. Semimajor Axis Error Bound. The orbit altitude varies due to atmospheric drag. Requirement 6 simply defines how much the altitude is allowed to drop before being raised. In order to control the ground track to within the ±20 km box stipulated by requirement 6, it is necessary for the spacecraft to perform frequent altitude raising
752
Operations Considerations in Orbit Design
14.4
maneuvers. An initial altitude is chosen that is above the nominal and which places the ground track 20 km east of the reference track, drifting westward as the altitude decreases. The altitude is chosen such that the nominal mean altitude is reached when the ground track error reaches -20 km. As the altitude continues to decrease, the orbit perturbations cause the ground track drift to change directions and drift eastward towards the +20 km edge. When that edge is reached, the next maneuver is performed to restart the cycle. To keep the burn frequency to a minimum, it is desirable to have as large a control box as possible. However, the larger the control limits, the greater the chance of vio lating the ground track limits, primarily due to the inability to accurately predict solar flux activity in the long term. The maneuver to set the initial altitude is chosen by prop agating the ground track over the cycle period using a prediction of the solar flux level. If the actual solar flux value is lower than expected, the western boundary of the control box will be violated. Therefore, a box must be chosen based on the ability to predict the solar flux level. This new box represents the error bound on the ground track control box and, hence, on the orbit altitude. In order to ensure that changes in flux level would not cause Terra to violate its ground track control box, analysis was performed to determine longitude errors in the ground track repeat cycle using different flux prediction levels to develop an under standing of the possible range in ground track errors. Simulations were run using a numerical propagator with worst-case low flux levels and average daily prediction er rors to choose the lower control box limit, since a lower than expected solar flux could cause the ground-track to violate the western boundary of the control box. To further mitigate the possibility that a decrease in the solar flux would cause a violation of the western ground-track error limit, average solar flux prediction errors were introduced. Using these models, a range of ground-track control sub-limits were evaluated to determine if the post-maneuver ground-track error would exceed the western limit of -20 km. In the case that a violation occurred, a new sub-limit was chosen and the targeter rerun to determine the new bum duration. This process was repeated until an acceptable ground-track resulted. Figure 14-10 shows the results for each flux level examined. As expected, the lower flux cases allowed more westward drift. Notice that the majority of the solar flux cases result in nearly identical eastern drift rates, as indicated by the similar upward positive slopes in the ground track error. Drift rates near the eastern edge of the box averaged approximately 1.5 km per day. Given this information and the desire to have a bum delay buffer of 2 to 3 days prior to violating the eastern ground track boundary, an east ern control limit of +15 km was chosen. In this manner, the Terra ground-track control box limits of +15 km and -5 km were chosen to prevent violating the ground-track error limits of ± 20 km. The upper control limit o f +15 km was chosen to allow a 2 to 3 day bum delay without violating the eastern boundary. The lower control limit o f-5 km was chosen to prevent a lower than expected solar flux from causing a violation of the western boundary. Eccentricity and Argument of Perigee Error Bounds. Requirement 7, a radial constraint, affects how tightly the orbit must be frozen, or the allowable variation in eccentricity. This requirements does not affect the altitude maintenance or error bounds on the semimajor axis because the ground track control (requirement 6) requires that the altitude at a given latitude be maintained within about 1 km. Thus, meeting requirement 6 ensures that requirement 7 wouldn’t be violated as long as peri gee remains fixed at the same latitude. However, perigee rotates to some degree as the
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
753
E poch
Fig. 14-10.
Effect of Various Solar Flux Levels on Terra GroundTrack Error [Wilkin, 1998].
eccentricity changes, even in a frozen orbit. Therefore, further analysis was required to determine the acceptable amount of rotation of the argument of perigee and the as sociated eccentricity variation. There was a desire among project scientists for Terra to minimize the science down time by minimizing the number of maneuvers. To ameliorate the situation, performing ground track control using one maneuver instead of the traditional two bum pair was proposed. To validate this operations scenario, it was necessary to investigate whether the frozen orbit would degrade after many ground track maneuvers had taken place, resulting in violation of the radial position constraint. Optimization of the near frozen orbit condition was examined by selective placement of a single ground track mainte nance maneuver at the mean anomaly that produced mean eccentricity and argument of perigee values closest to mission nominal (eccentricity = 0.00116, argument of peri gee = 90 deg). Results indicated that allowing a worst-case altitude deviation of ±5 km for these maneuvers corresponded to an eccentricity deviation from nominal of +0.0008/-0.0006 as shown in Fig. 14-11. This eccentricity results in the frozen orbit boundary indicated on Fig. 14-12. The corresponding argument of perigee variation is ±40 deg for the above nominal case and ±33 deg for below nominal [Noonan, 1996]. Therefore, conservative values of the mean argument of perigee error bounds of ±20 deg and mean eccentricity error bounds of ±0.0004 were chosen to maintain the alti tude to within ±5 km at a given latitude. It was determined that performing the required ground track control maneuvers in a place in the orbit chosen to optimize the frozen condition meets the radial constraint without imposing further restrictions on the orbit. Inclination Error Bound. The Terra inclination requirement did not specify error bounds; however, the inclination is implicitly bounded by maintaining the mean local time. Over the life of the mission, solar and lunar perturbations on the orbit will causc the inclination to gradually decrease. This will alter the node rate, which will cause the local time to change. The inclination can be controlled actively, using maneuvers per pendicular to the orbit velocity. These maneuvers are costly in fuel and operationally complex, often involving yawing the spacecraft in a manner not used for any other
754
Operations Considerations in Orbit Design
14.4
E c
o
fl>
■o
o
I Eccentricity Fig. 14-11.
Terra Altitude Variation vs. Eccentricity [Noonan, 1996].
Mean Argument of Perigee (Deg) Fig. 14-12.
Effect of Various Solar Flux Levels on Ground Track Error [Noonan, 1996].
mission activities. Therefore, inclination maneuvers are often avoided by spacecraft that can take advantage of passive control [Folta, 1992]. In passive control, the incli nation is offset from the optimal Sun-synchronous one by a small amount (on the order of hundredths of a degree). Lunar and solar forces then cause the inclination to move in a beneficial direction, similar to the philosophy of ground track control. The turn around period is on the order of several years, making it possible to complete an entire mission without performing an inclination maneuver if the initial inclination is suffi ciently accurate. Most launch vehicles only guarantee inclination accuracy to 0.03 to 0.10 deg. Changes from the nominal of this size would not achieve the passive control goals, so a small initial inclination change maneuver may be necessary. As discussed in Sec. 14.4.3, Terra had multiple inclination targets given a launch time within its 25-min window. Figure 14-13 shows the mean local time drift produced through passive inclination control assuming optimal inclination targets were achieved for 10:20 am and 10:40 am orbits.
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
755
Years of Mission Life Fig. 14-13.
Terra Mean Local Time Drift for Various Initial Values [McIntosh, 1998].
14.4.3 Launch Window Parameters The Terra launch window is predominantly defined by the requirement for a Sunsynchronous orbit with a mean local time of 10:30, ±15 min. In order to achieve a par ticular local time, the spacecraft much launch when that plane is in the right position with respect to the launch site. Terra launched on an Atlas IIAS, which had the capa bility to do guided targeting. This means that for each minute of the window, the vehicle could target different values of inclination that would maximize the time spent within the local time control box. This allowed the window to be widened to a 25-min block instead of only being able to target a 10:30 orbit. The Terra launch window times corresponded to achievable local times of 10:15 to 10:40 am. A buffer was left for pos sible injection errors as sized by the vehicle contractor. Landsat-7, which launched on a Delta II, did not have this flexibility, as the Delta could only use one target. There fore, the Landsat-7 window was much narrower, on the order of 5 min, despite having the same local time error bound requirement. To compute the launch vehicle targets, optimum inclinations were determined for every minute of a launch window opening at 10:15 and closing at 10:40. For each minute, mean orbital elements were propagated for 6 years and the inclination was varied until a value was found that resulted in a maximum mean local time of 10:44 (1 min short of the control limit) at some point during the 6 years. The mean elements were converted to osculating and a polynomial was fit to the osculating inclinations. The polynomial coefficients were then loaded into the vehicle flight software, where they were used after liftoff to compute the target inclination through the following process. When the vehicle senses liftoff, the GMT of the descending node can be calculated from: GMTdn = GMTLq +
+ A?2
(14-11)
756
Operations Considerations in Orbit Design
14.4
where GMTLo is the GMT of the software detected liftoff, At\ is the time of launch vehicle flight from liftoff to spacecraft separation, and Af2 is the time from spacecraft separation to the descending node. The mean local time of descending node, MLTDN, is then computed from: MLTDn = GMTDn - [(360 deg - LDN) x (86,400 sec/360 deg)]
(14-12)
where GMTDN is the GMT at the descending node crossing in scconds and LDN is the East longitude of the descending node in deg at the time of the descending node cross ing. Finally, the target inclination is computed using the polynomial y = C0 + C ^ + C2^2
(14-13)
where x is the MLTDN and y is the osculating inclination. Note that this method assumes constant values for LDN, A a n d Af2 for all Ter ra launch times. This is not exactly true given that the vehicle trajectory times may vary due to winds and launch date geometry. The constant values used were com puted by the launch vehicle contractor assuming a nominal powered flight trajecto ry. Analysis showed that variations in the times were sufficiently small to allow inclination targeting scheme within the desired accuracy, assuming no launch vehicle errors. Terra actually launched at the end of its launch window. The inclination targeting scheme successfully chose the target to maximize the time spent in the control box as shown in Fig. 14-14. However, a maneuver is necessary at about 3 years into the mis sion to change the mean local time drift direction and remain in the control limits for the rest of the 6-year mission. Figure 14-14 shows one possible local time profile which could be used to achieve mission goals.
Years From Apr 27,2000
Fig. 14-14.
Terra Mean Local Time Drift Following Mission Orbit Acquisition. Assumes an inclination maneuver performed at 3 years.
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
757
14.4.4 Achieving Mission Orbit Although Terra did not have any rendezvous or docking requirements, there was an orbit phasing requirement, which is similar to a rendezvous problem. Strategy for the ascent planning is based on the need to achieve a Sun-synchronous, frozen, ground track control, and Landsat-7 phasing requirements simultaneously. The World Reference System, the target ground track reference path system used by Landsat, consists of 233 equally spaced descending node “paths.” Achieving the mission orbit for a satellite attempting to control its ground track to this reference usually requires a series of maneuvers to raise the spacecraft to the proper altitude to synchronize with the system grid. The final trim maneuver is timed so that the spacecraft ground track will align with the nearest reference system path within the allowed control limits. Usually it does not matter which path is achieved at the start. Phasing with another spacecraft, however, requires that the ascent be targeted to a specific path. In addition, the Terra anomaly is dictated by the difference in mean local time between the two spacecraft. The nominal launch window for Terra could result in a range from 10:15am to 10:40 am. With the Landsat-7 mean local time near 10:00 am at the time of Terra launch, the two orbit planes could be separated anywhere from 15 to 40 min. The actual launch time and local time difference dictate where Terra must be positioned relative to Landsat-7. For example, if launch occurs at the end of the window, Terra must ascend to a target point 40 min behind Landsat-7 in order to achieve the same World Reference System path. In addition, the initial phase angle between Terra and Landsat-7 varies with the giv en launch date in a 16-day repeat pattern. The ascent maneuver plan had to be flexible enough to accommodate all possible configurations. Figure 14-15 shows the relative positions of Terra and Landsat-7 after the ascent has been completed; resulting posi tions for launches at the opening and closing of the window are indicated (spacecraft
Fig. 14-15.
Final Relative Positions Resulting from Launch at Opening and Closing of Window [McIntosh, 2000].
758
Operations Considerations in Orbit Design
14.4
are shown in the same plane for simplicity). While Terra sits on the ground waiting for launch, the relative phasing with Landsat-7 repeats on a 16-day cycle. Thus, a set of 16 maneuver sequences can be computed that will cover all nominal initial conditions. Once launch occurs and the nominal injection orbit is achieved, the synodic period is on the order of 10 days. Therefore, if the initial opportunity to begin the maneuver sequence is missed, the sequence can be started again 10 days later with little change. The synodic period is important in determining the frequency of opportunities for per forming the ascent (see discussion in Sec. 2.2). As the altitude is raised, this period becomes longer. If the ascent sequence is interrupted, it may be a long time before the two spacecraft return to the required phase angle. Figure 14-16 shows the synodic period as a function of altitude difference. While Terra remains in the nominal parking orbit, the synodic period is about 10 days. If Terra ascends to an altitude 5 km below Landsat-7, the synodic period becomes 65 days; this means there would have to be a wait of more than 2 months before completing the ascent if the final bum is missed. Phasing could be adjusted by lowering the orbit, but this wastes fuel, and would require an undesirable 180-deg yaw of the spacecraft.
0
5
10
15
20
25
30
35
Altitude Difference (km) Fig. 14-16.
Synodic Period as a Function of Altitude Difference for Terra and Landsat-7. Semimajor Axis is 7,078 km [McIntosh, 2000],
Until the end of the ascent phase, Terra is below Landsat-7 and flying faster. Each time the path difference returns to the same value, Terra has revolved one extra orbit relative to Landsat-7. A zero path difference is achieved at the end of the ascent phase. Since both spacecraft will perform periodic ground track maintenance maneuvers to keep their own ground track errors within small ranges (±20 km for Terra and ±5 km for Landsat-7), they will both remain on the same path and follow the same repeat cycle until end of life. The exact orbital period required to stay on the World Reference System is closely correlated with node rate. As the node rate changes, very small adjustments to the
14.4
Example 1: Defining Launch, Orbit, and Disposal Parameters for Terra
759
orbital period will be required to keep the spacecraft on track. As the mean local time of Landsat-7 and Terra evolve differently, separation between the two spacecraft will also change with time. As both spacecraft maintain their respective ground track errors within bounds, a natural byproduct will be to keep an along-track separation equal to the mean local time difference. If the orbit planes drift farther apart, the spacecraft along-track separation will also increase; if the orbit planes move closer together, the along-track separation will decrease. Ground track control and phasing are accomplished by varying the number of orbits between maneuvers, keeping the bum magnitudes fixed. This proved to be the sim plest method from a spacecraft operations standpoint. The frozen orbit eccentricity is achieved by positioning the maneuvers at two optimum locations on nearly opposite sides of the orbit. The number of orbits between bums is adjusted until Terra is on the same path as Landsat-7 and the ground track error is less than 20 km. The final bum is also fine-tuned to give the desired ground track drift that will keep the spacecraft within bounds for the longest possible time. The actual liftoff time (18:57:36 UT) was within 20 sec of the end of the window. This meant that the goal for the ascent was now set for a point nearly 40 min behind Landsat-7. After launch, analysts used the current orbital states of Terra and Landsat7 to plan ascent maneuvers. The ascent was completed on Feb. 23, 2000. The final ascent burn stopped the ground track drift at +17 kmf near the western boundary of the error box, and started the spacecraft on a normal drift cycle.
14.4.5 End-of-Life Disposal.* Terra was built at the time when the NASA guidelines on orbit debris were first written. Therefore, although a disposal analysis was completed, no requirement was placed on the spacecraft to be deorbited because it was determined that the amount of additional fuel required could not be placed onboard given the existing mass margins. In order to compare the three methods described in Sec. 14.4, the Terra spacecraft and orbit characteristics were used to perform a parametric end-of-life disposal analy sis. Analyzing the spacecraft orbit to determine guideline adherence as described in Sec. 14.4 involves first computing the natural decay lifetime of the spacecraft to deter mine if the 25 year constraint is met. The mass and frontal area used were 4652.5 kg and 42,5 m2, giving an area to mass ratio of 0.009 m2/kg. Using the NASA guidelines method plots [Reynolds, 1995], a 705 km circular orbit with a lifetime of 25 years cor responds to an area to mass ratio of 0.03 m2/kg. Since this is greater than the Terra val ue of 0.009 m2/kg, the spacecraft will not naturally reenter within 25 years. For 0.009 m2/kg and an apogee height of 705 km, a 25 year lifetime corresponds to a perigee height of 550 km. Thus, according to the NASA guidelines method, Terra would have to be maneuvered from its 705 km circular orbit to a 705 x 550 km orbit in order to comply. This would require a AV of 41.6 m/s. For comparison, a numerical analysis using full force modeling and expected solar flux conditions was then performed. With mean solar flux predictions, the spacecraft was found to require an end-of-life orbit of 705 x 571 km to meet the 25 year limit with decay beginning on the scheduled end-of-life date of 6/30/03. This would require a AV of 35.9 m/s, a savings of 5.7 m/s relative to the NASA quidelines method. However, a
* Adapted from Newman [1995].
760
Operations Considerations in Orbit Design
14.5
likely end-of-life date must be selected with the understanding that if the mission slips to a period of lower solar flux, disposal will cost more fuel. For comparison, fuel costs were determined for at solar maximum using a mean solar flux level and +2cr flux levels on both dates. Table 14-4 presents the results, as well as the percentage of the total fuel budget that must be used to place the spacecraft into the appropriate orbit for uncontrolled reentry. Obviously, the NASA guideline method yields the most conser vative and costly solution. At the time, it was determined that the fuel required for any of these methods, which ranges between 16% and 38%, would be prohibitive, since all of the budgeted fuel is needed to perform the mission. Hence, Terra was allowed to not adhere to the new NASA guidelines. TABLE 14-4. Terra Starting Orbits for Uncontrolled Reentry Based on Varying Model Assumptions. EOL=end-of-life. Fuel budget = 110 m/s. Change in Perigee Height from NASA Method (km)
Required Deorbit Fuel for lsp-200s (kg)
Position of Fuel Budget Required fo r Reentry (%)
Model
Disposal Orbit (km)
Change in AV from NASA AV Method (m/s) (m/s)
NASA Method
705 X 550
41.6
N/A
N/A
99.7
37.7
EOL on 6/2003, mean flux
705 X 571
35.9
5.7
21
85.9
32.5
705 x 587 EOL ©solar max, mean flux
31.6
10.0
37
75.5
28.6
EOL on 6/2003, +2cf!ux
705x614
24.3
17.3
64
58.0
22.0
EOL ©solar max, +2(7 flux
705 X 638
17.8
23.8
88
42,4
16.1
Reynolds [1995] states that “For cases where post-mission disposal is required, and for mission orbit altitudes below -1200 km, typically no more than 3%-5% of the final mass is required for this disposal maneuver. This mass fraction is comparable to flight performance reserves.” The final mass of Terra is 4652.5 kg, of which 3%-5% would be 139.6 kg-232.6 kg. While the fuel required for disposal given in Table 14-4 is less than these percentages, it is also a large portion of the fuel budget. Most spacecraft allot approximately 10% of their fuel budget to reserves, values more on the order of 25 kg. While this reserve amount, based on the Terra fuel budget of 241.5 kg, would suffice for the disposal cases which use the +2 a predictions shown in Table 14-4, it would not suffice if solar activity was on the level of the mean predictions. For com parison, the GSFC method was also used to scale the perigee height values. As shown in Table 14-2, the results are very similar to the numerical method results.
14.5 Example 2: End-of-Life Disposal of CGRO The Compton Gamma Ray Observatory (CGRO) was designed from inception to have the ability to reenter at die end of its useful life. A portion of the propellant was reserved for this purpose. When the fuel remaining reached this limit, the observatory would be reentered. Figure 14-17 shows the nominal predicted reentry scenario for CGRO, in which breakup of the spacecraft occurs at 122 km, and the remaining debris
Example 2: End-of-Life Disposal of CGRO
14.5
761
reenters over a pre-designated scatter zone in the Pacific Ocean. The CGRO debris footprint is shown in Fig. 14-18. Calculations must be performed to ensure that the debris scatter will remain within this footprint. 350 km
Fig. 14-17.
Reentry Trajectory for CGRO [Brown-Conwell, 1989].
30
20
10 r ~
0 £ ® *a ® ~
Notes: A—10% Hot and Max B.C. X—Max B.C.
H—10% Hot N—Nominal C—10% Cold W—Min B.C. B—10% Cold and B.C.
-20
t-30 -160
-150
-140
-130
-120
-110
-100
-SO
-80
-70
West Longitude (Deg)
Fig. 14-18.
CGRO Footprint and Predicted Debris Scatter. [Brown-Conwell, 1989]. B.C. = Ballistic coefficient.
Two factors affect the debris scatter—the altitude at which breakup of the space craft occurs, and the altitude at which the spacecraft is assumed to reenter. Higher breakup altitudes will produce more debris scatter, yielding a larger footprint. Lower ing the perigee height for reentry decreases footprint size but increases the required AV. An analysis of the debris footprint [CSC, 1986], assumed CGRO would break up at 122 km to be conservative, although actual breakup was predicted at 80 km.
762
Operations Considerations in Orbit Design
14.5
The acceptable level of casualty expectation resulting from a given piece of debris impacting the ground is a common parameter used in evaluating the disposal options for a spacecraft. This parameter may be calculated using the instructions given by NASA [1995], Flight Experience.* On June 4,2000, CGRO entered the Earth’s atmosphere over the Pacific Ocean target. This was the first controlled re-entry of an unmanned space craft from low Earth orbit by NASA. The complexity and criticality of this operation was enhanced by the 14,000 kg mass of the spacecraft and the loss of 2 of the 4 orbit control thrusters soon after launch in 1991. Figure 14-19 shows the spacecraft.
Fig. 14-19.
The Compton Gamma Ray Observatory.
In December 1999, it was recognized that a detailed plan needed to be developed for a controlled re-entry. A target date of summer 2000 was chosen as a balance between allowing enough time to plan the complex operation versus trying to accom plish the activities before hardware on the spacecraft failed. One of the gyros had failed already, causing mission planners to worry about the ability to control the point ing of the spacecraft during the maneuvers if another gyro were to fail. Approximately one month before re-entry, all procedures where frozen and rehearsed with the final mission profile. The re-entry maneuver scenario consisted of an engineering bum and four 26+ min bums, each centered at apogee, that dropped the spacecraft from it’s 510 km circular orbit to a 510 x 50 km terminal orbit. On May 28, 2000, 19:44:00 GMT, the engineer ing bums were executed. These consisted of short firings of the 20N and 220N thrust ers to verify performance. Re-entry bum # 1 was conducted on May 31,01:51:05 GMT * Adapted from Mangus [2000]
14.5
Example 2: End-of-Life Disposal of CGRO
763
and put the spacecraft in a 510km x 350km orbit. Burn #2 on June 1, 02:36:52 GMT placed the spacecraft in a 510km x 250km orbit. The final two re-entry burns were done on June 4 at 03:56:00 GMT and 05:22:21 GMT, dropping perigee to 150 km and finally to 50 km. Within 30 min after burn #4, the spacecraft had completed its re-entry into the Pacific. Figure 14-20 shows a simplified version of the burn sequence. Between Burns 1, 2 and 3, the spacecraft remained in a power positive “Parking Attitude” under wheel control. Six min before the bum, the spacecraft entered thruster control. At the bum minus 2 min mark, the spacecraft pitched at 1 revolution per orbit to maintain the thrusters parallel to the velocity vector. A command from the ground fired the 220 N Orbit Adjust Thrusters.
Burn Midpoint
Fig. 14-20.
Simplified Burn Sequence.
After Bum 3, aerodynamic torques would cause the controller to saturate the wheels at perigee. Therefore, the spacecraft remained under thruster control during a final perigee pass. Figure 14-21 shows the predicted footprint, represented as the three symbols con nected by a line in the ocean. The circles and line along the American coast represent the boundaries of the safety zone that the debris had to remain within. Since the bums were of nominal performance, the spacecraft objects fell around the star at the center of the line. The objects with a low mass to area ratio, i.e. the solar arrays, fell on the upper left side of the star and the objects with a high mass to area ratio, i.e. the titanium bolts, fell downstream from the star. All objects fell within the plus sign and the circle along the line. Visual contact from an U.S. Air Force plane, contracted by NASA to track re-entry, verified the proper time and location of impact.
764
Operations Considerations in Orbit Design
-Ai
._
4 .-*■■■ vUv- >iv ,
.
.'■'i
A
tW 'r i
GRO Impact Footprint - June 2000 Harris-Prlester Atmospheric Model Bum 3 Targeted to 1SO km Perigee Burn 4 - 3 0 Min. Duration ue- Nominal Burn/Intact BC ... o - 10% Cold/Heavy BC + -1 0 % Hot/Light BC
Fig. 14-21.
n - 1 0% Cold/Heavy BC B u r n 4 - 12 Min. Late
::5
CGRO Predicted Debris Footprint. BC = Ballistic coefficient.
References The Boeing Company. 1996. Delta II Payload Planner’s Guide, MDC H3224D. April. The Boeing Company. 1999. Delta III Payload Planner’s Guide, MDC 99H0068. October. The Boeing Company. 1999. Delta IV Payload Planner’s Guide, MDC 99H0065. October. Brown-Conwell, Evette R. 1989. GRO Mission Flight Dynamics Analysis Re port: Controlled Reentry o f the Gamma Ray Observatory, CSC/TM-90/6001. November. Chiulli, Roy M. 1994. International Launch Site Guide. El Segundo, CA. The Aero space Press. Computer Sciences Corporation. 1986. Gamma Ray Observatory (GRO) Flight Dynamics Report; Analysis o f Debris Impact Footprint fo r GRO Controlled Re entry, CSC/TM-86/6042. Folta, David and Lauri Kraft. 1992. “Methodology for the Passive Control of Orbital Inclination and Mean Local Time to Meet Sun-Synchronous Orbit Requirements,” AAS 92-143, AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, CO. February 24-26. International Launch Services, Inc. 1998. Atlas Launch System Mission Planner’s Guide, Revision 7. December. Isakowitz, Steven J. 1999. International Reference Guide to Space Launches (3rd ed.), American Institute of Aeronautics and Astronautics, Washington, D.C.
References
765
Kraft, J. Donald. 1996. “Launch Vehicle Selection, Performance and Use,” Applied Technology Institute. April. Mangus, David and Susan Hoge. 2000. “CGRO Reentry Summary Report” in the Flight Dynamics Analysis Branch End o f Fiscal Year 2000 Report, NASA GSFC Code 572. November. McIntosh, Richard J., Noonan, Paul J., and Lauri K. Newman. 2000. “Terra Ascent Planning to Meet Landsat-7 Phasing Requirements,” AIAA 2000-4342, AIAA Astrodynamics Specialist Conference, Denver, CO. August 14—17. McIntosh, Richard J. 1998. Computer Sciences Corporation, Analysis “Mean Local Time Drift Analysis,” April 2. Mrozinski, Richard B. 2001. “Entry Debris Field Estimation Methods and Application to Compton Gamma Ray Observatory Disposal,” NASA CP-2001-209986, 2001 Flight M echanics Symposium Proceedings. June 19-21. NASA Office of Safety and Mission Assurance. 1995. “NASA Safety Standard: Guidelines and Assessment Procedures for Limiting Orbital Debris,” NASA Head quarters, Washington D.C., NSS 1740.14, August NASA Code Q. 1997. Policy fo r Limiting Orbital Debris Generation, NPD 8710.3 (formerly NMI 1700.8). Newman, Lauri K., Brian P. Ross, et al. 1993. “Reentry Analysis for Low Earth Orbit ing Spacecraft,” AAS 93-307, Flight Mechanics/Estimation Theory Symposium, Greenbelt, MD. Newman, Lauri K. and David C. Folta. 1995. “Evaluation of Spacecraft End-of-Life Disposal to Meet NMI Guidelines,” AAS 95-325, Astrodynamics Specialist Con ference, Halifax, NS, Canada, August 14-17. Noonan, Paul. 1996. “EOS AM-1 Ground-Track Control and Frozen Orbit Analysis,” Computer Sciences Corporation, CSC/TM-56816-02. September 9. Orbital Sciences Corporation. 2000. Pegasus User's Guide, Release 5.0. Orbital Sciences Corporation. 1999. Taurus Launch System Payload User's Guide,
Release 3.0. Reynolds, R. C., Lockheed Corporation. 1995. Rationale for Guideline Limit o f 25 Years fo r Post-Mission Removal from Orbit, NASA Code Q, Johnson Space Cen ter, and Lockheed Engineering and Science. Richon, Karen V. and Lauri K. Newman. 1995. “Flight Dynamics Support For The Clementine Deep Space Program Science Experiment (DSPSE) Mission ” AAS 95-444, Astrodynamics Specialist Conference, Halifax, NS, Canada. August 14-17.
Schatten, Kenneth. 2001. “Solar Activity Forecasting for use in Orbit Prediction.” NASA CP-2001-209986, 2001 Flight Mechanics Symposium Proceedings. June 19-21. Sietzen, Jr., Frank. 2001. “Mir: Resting in Peace.” Aerospace America. May.
766
Operations Considerations in Orbit Design
Wertz, James R., and Wiley J. Larson (ed), 1999. Space Mission Analysis and Design, (3rd ed.). Torrance, CA: Microcosm Press and Dordrecht, The Netherlands: Kluwer Academic Publishers. Wilkin, Paul. 1998. EOS AM-1 Ground-Track Control Box Study, Computer Sciences Corporation, CSC/TM-28292-02. August 20.
APPENDICES
The appendices summarize the equations and data needed for mission engineering of spacecraft orbit and attitude systems and the analysis of payload and spacecraft mission geometry.
A. Spherical Geometry B. Coordinate Transformation C. Statistical Error Analysis ...........................................
D. Summary of Keplerian Orbit and Coverage Equations E. Physical and Orbit Properties of the Sun, Earth, Moon, and Planets F. Properties of Orbits About the Moon, Mars, and the Sun G. Units and Conversion Factors
Appendices A. Spherical Geometry
769
B. Coordinate Transformations
801
C. Statistical Error Analysis
807
D. Summary of Keplerian Orbit and Coverage Equations
835
E. Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
853
F. Properties of Orbits About the Moon, Mars}and the Sun
873
G. Units and Conversion Factors
889
Fundamental Physical Constants
Inside Front Cover
Spaceflight Constants
Inside Front Cover
Earth Satellite Parameters
Inside Rear Pages
These appendices provide the basic formulas and data needed to implement the algorithms in this book for nearly any space mission. Most of the spherical geometry and orbit formulas are standard ones that can be found in many references. The fullsky geometry formulas (Sec. A.7) and spherical-to-spherical coordinate transforma tions were developed by Microcosm and have not previously been published. All numerical data is given to its full available accuracy or, in die case of infinite series, to a minimum of 16 places so that in nearly all cases values can be used without accuracy being an issue. The fundamental physical constants and conversion factors based on them are those determined by the National Bureau of Standards using a least squares fit to the best available experimental data [Mohr and Taylor, 1999; 2000]. A complete listing is given on the NIST website at physics.nist.gov/constants. Their intent is to create a set of constants that is mutually consistent to within the experi mental accuracy. Other constants or conversion factors, such as the speed of light in vacuum or the conversion between inches and meters, are exact definitions of the units involved. (See, for example, Taylor [1991].) For astronomical and astronautical constants, values adopted by the International Astronomical Union are given. Many of these are quoted from Astrophysical Quan tities [Cox, 2000], which contains a great deal of additional astrophysical data. We highly recommend this volume for those who need additional quantitative detail about the solar system or other astronomical quantities. Similarly, those looking for precise definitions of coordinate systems, time systems, or the computational aspects of astro nomical ephemerides should consult the excellent reference by Seidelmann [1992].
767
768 Finally, the tabular summary of properties of orbits about the Earth (inside rear pages), Sun, Moon, and Mars (Appendix F) are derived from formulas and constants elsewhere in this volume. The formulas used are given at the front of each of the tables. We would, of course, appreciate any errors or omissions in any of the material being brought to our attention at [email protected].
References Mohr, Peter J. and Barry N. Taylor. 2000. “CORDATA Recommended Values of the Fundamental Constants.” Reviews o f M odem Physics. Vol. 72, No. 2, ______ t 1999, “CODATA Recommended Values of the Fundamental Constants.” Journal o f Physical and Chemical Reference Data. Vol. 28, No. 6. Cox, Arthur N, ed. 2000. Allen’s Astrophysical Quantities (4th Edition). New York: Springer-Verlag. Seidelmann, Kenneth P., ed. 1992. The Explanatory Supplement to the Astronomical Almanac. Mill Valley, CA: University Science Books. Taylor, Barry N. 1991. The International System o f Units (SI). National Institute of Standards and Technology (NIST), Special Publication 811, U.S. Department of Commerce: U.S. Government Printing Office.
Appendix A Spherical Geometry A,1 A.2 A.3 A.4 A.5 A.6 A. 7
Arc Length and Rotation Angle Formulas Equations of Great and Small Circles Area Formulas General Rules for Spherical Triangles Napier’s Rules Right and Spherical Quadrantal Triangles Oblique Full-Sky Spherical Triangles
Side-Side-Side Spherical Triangles; Side-Side-Angle Spherical Triangles; Side-Angle-Side Spherical Triangles; Angle-Angle-Side Spherical Triangles; Angle-Side-Angle Spherical Triangles; Angle-AngleAngle Spherical Triangles; atan2 and acos2 Functions; Middle Side Law and Middle Angle Law; Singularities and Degenerate Triangles A.8 Differential Spherical Trigonometry A.9 Common Relations Between Trig Functions A. 10 Bibliography of Spherical Trigonometry
Chapter 6 provides the basis for using spherical geometry for a great many problems in orbit and attitude analysis, space mission geometry, and angles-only mea surement analysis. Unfortunately, prior works have been oriented primarily toward manual computation or older computer systems in which the evaluation of trig func tions was painfully slow. Analytic approaches tended to be forced into vector terms, irrespective of whether this was analytically convenient, easily tested, or computa tionally efficient. The analyst choosing to use a correct spherical geometry solution for angles-only geometry was constantly reinventing the wheel (or at least the rounded triangle) by using the law of sines or cosines. Too often, a longer and more computa tionally complex vector solution or plane geometry approximation will be used rather than a simpler, exact formula because of the lack of an easily applied set of spherical geometry formulas. The purpose of this appendix is to eliminate that problem. Of course, many problems are conveniently worked using unit vectors or are moreor-less equally easy in vector or spherical geometry formulations. Consequently, arc length and rotation formulas are expressed both ways. However, the solution to general spherical triangles is not conveniently done in terms of unit vectors and no simple formula for angular areas comparable to the spherical excess rule exists in vector terms. Previously, angular measurements which included arc lengths or rotation angles greater than 1SO deg required keeping track of the quadrant of each of the components, which rapidly becomes very inefficient in either spherical or vector terms. This
769
770
Appendix A
A'l
problem is now entirely eliminated by introducing solutions to all possible full-sky triangles in which any of the sides or angles can range from 0 to 360 deg. These triangles are discussed at length in Sec. 8.1 and the full set of solutions is given in Sec. A.7, below. In turn, these solutions make possible the analysis of the dual-axis spiral, introduced in Sec. 8.2 and applied throughout much of the rest of the book. The formulas here are intended to provide practical, usable receipes for geometrical problems. We have attempted to provide formulas and solution conditions that can be simply plugged into a hand calculator, spreadsheet, or computer routine with little or no thought—i.e., allowing you to concentrate on the problem at hand and not the spherical geometry. If more background is needed, Chap. 6 provides a general intro duction to using spherical geometry in mission analysis and orbit and attitude prob lems, Chap. 7 discusses the theory of angular measurements, and Chap. 8 introduces full-sky geometry and provides a complete taxonomy of full-sky triangles.
A.l Arc Length and Rotation Angle Formulas Let Pi 7 i = 1, 2, 3, be three points on the unit sphere with coordinates (a*, <%).The arc-length distance, (?12, between F] and P i is given by; cos $i 2 = cos 021 = sin
0<
6
< 180°
S2 + cos
If P;• are unit vectors corresponding to
(A-la)
, then
cos 0 = P] *P2 0 < 0 < 180° (A-lb) The rotation angle, 0 (P\, P2, P 3), from Pi to P2 about P 3, is cumbersome to calculate and is most easily obtained from spherical triangles (see Secs. A.4 to A.7) if any of the triangle components are already known. To calculate directly from coordi nates, obtain as intermediaries the arc-length distances B y, between pairs of points /J , Pj. Then cosd>(P1,P2;P3) - cosai2 ~ cosgl3 COS023 sinf?i3 sin023
0 <<£<360°
(A-2a)
with the quadrant determined by inspection. For automated quadrant resolution use the side-side-side triangle solution in Sec. A.7.1 If the three points are expressed as unit vectors, then
p3-(p,xp2)
tan<s> p 2)-(p3.p1)(p3-p2)
with the quadrant of <2>determined by the signs of the numerator and denominator, as done by the ATAN2 computer function. (See Sec. A.6 and A.7.)
A .2 Equations of Great and Small Circles The equation for a small circle of angular radius p and centered at P0 = ( a 0, <50) in terms of the coordinates, X = (a , 5), of the points on the small circle is, from Eq. (A-la), cos p = sin S sin <50 + cos
6
cos <50 cos (a - ccQ)
(A-3a)
A.2
Spherical Geometry
771
If the center and points on the small circle are expressed as unit vectors, then from Eq. (A-lb), X •P0 = cos p
(A-3b)
The arc length, /?, along the arc of a small circle of angular radius p and between two points on the circle separated by the rotation angle,
(A-4)
The chord length, 7 , along the great circle chord of an arc of a small circle of an gular radius p is given by cos 7 = 1 - (1 - cos 0 )s in 2 p
0 < 7 < 180 deg
(A-5)
The equation for a great circle with pole at (Cfo> ^0) ^ fr°m Eq. (A-3a) with p = 90 deg, tan 5= - cot Sq cos
(A-6a)
( c e - ccq)
In terms of unit vectors, the equation for a great circle is, from Eq. (A-3b) X*P0 -O
(A-6b) A
a 0 and S0 are related to the coordinates of P0 by: Sq = a s i n z 0
a 0 = atan (y0/jc0)
(A-6c)
The inclination, j’o- and azimuth of the ascending node, 0 O, (point crossing the equator from south to north then moving along the great circle toward increasing azimuth) of the great circle are i0 = 9 o ° - a 0 4>0=90° + a 0
(A-6d)
A
In terms of the coordinates of the pole, Pq , the expressions for
and (pQ are:
i*0 = acos zq <j>o= -atan (x0/ y0)
(A-6e)
The equation for the great circle in terms of inclination and ascending node is tan S = tan t0 sin ( a - 0 0)
(A-6f)
The equation of a great circle through two arbitrary points is given by Eq. (A-8a). Along a great circle, the arc length, the chord length, and the rotation angle, 0 , are all equal, as shown by Eqs. (A-4) and (A-5) with p = 90 deg.
772
Appendix A
The direction of the cross product between two unit vectors associated with points Pi and P2 on the unit sphere is the pole of the great circle passing through the two points. The intermediary, fa, is cot fa -
tan 8n tan<5i
sin ( d j- c c i)
-co t(cr2 - a j)
(A-7a)
As shown in Fig. A -l, fa is the azimuth of point P\ relative to the ascending node of thejpreat circle through P\ and P2- The coordinates, (a c t Sc ), of the cross product Pj x P2 are given by a c = a 1-9 0 ° - fa
tan<5L =
sinffj tan5j sin (a 2
(A-7b) (S ^O ) (A-7c) (5 i= 0 )
In terms of unit vectors, the direction of the cross product, C , is: C - Pi XP2
P,XP2I
(A-7d)
Pc = Direction of Cross Product, P 1 x P2
= Pole of Great Circle
Fig. A-1.
Finding the Pole of a Great Circle. pc is the pole of the great circle passing through Pt P2 and is also in the direction of the cross product P, x P2 - f t is an intermediate variable used for computation.
Spherical Geometry
A.3
773
Combining Eqs. (A-7b) and (A^6f) gives the equation for a great circle through points P 1 and P2:
tanA sin tan <5 =
_ ai +^
* 0)
sin ft
(A-8a)
tan 2---- gin£a _ ^ + ^
^
_ o)
sin(a2 ~a\) /V
A
Finally, in terms of unit vectors, the equation of a great circle through Pi and P2 is: X ’(P ! x P 2) = 0
(A-8b)
A.3 Area Formulas All areas are measured on the curved surface of the unit sphere. See Sec. 8.1 and Table 8-3 for a discussion of angular area. For a sphere of radius R, multiply each area by R 2. In the area formulas, all arc lengths are in radians and all angular areas are in steradians fsr). where 1 sr = solid angle enclosing an area equal to the square of the radius = (180/71)2 deg2 * 3,282.806 350 011 743 794 deg2 The surface area of the sphere is Q, = 4n
(A-9)
The area of a lune bounded by two great circles which intersect at an angle of 0 radians is
= 20
(A-10)
The area of a spherical triangle whose three rotation angles are 0j, 02->an
+ ^2
^3 —^
(A-l 1)
The area of a spherical polygon of n sides, where S is the sum of its rotation angles in radians, is Qp = S - ( n - 2)71
(A-12)
The area of a small circle of angular radius p is Qc = 2n (1 -c o sp )
(A-13)
The area of a segment of rotation angle, 0 ,in a small circle of angular radius p (i.e., a “pie piece”) is ®sc = < H l-c o s p )
(A-14)
The area of a ring or annulus of inner radius p j and outer radius p 0 is Qr = 2n (cos p : - cos Pq)
(A-15)
774
Appendix A
A.4
The area of a segment of rotation angle, 0, in an annulus of inner radius p\ and outer radius p 0 is Qsr =
(A-16)
The overlap area between two small circles of angular radii p and e, separated by a center-to-center distance, a, is
£20 = 2 7 1 -2 cos p acos
- 2 cos £ acos
- 2 acos
cos £ - cos p cos a sin p sin a
cos p - cos s cos a
(A-17)
sm e sin a
cos a - cos e cos p sin £ sin p
(|p - e\ < a < p + e)
If e goes to 90 deg, then the above simpliefies to the formula for the overlap area between a great circle and a small circle of angular radius p, separated by a centerto-center distance, a: Q o = 27t - cosp acos (cot a cot p) - 2 acos (cos a I cot p) 90° - p | < a < 90° + p
(A-18)
Recall that area is measured on the curved surface.
A.4 General Rules for Spherical Triangles The laws of sines and cosines apply to all spherical triangles. For normal spherical triangles (as discussed in Sec. 8.1), all of the sides and angles fall in the range of 0 to 180 deg and inverse trig functions should be taken over this range. Quadrant ambigu ities within this range (i.e., the inverse sine function, sin-1, or asm) can be resolved by the rules at the end of the section. Solutions for the quadrants in full-sky triangles (i.e., all sides and angles ranging from 0 to 360 deg) follow automatically from the full-sky spherical triangle formulas in Sec. A.7.
Fig. A-2.
Standard Notation for a Spherical Triangle.
A.4
Spherical Geometry
775
The following rules hold for any spherical triangle: sin a sin A
The Law of Sines:
sinfr sinB
sine sinC
(A-19)
The Law of Cosines for Sides: cos a = cos b cos c + sin b sin c cos A cos b —cos c cos a + sin c sin a cos B cos c = cos a cos b + sin a sin b cos C
(A-20a) (A-20b) (A-20c)
The Law of Cosines for Angles: cos A = -cos B cos C + sin B sin C cos a cos B = -cos C cos A + sin C sin A cos b cos C = -cos A cos B + sin A sin B cos c
(A-21a) (A-21b) (A-21c)
Napier’s Analogies: tan
A + B) = (cot-i-C)
ta n ^ ( A - S ) = (c o t^ c )
tan ^ (a + b) - (tan j c)
tany(<2-&) = (tanyc)
cos^-(
cos^-0 + &)
(A-22a)
sin jr (a -b ) s in ^ o T fc ) c o s j( A - f l) cos^CA + 5) sin-j (A ~ B) sin-^(A + B)
(A-22b)
(A-22c)
(A-22d)
Gauss’s Formula: ri
i
s in [^ -B )] =
sin[£(a-&)| j
cos (y c )
(A-23)
Equations (A-22) and (A-23) can be extended to other components by cyclic conversion of all sides and angles. Cyclically changing a formula means that “a” is replaced by “£>” by “c”, and “c” by “a.” The pattern follows a cycle, as in Fig. A-3. These quadrant rules can be used to resolve quadrant ambiguities in the trigon ometric functions: • If one side differs from 90 deg by more than another side, it is in the same quadrant as its opposite angle. • If one angle differs from 90 deg by more than another angle, it is in the same quadrant as its opposite side. - Half the sum of any two sides is in the same quadrant as half the sum of the opposite angles.
776
Appendix A
A.5
For example, c = atan (tan b cos A) ± atan (tan a cos B) becomes b = atan (tan a cos C) ± atan (tan c cos A) and then a = atan (tan c cos B) ± atan (tan b cos C)
Fig. A-3.
Cyclic Conversion.
A.5 Napier’s Rules The law of cosines and law of sines reduce to simpler relations, known as Napier’s Rules*, for right spherical triangles in which one of the angles is a right angle and quadrantal spherical triangles in which one of the sides is 90 deg long. (Unlike plane triangles, spherical triangles can have 1, 2, or 3 right angles and 1, 2, or 3 sides that are 90 deg long.) Any two of the remaining components, including the two remaining angles, serve to completely define the triangle. Navigation based on spherical geometry was an important element of math in the early 20th century. At that time Napier’s rules were often included in high school texts. They are normally formulated in a way intended for easy memorization by putting each of the 5 unknown components in a circle and expressing the rules in terms related to their position in the circle (i.e., “the sine of any angle is equal to the product of the cosines of the two opposite angles,” and so on). The details can be found in many old high school texts or the references in the bibliography in Sec. A. 10. To make life easier for today’s students and engineers, we have written out explic itly all possible permutations of Napier’s Rules, including the quadrants, These are given in Sec. A.6. For right and quadrantal spherical triangles, the quadrant of the so lution can also be determined by three simple rules: • In a right spherical triangle or a quadrantal spherical triangle, a rotation angle and the side opposite are in the same quadrant. • When the hypotenuse of a right spherical triangle or the angle opposite the hy potenuse of a quadrantal spherical triangle is less than 90 deg, the legs are in the same quadrant; when the hypotenuse (or opposite angle) is greater than 90 deg, the legs are in different quadrants. • When the two given parts are a leg and its opposite angle, there are always two solutions.
* Introduced by 17th century Scottish mathematician and Baron of Merchiston, John Napier. Napier also invented logarithms and introduced the decimal point into mathematical notation.
A.6
Spherical Geometry
777
A.6 Right and Spherical Quadrantal Triangles
Right Spherical Triangles The line below each formula indicates the quadrant of the answer. Q(A) - Q(a) means that the quadrant of angle A is the same as that of side a. “2 possible solutions” means that either quadrant provides a cor rect solution to the defined triangle.
Given
Find
a, ft
cos h = cos a cos b Q(h)= {Q(a)Q(ft)}*
a, h
cos b = cos h / cos a sin A = sin a / sin h Q(b) = {Q(a)/Q(/i)}** Q(A) = Q(a)
b,h
cos a = cos h / cos b cos A = tan b / tan h sin B = sin 6 / sin h Q (A) = {Q( b)/Qtk)}** Q(B) = Q(b) Q(«) = {Q(*)/Q(A)}**
a, A
sin 6 = tan a /ta n A 2 possible solutions
sin h = sin a I sin A 2 possible solutions
sin B = cos A / cos a 2 possible solutions
a, B
tan b = sin a tan B Q(b) = Q(B)
tan h = tan a / cos B Q(A) = (Q(a)Q(B)}*
cos A = cos a sin B Q(A) = Q(a)
b, A
tan « Q(*)
tan A = tan b / cos A Q(A) - {Q(6) Q(A)}*
cos # = cos ft sin A Q(S) = Q(b) sin A = cos B / cos ft 2 possible solutions
= =
sin M an A Q(A)
tan A = tan a / sin b QCA) = Q(a)
tan S = tan ft / sin a Q(fi) = Q(ft) cos 1? = tan a / tan A Q(B) = {Q(a)/Q(/i)}**
b, B
sin a = tan b / tan B 2 possible solutions
sin A = sin b f sin B 2 possible solutions
h, A
sin a = sin h sin A Q(a) = Q(A)
tan b = tan h cos A tan 2? = 1 / cos A tan A Q(b) = {Q(A)/Q(h))** Q(i?)={Q(A)/Q(/0}**
h,B
sin b —sin h sin B
tan a
=
Q(b)
Q(a)
=
A, B
=
Q(B)
cos a = cos A / sin B Q(a) = Q(A)
tan h cos B
tan A = 1 /cos h tan £ {Q(B)/Q(h)}** Q(A)={ Q(B) / Q(/i) }**
cos b = cos 5 / sin A Q(ft) = Q(*)
cos h - 1 / tan A tan 5 Q(A) = {Q(A)Q(B)}*
* {Q(*) Q(v)} = 1st quadrant if Q(x) = Q(y), 2nd quadrant if Q(x) * Q(y)
** {Q(x) / Q(h)} = quadrant of x if h < 9 0 deg, quadrant opposite* if h > 9 0 deg
778
Appendix A
A.6
Quadrantal Spherical Triangles The line below each formula indicates the quadrant of the answer. Q(A) = Q(a) means that the quadrant of angle A is the same as that of side a. “2 possible solutions” means that either quadrant provides a correct solution to the defined triangle.
Given
Find
A, B
cos H = - cos A cos B Q(H) = {Q(A)Q(B)}*
A, H B, H
qq
tan a = tan A / sin B Q(«) = QW
cos B = —cos H / cos A sin a = sin A / sin H Q(B) = {Q(A)\fi(H)}** Q(«) = Q(A) cos A = - cos H / cos B
cos a = - tan B / tan H
tan ft = tan jff/sinA w ) =
cos b = - tan A /tan H Q(b)={Q(A)\Q(H)}**
sin b = sin B / sin
Q(A) = {Q (B)\Q (H )}** Q(a) = {Q(5) \ 2 (^0 }** Q(*) = Q(B)
A, a A, b
sin B = tan A /ta n a
sin tf = sin A /sin a
sin b = cos a /cos A
2 possible solutions
2 possible solutions
2 possible solutions
tan B = sin A tan b
tan H = - tan A / cos b < m = {Q(A)Q(b)}*
cos a = cos A sin b Q(fl) = QCA)
tan A = sin B tan a Q(A) = Q(a)
tan H = - tan B /cos a c a m = {Q(B)Q(a)}*
cos &= cos J5 sin a Q(W = Q(B)
sin A = tan B / tan b
sin H = sin B I sin b
sin a = cos b / cos B
2 possible solutions
2 possible solutions
2 possible solutions
sin A = sin H sin a Q(A) = Q(a)
tan B = - tan H cos a
tan b - - V cos H tan a
Q (B) = Q(6)
B, a B, b Hf a H, b a, b
Q(B) = {Q(a)\Q(H)}** Q(b)={Q(a)\Q(H)}** tan A = - tan H cos b
sin B = sin H sin b Q(B) = Q(b)
Q(A) = {Q (b)\Q (H )}** Q(a)={Q(b)\Q(H)}**
tana = -1/ cos H tan b
cos A = cos a / sin b Q(A) = Q(a)
cos B - co$ b /sina Q(B) = Q (b)
cos H - —1 / tan a tan b Q(fl) = {Q(a)Q(b)}*
* {Q(jt) QCy)} = 1st quadrant if Q(x) = Q(y), 2nd quadrant if Q(x) t- Q(y) **{Q ( x)\ Q(H) ) = quadrant of* if H > 90 deg, quadrant opposite * if / / < 90 deg
A.7
Spherical Geometry
779
A.7 Oblique Full-Sky Spherical Triangles* An oblique spherical triangle has arbitrary sides and angles. Sides and angles are typically defined over the range of 0 to 180 deg. However, in this section we provide general rules, not previously published, for full-sky spherical triangles in which any sides or angles can be anywhere in the full range of 0 to 360 deg. (See Sec. 8.1 for a complete list of properties of all of the full-sky triangles.) This full range of angles makes computer implementation particularly simple. We define the relevant triangle or triangles in a problem and set up computer routines based on the formulas below. The range of the input variables is unlimited^ and the output will automatically provide the correct angle, including the quadrant. The nomenclature for the sides and angles for full-sky triangles is the same as that shown in Fig. A-3 in Sec. A.5. Given any three components in an oblique spherical triangle, the remaining three components can be determined. AU of the components (both known and unknown) can lie in the full range of 0 to 360 deg. There are 6 possible combinations of known and unknown components, as listed in Table A -l. See Sec. 8.1 for a complete discussion of the taxonomy of full-sky triangles and specifi cally Table 8-4 in Sec. 8.1 for a list of solutions depending on how the triangles are defined. TABLE A-1. The 6 Categories of Oblique Spherical Triangles. Given
Find
No. of Solutions
Where Solved
Side-Side-Side
a, b, c
A, B, C
0 or 2
Sec. A.7.1
Side-Side-Angle
a, b, A
B, C,c
0 or 2
Sec. A.7.2
Side-Angle-Side
a, C, b
A, B ,c
2
Sec. A.7.3
Angle-Angle-Side
A, B, a
b, c, C
0 or 2
Sec. A.7.4
Angle-Side-Angle
A, c, B
a, C, b
2
Sec. A.7.5
Angle-Angle-Angle
A, B, C
a, b, g
0 or 2
Sec. A.7.6
Type
Full-sky solutions are given below for each of the categories. Conditions under which solutions exist and singularities which occur are also listed for each case. Note: the solution conditions represent those conditions for which a triangle exists and are not related to the approach used to solve for the missing components. For example, 3 known sides of 10 deg, 20 deg, and 60 deg can not form a spherical triangle because one side is longer than the sum of the other two. Similarly the singularity conditions represent singularities in the triangles themselves (such as a side going to 0), not in the solution approach. Whenever a solution exists in a full-sky triangle, a second solution also exists. Formulas for both are given in each subsection along with an illustration of how the * The full-sky formulas presented in this section are due to Rob Bell, Leo Early, Hans Meis singer, Herb Reynolds, and James Wertz of Microcosm. Trig functions are defined over an unlimited domain. However, for computers or calculators (which use series approximations), better precision will be obtained if the variables are first reduced to the range 0 to 360 deg with modulo arithmetic. In some cases, the modulo function may be a part of the definition of the trig series.
780
Appendix A
A.7
second solution arises. When the solved for component can range from 0 to 360 deg, there must be some method, such as the FORTRAN atan2 function, to allow inverse trig functions to uniquely cover this full range. The approach used in this appendix is the newly created acos2 function, defined in Table 8-5 in Sec. 8.1.
A.7.1 Side-Side-Side Spherical Triangles For these triangles, the three sides are given and we wish to determine the three rotation angles. That is, Given a, b, c
Find A, B, C
The requirement for a solution is that if one end of two of the sides, say a and b, are attached to the ends of side c, then the remaining ends of a and b must be able to meet. For each solution, the second solution is the inverse spherical triangle in which each of the angles is replaced by 360 deg minus the angle. This simply reverses the roles of “inside” and “outside” for the triangle as shown in Fig. A-4.
(A)
(B)
Fig. A-4. The Two Solutions for the Side-Side-Side Triangle. For ease of identification in Figs. A-4 to A-9, The “given” variables are bold italics and the “given” lines and angles are bold on the figure.
Solution Conditions: The three given sides form a triangle if, and only if: 1180° —<21—1180° —^11 <
|l8 0 ° - c j
<
||l 8 0 ° - a | + |l8 0 ° - f r ||
(A-24)
If this condition holds (or does not hold) for any permutation of a, b, and c, then it will hold (or not hold) for the two remaining permutations.
A.7
Spherical Geometry
781
As listed in Table A-2, a singular case occurs whenever one of the specified sides is 0,180 deg, or 360 deg. For the side-side-side triangle, all of the singularities are, of course, cyclic permutations of each other. Nonetheless, they are all listed in the table for completeness. The various degenerate spherical triangles that result from singular ities are shown in Fig. A -10 in Sec. A.7.9. TABLE A-2. Singular Cases for Side-Side-Side Spherical Triangles. As with all of the solu tions in Sec. A.7, the singularities are due to the nature of the triangle, not the fact that it is being solved via spherical geometry. See Sec. A.7.9 for singularity types and how to handle them in practice. If
Then
a —>0°
£>-» c or
b —> c a -? c
b~> 0° or C-»0° or a -? -180°
b^> 180°
a -» c
a-> b a-> b b + c ^ 180° 3 + c —> 180°
A —»0° A -> 360° 0°
B 360® C ^0° C —»360° A 180°
C—>• 180°
a + b -> 180°
a -> 360°
£>-» C
6-> 1 80° C 180° /A —> 0°
b-> c
A -» 360°
or
a -» c
b ->300°
0°
or
a -> c
S -> 360°
or
a b a —>b
C —?■360°
c —> 360°
0°
Type e + c - > i 80 °
Type 1 Arc
B + C —» 540°
Type 1 Arc Type 1 Arc
A + C —t ‘\ 80° A + C-J-540* A + S —»180° A + B —>540° C
Type 1 Arc Type 1 Arc Type 1 Arc
A^C A^B
Lune Lune Lune
B + C - * 540° B + C —>180°
Double Arc Double Arc
A + C ^ 540° A + C ^ 180° A + B —>540° A + B-> 180°
Double Arc Double Arc Double Arc Double Arc
First Solution: If a non-singular solution exists, then there are two solutions. One is given by: Aj - acos2
By - acos2
Q = acos2
cos a - cos b cos c ,H (a) sin b sin c cos b - cos a cos c
(A-25 a)
,H (2>)
(A-25b)
cos c - cos a cos b ,H (c) sin a sin b
(A-25c)
sin a sin c
The acos2 function is defined in Sec. A .7.7 and discussed further in Sec. 8.1.
Second Solution: The second solution is given by: ^2= 360°^ B2 - 360° - Bi C2 = 360° - Cx
(A-26a) (A-26b) (A-26c)
782
Appendix A
A.7
A.7.2 Side-Side-Angle Spherical Triangles For these triangles, we know two sides and one of the two angles not included
between the sides. That is, Given a, b, A
Find B, C, c
For simplicity, assume unknown side c\ lies along the equator of a coordinate sys tem as shown schematically in Fig. A-5. (See also Fig. 8-6 in Sec. 8.1.) Known angle A and side b bring us to a vertex at Cj. We then construct a circle centered on Q of radius a, the length of the remaining known side. The solution condition [Eq. (A-27)] corresponds to the requirement that the circle touch the “equator” that contains side C]. If a is either too short or too long, then there is no solution. The two solutions for this case correspond to the two locations at which the circle crosses the equator. Note that by the way we have defined the problem, side C\ starts out to the right. Therefore if one or both of the crossings is to the left of A, then side cj will be a long arc greater than 180 deg.
Because one or both of the solutions may be greater than 180 deg, the solution set for the side-side-angle triangle is substantially simpler in full-sky geometry than in classical spherical geometry in which all of the sides and angles are constrained to be less than 180 deg. In full-sky geometry, the long arcs are acceptable and the two solution can be simply written down. In classical spherical trig, a long arc is not an “allowed” solution. Therefore, in traditional solutions we must compute an intermedi ate variable and potentially throw away one or both of the solutions.
Solution Conditions: The three given components form a triangle if, and only if:
I sin b sin A I < i sin a I
(A-27)
As listed in Table A-3, a singular case occurs whenever one of the specified sides or angles is 0, 180 deg, or 360 deg. The various degenerate spherical triangles that result from singularities are shown in Fig. A -10 in Sec. A.7.9.
A.7
Spherical Geometry
783
TABLE A-3. Singular Cases for Side-Side-Angle Spherical Triangles, a, b, and A are given. B, C, and c are unknown. See Sec. A.7.9 for singularity types and how to handle them in practice. If
Then
a -> 0° or
A -> 0° A -> 360° c —> a
or
c -> a
b^0°
A -5-0°
c~> b c —> b B -» 0° 6 -» 360°
Type
B+ C —> 180° B + C - + 5 40° A + C -> 180° A + C -» 540°
Type Type Type Type
e+
b
1 Arc 1 Arc 1 Arc 1 Arc
180°
Type 1 Arc
360° b + c ^ > 180°
c -> b
S + C -> 5 4 0 °
Double Arc
A
a+ c 180° c -» a - £>
180° C -> 0°
B ^C C —> A
Lune Lune
A -> 0°
Type 2 Arc
or
c -> a - £>
C -» 360°
or or
fo+ c-> 180° a+£> + C-»360° c -» b
a -s-180° 180° 4 -»0°
>1 -> 360° B -> C C -» 180° B + C -> 540°
Type 2 Arc Lune Great Circle Double Arc
B + C -» 1 80° A + C— >540°
Double Arc Double Arc
A +C— >180°
Double Arc Type 1 Arc
0° or
a -» 180° 180°
A -=► 180°
a -* 360° or
a
O— ^ £>
360°
c —>a
b -> 360° or
c -> a
or
b c -» b
A -> 360°
180®
6 —»0° B -> 360° a -» 0° a —>360°
B + C -> 540° B + C -M 80°
Double Arc
First Solution: If a non-singular solution exists, then there are two solutions. One is given by: . , sin b sin A mod 360® J?! = asm sin a j
(A-28a)
c, = M S L (a ,* ,A ,5 1)
(A-28b)
Cj = MAL (A, B h a, b)
(A-28c)
where the asin function is defined over the range -90 deg to +90 deg and MSL and MAL are the Middle Side Law and Middle Angle Law defined in Sec. A.7.8.
Second Solution: The second solution is given by: B2 = (180°-B |) mod 360°
(A-29a)
c2 = MSL (a, b, A, B2)
(A-29b)
C2 = MAL (A, B2i a, b)
(A-29c)
where, as above, MSL and MAL are defined in Sec. A.7.8.
784
Appendix A
A.7
A.7.3 Side-Angle-Side Spherical Triangles In this case, we know two sides and the included angle. Thus, Given a, C, b
Find A, B, c
In this case there are two solutions for all possible combinations of known com ponents. As shown in Fig. A-6, one solution corresponds to connecting the known endpoints with a short arc (less than 180 deg) and the other solution corresponds to connecting them with a long arc of greater than 180 deg.
Solution Conditions: There are two solutions for all possible combinations of known components cover ing the full range of 0 to 360 deg. As listed in Table A-4, a singular case occurs whenever one of the specified sides or angles is 0, 180 deg, or 360 deg. The various degenerate spherical triangles that result from singularities are shown in Fig. A-10 in Sec. A.7.9.
First Solution: If a non-singular solution exists, then there are two solutions. One is given by: Ci= acos2 [cos a cos b + sin a sin b cos C, H(C)]
(A-30a)
Ai - acos2
cos a - cos b cos c ,H (a) sin b sin c
(A-30b)
Bi = acos2
cos b - cos a cos c ,H (b) sin a sin c
(A-30c)
where the acos2 function is defined in Sec. A.7.7 and further discussed in Sec. 8.1.
A.7
Spherical Geometry
785
TABLE A-4. Singular Cases for Side-Angle-Side Spherical Triangles, a, C, and b are given. A, c, and B are unknown. See Sec. A.7.9 for singularity types and how to handle them in practice. If
Then
a -> 0° or
b-+ 0° or 0°
A->Q° A -»360°
c —> b c -» b 3 —> 0° 5 -»360°
c —> a c -> a c —> 0°
a —> b
B + C -» 180° B + C 540° A + C -> 180s A + C-> 540° A+ 180°
Type Type 1 Arc Type 1 Arc Type 1 Arc Type 1 Arc Type 1 Arc
C—} 360° b + c -> 180° a + c -» 180° c —» a + b
a-> b A 180°
A + B->540° e -> c
Double Arc Lune
B -> 180°
A -* C B -+ 0 °
Lune Type 2 Arc
or
ch> a + b
A -> 360°
e - * 3S0°
Type 2 Arc
or or
c 180° >*->180°
A -) B B-> 180°
Lune Great Circle
A —»0° A -» 360°
B + C -* 540°
B+ C-> 180°
Double Arc Double Arc
B -» 0°
A + C->540°
Double Arc
or
a + b -> 180° a + b + c —> 360° c-> b c-> b c —> a c-> a
B —*■360°
A + C-> 180°
Double Arc
>4 + B -> 540°
Type 1 Arc
or
a-> b a^b
+ B -» 180°
Double Arc
or a
180° b -> 180° C -> 180°
a -> 360° or
b ->•360° C -> 360°
>4—> 0°
c -> 360°
Second Solution: The second solution is given by: c2 = 360° - c
(A-3 la)
A2 = (A + 180°) mod 360°
(A-3 lb)
B2 = (5 + 180°) mod 360°
(A-31c)
A.7.4 Angle-Angle-Side Spherical Triangles For these triangles, the known components are two angles and one of the sides not between them. That is, Given A, B, a
Find b, c, C
For convenience of illustration, assume unknown side q lies along the equator of a coordinate system as shown schematically in Fig. A-7. Known angle B and side a bring us to a vertex at C\. To complete the triangle, we find vertex A along the equator, such that the remaining unknown side b x intersects Cj. There will be no solution if the vertex Cx is far enough above the equator that great circles making an angle A with the equator do not come that close to the pole. The two solutions correspond to intersect ing C\ on the ascending or descending portions of the arc as illustrated in Fig. A-7. As with the case of the side-side-angle triangle described in Sec. A.7.2, either or both solutions for some of the unknown components may be greater than 180 deg. This
786
Appendix A
A.7
(A)
P2 (B) Fig. A-7.
The Two Solutions for the Angle-Angle-Side Triangle.
is not a problem for the full-sky triangles. However, in classical spherical trig, in which all of the angles and sides are required to less than 180 deg, some of the real solutions will need to computed and then discarded, because they are not “allowed.” This means that the full-sky solutions will be simpler to both compute and interpret than the more restrictive classical solutions.
Solution Conditions: The three given components form a triangle if, and only if: sin B sin a \ <
sin A
(A-32)
As listed in Table A-5, a singular case occurs whenever one of the specified sides or angles is 0,180 deg, or 360 deg. The various degenerate spherical triangles that re sult from singularities are shown in Fig. A-10 in Sec. A.7.9. First Solution: If a non-singular solution exists, then there are two solutions. One is given by: /
b\ - asin
\ sin B sin a mod 360° sin A J \
(A-33a)
Cj = MSL (a, b\, A, B)
(A-33b)
Cj = MAL (A, B, a, bx)
(A-33c)
where the asin function is defined over the range -90 deg to +90 deg and MSL and MAL are the Middle Side Law and Middle Angle Law defined in Sec. A.7.8.
Spherical Geometry
A.7 TABLE A-5.
787
Singular Oases for Angle-Angle-Side Spherical Triangles. A, 8, and a are given, b, c, and C are unknown. See Sec. A.7.9 for singularity types and how to handle them in practice.
If a
Then £?—> c b c
B + C -> 180° B + C 540°
c
S + C-> 180°
c
S + C -» 5 4 0 °
c
4+ 180° 4 + C -> 5 4 0 °
or
A 0° A -> 360° a -> 0° a —^ 360°
b
or
b ^ 0° b -»360° b+ 180° b + c-> a b+ c -j- a b + c 180° a + b + c —^ 360° £>-» a + c >a + c a+ c^> 180° a + b + c —y 360°
a
0°
A^0C 6 —> 0° or a -> 180° A - * 180° or or or e -> 180° or or or a —^ 360°
a -» c 180° 0-»O °
c -» e c -> o °
e -»> 360°
C -> 360°
a -4 180°
C -*B C -4 180°
Type Type 1 Arc Type 1 Arc Type 1 Arc Double Arc Type 1 Arc Double Arc Lune Type 2 Arc Type 2 Arc Lune
180° ^ -> 0 ° ^ 360°
C -> 0 ° C -> 360°
Great Circle Type 2 Arc Type 2 Arc
£?-> 180° /4 180°
C -» A C -4 180°
Lune Great Circle
e + C -> 540°
Double Arc
or
c
A -» 360°
B + C~> 180°
Double Arc
c
c
or
a —► c a^ c
a^0° a -» 360° b-> o° b _> 360°
8 + C -»■ 540°
or
Type 1 Arc Double Arc Type 1 Arc Double Arc
£ > -> C
A -> 360° B -»• 360°
B + C -> 180* 4 + C -» 540° 4 + c^> 180°
Second Solution: The second solution is given by: b2 = (180° - bx) mod 360°
(A-34a)
c2 = MSL (a, b2, A, B)
(A-34b)
C2 = MAL (A, B, a, b2)
(A-34c)
where, as above, MSL and MAL are defined in Sec. A.7.8
A.7.5 Angle-Side-Angle Spherical Triangles For these triangles, the known components are two angles and the included side. That is, Given A, B, c
Find a, b, C
As in the ease of the side-angle-side triangle from Sec. A.7.3, there are two solutions for all possible combinations of known components. As shown in Fig. A-8, the known components define the two planes of the unknown sides a and b. These planes will intersect in two locations, 180 deg apart. Either intersection can be the unknown vertex, C(.
788
Appendix A
A.7
c (A) Fig. A-8.
(B)
The Two Solutions for the Angle-Side-Angle Triangle.
Solution Conditions: There are two solutions for all possible combinations of known components cover ing the full range of 0 to 360 deg. As listed in Table A-6, a singular case occurs whenever one of the specified sides or angles is 0, 180 deg, or 360 deg. The various degenerate spherical triangles that result from singularities are shown in Fig. A -10 in Sec. A .I.9. First Solution: If a non-singular solution exists, then there are two solutions. One is given by: C; = acos2 [- cos A cos B + sin A sin B cos c, H(c)j
(A-35a)
aj = acos2
cos A + cos B cos C\ ,H (A) sin B sin C\
(A-35b)
b\ - acos2
cos B + cos A cos Q ,H (B) sin A sin Q
(A-35c)
where the acos2 function is defined in Sec. A.7.7 and discussed further in Sec. 8.1. Second Solution: The second solution is given by: C2 = 360° - Ci
(A-36a)
a2 = {ax + 180°) mod 360°
(A-36b)
b2 = (&i + 180°) mod 360°
(A-36c)
A.7
Spherical Geometry
789
TABLE A-6. Singular Cases for Angle-Side-Angle Spherical Triangles.
A, c, and B are
given, a, C, and b are unknown. See Sec. A.7.9 for singularity types and how to handle them in practice. If
Then
C—> 0°
or or 0° or /4 —*• 180°
a ^b
c a —>c
B + C -> 180° B+ C —>540° A + C-> 180°
c C —> 180°
A+ C->540° a ^ b
c
180°
a - b-> c
0°
or or
a-b-> c
B -> 360°
b + c 180°
a —> 180°
or
a + b + c -» 360°
B -» 180°
B -> 180°
c —^ 360° or
A - * 360°
Double Arc Lune Type 2 Arc Type 2 Arc Lune Great Circle Type 2 Arc Type 2 Arc Lune Great Circle Double Arc
A -> 360°
a + c -> 180° a + b + c —>360°
A-> 180°
C -> 0° C -> 360° C —> A C -> 180°
a^b
C -> 0°
/A + S
a^> b
C -> 360° a ^ 0° a —$360° 0° b —>360° '
A + B^> 180*
Double Arc
B + C->540° B + C-> 180° A + C —>540° A + C ^ 180°
Type 1 Arc Double Arc Type 1 Arc Double Arc
0°
c or
£>-» c
or
a-> c a-> c
360°
0°
C 360° C —> B C 180°
Type Type 1 Arc Type 1 Arc Type 1 Arc Double Arc Type 1 Arc
b-a->c
b-a-> c
or or or
A + B^> 180p A + B^> 540°
a-* b
b —> 360° a+b
C-> 180°
B
C ^0° C —»360° a -> 0° a -> 360° 0°
f>->180°
540°
A.7.6 Angle-Angle-Angle Spherical Triangles In this case, the 3 rotation angles are given and we wish to determine the 3 sides. That is, Given A, B, C Find a, b, c Analogous to the side-side-side triangles, there is an inverse solution in which each of the sides is replaced by 360 deg minus the side. (See Fig. A-9.) This transforms a reg ular spherical triangle into a star and a notch into a fish. (See the discussion in Sec. 8.1) This case has no analogy in plane geometry. If the three angles are known, then only the ratios of the sides can be determined. However, in spherical geometry the sum of the rotation angles minus 180 deg is proportional to the area of the triangle. (See Sec. A.4.) Therefore, the three rotation angles uniquely determine the sides to within the twofold ambiguity of all full-sky triangles. Solution Conditions: The three given rotation angles form a triangle if, and only if: |1 8 0 ° -W -|B ||
*
\C\ <
180“ -||A |- |B ||
(A-37)
If this condition holds (or does not hold) for any permutation of a, b, and c, then it will hold (or not hold) for the two remaining permutations.
790
Appendix A
A.7
(A ) Fig. A-9.
(B )
Th e Tw o Solutions for the Angle-Angle-Angle Triangle.
As listed in Table A-7, a singular case occurs whenever one o f the specified angles is 0, 180 deg, or 360 deg. The various degenerate spherical triangles that result from singularities are shown in Fig. A -10 in Sec. A.7.9. T A B L E A-7.
Singular Cases for Angle-Angle-Angle Spherical Triangles. A, B, and C are given, a b, and c are unknown. S e e S e c. A .7 .9 for singularity types and h ow to handle them in practice.
If
Then
A^> 0 °
Typ e
b- > c b —=►c
S + C - > 180° 540°
Double Arc
6 —» 0°
a -)C
^ + C - » 180°
Type 1 Arc
b —» 360°
a -» c
>4 + C
c->Q °
a -* b
+
a -» b B -> 0 °
A + B —> 540°
Double Arc
C -> 0 °
Type 2 Arc
or
c —) 360° b+ c ^ a b + c -» a
B ->360°
C —» 360°
Type 2 Arc
or
fo + c -> 180°
a
180°
B
or
B -> 180°
C
4 -> 0 °
C ^ 0 °
Type 2 Arc
/4 - > 360°
C 4
Type 2 Arc
or
3 + b + c —> 360° a + c- > b a + o —^ b a + c^> 180° a + b + c — 360° a + b —> c a+ c
or
a + b -»1 8 0 °
or
a+ b
or
b —>c b^> c
a -»0 °
or
B^> 0 ° or C -> 0 ° or
A^> 180°
B - » 180° or or or C - » 180°
a
->
360°
180° —> 180° /4 —» 0° 4
->
360°
6
+
C —>
Type 1 Arc
540°
Double Arc
180°
Type 1 Arc
Lune 180° —» ->
360° C
Great Circle
Lune Great Circle
C -> 180° S^0°
Type 2 Arc
B - > 360°
Type 2 Arc
C^180°
4 -> B
Lune
4
B - » 180°
Great Circle
a —»0°
B + C - > 540°
Type 1 Arc
a —►360°
B + C -* 180°
Double Arc
a —> c
b -> 0 °
4 + C
Type 1 Arc
or
a —> c
b -> 360°
/4 + C ^ 180°
Double Arc
C->0°
4 + B —» 540°
Type 1 Arc
or
a- > b a —>b
c —» 360°
4 + B - » 180°
Double Arc
A -> 360° B - » 360° C - » 360°
+ c —> 360°
- »
180°
540°
A.7
Spherical Geometry
791
First Solution:
If a non-singular solution exists, then there are two solutions. One is given by: a\ - acos2
b{ = acos2
C\
= acos2
cos A + cos B cos C sin B sin C cos B +cos A cos C sin A sin C cos C + cos A cos B sin A sin B
.H(A)
(A-38a)
,H (B)
(A-38b)
,H (0
(A-38c)
where the acos2 function is defined in Sec. A.7.7 and discussed further in Sec. 8.1. Second Solution:
The second solution is given by: <22 = 360° - a
(A-39a)
b2 = 360° - b
(A-39b)
ct = 360° - c
(A-39c)
A.7.7 atan2 and acos2 Functions The atan2 function has been used in computer programming languages for many years and is intended to resolve the problem of quadrant ambiguities in inverse trig functions. Specifically, if 0 is an angle defined over the range 0 to 360 deg and sin 0 and cos 0 are known, then the atan2 function is defined by: atan2 (sin 0, cos <J>) = 0
(A-40a)
= [atan (sin 0 / cos 0) + 90° (1 - sign (cos 0))]mod 360°
(A-40b)
where atan is defined over the usual range o f -90 deg to +90 deg and the atan2 function is defined over the range 0 to 360 deg. While the above expression is correct, it is not well-defined when cos 0 goes to 0 and can result in mathematical singularities when used in practice. An alternative def inition which avoids these singularities and is well-defined over the entire range is as follows: atan2(x, y) = atan(y/x)
if 0 < y < \x\ and x > 0
= 360° + atan(y/x)
if - 1*|< y < 0 and x > 0
= 180° + atan (y/x)
if |y|< \x\ and x < 0
= 90° - atan (x/y)
if y > |jtj
= 270° - atan (x/y)
i f y < -|jcj
(A-41)
where, as above, atan is defined over the usual range o f —90 to +90 deg and atan2 is defined over the range 0 to 360 deg.
792
Appendix A
A.7
The acos2 function has been introduced in this volume to serve the same function as atan2 (i.e., be uniquely defined over the range 0 to 360 deg) when using the law of cosines. As discussed in Sec 8.1, the acos2 function is defined by: acos2[cos0, H(0)] = [H(0) acos(cos0)]mod36Odeg
(A-42)
0 deg < acos(0) < 1 8 0 deg
(A-43)
where
and H(0) = +1 if 0 deg <
H (<j>)
(A-44)
Properties o f the acos2 function are summarized in Table 8-5 in Sec. 8.1.
A.7.8 Middle Side Law and Middle Angle Law* To simplify the solution o f angle-angle-side and side-side angle triangles, it is con venient to define middle component laws when two sides (a, b) and their opposite an gles (A, B) are known. Specifically, the Middle Side Law, MSL, is: sin a cos b cos B + sin b cos a cos A sin c = -----------------------------------------------1- sin a sin b sin A sin B cos a cos b - sin a sin b cos A cos B cos c ---------- :-----:-------:—— — r - :— - ------1 - sin a sin b sm A sin B c = atan2 (cos c, sin c)
.. . (A-45a)
(A-45b) (A-45c)
Similarly, the Middle Angle Law, MAL, is: .
_
sin A cos B cos b + sin B cos A cos a
:------ --------------------------sin C --------- 1;— - sin a sin b sin A sin B
(A-46a)
„ —cos A cos B + sin A sin B cos a cos b cos C -----------------------------------------------------1 - sin a sin b sin A sin B
(A-46b)
C = atan 2 (cos C, sin C)
(A-46c)
where the notation is the same as that used throughout the section as defined in Fig. A-2 and the atan2 function is defined in Sec. A.7.7.
A.7.9 Singularities and Degenerate Triangles As listed in the tables throughout Sec. A.7.1 to A.7.6, a large number o f singulari ties can arise in spherical trigonometry. An important point to keep in mind is that these singularities arise from the nature of the problem or measurement set, not because o f the fact that it is being solved using spherical trig rather than vectors or * These relationships are due to Leo Early.
A.7
Spherical Geometry
793
some other approach. To see this, consider the airplane problem introduced at the beginning o f Sec. 8.1. Suppose that our pilot takes off from Los Angeles and flies 100 deg along a great circle arc in some arbitrary direction. For the second leg o f his flight he simply proceeds along the same great circle path that he has been following for another 79 deg. How does he turn so as to go directly home? He has two choices— he can turn around through 180 deg and go back the way he came or he can proceed ahead (0 deg turn) and go 181 deg. These are his only options if he wishes to fly a great circle route. Now suppose his second leg of the trip had been exactly 80 deg. He will have arrived at the antipoint of Los Angeles. The solution is no longer well-defined. He can turn in any direction and fly 180 deg on a great circle arc to get home. It doesn’ t matter whether we do the problem using vectors, spherical trig, or any other mathematically correct approach. There are an infinite number o f directions the pilot can use to get back. It’ s the nature o f the problem, not the process we're using to solve it, that created the singularity. Fig. A -10 illustrates all o f the singular solutions and the corresponding degenerate spherical triangles. The fundamental problem for each of them is that two or more of the components are not well-defined. For example, if one of the known sides is 0 and the two remaining sides are both unknown, then the triangle reduces to a segment of great circle. So long as the two remaining sides are equal, they can be arbitrarily long. We first discuss each o f the 6 types of degenerate triangles and then the general approach to solving them.
Type 1 Arc Segment If one (and only one) o f the known sides becomes 0, then the remaining two sides collapse into a single great circle arc. This means that the remaining two sides must be of equal, but arbitrary, length. The angle opposite the 0 length side must also be 0 and the sum o f the two remaining angles (the ones adjacent to the 0 length side) must be 180 deg. There is also an inverse angle solution in which the roles o f “ inside” and “out side” are reversed. In this case, the angle opposite the 0 length side becomes 360 deg and the sum o f the two adjacent angles is 540 deg.
Double Arc If one (and only one) o f the known angles becomes 0, then the two adjacent sides collapse into a single great circle arc. This means that the remaining two sides must be of equal, but arbitrary, length. If the side opposite the 0 angle is also 0, then the figure is the Type 1 Arc Segment. However, if the side opposite is 360 deg, then the figure is a Double Arc, effectively a great circle with an arc segment attached to it. (See Fig. A-10.) The remaining two angles must sum to 540 deg. In the inverse angle solution, the singular angle becomes 360 deg and the sum of the remaining angles becomes 180 deg.
Type 2 Arc Segment If one of the known angles becomes 180 deg, and the side opposite is not 180 deg, then the triangle again collapses into a great circle arc. The two remaining angles must be 0 (or 180 deg in the Great Circle case below), and the sum o f the two adjacent sides must be equal to the third side. However, that length is arbitrary. In the inverse angle solution, the “ inside” and “outside” are reversed, and the remaining angles are both 360 deg.
794
Appendix A
A.7
S m a ll T ria n g le
a=b c =0
4 + 6 = 180° C =0
L a rg e T ria n g le
<'r'B - 3 A
-
cr
A + B = 540° C = 360
a-b
A + B = 180° C = 360°
a= b c = 360°
C= 0
b T y p e 1 A rc S egm e nt
a-b C = 360°
A + B = 540°
C
C =0
^
L
j
.
B A
.
A - —=
----------- - ___
___ ___
i s o
A = B= 0 C = 180°
O II II CD
D oub le A rc
a+ b = c
a+b = c
'
Ty p e 2 Arc segm ent
a + b + c - 360°
A= B=C = 180°
1 ________ b r
a
c
A^B =C
a + b + c - 360°
= 180°
G r e a t C ir c le
a + b = 180° c =' 180°
A=B C - 180°
A=B C = 180°
a + 6=180° C = 180°
A+B+C=
a=b=c=0
• > c Lune
II
01
O u 0 n -G
A+B+C= 180°
Fig. A-10.
• Point
900°
Singularities and Degenerate Spherical Triangles. See text for discussion of each and how to best handle the resulting singularities.
Great Circle If one o f the known angles becomes ISO deg, then an alternative solution to the Type 2 Arc Segment is for the two adjacent sides to sum to 360 deg minus the opposite side. In this case, the 3 sides are again arbitrary, but sum to 360 deg and the figure is simply a great circle. All three angles are 180 deg. The inverse angle solution is now identical and is simply a matter of which side of the great circle we choose to pick as the “inside.”
Lune If one of the known sides becomes 180 deg, then the triangle becomes a lune with two of the vertices at antipoints o f each other (i.e., opposite poles of a spherical coor dinate system). The remaining two sides are arbitrary, but must sum to 180 deg and the angle opposite the 180 deg side must also be 180 deg. The two angles at the anti points are also arbitrary, but must be equal. Unlike the other degenerate triangles, the sum o f the angles and, therefore, the area o f the triangle, is arbitrary and can range from 0 to the full sphere. The inverse angle triangle for a lune is simply another lune.
A.8
Spherical Geometry
795
Point Finally, if two o f the known sides become 0, then the third side must either be 0, reducing the triangle to a point, or 360 deg, creating a Great Circle. In the point, the three angles are arbitrary but sum to 360 deg. The inverse of the point is the full sky in which the three sides remain 0 and the three angles sum to 900 deg.
Dealing with Singularities and Solving Degenerate Triangles The most important issue in dealing with singular solutions is to recognize that they are real. Thus, if one side o f a spherical triangle goes to 0 and the other two sides are unknown, then those two sides must be equal, but can be arbitrarily long. There are infinitely many solutions for the specified problem. It is generally inconvenient to write out all o f these solutions. Nonetheless, we need some practical approach to deal ing with this circumstance because it can arise in realistic problems, such as finding the spacecraft attitude using the Sun and Earth as reference vectors when they are 180 deg apart in the sky. There are three reasonable alternatives: Set a Flag. One alternative is to simply set a flag indicating the presence o f the singularity. For completeness, that flag should also convey what is known about the solution, i.e., that the unknowns are equal or that they sum to a specific value. Set Solutions to a Representative Value. For discrete solutions (i.e., where individ ual triangles are being solved) the only reasonable alternative to a simple flag is to pick a reasonable value for the unknowns which is consistent with the solution conditions. Thus, if two unknown sides are arbitrary, but must be equal, we could set them both to 90 deg. The same approach would work if they were arbitrary, but must sum to 180 deg. Preserve Solution Continuity. There is an additional option if the problem involves either analytic or numerical functions such that one or more of the components approaches a singular state (0 or 180 deg) along a continuous, smooth curve. In this case, the best approach is to allow the solved for sides or angles which become arbi trary at the singularity to remain continuous through the singularity. For example, if a vanishing side approaches 0 along a curve for which the remaining solved for sides both approach 60 deg, then they should retain this value when the third side vanishes. As a practical example, assume that we are using the Earth and Sun for spacecraft attitude determination and control, with the Sun being used to measure the yaw angle about nadir. If the Sun is in the orbit plane, then it will pass through the zenith as the spacecraft goes around in its orbit and the yaw solution will be undefined at that point. The most practical approach is to assume that the yaw angle does not change (or, if it is changing, that it continues to change at a uniform rate) during the brief period that the Sun is at the zenith. While this may not be mathematically “correct,” it is a reason able solution for most practical applications.
A.8 Differential Spherical Trigonometry The development here follows that o f Newcomb [1960], which contains a more extended discussion o f the subject.
A.8.1 Differential Relations Between the Parts of a Spherical Triangle In general, any part of a spherical triangle may be determined from three other parts. Thus, we may determine the error in any part produced by infinitesimal errors
796
Appendix A
A.8
in the three given parts. This may be done by determining the partial derivatives relat ing any four parts of a spherical triangle from the following differentials, where the notation o f Fig. A-2 is retained. Given three angles and one side: - sin b sin A dc + dC + cos a dB + cos b dA = 0
(A-47)
Given three sides and one angle: - d c + cos A dB + cos B da + sin a sin B dC = 0
(A-48)
Given two sides and the opposite angles: cos c sin A dc - cos b sin C da + sin c cos A <M - sin a cos C dC = 0
(A-49)
Given two sides, the included angle, and one opposite angle: - sin A da + cos b sin C da + sin b dC + cos A sin c d£ = 0
(A-50)
As an example o f the determination of partial derivatives, consider a triangle in which the three independent variables are the three sides. Then, from Eq. (A-48), dC_ da
b,c
cos B
cot B
sin a sin £
sin a
(A-51)
A.S.2 Infinitesimal Triangles The simplest infinitesimal spherical triangle is one in which the entire triangle is small relative to the radius o f the sphere. In this case, the spherical triangle may be treated as a plane triangle if the three rotation angles remain finite quantities, If one of the rotation angles is infinitesimal, the analysis presented below should be used.
Fig. A -1 1. Spherical Triangle With One Infinitesimal Angle.
Figure A -11 shows a spherical triangle in which two sides are of arbitrary, but nearly equal, length and the included rotation angle is infinitesimal. Then the change in the angle by which the two sides intercept a great circle is given by &A = A ' - A = 180° ~ ( B + A ) = 8C cos b
(A-52)
The perpendicular separation,
(A-53)
A.9
Spherical Geometry
797
If two angles are infinitesimal (such that the third angle is nearly 180 deg), the triangle may be divided into two triangles and treated as above.
A.9 Common Relations Between Trig Functions The following relationships hold between the trigonometric functions and, conse quently, are true in either plane or spherical geometry.
A.9.1 Relations Among the Functions tan * = 1 / cot x = sin x / cos x
(A-54a)
sin2* + cos2* = 1
(A-54b)
1 + tan2 x ~ 1 / cos2 x
(A-54c)
1 + cot2 x = 1 + 1/ tan2 x = 1 / sin2 x
(A-54d)
sin x = cos (90° - x) = sin (180° - jc)
(A-54e)
cos x = sin (90° - *) = -cos (180® - x)
(A-54f)
tan x = cot (90° - x) = -tan (180° - *)
(A-54g)
cot x = 1 / tan x —tan (90° —x) = -cot (180° 1 / sin x = cot
( jc /
2) - cot x
—
jc )
(A-54h) (A-54i)
A.9.2 Sums and Differences of Angles sin ( jc + y) = sin jc cos y + cos x sin y
(A-55a)
sin ( jc - y) = sin x cos y - cos jc sin y
(A-55b)
cos (jc + y) = cos x cos y - sin x sin y
(A-55c)
cos
(A-55d)
( jc -
y) = cos jc cos y + sin x sin y
tan(x + y) =
tan(jc - y) =
cot(* + y) =
cot(* - y) =
tan* + tany
1—tan* tan_y tan*-tany 1+ tan * tan y cot jc cot y - 1 cotx + coty cot* coty + 1 cot x - cot y
(A-55e)
(A-55f)
(A-55g)
(A-35h)
Appendix A
798
A.9
A.9.3 Double and H alf Angle Formulas
sin 2x = 2 sin * cos * =
2 tan*
(A-56a)
1 + tan2 *
cos 2x = cos2 * - sin2 * = 2 cos2 * - 1 = 1 - 2 sin2 * = -—*an_ x 1+ tan2 *
(A-56b)
_ tan 2* =
(A-56c)
2 tan * 1 - tan2 *
_ cot2 X - 1 cot 2* = 2 cot* sin (* / 2) = ±
c o s ( * /2 ) = ±
tan (* / 2) = ±
(A-56d)
1 — COS*
(A-56e)
1 + COSJC
(A-56f)
1 -c o s *
1 -c o s *
sin*
1+ cos*
sin*
1+ cos*
/ /ox 1+ cos* 1+ cos * sin * cot ( x / 2 ) = ± ----------- = — --------= -----------V1 -c o s x sin* 1 -c o s *
(A-56g)
(A-56h)
A.9.4 Reduction Formulas sin * = + cos (* - 90°) = - sin (* - 180°) - - cos (* - 270°)
(A-57a)
cos * = - sin (* - 90°) = - cos (* - 180°) = + sin (* - 270°)
(A-57b)
tan * = - cot (* - 90°) = + tan (* - 180°) = - cot (* - 270°)
(A-57c)
cot * = - tan (* - 90°) = + cot (* - 180°) = - tan (* - 270°)
(A-57d)
See Table A-8 for additional reduction formulas.
A.9.5 Series Expansions sin*
*3 *5
*7
3!
7!
= x ------ + ------------+ . 5!
A.10
Spherical Geometry
T A B L E A-8.
799
Reduction Formulas. Example: Find sin (270° - x): Look in the row marked (270° - x), and the column marked “sine,” and get: - cos x. tangent
-X
-s in x
+ cos x
-t a n x
-c o t
90° + x
+ COS X
- sin x
- cot X
-t a n x
+ cos
+ sin x
+ cot X
+ tan x
1 o 0>
H
cosine
o
sine
X
cotangent X
180° + x
-s in x
-c o s
X
+ tan x
+ cotx
180° - x
+ sin x
-C O S X
- tan x
-c o t x
270° + x
-c o s
+ sin x
- cot X
-ta n x
270° - X
- cos X
J
- sin x
+ cot X
+ tan x
360° + x
+ sin x
|I
+ cos
X
+ tan x
+ cotx
360° - X
-s in x
+ cos x
-t a n x
-c o t x
X
x3 | *5
6
X
120
x7
(A-5 8a)
5,040
cos x
_ ^
X
2
^ X
X
4
6
+
(A-58b)
~ T + 2 4 ~ 7 2 0 + ''' x + —
2x3
17 x
62x
315
2,835
ta n x
=
COt x
1 x x 2x~ = ----- T - "77 “ x 3 45 945
3
+ -------- + -----------+ -------------+
15
4,725
\x\ < 90c
\x\ < 180c
(A-58c)
A. 10 Bibliography of Spherical Trigonometry Bell, Clifford and Tracy Y. Thomas. 1946. Essentials o f Plane and Spherical Trigo nometry. New York: Henry Holt and Company. Beyer, W illiam H., ed. 1984. CRC Standard Mathematical Tables, 27th Edition. B oca Raton: C RC Press, Inc. Bowditch, Nathaniel. 1966. American Practical Navigator. W ashington, D C : US Government Printing O ffice. Granville, W illiam A. 1942. Spherical Trigonometry. Rev. by Percey F. Smith. B os ton: Ginn and Company.
800
Appendix A
A.10
Green, Robin M. 1985. Spherical Astronomy. Cambridge: Cambridge University Press. Kells, Lyman M., William F. Kern, and James R. Bland. 1942. Spherical Trigonome try with Naval and Military Applications. New York: McGraw-Hill Book Compa ny, Inc. Newcomb, Simon. 1960. A Compendium o f Spherical Astronomy. New York: Dover. Taff, Laurence G. 1991. Computational Spherical Astronomy. Florida: Krieger Pub lishing Company.
Appendix B Coordinate Transformations B.l
Transformation Between Cartesian and Spherical
B.2
Coordinates Transformation Between Cartesian Coordinates
B.3
Transformation Between Spherical Coordinates
For problems in mission geometry, the most common representations of directions as seen from the spacecraft (or other observer) are in terms o f either spherical coo r dinates or unit vectors in a Cartesian (rectangular) coordinate frame. This appendix provides the four relevant coordinate transformations, i.e., Cartesian —> Spherical (Sec. 1), Spherical —>Cartesian (Sec. 1), Cartesian —> Cartesian (Sec. 2), and Spherical —> Spherical (Sec. 3).
B.l Transformation Between Cartesian and Spherical Coordinates The components o f a vector, R, in cartesian (x y, z) and spherical (r, Q, 0) coordi nates are shown in Fig. B -l and listed below. (See also Fig. 6-9 in Sec. 6.1.) Z
X Fig. B-1.
Components of a Vector, R, of length rin Cartesian (x, y, z), and Spherical (r, 0,
801
802
Appendix B
B-2
The declination, 5, used in celestial coordinates is measured from the equatorial (x-y) plane and is related to 9 by 8 = 90° - 9
(B-l)
The components in cartesian and spherical coordinates are related by the following equations: Arbitrary Vector
Unit Vector (1RI = 1)
x = r sin0cos 0
x = cos 0 sin 9
(B-2a)
y = rsin 0sin 0
y = sin 0 sin 9
(B-2b)
z - r cos 9
z = cos 9
(B-2c)
r = ( x 2 + y 2 + z2)112
r= 1
(B-2d)
9 = acos{z /(x2 + y2 + z2) 1/2}
9 = acos z
(B-2e)
0 = atan2 (y, x)
0 = atan2(y, x)
(B-2f)
The correct quadrant for <j) in Eq. (2f) is obtained from the signs o f x and y by, for example, using the FORTRAN atan2 function. See Sec. A.7.7 for nonsingular formu las for atan2.
B.2 Transformation Between Cartesian Coordinates
**
l
t
l
Given a cartesian rectangular coordinate frame (x, y, z) as defined in Sec. B -l, we can define a second cartesian coordinate frame (u, v, w) by defining three mutually perpendicular unit vectors along the axes of the new coordinate frame. (Both frames are assumed to be orthogonal, right-handed reference frames.) The unit vectors defin ing the new frame in terms o f the original coordinates are: “ x
1
1
1
wy 1
vy >N
_u z_
in
in
<>
ill
<3
uy
Given these vectors, we can form the direction cosine matrix*, or attitude matrix, A, as follows: u x
A = Uy _U2
vx
w x
vy
Wy
V2
W2 .
(B-4)
So long as u, v, and w form a right-handed triad of mutually perpendicular unit vectors, A will be a proper real orthogonal matrix. Only three of the nine elements are independent and det A - I. The attitude matrix is a coordinate transformation that maps * The matrix A is called the direction cosine matrix because each of the elements is the cosine of the angle between the two relevant vectors. For example, ii* - u •x - cos 0, where 9 is the angle between u and x .
B-3
803
Coordinate Transformations
vectors from the original frame to the new frame. Thus, if b is an arbitrary vector with components bx, by, and bz, then the components in the new frame are given by: V
u« b
bv =
v »b
b w
w •b
’u x
vx
uy
vy
_u z
vz
= Ab =
wx "
”bx"
W y
b y
WZ _
b z.
(B-5)
The inverse matrix, A-1, is the coordinate transformation that maps vectors from the new frame (u, v, w) back into the original frame (x, y, z). For the attitude matrix, the inverse is simply the transpose. That is: u* A~ l = A T =
Vx _w x
uy vy
V2
Wy
wz
(B-6)
and bu " :A -!
bv
(B-7)
_bw_
B.3 Transformation Between Spherical Coordinates We can use the formalism developed for full-sky geometry to significantly reduce the complexity o f transforming between two spherical coordinate systems. Specifical ly, assume that two spherical coordinate systems are related as shown in Fig. B-2, with the variables defined as follows: p
=
y/r =
arc length between the two positive poles (0 < p < 180 deg) azimuth o f the second pole in the first coordinate system ( 0 < y /l < 360 deg)
1^2 = azimuth o f the first pole in the second coordinate system (0 <
< 360 deg)
Given the coordinates (0 2><5?) o f a point, P, in the second coordinate system, we wish to find the coordinates (0 j, S{) of P in the first system, where <5'is the co elevation angle = 90 deg ~ 8. We first define the triangle shown in Fig. B-3, with: A01= y i - 0 !
(B-8)
A02= Wi —02
(B-9)
and
From the general solution for side-angle-side spherical triangles [see A.7.3 and Eqs. (8-4) to (8-6) in Sec. 8.1], we have two solutions: 8{ = acos2[cospcos(?2 +sinpsin^2 cosA02>tf(A02)l
A0! = acos2
COS(?2 —cos p COS(5f sin p sin
(B -10)
(B -ll)
B-3
Appendix B
804
■Pole 1.
R ef 2
Wo i
.
i P
j Ref 1
I \ A T--
Fig. B-2.
I I' ' t 1~ — i__
1 I
iX -
Relationship Between the Coordinate Systems.
Pole 1
Fig. B*3.
Definition of the Spherical Triangle for Coordinate Transformations.
Coordinate Transformations
B-3
805
and 8 { * = 360 deg- 5 /
(B-l 2)
A(j)]* - A0j + 180 deg
(B-l 3)
If we allow the angles to range from 0 to 360 deg, but restrict the sides to the range of 0 to 180 deg, then we have a unique solution [see Eq. (8-6) in Sec. 8.1)], except in the singular case in which the poles are identical or 180 deg apart: S[ = acos[cos pcos
A<j)j = acos
+ sin p sin 82 cos A02 ]
cos^2 -cospcos<5f H(5[)sinpsin<5{
9 0 d cg [H (5 f)-l]
(B-14)
(B-15)
This can be further reduced to the normal coordinate expressions ( a, 5), as: 8\ - acos(cosp sin<52 + sinp cos<52 cosA02) - 90 deg
A(j>i = acos
sin S2 - cos p sin H( A02) sin p cos
- 9 0 deg [H (A02)-1]
(B-l 6)
(B-l 7)
By symmetry, the reverse transformation will be: 52 = acos (cos p sin 5j + sin p cos <5i cos A02) ™90 deg sin
- cos p sin t?2
H(A0j)sin p cos<52
- 9 0 deg [H (A 0n)-l]
(B-18)
(B 19)
where acos is the normal inverse cosine function defined on the range o f 0 to 180 deg and H is the hemisphere function defined in Secs. A.7.7 and 8.1.
Appendix C Statistical Error Analysis Geoffrey N. Smit, The Aerospace Corporation C. 1 Probability Considerations C.2 Addition o f Random Variables C.3 Expectations and Moments C.4
Example: Minimizing Cost
This appendix is an analytic introduction to error analysis. The flow o f topics is summarized in Fig. C-l. Section 5.4 provides an introduction to error analysis and a general recipe for adding errors in an error budget. The process of creating an error budget is introduced in Sec. 5.3 and discussed in detail with tables o f error sources in Sec. 5.5. Chapter 7 provides an introduction to spacecraft position and attitude mea surements. Specifically, Sec. 7.2 provides a detailed assessment o f the evaluation of measurement uncertainty on the celestial sphere and Sec. 7.6 describes good and bad measurement sets (in terms o f uncertainty) and a set o f practical tests to determine them.
Fig. C-1.
Sequence of Topics in this Appendix.
807
80S
Appendix C
C-1
C.l Probability Considerations The basic problem to be addressed is: Given : a set o f components with errors e\, e2, -- - with known probability distributions.
The components are combined in a system such that the output error is e = e } + e2 + ... + en
(C-1)
Find: the probability distribution o f e.
As noted in Sec. 5.4, the more general problem is
e = F{el,e 2, . . . , e n)
(C-2)
Fmay, in general, depend on the state o f the system— i.e. if the component contribu tions are
xi = di + eiA i = h ...,n )
(C-3)
then the system output is some function, G(x), which can be decomposed into the nominal output, G(d), and the error, G(jc) = G(d) + F(x)
(C-4)
G can usually be expanded in a Taylor series. When this is the case, we have
G(x) = G(d) + eG\x) + higher order terms
(C-5)
This gives the system error,
F = G(x) - G{d) = eGXx)
(C-6)
which is a weighted sum o f the eit with coefficients which are functions o f the state, x. A weighted sum can always be scaled to be a pure sum. Therefore we will concen trate on the case o f a pure sum, since we have seen that it covers the majority of cases o f interest. Also, it allows us to derive a number o f concrete results which would be less understandable in a more general setting. If, for some reason, it is desired to retain a functional form other than a sum, this does not impact the conceptual basis for much that follows. For example, a worst case es timate o f the system error can readily be produced (and it typically will be too conser vative), and computational schemes can be devised to estimate the probability distribution o f e or quantities related to this probability distribution.
Note that e is a function of several random variables. To proceed, we first summa rize some of the concepts and definitions o f probability distributions involving several variables in order to fix notation and terminology. Consider first probability on the real line (i.e. we consider “events” labelled by real numbers— such as a measurement, x). The probability that x < y , Prob({x < y }), is de noted P(y). P(y) is defined to be the probability distribution. Assuming that the deriv ative exists d P / dy = p(y)
(C-7)
is called the probability density function, or pdf. p(y) dy is the probability that * lies between y and y + dy and r J — oo
P(x) j x = i
(C-8)
C-1
Statistical Error Analysis
809
In the definition o f the probability distribution, there are two variables (I’ ve called them x and y). The variable y is the actual argument o f the probability distribution, x is a “ dummy variable” , introduced for explanatory reasons. Thus, P(y) = Prob{x < y }, means that any “experiment” performed on the system described by P will yield an outcome less than or equal to y on P fraction of the trials. Thus, as shown in Fig. C-2, P is monotonically increasing. In the case o f the probability density, we are focussing attention on the neighborhood o f a single point, “y” . It is not uncommon to see termi nology like “ the probability that y lies between y and y + dy is given by p(y)dy,\ PM
Fig. C-2.
Probability Distribution P {y ) vs. y.
The variable, x, used to label points or “events” in our probability sample space is called a random variable or variate. I use the terms “ variable,” “random variable,” and “ variate” interchangeably.* The sample space need not be the real line. It could be ndimensional space (i.e. n copies of the real line), or, in general, any suitable set. Measurements of angles, distances, brightnesses, or frequencies are all random variables. The result o f any particular measurement would be a single real number, the value of which includes some element of chance. The associated probability densities would usually be peaked around the true value o f the item being measured, and would be negligible far away from the true value as shown in Fig. C-3. Probability of M e a su re m e n t
_r Fig. C-3.
Measurement
A Discrete Probability Density Function.
* Intuitively, the idea of a random variable is fairly clear— there is an element o f chance in any given “ reading” o f x> i.e. jc is drawn from some probability space (“black bag”) at random, and the likelihood that it will have any particular value is given by its probability density. The mathematical theory was originally developed to describe various games o f chance. The sub ject only became mathematically respectable after its foundations had been worked out inde pendent o f references to these intuitive notions. A rapid increase in the rate o f progress occurred at this point. (See [Kac, 1989].)
810
Appendix C
C-1
Examples o f multidimensional random variables include measurements o f “physi cal” vectors such as the 3-dimensional velocity vector of an object and the arguments of functions o f several 1-dimensional random variables. For example, to determine the area of a rectangle, we have to measure two lengths. This input “ vector” would be a 2dimensional random variable. The pointing errors of a spacecraft typically depend on the value o f numerous parameters (such as sensor readings, temperatures, structural deformations) and this input vector would be a multidimensional random variable. Random variables can also be discrete, i.e. limited to a finite number o f values. Exam ples are the outputs o f coin or die tossings. The output from a digital sensor is discrete. For a multi-dimensional random variable, the probability density has the form p ( x u x 2, - ; X n)
(C-9)
This gives the probability that x will be found in the hypercube {(* !,* ! + dx{), (x2,x 2 + dx2), ...,(* „ + <&„)}
(C-10)
The multi-dimensional probability distribution is defined as P ( » = Prob{*1< y 1)
(C -ll)
and we have p(x) p(x) = ------ —
------
(C-12)
dxidx2... dxn
The system errors that we are trying to estimate depend on the input errors contrib uted by numerous components, i.e. the system error depends on a multivariable prob ability distribution. In general, we don’ t have any data on this multivariable distribution, per se. What we have is data on the distribution of each of the inputs. We may also have some information on whether some o f these inputs interact with each other so that the value measured by one input may be coordinated with that measured by another. We are thus faced with the task o f building up the multivariable distribu tion from its “projection” on each o f the input axes. This brings us to the topic of such projections (marginal distributions), and the topic of interdependence between the in puts (conditional probabilities).
Marginal Distributions One can “ integrate out” some o f the variables o f a multivariate distribution func tion, leaving a probability distribution in the remaining variates.* For example,
J ^ p ( x i , . . . , x n)d x 2 dx3...dxn
(C-13)
leaves a density
* The significance o f the integration is as follows. For simplicity consider the two variable case. The multivariate probability distribution function, p(xh x2)dxldx2 is the probability that*! and *2 will lie in {(*!, x{ + cfcq), (x2, x2 + dx2)}. x2)dx2\}dxi is the probability that*] and *2 will lie in the strip {(*b xi+dx{), any x2}, i.e. we’ ll “accept” any measurement which has *1 in (* ],* ! + dx{) regardless of the x2 value.
C-1
Statistical Error Analysis
811
(C-14) Pi and pi are called the marginal distribution function and marginal density o f x\ associated withp(x^, . .. , x n) and P(x\, . . . , x n). Marginal distributions o fx 2,x 3, ... can be similarly defined. The figure below illustrates the marginals o f a bivariate probabil ity density. (The geometry o f the figure also illustrates why they are called marginals.)
Marginal on y-axis P
Bivariate Density
Marginal on x-axis x Fig. 0-4.
Joint (Bivariate) Probability Density Showing Marginals on Each of the Axes.
The multivariate density, p, is called thejoint density associated with these variates. Similarly we can define the joint distribution. While a given multivariate distribution function (or density) will have unique mar ginals, the inverse problem o f finding a joint distribution given the marginals does not have a unique solution. In the error analysis problem being addressed in this section, the information that we have about the joint distribution o f the errors, eb comes from its marginals, thep it and the correlations between the e^* In addition to the uniqueness issue mentioned above, for some marginal distributions not all correlation coefficients (between -1 and +1) are possible, so that we must also observe bounds on the correla tions we input to the problem. Assuming that we have posed the problem properly, we can in principle generate a joint distribution (e.g. via a Monte Carlo simulation) con sistent with the marginals. Table C-1 lists a number of common probability distribu tions used in statistics.
Conditional Probabilities and Independence Random variables can be coupled, so that the values taken on by one may give us information about the others. For example, a spacecraft structural distortion may be re lated to a temperature change. The probability that an event A occurs, given that event B has occurred, is written P(A\B) and said “probability o f A, given 5 .” From the defi nition,
P(A,B) = P(A\B)P(B)
(C-15)
* The word “correlation” has a technical meaning which will be defined later. It refers to the degree to which the values o f random variables are interrelated.
812
T A B L E C-1.
Appendix C
Som e Common Probability Distributions.
Name, Comments
Probability Density Function p(x) = 1J(b— a), a<x
Arises during
quantization - a )2 / 2s2j
exp(af+ &!2)
J
normal variates. Also, geometric meaning (tan of a uniformly distributed variate)
Complicated
M = 8 i ] ( x / 2)
| x / s 5je | -x 2 / 2s2j
V = ((4 - ny2)s2
K
(* > 0 ) (zero for
x
< 0)
a/
- b)2 + a2j j
M=b
M =A
Ake (-X ) i k\ (prob of a negative number of occurrences = 0)
h
,
^
V=X
exp[A{ expo's) 1)1
k
M ~mp
Binomiar Prob. of getting k wins in m trials, each with prob of success,
exp(-bs-a|s|)
Variance diverges
Poisson Prob. of getting k of independent events (discrete)— model of shot noise, etc.
a)s
k
Cauchy Ratio of two
(exp(sb) -
II
of radial error with normal components
exp(sa))/(b-
II
Rayleigh Distribution
^ l / V 2ns2
Characteristic Function
M - (a+b)/2 V = (f>-a)2/l2
t rn.
Normal Ubiquitous: arises via sums, also as the limit of some distributions
Mean, M Variance, V
Graph
5 >
Uniform Distribution
C-1
cj^pkq m~k
V
=mpq
[1 + p(exp(s) u r
till,. k
p=1 - q
*The Bernoulli distribution is a binomial distribution with m = 1
i.e. the probability of A and B is the probability o f B times the probability o f A given B. The events A and B are called independent if P(A|B) = P(A)
(C-16)
P{A,B) = P{A)P(B)
(C-17)
which is the same as
In this case, the densities can also be written as products, e.g.*
* Independent variates have many mathematical similarities to orthogonal coordinate systems. In particular, they tend to make the algebra much simpler than coupled variates and non-orthogonal coordinates, respectively.
C-1
Statistical Error Analysis
/>(*!, x2) = P(x! )p(x2)
813
(C -18)
Later, we will introduce the moments o f probability densities. These provide simple measures o f various attributes o f the density function. The “correlation” will be a mo ment quantifying the extent to which the variates in a multivariate density are indepen dent.
Functions of Random Variables Given a random variable, x, with density p(x) or a set o f random variables, jcl5 x2, ... , x n, with density p (x i,x 2, ■■■, xn), we frequently need to determine the density cor responding to some function of * or o f Xj, x2, xn. This is the central issue o f the error combination problem. First look at the one dimensional case. Let y = y(*)
(C-19)
If y is a monotonic function, we have Py (y)=P x[x
(C-20)
and (C-21) i.e. Py(y) =^C*(y)) \dx/dy\
(C-22)
For example, if x has density function p(x), and y = 3x+l
(C-23)
q ( y ) = p ( ( y -l ) /3 ) /3
(C-24)
then y has density function
As a second example, if * has density function p{x), and y = x2, then y has the den sity function
?(y)=[p(V7)+p(-V7)]/[2V?]
(c-25)
The pairing o f terms here derives from the fact that y is not monotonic. Similarly* in the multivariate case, if x = ( x h . . . , x n)
(C-26)
y = (yi*
(C-27)
and
are related by a one-to-one transformation, then P,(y) = P,[jc(y)]
(C-2 8)
Pyiy) =Px[x(y)]\fcfdy\
(C-29)
and
814
Appendix C
C-1
where |dx/
(C-30)
* = (*!, ...^ „),w < n .
(C-31)
and
A special case o f this is the sum, y = x1 + ...+Ain. We are taking a generalized sort of marginal, and we need to integrate out some of the variables. The problem is to deter mine what needs to be integrated out. This is usually more easily done in the case of the probability distribution, as opposed to the density. For example, assume we have: yi = yfai* ••■*„), i = 1.
m
(C-32)
The marginal distribution, Py^ is, by definition, the probability that iiy l < Y{* as shown in Fig. C- 5, for 2-dimensions. In “ y - space” , the set o f all appropriate y ’ s is the shaded area.
Fig. C-5.
Shaded Region Represents the Set of Points for Which the y 1 Component is Less than or Equal to V^.
To find Py^ we need to associate a probability density with each part o f the shaded region, and then integrate (see Fig. C-6). Since we only know the density func tion in the x-space, we must find the corresponding region there, and integrate there. Thus, for the distribution, py (yj) = probability thatyj < Y±, we need to integrate over all combinations o f the * ’ s which yield a >'] < Y\.
p>i
M=J..J ,
xn\dxv ..dxn ^
(similarly for y 2, ^3,...). The corresponding joint probabilities are
(c -33)
C-2
Statistical Error Analysis
= J
815
px(xu ...,x n)dxl ...dxn
y fa i ~ Xn)
(C-34)
yk(x\ - xn ) ^ yk
Al! x’s such that
>*1
Fig. C-6.
Shaded Region is the Set of Points for which y 1(jr1...xn) < Y^.
Depending on the nature o f the function y, this can take some effort. Similarly, the densities can be found by either (1) differentiating the P’ s, or (2) integrating the px's over suitable subsets o f x space. The next section illustrates this for the special case where y = x^ + x2 + ...+ xn.
C.2 Addition of Random Variables Here we have a single function, y = y (jq, x 2>.. .xn) given by y = x t + x 2 + ...+ xn.
(C-35)
We shall first reproduce the above analysis for this special case. It turns out that when the variables are independent, the analysis can be significantly simplified— in fact, all the standard “textbook” results for the addition o f random variables are for independent random variables. We will point out below how these simplifications occur. For simplicity, first consider the case o f two variables. Let the joint distribution o f Xj and x2 be Px(JCj, x2) and the density be p x(xx, x2). As shown graphically in Fig. C-7, the probability that x x + x2 < Y say is given by the integral o f px over the shaded region, i.e. over all points x2 such that xj + x2 ^ Y. Hence,
(C-36)
The density is given by:
816
Appendix C
C-2
*2
Fig. C-7.
Shaded Region is the Set for Which x-f + x2 < Y
P y(Y) = j j dxidx2 p(x1,x 2) 8 ( Y - x 1 - x 2) = j p(x1, Y - x 1)dx1
(C-37)
This type o f analysis is easily extended to more than two variables. In general, the formulas get very complicated unless the variables are independent. The main excep tion is if Px is multivariate Gaussian. The underlying reason for the normal distribution retaining its tractability, even in the presence of correlations, is the fact that we can al ways transform to a new set of variates which are independent. In the case where the xt are independent, the analysis can be simplified as follows. Starting again with the two variate case, we have p(xx, x2) = Pi(x{) p 2(x2)
(C-38)
y = x l + x2
(C-39)
and
py (y ) = J P x (x i * y - x 1 ) ^ 1
= J p\ {xi ) p 2 { y -
(C-40)
This is suggestive o f the convolution of p\ and p 2, and invites the use of Fourier transforms. In probability theory, the Fourier transform o f a density is called its char acteristic function .*
* Characteristic functions or Fourier Transforms are ubiquitous in probability theory and statis tics. In fact it has been said that knowledge of statistics, like Electrical Engineering, reduces to a knowledge of the Fourier Transform.
Statistical Error Analysis
817
The Fourier Transform o f a convolution is the product o f the Fourier Transforms.
F7\Py(y)] = FT j Piixi )P 2 { y ~ X i ) dx l =
i)5 pi(xl) p 2(q)dx1 dq
= F T {Pl(x)) FT(p2(x)) = Gxj ( s ) G
X2
(s) = Gy (s)
(C-4 1)
Hence the distribution of y can be found by inverting the Fourier Transform. This is easily extended to n independent variables— we simply get more terms in the prod uct (C-42)
G = G1G2G3..... Gn
Examples (1) The sum of n normal independent random variables is a normal random variable: (C-43)
X = X 1 + X 2 + •■■+Xn
^,
4 0 expl 2 p(x) =
•JlTT 1
.
exp
a;
<7
n
(C-44)
where
C72 = (7 ?+ ... + <J%
(C-45)
The sum o f n statistically independent random variables is normal if and only if each of the variables is normal. (2) The binomial, Poisson and Cauchy distributions also “reproduce themselves” under addition o f independent variates: If
(C-46) Then pW = ( * ) qX
“ q)N~X* N = n1 + ... + nn
(C-47)
If
Pi{*i) = e~4i 7 7
(C-48)
818
Appendix C
Then p(x) = e i ^ j
(C-49)
x\
If
2
(C-50)
Then p(x) =
1
1 (C-51)
(3) If we add an arbitrary random variable, jc , to a uniformly distributed random vari able, y, the sum, z ~ x + y yields the moving average o f jc, assuming jc and y are in dependent. Let
(C-52) - 0
otherwise
Then pz (z) = \ P x { x ) P y { z ~ x ) dx = ^ i \ Z* “a P x ^ d x
(C-53)
= - ^ [ px ( z + a ) - px ( z - a ) ]
In general, the probability distributions o f sums x l + x2 + ..., where the xi come from different distribution types, can yield messy algebra.
Other Functions of Random Variables We have already presented the general approach to finding the probability distribu tions o f functions o f random variables other than sums. We have emphasized sums be cause they are important, particularly since we have indicated that most error combination problems can be reduced to sums, and also because they are tractable. In a few cases, analytical results can be obtained for other functions, as the following ex amples show. To find the distribution o f a product of two independent random variables, let the variables be x and y, with joint density f(x,y) —p(x)q(y) since they are independent. We want to find the density, w(z), of z = xy. We have
(C-54) To find the distribution o f a quotient o f two independent random variables, with x, y, p(x), q(y) as above, and with z = xfy and the density of z being w(z), we have:
C-3
Statistical Error Analysis
W{z) =
p{yz)q(y)\y\dy
819
(C-55)
For p and q normal, W reduces to a Cauchy distribution. Once we have the probability distribution of the output (i.e. the system perfor mance), we are in a position to estimate the probability that the system error will lie within the required bounds. The generation o f this probability cannot be accomplished analytically, except for some special cases, such as independent, identically distribut ed with the pi given by some o f the standard types. It will in general require the use o f a Monte Carlo simulation plus some insight to ensure that the basic problem is well defined. This can be time consuming. Also, often the input data (e.g. the p$ are not particularly well known, and it might not make sense to expend tremendous amounts of computational effort analyzing their consequences. For these reasons, it is often of interest to find simplified ways o f assessing the system performance. It turns out that the moments o f e can be calculated quite simply in many situations as discussed in the next section.
C.3 Expectations and Moments It is often convenient to find a few numbers which summarize the key features of a probability distribution. For example, consider the following density functions. In Fig. C-8(A), the density is centered about the origin and is fairly spread out. This indicates that “on average” readings of this variable will give zero, but that there will be a lot of scatter. In Fig. C-8(B), we also have a variable centered about the origin, but in this case there will be much less scatter. In Fig. C-8(C), the density is centered about a non zero value— i.e. on average, we expect these measurements to give a reading displaced from zero. In Fig. C-8(D), the probability density is “ skewed”— there is more scatter for positive than for negative values. The basic point is that the behavior o f the random variable can be summarized via a few general features o f the density— i.e. measures of location, scatter and symmetry. Invoking the analogy o f the mass moments o f a rig id body, these features are usually quantified by taking the “moments” o f the density function.
(A ) Large Scatter
Fig. C -8 .
(B ) Small Scatter
(C ) Non-zero Average
(D ) Asymmetric
Q u a lita tiv e F e a tu re s o f D is trib u tio n s .
We start with some definitions. The expectation o f a function/of a random variable x with probability density p(x) is defined to be: (C-56)
820
Appendix C
C-3
The nth moment o f the probability density pix) is then defined as (C-57) The following rules for expectations are easily established: (C-58) (C-59) (C-60)
(C-61) for all x then, E(fx) < E ( f 2)
(C-62)
|(E(/(x))[ < EQ/(x)\)
(C-63)
These results are completely independent of the probability distributions o f the ran dom variables. The first moment, p, given by (C-64) corresponds to the “center o f mass” of the probability density. It is called the mean or expected value o f the random variable, x. The second moment is called the variance (C-65) The variance is the square of the standard deviation, a. The variance corresponds to the moment o f inertia of p, and the standard deviation to the radius o f gyration. They give a measure o f how spread out p is, and hence how scattered the observations of the random variable, x, will be. Specifications of error bounds are often given in terms of standard deviations i.e. “n times the standard deviation should not exceed a certain val ue,” where n is typically 1, 2 or 3. Implicit in such a specification is a probability fig ure. For a given density, p, the probability that a measurement lies within n standard deviations o f the mean is well defined. For example, for a normal distribution, these values are 68.3%, 95.5%, and 99.7% respectively as given in Table 7-3 in Sec. 1.2.2.2. We discuss later the fact that these probabilities will change if a different distribution is used. We now return to the problem o f characterizing the probability distribution o f the system error, given those o f the components. We can use the above rules to calculate some moments o f sums: £ [£ * ;] = # [ z x? ] = 1L{ex} J + 2 s S e(xi
j
(C-66)
Statistical Error Analysis
C-3
821
Hence, V a r(£ xt) = X(Var(*,)) + 2 £ S C o v ^ x ;-J
(C-67)
If the xt are independent, the covariances are zero, and we have Yar(S(xI)) = 2(V ar(^))
(C-68)
Taking square roots, we have
If we have perfect correlation, the standard deviations sum: Var(^i + x2) = Var(x1) + Var(x2) + 2
+ 2ai<72 = (<7j + 0*2)2 (C-70)
Hence crX)+^2 = crXi + <JXi (for r = -1, this becomes 1^ - C72|)
(C-71)
Similar analyses can be carried out for more variables. Hence, if we know the means and standard deviations (or variances) o f our inputs, it is easy to calculate the mean and standard deviation o f the output, particularly if the inputs are either indepen dent or perfectly correlated. When the variables are uncorrelated, the system standard deviation is the square root o f the sum of the squares, called the root sum square or RSS, of the individual standard deviations. When the variables are correlated, the system error starts to take On more o f the appearance of a sum of the errors of the input variables. The interpre tation is that the statistical “ smoothing” no longer takes place, and the variables add algebraically. Between these two extremes lie systems with partial correlations. The results also lie in between, and the algebra becomes messier. If the output is a weighted sum o f the individual inputs, we can perform a similar analysis. For uncorrelated inputs, we get a weighted sum o f squares for the variance, and for perfect correlation, we get a weighted sum. Note that in this case we can always rescale the inputs so that the weights are all equal to one— i.e. the weightings really introduce no new concepts or complications. These results form the basis for some powerful ideas in probability theory and its applications: (1) Law of Large Numbers Take the average of a large number of samples, Hn = (xi+X2 + .. . + x nyn
(C-72)
V a r ( / y = V a r {x)/n
(C-73)
then
Thus, the variance of the sample means is (1 / m ) times the variance of the population. By taking large enough samples we can cluster as close to the mean as desired. However, there are some bounds on “how close". For example, Chebychev 's inequality says that the prob ability that we lie within ccr^4n of the mean is greater than 1 - 1/c2, for any c. This idea underlies a number of filtering approaches.
822
Appendix C
C-3
(2) Central Limit Theorem If the xi are independent and identically distributed, each with mean \x, and variance a2, then zn =(*1 + x 2 +... + xn) 1 4 n o ^
(C-74)
is normally distributed as n - » » regardless of the distributions of the Xj. This is used, for example, to justify the use of normal distributions in the theory o f errors. The basic idea is that errors o f measurement in most physical systems are due to a large number o f small in fluences, so that one can argue that their combination is normally distributed. Experimen tally, this is borne out in a wide variety o f situations. It is somewhat o f a miracle, because the normal distribution also happens to be extremely convenient mathematically. A very large portion o f all that has been written on statistics assumes that samples are drawn from normally distributed populations.
(3) Stable distributions We have already noted that the binomial, Cauchy, Poisson and normal distributions are self reproducing under summation (and hence averaging)— i.e. sums o f such variates are also so distributed. This concept can be generalized to that o f a stable distribution. Let x, xh .... , xn be independent identically distributed random variables with density />(*,■)- Let
sn = x{ + x 2 + ...+ x n
(C-75)
The density p(x) is stable if there exist constants cn> 0 and yn (for all n > 2) such that sn has the same density as c^x + yn. If yn = 0, p(x) is called strictly stable. For the normal distribution, c„= nm . For the Cauchy, cn = n. One can show [Feller, 1957, 1966]: that cn can only be o f the form (C-76) b
0
p(*)=
/ 1 - g 1/2jc
(*> 0 )
ylTix
=0
(jc<0)
(C-77)
is stable with a = 1/2. The stable densities with a < 2 do not have variances, i.e. their second moments are unbounded. (4) Computing Higher Moments The calculation of the higher moments of sums of independent random variables is usu ally accomplished using the characteristic function. The log of a characteristic function is easy to differentiate. We define the nth cumulant of a random variable, x, as *n M = - 7 7 7 l0§ G» i du
«= o
The cumulants of independent random variables are additive, i.e. Kr[xl + x2 + ... + xn] = Kr[xl] + Kr[x2) + . . . + K r[xn]
(C-79)
Even if the Fourier Transform of the sumcannot be inverted, we can evaluate the cumulants and express the moments in terms of them. In fact, the first three moments about the mean equal the first three cumulants. This also shows that the first three moments are additive for independent summands. If we take the Laplace transform of a density function (analogously to taking the Fourier transform to form the characteristic function) the result is called the moment generating function (MGF). The reason for this is that the moments turn out to be the coefficients in
Statistical Error Analysis
823
the Taylor series for the MGF. Let G(s) be the MGF of the probability density p(x). Then G(s) =
= £ [e s*]
(C. 80)
Now note that G' ( 5) =
G"(.s) = £|;t2e " :J,--.,G(,0(.y) = £A:'leJxj
(C-81)
Hence <7(0) = 4 4 G "(0 )-£ [^ ],...,G (»)(0 )-[*»]
(C-82)
Finally note that G(s) = l + G (0)5 + G"(0)$2 / 2! +... = 1+ £[x]j + e [* 2]s2 / 2! +...
(C g3)
Two and Three Dimensional Target Spaces We have seen that if the target location must be specified as a 2 or 3 dimensional vector, we can readily calculate the standard deviation o f the error in terms o f the standard deviations o f the 1-dimensional components. We now discuss the signifi cance of the fact that this system error will have a different probability density from those o f the components. To illustrate the ideas involved, we shall present the simplest possible case— a bivariate normal distribution— and then briefly indicate what hap pens in general. Suppose the target is nominally at (0,0) in the (x, y) plane, and that the x and y error densities are given by normal densities; p(x ) = - — = —
e -x 'H a l
(C-84)
^
(C-85)
Suppose, further, that these are independent, and that ax - a y - cr. Then the probability of making a measurement in an element o f area, dxdy is given by p(x,y)dxdy = p(x)p(y)dxdy = ^ - ^
eJ *2+y1^ ^ 1 dxdy
(C-86)
We can convert to polar coordinates (r, G) and integrate out the ^-dependence, leaving the distribution of radial errors: p(r)dr = J T
e
ar
(C-87)
This is called a circular normal or Rayleigh probability density. It is usually tabu lated in terms of the circular error probability , or CEP. Assume that the cumulative distribution function, P {r < /?} is given by
(C-88)
824
Appendix C
C-3
The CEP is defined as the R such that P(R) = 0.5. Hence CEP = crV2!n2 = \ .\ llo
(C-89)
We have also seen that the standard deviation of r is a r =V2cr
(C-90)
(C-91)
p (V2 ct)
= 1 -* H = 0.6321
(C-92)
Similarly the “2o” probability is 1 - e4 = .9817, and the “ 3a” probability is 0.9998. Numerical values for 1,2,3, and 4 c variations in 1,2, and 3-dimensional problems are given in Table 7-3 in Sec. 7.2.2. The large print is that the probabilities associated with n-sigma errors vary with dimension and with the probability distributions. Note that in two dimensions there is an extra V * factor in the area element. For small r we enclose fewer points, while at large r we enclose more. In three dimensions we get a similar effect, except that now the extra factor involves r2. The main message of this discus-
sion is that the probabilistic significance of n-sigma error bounds changes with the dimensionality of the problem and with the assumed probability distribu tions. Adding Errors of Different Frequencies Any measurement on a system can be regarded as drawing a sample from an en semble o f systems, a population with the statistical properties we have derived. Up to now we have not said anything about how these samples are drawn. We now consider to what extent the sampling procedure can affect the errors. In an actual system* this sampling procedure is realized as a time series of measurements. Apart from providing us with a string o f readings (and hence the opportunity to do statistics) a time series has extra structure. Readings taken close together in time can be more closely related than those taken far apart. As a result, the probability distributions o f the output will depend on the time scales used in taking the readings. Since time domain and frequen cy domain behavior can be related through the Fourier transform, the spectral structure of the errors can influence the way in which we combine them. From a systems engineering point o f view, one of our goals in error budgeting is to eliminate excessive conservatism. We don’ t want to impose unnecessarily strict re quirements on the subsystems, since this drives up cost. The “extra degree o f freedom” obtained by considering frequency can sometimes be exploited to reduce conserva tism. The basic idea is that at any given time only a limited fraction o f the total error variance may actually be accessible. This is illustrated schematically in Fig. C-9. We may think o f the shaded area (the short term available error variation) as wandering around the total area. At small time scales, our error variance is defined by the shaded area. At long time scales, we can observe errors anywhere in the larger, unshaded area. For this to make sense, there must be two time scales in the problem— a short time scale associated with activity within the shaded area, and a longer one associated with
C-3
Statistical Error Analysis
825
the motion o f the shaded area. We also require that the motion o f the shaded area be classified as “noise”— i.e. not as a deterministic motion which we should have ac counted for in the measurement process.
Fig. C-9.
Th e Amount of Variation Available to a System in the Short Term (Shaded Region) May (1) be Smaller than the Total Amount Available in the Long Term (Unshaded Region), and (2) May Move Around with Time.
For a simple illustration, consider adding two pure sinusoids. If the frequencies are far apart, the sum will look like the curve in Fig. C-10.
Fig. C-10. Sinusoidal Example of Short Term Variation (Shaded) vs. Long Term Variation (Unshaded).
Measuring over a long time scale, we’ ll pick up the total variation. Over a short time scale we’ ll get a smaller “ local” variation, plus a slowly varying bias. However, if the frequencies are close together, we’ ll get a beat pattern as shown in Fig. C -11. In principle, if the frequencies were very steady (in which case they wouldn’ t be “noise” ) we could take advantage o f the small amplitudes at the nodes o f the beats. In practice, there is a band o f frequencies, and the beats don’ t occur. In both these examples, the total sample space available is large (essentially the total variation o f the combined signal). In the first case, however, because o f the spread in frequencies, the perceived variance can be less than the total variance if an appropriate time scale is used. It is possible to have “random analogs” to the sinusoids used in the discussion above— an example would be integrated white noise as it appears on a gyro angle out-
826
Appendix C
C-3
Fig. C -1 1. Beat Pattern in Mixture of Nearby Frequencies.
put. Here the bias can shift slowly with time (and can be tracked and eliminated from the error budget), but it is not deterministic.’1'
Stochastic Processes If we consider our system to evolve in time but that it is influenced by some random effects, then the state o f the system will be given by a time series o f random variables. Also, the series o f random disturbances, and the measurements with random errors that we use to describe our system will also yield time series of random variables, Such se ries are called random processes. We will show how the frequency content o f such processes can be measured, and that processes with various frequency contents can be constructed. We will then give an analog of our sinusoidal example. A stochastic process (stochastic means random) is a family {X t\t £ T) of random variables. For our purposes, the index set, T, will either be a continuous or a discrete time variable. The Xt will typically represent the state of our system. A random process is strongly stationary if its probability distributions are invariant under time shifts. It is weakly stationary if the first two moments (i.e. the means and covariances) are in variant under time shifts. For a discrete time variable, a stationary random process is easy to visualize. For each time step, f,-, we draw an X from its probability distribution, p(X), and plot it as shown in Fig. C-12.
,
I'
I
I
I
I'
I I
I
Fig. C-12. A Discrete Tim e Stationary Random Process, x = x[f). See text for definitions.
If we observe the process for a long time, and count the number o f times the ordi nate lies between X and X + dX for various X we will find that the probabilities of the various X values match p(X). Next, we want to see how a frequency spectrum can be associated with such a pro cess. Start in the time domain, and introduce a measure o f how much the X’ s are cor related with their neighbors. A high positive correlation would imply that X does not * There are also a number o f deterministic situations in which a slowly varying bias might occur. Typical examples include disturbances that are periodic with the spacecraft orbit, such as ther mal or lighting effects.
C-3
Statistical Error Analysis
827
change rapidly with time. The autocorrelation function o f a univariate process is de fined as
« (0 -^ {x (i-* )X * W }
(C-93)
It measures the correlation between readings taken a time interval t apart. Given two processes, x(t) and y(t), their cross-correlation is rv
W 3 E { * ( '+ j> * ( ') }
(C-94)
We now look at how this temporal behavior translates into the frequency domain. The power spectral density o f a process x{t) is the Fourier Transform o f its autocorrelation: S(
e~lMR(t)dt
(C-95)
R(t) =
S(a>)e‘w d 0
(C-96)
The cross-power spectrum, or coherence, of two processes, x(t) and y(t) is
S | R ^ e -m 'd t
(C-97)
=
(C-98)
As the name implies, this measures the power in our signal as a function o f frequency, and hence indicates its frequency content. For example, a theoretical model of white noise, W(f), is given by (C-99)
Sww(c o )= l,R ww(t) = $(t)
There is no correlation over time. For Gaussian white noise, the probability density of each W(f) is Gaussian.* The total power in white noise is
Power =
~
(c-ioo)
Any real physical system would attenuate the high frequencies. Modifications o f white noise by filtering are termed colored noise. As a second example, a random walk, also called Brownian motion is integrated white noise. Many physical processes can be modeled as linear systems driven by white noise. Autoregressive models are an example of this. They have the form 'Zan x , . n = a >,
(C-101)
Moving average models are another example. They have the form x , ^ b n co,.n
* In the noise literature, “Normal” distributions are usually called “Gaussian1
(C-102)
828
Appendix C
C-3
These can be combined to yield moving average autoregressive models: ?.a n xt_ n = 'L b n a>t_ n
(C-103)
Similar definitions involving stochastic differential equations apply in the continuous case. It should be noted that the “ stochastic calculus” involved in the continuous case involves some subtleties. See, for example, Karatzas and Shreve [1997], and Oksendal [1998]. Since many stochastic processes can be built up by passing white noise through var ious kinds of filters, the inverse problem suggests itself—finding a filter which will reduce a given stochastic process to white noise. Such filters are commonly used in estimation theory— once we get down to white noise, we have essentially squeezed all the information out o f the process. We now return to the “stochastic analog” o f the sinusoidal example given previous ly. We first need to generate a pair o f stochastic processes which have very different frequencies, and then discuss how they can be combined. In a real situation, the sto chastic processes would be provided by nature. We shall use a pair o f mathematically very simple autoregressive processes. (See Box and Jenkins [1970].) Let x, = a X{_i +
(C-104)
and y ( = - a y i - i + Vj
(C-105)
be two stationary processes. It can be shown that this requires \a\< 1. u and v are sup posed to both be white noise with variance a2. As shown in Fig. C-13, heuristically, x tries to match itself, and hence is relatively smooth, with slow meanderings. y reverses sign at each time step, and hence has a much higher frequency appearance. This is con firmed by calculating the power spectra (the Fourier transforms o f the autocovariance functions). See Fig. C-14.
Fig. C-13. (A ) Realization of x f = axM +u,- (o
If we sum x and y we obtain a process with ostensibly the sum of their variances. This is most easily seen by transforming these autoregressive processes into moving average processes. We have
829
Statistical Error Analysis
Fig. C-14.
(A ) Power Spectra of X/= axw +i/^.(B) Power spectra // = ayM + v { .
(C-106)
(1 -a B )x i = u i where B is the back-shift operator,
(C-107)
Bxi = kt_ :
Hence xi = «j (1 + aB + [aB]2 + ....)
(C-108)
Var(jt() = Var(u)(l + a2+ a4 + ...) = 02/(1 - a2)
(C-109)
yi = vi( l - a B + [ a B ] 2 - ...)
(C-110)
Var (y$ = Var (v)(l + a2 + a * + . . . ) = 0^1(1 -
(C-111)
Z= x + y
(C-112)
Zi - «,-(1 + aB + [aB]2 + ...) + v,
(C-113)
and
Similarly,
and
If we call
we have
and Var(z<) = Var («)/(1 - a2) + Var (v)/(l - a2) =
+ tryf2
(C-114)
This corresponds to the unshaded area alluded to in the introduction to this section. To see quantitatively that the short term variance is lower we can proceed in various ways. First, we can simply add the two plots, obtaining the results shown in Fig. C-15. We essentially get y modulated by x. For short time periods, the local variation is predominantly that o f y. Alternatively, we can note that the spectrum at high frequen cies is essentially that o f y as shown in Fig. C-16.
830
Appendix C
C-3
Time
Fig. C-15. Sum of the T w o Processes Xj - ax{ _-j + ut and yt - -ayj_-| + Vt
Fig. C-16. Spectrum of Combined Process.
This again implies that the short term fluctuations are mainly those due to y. Finally, we can look at the autocorrelation function, as seen in Fig. C -17, and note that for short time differences, deviations of the autocorrelation function from unity (i.e. “smooth ness” ) are due to y:
Fig. C-17. Autocorrelation Function for Combined Process.
In summary, the basic point is that we can “ achieve” <jy even though the net vari
ance of the system is a\ + Gy, over time scales short compared with the main frequen cy content o f “x” . Hence, in coming up with an error budget (for short time scales) we can reduce the variance from the “ nominal value” . For example, if we return to Table 5-15, we would expect errors (2) and (7) to vary much more slowly than the others. On the shorter time scale of the other errors we would thus expect to be able to estimate these “biases” and correspondingly reduce the system error. The effect of time scale on variance can be conveniently summarized in terms o f the Allen Variance a^it), which gives the variance in output measurements when those measurements are averaged over a time scale, t. If the output is a sum of inputs with different spectra, these can be identified in a plot o f <7^(0. At different time scales, different spectra become dominant. There is an optimum time scale that in volves a minimum variance. From the standpoint o f error budgeting, we need to iden tify an appropriate frequency band. This, in turn will yield the corresponding output variance.
C-4
Statistical Error Analysis
831
C.4 Example: Minimizing Cost Up to this point, we have not discussed how the error budget is distributed among the various components. We have only shown how a given set of errors contributed by the components can be added. Normally the flowdown o f requirements to the compo nents is accomplished by simply spreading the pain as evenly as possible. In this sec tion we briefly describe how the budget can be optimized in terms o f cost. An error budget should be formulated with an eye toward the sensitivity o f the cost o f each component to its accuracy requirement. However, cost figures are hard to come by. If an item can be bought off the shelf, there should be no problem. If a new item has to be developed, the risk and cost go up. The more challenging the require ment, the more surprises can arise which tend to add to the cost. For costing analyses, these distributions are often approximated by triangular dis tributions as shown in Fig. C-18. As the annotations in the figure indicate, the shape o f the triangle will correspond to the amount of risk, and the location to the expected cost. As the requirements on a given object are made more stringent, its triangle will start looking more risky. Fig. C-19 shows a hypothetical family o f distributions for a given type o f instrument, as the error bounds are tightened. Note that the highest pos sible cost grows much more rapidly than the “expected cost” , and the corresponding distributions become more and more skewed. The idea is to come up with a family o f such curves for each of the components. Having compiled all this information, we now must come up with our error budget— i.e. we must apportion out the total allow able system error to the various components in such a way as to minimize the cost. Conceptually, we want to superimpose curves of constant system cost onto a plot of the error budget constraint (Fig. C-20), and find the lowest cost consistent with the constraint, i.e. the point o f tangency. From this we can read off the optimal error bud get process as follows: 1.
For each component, we can translate the family o f cost curves into a curve o f accuracy vs cost. We then sum these to obtain curves of system cost. The sum mation procedure will require care, since it should account for correlations, and, perhaps more important, account for risk. This amounts to deciding which curves to add. A conservative approach would be to use the upper bound curves. This is usually too conservative. Note, for example, that if we push the performance o f one item, we may be able to back off on another— i. e. the costs tend to be correlated. We have seen that it is simple to add mean values, and standard deviations. Hence, we can find “mean plus n-sigma” curves for the system cost.
2.
Finding the error budget constraint curve, i.e. combinations o f component errors which are compatible with the allowed system error, is straightforward assuming we characterize the errors via their standard deviations. For exam ple, if we have n components, then the surface of all combinations o f sigmas which will RSS to a given system sigma is an n-sphere. For example, in 2dimensions we have a circle. This can be compared with the more conserva tive sum, given by the straight line in Fig. C-21. Other correlation values will give intermediate curves. These can now be overlaid on the cost curves to find the optimum vector of sigmas to create the error budget.
In a cost setting it is common to have to provide a probability distribution o f system cost— i.e. not simply an expected value or a mean plus n-sigma. Such distributions are
832
Appendix C
C-4
Triangle for a given object joins lower bound on cost estimates, most likely, and upper bound
Low risk object
High risk object A long tail {high upper bound) signifies high risk
Cost
Fig. C-18. Idealized Probability Density Functions for Low and High Risk Items.
Fig. C-19. Family of Probability Density Functions Parametrized Accordingly to Risk (or Performance)
Fig. C-20. Curves of Constant System Cost as a Function of Com ponent Performance Requirements— and Hence Risk— as Derived from the Cost-R isk Curves in Fig. C-19. Superimposed on a Curve of the Error Budget Constraint (Combination of Component Performances which Yield a Given System Performance). The opti mal error budget can be read off from this plot.
C-4
Statistical Error Analysis
833
Fig. C-21. Curves of Constant System Error Budget (Sum and RSS).
usually generated using Monte Carlo techniques based on the triangular distributions described above. The cost portion of the error budgeting problem described in this ex ample can also be handled directly via Monte Carlo methods. It should be noted that error budgets are not typically developed as outlined in this example. The error is often uniformly distributed among the components, or else is al located based on the known capabilities of existing components. We will generally have some awareness of the state o f the art, and will avoid setting unreasonable and, hence, expensive requirements. The error budgeting process usually involves some it erations and negotiation of “terrible injustices” [Williams, 1992].
References Feller, W. 1957, 1976. An Introduction to Probability Theory and its Applications. Vol.l (1957), Vol.2 (1966), Wiley. Meyer, P.L. 1972. Introductory Probability and Statistical Applications■ AddisonWesley. Papoulis, A. 1965. Probability, Random Variables and Stochastic Processes. McGraw-Hill. Parzen, E. 1960. Modern Probability Theory and its Applications. Wiley . Kac, M. 1989. Statistical Independence in Probability, Analysis and Number Theory. MAA. Box, G. and G. Jenkins. 1970. Time Series Analysis. Holden-Day. Karatzas, I. and S. Shreve. 1997. Brownian Motion and Stochastic Calculus. Springer. Oksendal, B. 1998. Stochastic Differential Equations. Springer. Williams, Michael. 1992. “Requirements Definition.” In Larson, Wiley J. and Wertz, James R. (eds.). Space Mission Analysis and Design, 2nd ed. Dordrecht, The Neth erlands and Torrance, CA: Kluwer Academic and Microcosm, Inc.
Appendix D Summary of Keplerian Orbit and Coverage Equations D. 1 D.2 D.3 D.4
Equations for Circular Orbits Equations for Elliptical Orbits Equations for Parabolic Orbits Equations for Hyperbolic Orbits
This appendix provides summary formulas in addition to those given in Table 2-4 in Sec. 2.1. Formulas are provided for all four types of Keplerian orbits— circular, el liptical, parabolic, and hyperbolic. See Sec. 2.1 for definition of orbit variables, deri vations, and conditions. See Table D -l for values of the gravitational constant, /x, for the Earth, Moon, Sun, and Mars. See Table E-l in App. E for values o f for all o f the major bodies in the solar system. See Sec. 9.1 for definition o f most of the geometric variables. Finally, numerical evaluations o f most o f the characteristics of the motion are contained in Appendix F for orbits about the Sun, Moon, and Mars and on the pag es preceding the inside rear cover for orbits about the Earth. fj, -
GM is the gravitational constant of the central body and is much more accurately known than either the mass, M, or the universal gravitational constant, G, the least accurately known of the funda mental physical constants.
For all of the orbit types, v is the true anomaly = the angle from perifocus to the orbiting body and M = n (t - T) is the mean anomaly, where t is the time of observation, T is the time o f perifocal passage, and n is the mean angular motion. Other parameters are defined in the table and discussed at the text reference cited in the table. T A B L E D-1.
Values of the Gravitational Constant, & for the Earth, Sun, Moon, and Mars. Values are those given by Cox [2000], which are in close agreement with those in use by the International Astronomical Union. See Table E-1 in App. E for other central bodies. Central Body
f* (m 3/s2)
& (mi-5/s)
Earth
3.986 004 41x1014
19,964,980.4
Moon
4.902 798 98x1012
2,214,226.5
Mars
4.283 200 00 x1013
6,544,616.1
Sun
1.327 124 38 x 1020
11,520,088,436
835
836
Appendix D
D-1
D.l Equations for Circular Orbits See Sec. 2.1 for definition of orbit variables, Sec. 9.1 for definition o f geometric variables, and Sec. 8.3.3 for application of the Euler axis formula as to satellite ground tracks. See rear end-papers for numerical evaluation of most Earth satellite formulas and App. F for properties o f orbits about the Moon, Sun, and Mars. Note: A bold “ C” is put in front o f each descriptor to denote formulas for circular or bits. C. Defining parameter:
a = semimajor axis = r - radius
(D -1)
(For a discussion of Keplerian orbits, see Sec. 2.1; for transformations between orbital elements and position and velocity, see Sec. 2.7.1.1 and 2.7.1.2; for the apparent shape of a circular orbit viewed from near-by, see Sec. 6.3.4.) C. Parametric equation:
x2 + v2 - a2
C. Distance from focus, r :
(D-2) (D-3)
r -a
C. Specific energy, £ (= Total energy per unit mass): (D-4) 2(3
2
a
C. Specific angular momentum, hi
h = |h| - *JJui
(D-5)
C. Angular momentum vector, h:
h= r x V
(D-6)
C. Nodal vector, N: C. Inclination, i:
N = zxh = zxh Ih i = acos (hz / h)
(D-7) (D-8)
(For a discussion of apparent inclination, see the Boxed Example at the end of Sec. 9.3.1.) C. Right ascension o f the ascending node, £2 : C. Flight path angle,
Q. - atan2 (Ny ,NX)
(D-9)
0
(D-10)
C. Perifocal distance, q :
q = a = r
(D-11)
C. Semi-parameter, p :
p = a
(D-12)
C. Semimajor axis, a:
a -
r
(D-13)
C, Eccentricity, e:
e = 0
(D-14)
< P jp a
=
C. Mean motion, n:
(D-15)
C. Mean anomaly, M :
M = M q + nt
(D-16)
C. True anomaly, V:
v = M
(D-17)
v = n
(D-18)
C. Rate o f change o f true anomaly, v :
D-1
Summary of Keplerian Orbit and Coverage Equations
C. Period, P :
(D-19)
P = 2 n / n = 2 n ^ a 3 I jil
= = = =
1.658 669 010 1.495 569 413 5.059 429 211 1.996 228 781
x x x x
V =
C. Range • range rate, r r
:
C. Areal velocity, A • C. Escape velocity, VE :
(for Earth; P in min, a in km) (for Moon; P in min, a in km) (for Mars; P in min, a in km) (for Sun; P in days, a in km)
1 0 a312 IO-3 a3/2 IO-4 a3/2 IO-10 a3/2
C. Velocity, V :
837
TTa = an
(D-20)
rr = 0
(D-21)
A = 0.5.^Jui = 0.5 a2n
(D-22)
VE = ^ 2 ji f a = ^ 2 V
(D-23)
C. Euler axis co-latitude, S'E :
8E - atan
rtsin* -ft)r + ncosi
(D-24)
d)E = inertial rotation rate of central body
= 0.004 178 074 deg/sec = 0.000 152 504 deg/sec - 0.004 061 249 deg/sec
(Earth) (Moon) (Mars)
(For a complete set of closed-form ground track equations, see Table 8-9 in Sec. 8.3 and Sec. 9.3.1.)
C. Euler rotation rate,
(Oe = nsin i / sin SE
:
(D-25)
= ^ w E + n1 + 2nCi)E cos i
C. Angular radius of the Earth, p : /?£ = = = -
p = asin (Re /a)
(D-26)
radius o f central body 6378.140 km (Earth) 1738.2 km (Moon) 3397 km (Mars)
(Assumes a spherical Earth. For the effect of oblateness, see Sec. 9.1.5.)
C. Distance to the horizon, D :
D = a cos p
(D-27)
= ifa2 - Re (For calculation of angles and distance for targets not on the horizon, see Sec. 9.1.1.)
C. Maximum Earth central angle, C. Instantaneous Access Area, IAA :
:
Xmax = 90 deg - p = acos (RE/a) IAA - KA( 1 - cos Xmax)
Ka = 2 k ~ 6.283 185 311
= 3602 / (2ti) * 20,626.480 62 = 2.556 041 87 x IO8
for area in steradians for area in deg2 for area in km2 (Earth)
(D-28) (D-29)
Appendix D
838
= 1.898 363 x IO7 = 7.250 550 X IO7
D-1
for area in km2 for area in km2
(Moon) (Mars)
(For alternative ways to calculate ground coverage, see Sec. 9.5.1; for other ground area formulas, see Table 9-7 in Sec. 9.5.1; for transformations between geocentric and access area coordinates, see the Boxed Example in Sec. 9.1; for projections of sensor fields of view onto the central body, see Sec. 9.1.4.) C. Area Access Rate, AAR :
AAR = 2 KA sin* max
C. Maximum time in view,
:
Tmax = PXmax /180 deg
(D-30) (D-31)
(For a complete set of target or ground station coverage formulas, see Table 9-4 in Sec. 9.4.1.) C. Maximum angular rate seen from ground, 6 ^ ^ : G
=
a 36Q° P(a - Re )
(D-32)
C. Node precession rate due to J2. AQJ2: f o f cos/ = Kj2 & nn cose
~
- —2.064 74 x IO14 ar11- cos i - -3.220 x IO11 ariri cos i = -3.483 x IO13 cr111 cos i
(D-33)
(Earth, a in km) (Moon, a in km) {Mars, a in km)
(In the numerical forms, ADJ2 is in deg / calendar day) C. Sun-synchronous inclination, iss: i„ = acos
JOucJ / 2 \
(D -34)
K n SP
= acos(-4.773 7 X lO"^ <j7/2) = acos (-3.061 x IO-12 a112) = acos (-1.505 x IO-14
(Earth, a in km) (Moon, a in km) (Mars, a in km)
(Where SP is the sidereal period o f the planet about the Sun in calendar days. For a discussion of Sun synchronous orbits, see Sec. 2.5.3.) C. Revolutions per day, Revfd: Rev/d = Day / P = 1,436.07 / P = 39,343 / P = 1,477.38 / P
(D-35) (Earth, P in min) (Moon, P in min) (Mars, P in min)
(This is the number of orbit revolutions for each rotation of the planet on its axis rela tive to the fixed stars (i.e., the inertial rotation period). “Day” is the length of the sidereal day for the given central body. For more accurate expressions, see the discus sion o f repeating ground tracks in Sec. 2.5.2.)
D-2
Summary of Keplerian Orbit and Coverage Equations
C. Node spacing, A/V:
AN - 360 deg (P / Day) = 0.250 684 P (Earth, P in min) = 0.009 150 3 P (Moon, P in min) = 0.243 675 P (Mars, P in min)
839
(D-36)
(For the equations for repeating ground track orbits, see Sec. 2.5.2.)
C. Maximum eclipse, TE:
TE = P(p / 180 deg) = P asin(i?g/ a)/180 deg
(D-37)
(For computation o f eclipse duration for any Sun angle, see Sec. 6.3.2; for the formu las for eclipse conditions, see Sec. 11.4.)
C. Sun angle constraints for terminator visibility, P 90° + % - p
< P ’ < 90° - $ + p
(D-38)
(£ is the dark angle defined in Sec. 11.5.2. For general conditions of terminator visi bility, see Table 11-3 in Sec. 11.5.2.)
C. Transit time for a spacecraft whose celestial coordinates are known, T: T = a + L ~ GST
(D-39)
(where a is the right ascension, L is the observer East longitude, and GST is the Green wich Sidereal Time. For computation o f observability times of spacecraft for which the celestial coordinates are known, see Table 9-6 in Sec. 9.4.4.
D.2 Equations for Elliptical Orbits See Sec. 2.1 for definition of orbit variables, Sec. 9.1 for definition of geometric variables, and Sec. 8.3.3 for application of the Euler axis formulas to satellite ground tracks. See rear end-papers for numerical evaluation of most Earth satellite formulas and App. F for properties o f orbits about the Moon, Sun, and Mars.
Note: An “ E” is put in front of each descriptor to denote formulas for Ecliptical Or bits. Subscript A stands for evaluation o f parameters at apogee; subscript P stands for evaluation of parameters at perigee.
E. Defining parameters:
a = semimajor axis b = semiminor axis
(D-40)
(For a discussion o f Keplerian orbits, see Sec. 2.1; for transformations between orbital elements and position and velocity, see Sec. 2.7.1.1 and 2.7.1.2.) 1
E. Parametric equation:
?— + ~
2
= \
(D-41)
E. Distance from focus, r: r =
+ = a( 1 - ecosE ) = ------ £------ = — — (1 + ecosv) 1 + ecosv 1 + ecosv
(D-42)
$40
Appendix D
D-2
E. Specific energy, £ (= Totai energy per unit mass): £ =
2
•—— = —— < 0
(D-43)
2a
r
E. Specific angular momentum, h: h = \h\ =
= rVco&Qjpa = r2v = rAVA = rpVp
E. Angular momentum vector, h: h = r x V
(D-45)
E. Nodal vector, N:
(D-46)
N = z x h = zxh//s V x h
r '
e = -------------- ----------------------------- (D-47)
E. Eccentricity vector, e:
//
E. Inclination, i:
r
i = acos (hz / h )
(D-48)
(For a discussion of apparent inclination, see the Boxed Example at the end of Sec. 9.3.1.)
E. Right ascension of the ascending node, Q:
Q = atan2 (Ay Nx)
(D-49)
E. Flight path angle, typaz fyfpa = atan2 (cos<j)jpa, sin0^a) = acos^ f j p / rV) .
(D‘ 50)
e sin E
s m (ftfpa =
1 - e 2 cos2 E 1 - e2
E. Perifocal distance, q = radius of perigee, rP : r= rp = a ( \ - e ) = = rA (l7 7 ) y 1+ e
= £ = * ( v* /v')
E. Semi-parameter,pi p = a ( l -e 2) ~ r (l + e cos v) = q ( l + e ) = b2/a = h2l\i
(D-52)
E. Radius of apogee, rA: rA — a(l -t e) — 2a - rp = b f rp = q
1+ e 1- e I
(D-53) a - c
E. Semimajor axis, a: _ rA + rp = _ _ vrpi _ = _ rA = _ p a = 2 1- e 1+ e 1- e
=_ _ rfi_ =_ i r/z \ 2e In 2
1/3 (D . 5 4 )
D-2
841
Summary of Keplerian Orbit and Coverage Equations
E. Semiminor axis, b :
b = a^l - e 2 - ^jap - yjrArP -
(D-55)
]ip I h
E. Eccentricity, e : e = let =
1+
2 eh*
a2 ~ b 2
_ rA ~ rp _
=
rA + r F
a
a
a
(D-56)
0<e< 1 3
i t
E. Mean motion, n:
n-
E. Mean anomaly, M:
M = M 0 + nt = E - e sin£
a
(D-57)
= —
(D-58)
E. Eccentric anomaly, E: 1- e tan(v / 2) 1+ e
E = atan2(sin £, cos£ ) = 2 atan
(D-59)
/ (a -
. fr . ) = acos -------- = asm —sin v = acos I
cos E =
)
ae
e + cos v
J
U
sinv-\/l - e ‘
sin£ =
1 4-
a -r
1 + *?cosv
as an iterative equation with successive estimates Ef. M - Ei + e sin Ei
^if+l = Ei +
1 - e c o s£ ; 1+ e
E. True anomaly, v : y = atan2 (sin v, cos v) = 2 atan
tan (£/2)
(D-60)
\\-e
= acos cos E - e cosv ----------------1 - ecosE
= acos
( cos E - e
1 - ecos E
_ sin ji-Jl - e2 smv = 1 - ecos E
as a power series in M\ S '>
v?»M -|-2esinAf + - e
4
1 * sin2A/— e sinM
13 + — e3 sin3M +
4
(D-61)
j
E. Rate of change of true anomaly, v : v• _- —
_ Vprp _
VA rA
= -na j -2y ir -
(D-62)
842
D-2
Appendix D
E. Period, P :
(D-63)
P = 27c / n = 2n ^ a 3 / fi
- 1.658 669 010 x IO-4 a3/2 = 1.495 569 413 x IO"3 a3/2 = 5.059 429 211 x IO 4 *3/2
(for Earth; P in min, a in km) (for Moon; P in min, a in km)
= 1.996 228 781 x IO-10 a3/2
(for Sun; P in days, a in km)
(for Mars; P in min, a in km)
E. Velocity, V: 1-
V =
(D-64)
1 + ecosv
=
fpa)
Ya = ! > Vp rA
5? II
A*
frl 1
2 nrA
a J
KrP,
■M
E. Radial velocity, Vr:
Vaz =
™
Vsin
1 -t- ecosv
E. Range • range rate, r r ;
;
rrr = e-Jajl sm E
E. Areal velocity, A '
A = 0.5^afl(/-e2') =
E. Escape velocity, VE:
VE = ^IfT Tr
E. Euler axis co-latitude,
S------------------------------------
£ ii
v a
E. Azimuthal velocity, Vaz:
IjA.
8 e = atan
(D-65) (D-66)
(D-67) 0.5 r 2v
(D-68) (D-69)
vsin i
-(Oi + vcosi
(D-70)
inertial rotation o f central body = 0.004 178 074 deg/sec (Earth) = 0.000 152 504 deg/sec (Moon) = 0.004 061 249 deg/sec (Mars)
(Og -
(For a complete set of ground track equations, see Table 8-9 in Sec. 8.3 and Sec. 9.3.1, which also includes a discussion of approximations for small eccentricities.)
D-2
Summary of Keplerian Orbit and Coverage Equations
E. Euler rotation rate, 0)E:
0)E = vsin i / sin SE
843
(D-71)
= ^]coE + v 2 + 2 vcqe cosi
E. Angular radius of the Earth, p:
p = asin (REl r)
(D-72)
RE = radius of central body
= 6378.140 km
(Earth)
= 1738.2 km
(Moon)
= 3397 km
(Mars)
(Assumes a spherical Earth. For the effect o f oblateness, see Sec. 9.1.5.)
E. Maximum angular radius of the Earth, p , ^ :
p max = asin(/?£ / rP)
(D-73)
E. Minimum angular radius of the Earth, Pmin'•
Pmin - asin(/?£ / rA)
(D-74)
E. Distance to the horizon, D:
D = r cos p = -\jr2 - RE
(D-75)
(For calculation o f angles and distance for targets not on the horizon, see Sec. 9.1.1.)
E. Maximum distance to the horizon, Dmax = rA COSPmin =
VrA
~ RE
(D-76)
E. Minimum distance to the horizon, Omi„: Dmin = rP cos Pmax = ^ rP ~ RE
(D-77)
E. Maximum Earth central angle, ^max = 90 deg - p = acos(J?£ / r)
E. Instantaneous Access Area, IAA:
IAA = KA{ 1 - cosAmax)
- 2n ~ 6.283 185 311 = 3602 / (2tc) - 20,626.480 62 = 2.556 041 87 x 10^ = 1.898 363 x IO7 = 7.250 550 x IO7
(D-78) (D-79)
for area in steradians for area in deg2 for area in km2 (Earth) for area in km2 (Moon) for area in km2 (Mars)
(For alternative ways to calculate ground coverage, see Sec. 9.5.1; for other ground area formulas, see Table 9-7 in Sec. 9.5.1; for transformations between geocentric and access area coordinates, see the Boxed Example in Sec. 9.1; for projections of sensor fields of view onto the central body, see Sec. 9.1.4.) E. A re a A ccess R ate, A A R i
AAR =
^ I s i n ^
7t
E. Maximum time in view, Tmax:
Tmax = 2Xmax / v
(D -80)
(D-81)
(For a complete set of target or ground station coverage formulas for elliptical orbits, see Table 9-5 in Sec. 9.4.2.)
844
Appendix D
D-2
E. Maximum angular rate seen from ground, 6^^:
&max ~ /
vr
D \ (D-82)
[r ~ % )
E. Nude precession rate due to J2, AI2j2: AQj2 = -1.5 J2n {Rg/ a)2 (1 - e2)~z cos i = KJ2 cr112 (1-e2) ~2 cos i = -2.064 74 x IO14 a~7/2(l - e2)-2 cos i = -3.220 x IO11 crlfe (1 - e2)-2 cos 1
(Earth, a in km) (Moon, a in km)
= -3.483 x IO13
(Mars, a in km)
(D-83)
(In the numerical forms, AQJ2 is in deg/calendar day.) E. Sun-synchronous inclination, i$5: 360 a K
j
2 \\
-
7 /2
*2)
(D-84)
spj
= acos (-4.773 7 x IO-15 a7/2(l - e2)2 ) = acos (-3.061 x 10~12 a7/2( 1 - e2)2 )
= acos (-1.505 x IO" 14 a7/2(l - e2)2 )
(Earth, a in km) (Moon, a in km) (Mars, a in km)
(Where SP is the sidereal period o f the planet about the Sun in calendar days. For a discussion of Sun synchronous orbits, see Sec. 2.5.3.)
E. Perigee rotation rate due to J2, Aa>j2 : A
(D-85)
= K/2 a -7/2( l - e2)‘ 2^ 2 -| s in 2 ij where Kn is the same as for AHJ2 (In the numerical forms, A coJ2 is in deg/calendar day. Note that the perigee rotation rate depends on the central body. However, the fact that perigee rotation goes to 0 at the critical inclination o f 63.4 deg is independent o f the central body.)
E. Revolutions per day, Rev/d: Rev/4 = DayIP = 1,436.07 I P = 39,434 / P = 1,477.38 / P
(D-86) (Earth, P in min) (Moon, P in min) (Mars, P in min)
(This is the number o f orbit revolutions for each rotation of the planet on its axis rela tive to the fixed stars (i.e., the inertial rotation period). “Day” is the length of the side real day for the given central body. For more accurate expressions, see the discussion of repeating ground tracks in Sec. 2.5.2.)
D-3
Summary of Keplerian Orbit and Coverage Equations
E. Node spacing, AN:
845
AN = 360 deg(P / Day)
= 0.250 684 P = 0,009 150 3 P = 0.243 675 P
(D-87) (Earth, P in min) (Moon, P in min) (Mars, P in min)
(For the equations for repeating ground track orbits, see Sec. 2,5,2.)
E. Maximum eclipse at a specified true anomaly, TE ~ 2p / v = (2 / v) asin(RE / r)
(D-88)
(For computation of eclipse duration for any Sun angle, see Sec. 6.3.2; for the formu las for eclipse conditions, see Sec. 11.4.)
E. Sun angle constraints for terminator visibility, ft 90° + | - p
< p < 90° - % + p
(D-89)
(£ is the dark angle defined in Sec. 11.5.2. For general conditions o f terminator visi bility, see Table 11-3 in Sec. 11.5.2.)
E. Transit time for a spacecraft whose celestial coordinates are known, T: (D-90)
T = a + L - GST
(where a is the right ascension, L is the observer East longitude, and GST is the Green wich Sidereal Time. For computation o f observability times o f spacecraft for which the celestial coordinates are known, see Table 9-6 in Sec. 9.4.4.)
D.3 Equations for Parabolic Orbits As discussed in Sec. 2.1 which defines the orbit variables, parabolic orbits represent the boundary between closed, negative total energy elliptical orbits and unbounded, positive total energy hyperbolic orbits. Sec. 9.1 defines the geometric variables in volved.
Note: A “P” is put in front o f each descriptor to denote formulas for Parabolic orbits. Subscript P stands for the value o f the parameter at perigee.
P. Defining parameters:
p = semi-latus rectum = semiparameter q = perifocal distance
(D-91)
(For a discussion o f Keplerian orbits, see Sec. 2.1; for transformations between orbital elements and position and velocity, see Sec. 2.7.1.1 and 2.7.1.2.)
P. Parametric equation: P. Distance from focus, r:
r2 = Aqy r = ----- ------ =
1 + cosv
(D-92)
+ ^ ~ q + D2 I 2 1 + ecosv
P. Specific energy, £ (= Total energy per unit mass):
£ = 0
(D-93) (D-94)
P. Specific angular momentum, h: h = |h| =
= r V cos (j>fpa = r2 v = rPVP
(D-95)
846
Appendix D
D-3
h= rxV
(D-96)
P. Angular momentum vector, h: P. Nodal vector, N:
N=zxh-ixh/A
(D-97)
P. Eccentricity vector, e:
vxh r e = —------ —
(D-98)
P. Inclination, i:
i
= acos (hz I h )
(D-99)
(For a discussion of apparent inclination, see the Boxed Example at the end of Sec.
9.3.1.) P. Right ascension of the ascending node, Q:
P. Flight path angle, <^a:
.Q = atan2(;V>,, N x)
(p^a = v /2 = acos (^/JuP/rV)
(D-100)
(D-101)
P. Perifocal distance, q = radius of perigee, rP : q - rp = p / 2
(D-102)
P. Semi-parameter,p:
p = 2q = hl lpL = r(l+cosv)
(D-103)
P. Semimajor axis, a:
a = »
(D-104)
P. Semiminor axis, b:
b - <=>°
(D-105)
P. Eccentricity, ei
e = 1
(D-106)
P. Mean motion, n:
n = 2■yjfi/ p3 = Q j tan(
+ \ tan3( ^ )
(D-107)
where t = time from perigee passage.
P. Mean anomaly, M:
M = M0 + nt = qD + D^I6
(D-108)
P. Parabolic anomaly, D:
D - -J2q tan(v / 2)
(D-109)
v = atan 2 (cos v, sin v) co sv = ( p - r ) l r
(D - 110)
P. True anomaly, v :
sin v = Dp / r^Jlq
P. Rate of change of true anomaly, v:
v ^ -Jup l r 1 - VPrP / r2
(D-111)
P. Period, Pi
P - ~
(D-112)
P. Velocity, V:
V = VE = yj2pLlr = J w / ( r cos#fpa) (D-113)
P. Velocity at perigee, Vp:
Vp =
[2 ~LL
—
(D -114)
V rP
P. Parabolic velocity = Escape Velocity, VE: P. Azimuthal velocity, V^:
Vp = *J2p. / r
V^ = Vcos(j>jpa = rv
(D -115) (D-116)
D-4
Summary of Keplerian Orbit and Coverage Equations
P. Radial velocity, Vr :
Vr = V s m ^ = Va z —
m
P. Range • range rate, rr'.
847
= r v esm V
q
1 +
(D-118)
rr = J j i D
P. A real velocity, A '
(D-117)
ecosv
2 = 0.5 r 2 v
A =
(D -119)
(For a complete set of ground track equations, see Table 8-9 in Sec. 8.3 and Sec. 9.3.1.)
All Euler Axis and coverage formulas for Parabolic Orbits are the same as for Hyperbolic Orbits
D.4 Equations for Hyperbolic Orbits As discussed in Sec. 2.1 which defines the orbit variables, hyperbolic orbits are unbounded with positive total energy. As shown in Figs. 2-1 and 2-3 in Sec. 2.1.1, the hyperbola has two discrete segments, only one of which represents the orbit. Section 9.1 defines the geometric variables involved. Section 12.7.3 discusses hy perbolic orbit parameters with respect to planetary arrival and departure.
Note: An “H” is put in front of each descriptor to denote formulas for Hyperbolic or bits. Subscript P stands for the value of the parameter at perigee. H. Defining parameters:
= semi-transverse axis (a < 0) b —semi-conjugate axis
(D-120)
a
(For a discussion of Keplerian orbits, see Sec. 2.1; for transformations between orbital elements and position and velocity, see Secs. 2.7.1.1 and 2.7.1.2; for planetary arrival and departure, see Sec. 12.7.3.)
H. Parametric equation:
(D -121) 2
2
- - ^ = 1 a2
b2
H. Distance from focus, r: q ( l + e)
(
e
1
+ ecosv
ycosH
1
+ ecosv
p?
r = —----------- =
J
7 ta n H
- 1 ---------
sinv
(D-122)
H. Specific energy, £ (- Total energy per unit mass): e
= -J L = Y 1 _ E > 0 2a 2 r
(D-123)
H. Specific angular momentum, h: h = |hj = J j]p = rVcos<j)fra - r 2 V = rP VP
H. Angular momentum vector, h:
h= r x V
(D-124) (D-125)
848
Appendix D
H. Nodal vector, N:
N = z x h = zx h /h
(D-126)
H. Eccentricity vector, e:
e = —* h - -
(D-127)
H. Inclination, i :
i = acos {hz l h )
(D-128)
D-4
p
r
(For a discussion of apparent inclination, see the Boxed Example at the end of Sec. 9.3.1.)
H. Right ascension of the ascending node, C2: H. Flight path angle, f a :
Q = atan2 (Ny, Nx)
/ rV]
(D-129) (D-130)
H. Auxiliary angle of the hyperbola, H: H = acos ( —— I = acos [ ----- ------ ] I a + rJ
(D-131)
I 1+ rh/pJ
H. Turn angle, yr. ^ = 2 a s i n f - l = 2asin ------- \ ------
Ve)
=
2
[l + v l q l f i .
- 2 a t a n f —1
(D-132)
Kb)
atan j — —— = 2 atan ^ VPV„ rP ) [pv^rj COS
)
(For a discussion of planetary fly-bys, see Secs. 2.6.3 and 12.7.1.)
H. Perifocal distance, q = radius of perigee* rP: _ ^M( h ^ e) = -£-(.e M q = rF = a( 1 - e ) = y P£ - = l + - l)
(D-133)
H. Semi-parameter,/>: y2
^2
p = ail - e2) = r(l +
a - - — - f-^-1 2s
a<
W )
=
Vi
- —— = e-l
(D-134) —
e2 - l
(D-135)
0
H. Semiminor axis = semi-conjugate axis, b : =
H. Eccentricity, e:
—
(D-136)
D-4
Summary of Keplerian Orbit and Coverage Equations
e = lei =
e>
1+
l£h2
a2 + b 2
1 + UrPL =-
=
1+
a
M
1+
JL =
a
849
U L y tI
(D-137)
fl
1
H. Mean motion, n:
(D-138)
n = ^J-jU / a
H. Mean anomaly, M:
M = Mq + nt = e sinh F - F
H. Hyperbolic anomaly, F:
F=
2 atanh
c - 1
V
e+l
2
(D-139) (D-140)
-------- tan —
. sin v ^ e 2 - 1 sinh F = ------ ---------1 + e c o sv
cosh F =
e + cosv 1 + e cos v
as an iterative equation with successive estimates Ff: F.
= F. + M ~ esinh Fi + Fi
1+ 1
(?cosh Fi
1
-1
H. True anomaly, V: v = atan2 (sinV, cosV) =
2 atan
r~2—r ^
e+l L F ------ tanh — e~ \
*2 asm ■(— a ■yJe - 1 tan H = 2atan o )
co s v =
cosh F - e 1 - ecosh F
sm v =
-sinh F«Je2 - 1 1 - ecosh F
(D-141)
2
Tie-----+1 tan —h p -1
2
H. Rate of change of true anomaly, v : v = ^[Jjp / r2 = Vprp / r~
(D-142)
H. Period, P:
(D-143)
H. Velocity, V:
P = V = j- r L _ !~ = r
a
(rcos$fya)
Vr
2fi
H. Velocity at perigee, Vp: Vr = ^
(D -144)
(D-145)
+ V~ = i t il + e)
D-4
Appendix D
850
H. Hyperbolic excess velocity, V^ : (D-146)
(The role of C3 , and related parameters in interplanetary mission analysis is dis cussed in Secs. 12.3.2 and 12.7.3 to 12.7.5.)
H. Reference launch energy, C3 ; H. Azimuthal velocity, V^: H. Radial velocity, Vr:
C3 =
(D-147)
Vaz - rv - V cos
(D-148)
Vr - Vsin <jfjpa _= I Vgze sin v =
H. Range • range rate, r r :
rV g s in V 1 + ecosv
(D-149)
sinh F
(D-150)
H. Areal velocity, A :
A = U a n ( \ - e 2)
(D-151)
H. Escape velocity, Vg;
VE = ^ 2 Hi r
(D-152)
H. Euler axis co-latitude, S 'E\
8 p = atan
G>£ = =
= =
rr =
vshh
~0)E -1- vcos* inertial rotation of central body 0.004 178 074 deg/sec (Earth) 0.000 152 504 deg/sec (Moon) 0.004 061 249 deg/sec (Mars)
(D-153)
(For a complete set of ground track equations, see Table 8-9 in Sec. 8.3 and Sec. 9.3.1.)
H. Euler rotation rate, £%:
(D-154)
(0E = vsinj'/sin<$£
= ^coE + v2 + 2vo)E cos i
H. Angular radius of the Earth, p: p = asin (% / rP) Re = radius of central body = 6378.140 km (Earth) = 1738.2 km (Moon) = 3397 km (Mars)
(D-155)
(Assumes a spherical Earth. For the effect of oblateness, see Sec. 9.1.5.) H. Maximum angular radius of the Earth, Pm^: p ^
H. Distance to the horizon, D:
= asin (Re I rP) (D-156)
D - rc os p = ^ r 2 - R e2
(D-157)
(For calculation of angles and distance for targets not on the horizon, see Sec. 9.1.1.)
D-4
Summary of Keplerian Orbit and Coverage Equations
851
H. Minimum distance to the horizon, Dmini Dmm = rP a »Pmai = ^ P ~ RE
H. Maximum Earth central angle, Xmax'.
?imax - 90 deg - p = acos (R^/r)
H. Instantaneous Access Area, IAA :
IAA = KA( 1 - co$Xmax) Ka = 2n * 6.283 185 311 for area in steradians = 3602 / (27c) - 20,626.480 62 for area in deg2 = 2.556 041 87 x 10* for area in km2 (Earth) = 1.898 363 x IO7 for area in km2 (Moon) = 7.250 550 x 107 for area in km2 (Mars)
(D-158) (D-159) (D-160)
(For alternative ways to calculate ground coverage, see Sec. 9.5.1; for other ground area formulas, see Table 9-7 in Sec. 9.5.1; for transformations between geocentric and access area coordinates, see the Boxed Example in Sec. 9.1: for projections of sensor fields of view onto the central body, see Sec. 9.1.4.) H. Area Access Rate, A A R :
AAR =
H. Maximum time in view, Tmax:
Tmax =
sin Xmax
iv
(D-161) (D-162)
(This approximation holds only when the rate of change of true anomaly and the dis tance to the Earth do not change strongly over the time in view. For more precise cal culations the start time and stop time of a viewing event should be calculated independently.) H. Maximum angular rate seen from ground, 0max •
<W
(0163)
H. M aximum eclipse at a specified true anomaly, Tg: Te = 2 p / v = ( T ) a s i n ( ^ )
(D-164)
(This approximation holds only when the rate of change of true anomaly and the dis tance to the Earth do not change strongly over the eclipse duration. For more precise calculations, the start time and stop time of the eclipse should be calculated indepen dently. For computation of eclipse duration for any Sun angle, see Sec. 6.3.2; for the formulas for eclipse conditions, see Sec. 11.4.) H. Sun angle constraints for terminator visibility, fi 90°+ £ - p < p < 9 0 ° - £ + p
(D-165)
(£ is the dark angle defined in Sec. 11,5.2. For general conditions of terminator visi bility, see Table 11-3 in Sec. 11.5.2.)
852
Appendix D
D-4
H. Transit time for a spacecraft whose celestial coordinates are known, T : T = a + L - GST
(D-166)
(where CCis the right ascension, L is the observer East longitude, and GST is the Green wich Sidereal Time. For computation of observability times of spacecraft for which the celestial coordinates are known, see Table 9-6 in Sec. 9.4.4.)
Appendix E Physical and Orbit Properties of the Sun, Earth, Moon, and Planets E. 1 Gravitational Constants of Major Solar System Bodies E.2 Physical Properties of the Sun E.3 Physical and Orbit Properties of the Earth Geocentric and Geodetic Coordinates on the Earth
E.4
Physical and Orbit Properties of the Moon Planetary and Natural Satellite Data
E.5
Planetary and Natural Satellite Data
This appendix provides physical and orbit data for the Sun, Moon, and planets. Properties of orbits about these bodies are listed in tables throughout the text. Most of these tables are listed on page facing the inside front cover. Numerical properties of orbits about the Sun, Moon, and Mars are given in Appendix F and for orbits about the Earth in the pages preceding the inside rear cover. For additional data on virtually all aspects of the Solar System, astrophysics, and astronomy, we highly recommend that the reader consult Cox [2000]. For a detailed explanation and high accuracy numerical methods and tables for computing planetary orbit ephemerides, see Seidelmann [1992]. For lower accuracy, but computationally convenient techniques, see Meeus [1998].
£.1 Gravitational Constants of Major Solar System Bodies In Table E-l, values of jU = GM are given to their full available accuracy. Other values are rounded. Values for the smaller satellites are estimates based on a typical density and the object’s size. Except for the Sun, the last 3 columns are evaluated at the object’s surface (or at the largest dimension for irregular bodies). Data from Cox [2000] and Seidelmann [1992]. TABLE E-1. Object
Gravitational Parameters fo r Orbits About Major Solar System Bodies. GM (m3/s2)
Orbital Velocity (km/s)
Sun (at surface) 1.327 124 3 x 1020 1.327 124 3 x 1020 Sun (at 1 AU)
436.822 29.716
Orbit Period (min)
Escape Velocity (km/s)
167.0 529,642
617.760 42.024
85.0
4.250 10.362
PLANETS AND SATELLITES
Mercury
2.203 4 x 1013
Venus
3.249 x 1014
3.005 7.327
853
86.5
854
Appendix E
TABLE E-1.
E-1
Gravitational Parameters for Orbits About Major Solar System Bodies.
Object Earth Moon
Mars
fi - GM (m3/s2) 3.986 004 41 x 1014 4.903 226x1012 4.283 2 x 1013
Phobos
7.093 x 105
Deimos
1.59 x 105 1.2669x1017
Jupiter Io
Orbital Velocity (km/s)
84.5 108.3
11.180 2.376
3.551 0.007
100.2
5.022
192.9 170.7
0.010 0.007
177.8
59.534
0.005 42.097 1.805 1.431
9.88786 x 1 0 12 7.17925x 1012
Amalthea
4.8 x 1 0 8
Himalia
6.3 x 108
Elara Pasiphae
5.3 x 107 1.3 x 107
Sinope Lysithea Carme
Ganymede Callisto
Escape Velocity (Km/s)
7.905 1.680
5.9597x1012 3.202727 x 1012
Europa
Orbit Period (min)
106.2
2.552
114.6
2.023
1.938
142.4
1.728
145.6
2.740 2.444
0.061
226.5
0.086
0.086 0.037 0.027
103.1 114.7 69.2
0.122 0.052 0.039
5.3 x 106 5.3 x 106
0.020 0.021
75.1 59.6
0.028
6.0 x 106
0.020
78.5 64.1
0.030 0.028
Ananke
2.7 x 106
0.016
Leda Thebe
4.0 x 10s 5.3 x 107
0.009
Adrastea
1.3 x 10s 6.0 x 106
0.010 0.017
134.4 120.9
0,044 0.014 0.025
25.088
251.6
35.480
Metis
Saturn
3.7934 x 1016
0.031
58.5 184.9
0.023 0.013
Mimas
2.50 x 109
0.109
200.2
0.155
Enceladus Thethys
4.7 x 109 4.18 x 1010
0.135 0.279
198.8
0.191
200.7
0.395
Dione
7.3 x 1010
0.362
162.0
0.512
Rhea
1.54 x 1011
0.449
Titan
8.9780 x IO12
1.867
178.1 144.4
0.635 2.641
Hyperion
1.3 x 109
0.086
218.9
0.122
Iapetus Phoebe
1.1 x 1011 2.7 x 107
0.386 0.016
195.0
0.545 0.022
Janus Epimetheus
1.28 x 10s 3.6 x 107
0.036
Helene
1.3 x 106 7.7x105
0.009
218.9
0.012
0.007
7.7 X 10s 1.45 x 106
0.007 0.009
218.9 218.9
0.010 0.010
218.9
0.013
9.34 x 106 8.67 x 106
0.011 0.013
689.7 458.6
0.016 0.018
2.3 x 105
0.005
218.9
0.007
Telesto Calypso Atlas Prometheus Pandora Pan
0.023
739,5 279.5 316.2
0.051 0.032
£-1
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
TABLE E-1.
855
Gravitational Parameters for Orbits About Major Solar System Bodies. (m3/s2)
Orbital Velocity (km/s)
5.7951 x 1015
/*= GM
Object
Orbit Period (min)
Escape Velocity (km/s)
15.058
177.8
21.295
Ariel Umbriel
9.01 x 1010 7.81 x 1010
0.394 0.365
154.6 167.6
0.557 0.517
Titania Oberon
2.36 x 1011
0.546
151.2
0.773
2.01 X 1011
0.514
155.2
0.726
Miranda
4.4 X 109
0.135
186.0
0.191
Cordelia
9.2 x 105
0.008
161.5
0.012
Ophelia Bianca
1.4 x 106
0.010
161.5
0.014
3.9 x 106
0.014
161.5
0.019
1.3 x 1 0 7 8.3 x 106
0.020
161.5
0.018
161.5
0.028 0.025
Juliet
3.1 x 107
0.027
161.5
0.039
Portia Rosalind Belinda
6.6 x 107
0.035 0.018
161.5
0.050
0.021
161.5 161.5
0.025 0.030
Puck
1.9 x 1 0 8
0.050
161.5
0.071
Caliban
1,1 x 1Q7
0.019
161.5
0.028
Sycorax
9.1 x 107 6.8354x1015
0.039
161.5
0.055
Uranus
Cressida Desdemona
Neptune
8.3 x 106 1.5 x 107
16.614
156.1
23.496
Triton
1.43 x 1012
1.027
137.9
Nereid Naiad
1.3 x 109 1.0 x 107
0.089 0.019
200.9 160.8
1.453 0.125 0.027
Thalassa
2.7 x 107
0.026
160.8
0.037
Despina
1.7 x 10®
Galatea
2.1 x 108
0.048 0.051
160.8 160.8
0.068 0.073
Larissa
4.77 x IO8
0.068
160.8
0.096
Proteus
4.40x109
0.142
160.8
0.201
8.7x1011
0.852
146.9
1.205
1.08 x 1011
0.427
145.4
0.604
115.4
0.586
115.9 104.0
0.334 0.357
Pluto Charon
ASTEROIDS LARGER THAN 300 KM DIAMETER
Ceres Pallas Vesta
7.83 x 1010 1.46 x 1010
0.414: 0.236
1.59 x 1010
0.252
856
Appendix £
E-2
E.2 Physical Properties of the Sun TABLE E-2.
Physical Properties of the Sun. Data from Cox [2000].
Radius of the photosphere
6.960 33 x 105 km
Angular diameter of the photosphere at 1 AU
0.530 69 deg
Mass
1.989 1 X 1030 kg
Mean density
1.409 g/cm3
Gravity at surface
2.740 X 104 cm/sec2
Moment of inertia
5.7 x 1053 gcm 2
Angular rotation velocity at equator
2.85 x 10-6 rad/sec
Angular momentum (based on surface rotation)
1.63x 1048 gem2 /sec
Escape velocity at solar surface
6.177 x 107 g cm/sec
Total radiation emitted
3.845 X 1026 W
Total radiation per unit area at 1 AU
1367 W/m2
Apparent visual magnitude at 1 AU
-26.75
Absolute visual magnitude (magnitude at distance of 10 parsecs)
+4.82
Color index, B-V
+0.65
Spectral type
G2 V
Effective temperature
5777 K
Inclination of the equator to the ecliptic
7.25 deg
Adopted period of sidereal rotation
25.38 days
Corresponding synodic rotation period (relative to Earth)
27.275 days
Oblateness: semidiameter equator-pole difference
0 ”0086
Velocity relative to nearby stars
19.7 km sec
TABLE E*3.
Sunspot Cycles, Maxima, and Minima. Largest Smoothed Year of Monthly Meant Maximum*
Year of Minimum
Smallest Smoothed Monthly Mean
-1 2
1610.8
—
1615.5
-11
1619.0
—
1626.0
-1 0
1634.0
—
1639.5
-9
1645.0
—
1649.0
-8
1655.0
—
1660.0
-7
1666.0
—
1675.0
—
9.0
4.5
15.0
-6
1679.5
—
1685.0
—
5.5
4.5
10.0
Sunspot Cycle Number
Cycle Length (Years)
Rise to Max (Years)
Fall to Min (Years)
—
4.7
3.5
—
7.0
8.0
10.5
—
5.5
5.5
13.5
—
4.0
6.0
9.5
5.0
6.0
11.0
—
8.2
E-2
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
TABLE E-3.
857
Sunspot Cycles, Maxima, and Minima. (Continued) Smallest Smoothed Monthly Mean
Largest Smoothed Monthly Meant
Cycle Length (Years)
Rise to Max (Years)
Fall to Min (Years)
—
3.5
5.0
1705.5
—
7.5
6.5
12.5
1718.2
—
6.2
5.3
12.7
—
1727.5
—
4.0
6.5
9.3
—
1738.7
—
4.7
6.3
11.2
Sunspot Cycle Number
Year of Minimum
-5
1689.5
—
1693.0
-4
1698.0
—
-3
1712.0
—
-2
1723.5
-1
1734.0
Year of Maximum*
8.0
0
1745
—
1750.3
92.6
5.3
4.9
11.6
1
1755.3
8.4
1761.5
86.5
6.2
5.0
11.1
2
1766.5
11.2
1769.8
115.8
3.3
5.7
8.3
3
1775.5
7.2
1778.4
158.5
2.9
6.4
8.6
3.4
10.2
9.8 17.0
4
1784.8
9.5
1788.2
141.2
5
1798.4
3.2
1805.2
49.2
6.8
5.5
6
1810.7
0.0
1816.3
48.7
5.6
7.1
11.1
7
1823.4
0.1
1829.9
71.7
6.5
4.0
13.6
8
1833.9
7.3
1837.3
146.9
3.4
6.3
7.4
9
1843.6
10.5
1848.2
131.6
4.6
7.8
10.9
10
1856.0
3.2
1860.2
97.9
4.2
7.1
12.0
11
1867.3
5.2
1870.7
140.5
3.4
8.3
10.5
12
1879.0
2.2
1863-9
74.6
5.0
6.3
13.3
13
1890.3
5.0
1894.1
87.9
3.8
8.0
10.1
14
1902.1
2.6
1906.2
64.2
4.1
7.5
12.1
15
1913.7
1.5
1917.7
105.4
4.0
6.0
11.5
16
1923.7
5.6
1928.3
78.1
4.6
5.5
10.6
17
1933.8
3.4
1937.3
119.2
3.5
6.9
9.0
151.8
3.2
6.9
10.1
201.3
4.0
6.5
10.9
18
1944.2
7.7
1947.4
19
1954.3
3.4
1958.3
110.6
4.1
7.6
10.6
164.5
3.5
6.8
11.1
7.2
9.6
20
1964.8
9.6
1968.9
21
1976.5
12.2
1980.0
22
1986.8
12.3
1989,6
158.5
2.8
23
1996.8
5
2000.6
170.1
3.8
—
11.0
112.9
4.7
6.3
10.9
Mean Cycle
6
* When observations permit, a date selected as either a cycle minimum or maximum is based in part on an average of the times extremes are reached in the monthly mean sunspot number, in the smoothed monthly mean sunspot number, and in the monthly mean number of spot groups alone. Two more measures are used at time of sunspot minimum; the number of spotless days and the frequency of occurrence of “old'1and “new" cycle spot groups. t The smoothed monthly mean sunspot number is defined here as the arithmetic average of two sequential 12-month running means of monthly mean numbers.
858
Appendix E
3 00
E-2
r
Year
Fig. E-1.
Historical Monthly 10.7 cm Radio Flux from the Sun (F10.7 Index) Since January 1947. For daily variations, see Fig. 2-19 in Sec. 2.4.
250 r
Fig. E-2.
Historical Smoothed Sunspot Values from the 18th Century to Present.
E-3
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
859
E.3 Physical and Orbit Properties of the Earth TABLE E-4.
Physical and Orbit Properties of the Earth. Data from Cox [2000], Muller and Jappel [1976]; Astronomical Almanac [1999],
Equatorial radius, a
6,378.136 km
Flattening factor (ellipticity), / = (a - c) la
1/298.257 = 0.003 352 81
Polar radius,’ c
6.356 753 x106 m
Mean radius,* (a2c)1/3
6,371.00 km
Eccentricity,* (a2- c2)V2/a
0.081 818
Surface area
5.100 657 x 108 km2
Volume
1.083 207 x 1012 km3
Ellipticity of the equator (amax- amin)/a mean
- 1.6 x IO"5
Longitude of the maxima
2 0°W, 160°E
Ratio of the mass of the Sun to the mass of the Earth
332 946.038
Geocentric gravitational constant, GM^ = txE
3.986 00441 x 1 0 l4 m3/s2
Mass of the Earth
5.973 7x1024 kg
Mean density
5.514 8 g/cm3
Gravitational field constants (See Eq. (2-24) in Sec. 2.4.) J2 = +1.082 626x 10- 3 J3 = -2.533 x 10“ 6 J 4 - - 1.618 6 x 10-6 Mean distance of Earth center from Earth-Moon barycenter
4,671 km
Average lengthening of the day (See Fig. 4-1 in Sec. 4.1.)
0.0015 seo/century
General precession in longitude (i.e., precession of the equinoxes) per Julian century, at epoch 2000
1.396 971 28 x 10-2 deg/century
Rate of change of precession
+6.184 x 10“ 4 deg/century2
Rate of change of the obliquity (Tin Julian centuries)
(-1.301 25 x IO” 2 7 )(1.64x10-6 7 2 + 5 .0 x1 0 -7 73) deg
Amplitude of the Earth’s nutation
2.556 25x10^-3 deg
Sidereal period of rotation, epoch 2000
0.997 269 632 3de= 86 164.100 4 s = 23h56m4.100 4 s
Length of tropical year (ref. = T), epoch 2000
3.155 692 5 x 107 S = 365.242 189 7de
Length of anomalistic year (perihelion to perihelion), epoch 2000
3.155 843 322 2 x 107 S = 365.259 643 77 de
* Based on adopted values of / and 3-
860
Appendix E
E-3
Altitude (km)
Fig. E-4.
Mass Spectrometer Incoherent Scatter (MSIS) Atmospheric Species Percentage Composition vs. Altitude. [Hedin, 1987]. For atmospheric density, see Fig. 2-20 in
Sec. 2-4.
TABLE E-5.
Atmospheric Layers and Transitions. Data from Cox [2000].
Layer Troposphere
Height, h (km) 0-11
Tropopause
11
Stratosphere
11-48
Stratopause
48
Mesophere
48-85
Mesopause Thermosphere Exobase
85 85-exobase
Characteristics Weather, T decreases with h, radiative-convective equilibrium Temperature minimum, limit of upward mixing of heat 7 increases with h due to absorption of solar UV by 0 3, dry Maximum heating due to absorption of solar UV by O3
T decreases with h Coldest part of atmosphere, noctilucent clouds T increases with h, solar cycle and geomagnetic variations
500-1000 km
Exosphere
> exobase
Region of Rayleigh-Jeans escape
Ozonosphere
15-35 km
Ozone layer (full width at e-1 of maximum)
Ionosphere
> 70 km
Homosphere Heterosphere
< 85 km > 85 km
Ionized layers Major constituents well-mixed Constituents diffusively separate
E-3
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
TABLE E-6 .
Altitude 0 1
2 3 4 5 6 S 10 15 20 30 40 50 60 70 80 90 100 110 120 150 220 250 300 400 500 700 1000
861
Atmospheric Layers and Transitions. P = pressure, T = temperature, p = mass density, N = number density, H = scale height, and I = mean free path. Data from U.S. Standard Atmosphere. [COESA, 1976], See also tables inside the rear cover for additional atmospheric data and related orbit decay parameters.
log P (Pa)
(K)
i°9 p (kg m~3)
log N (m-3)
(km)
log I (m)
+5.006 +4.95 +4.90 +4.85 +4.79 +4.73 +4,67 +4.55 +4,42 +4.08 +3.74 +3.08 +2.46 +1.90 +1.34 +0.72 +0.022 -0.74 -1.49 -2.15 -2.60 -3.34 -4.07 -4.61 -5.06 -5.84 -6.52 -7.50 -8.12
288 282 275 269 262 256 249 236 223 217 217 227 250 271 247 220 199 187 195 240 360 634 855 941 976 996 999 1000 1000
+0.088 1 +0.046 0 +0.002 86 -0.041 3 -0.087 -0.133 -0.180 -0.279 -0.384 -0.71 -1.05 -1.73 -2.40 -2.99 -3.51 ^4.08 -4,73 -5.47 -6.25 -7.01 -7.65 -8.68 -9.59 -10-22 -10.72 -11.55 -12.28 -13.51 -14.45
25.41 25.36 25.32 25.28 25.23 25.19 25.14 25.04 24.93 24.61 24.27 23.58 22.92 22.33 21.81 21.24 20.58 19.85 19.08 18.33 17.71 16.71 15.86 15,28 14.81 14.02 13.34 12.36 11.74
8.4 8.3 8.1 7.9 7.7 7.5 7.3 6.9 6.6 6.4 6.4 6.7 7.4 8.0 7.4 6.6 e.o 5.6 6.0 7.7 12.1 23.0 36.0 45.0 51.0 60.0 69,0 131.0 288.0
-7.2 -7.1 -7.1 -7.0 -7.0 -7.0 -6.9 -6.8 -6.7 -6.4 -6.0 -5.4 -4.7 -4.1 -3.6 -3.0 -2.4 -1.6 -0.85 -0.10 +0.52 +1.52 +2.38 +2.95 +3.41 +3.80 +4.89 +5.86 +6.49
T
H
E.3.1 Geocentric and Geodetic Coordinates on the Earth The geocentric latitude, 0', of a point, P, on the surface of the Earth is the angle at the Earth’s center between P and the equatorial plane. The geodetic or geographic lat itude, 0 , is the angle between the normal to an arbitrarily defined reference ellipsoid (chosen as a close approximation to the oblate Earth) and the equatorial plane. Astro nomical latitude and longitude are defined relative to the local vertical, or the normal to the equipotential surface of the Earth. Thus, astronomical latitude is defined as the angle between the local vertical and the Earth’s equatorial plane. Maximum values of the deviation of the vertical, or the angle between the local vertical and the normal to a reference ellipsoid, are about 1 minute of arc. Maximum variations in the height be tween the reference ellipsoid and mean sea level (also called the equipotential surface) are about 1 0 0 m. The shape of the reference ellipsoid is most commonly defined by the ellipticity or flattening factor, f = ( a - b)/a ~ 1/298.256 42 = 0.003 352 819, where a is the equa torial radius of the Earth and b is the polar radius. Also used is the eccentricity of the reference ellipsoid, e = (a2- b2)m la ~ 0.081 819 301. These are related by:
862
Appendix E
E-3
(E-1) f = 1- V l - e 2
(E-2)
On the surface of the Earth, the geodetic and geocentric latitude are related by: tan0 = tan
Fig. E-5.
Relationship Between Geocentric Latitude, 0\ and Geodetic Latitude, 0.
Geocentric coordinates are commonly expressed as Cartesian coordinates, (x, y, z). We then define the geocentric latitude,
tan 0 ' = z f p
(E-5)
Given the geodetic coordinates, 0and h, as defined in Fig. E-5, we can immediately determine the geocentric coordinates from: P = (w0 +/i)cos
z =
(E-6 ) (E-7)
where the radius of curvature o f the ellipse, N q, is given by: AL =
*
y j l - e 2 sin2 <j)
(E-8 )
E-3
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
863
Determining the geodetic coordinates from the geocentric coordinates is more com plex and requires an iterative technique. The approach used here is that of Nievergelt and Keeler [2000], which includes references to a number of earlier, less satisfactory methods. W ith a single iteration, this approach is good to 2 x 1 0 ^ deg for the geodetic latitude and 1 mm in geodetic altitude. Successive iterations can improve this, al though that would rarely be needed. The 4-step iterative approach is as follows: _______
Step 1
12
Set the iteration counter, n - 0. Compute p = V* +y timate: h0 = 1 - 1 / ^ j ( p / a ) 2 + ( z / b ) 2
2
and the initial altitude es
VP 2 + z 2
(E-9)
Step 2 00
Compute the cosine, u0 = cos(A), and sine, v0 = sin(A), of the initial latitude estimate = X and the initial value of the intermediate variable, w0: C 2P
Z
? *
uo ~ I d ?
- > *\(J *p i + z L
z
p
f.
v0 = j , ?
2 2
w Q- y j l - e v0
(E-10)
where
a = b la
(E -ll)
Step 3
Compute the cosine, un+ j, and sine, v„+1 , of the improved latitude estimate the corresponding value wn+1, and the improved altitude estimate hn+\. \ a a 2 + h nw n ]p
“»+i = )
?
1
^ [^ c r2
J
(E-12)
p z + [ a + hn w n ] \ 2
[a + knw n ]z vn+i =
L , ............ ......................
|
y[acr2 +Vv>„]
K
+
1 =
J
p 2 + [ a + hn w n ] 2 Z 2
2 p
a u n+ 1 ; w n+1 _
+
_
av \+ \ w n+ 1
Step 4 Compute 0W+j = arctan(vn+1/«;i+|) with a standard algorithm that is stable near ±90
864
Appendix E
E«4
E.4 Physical and Orbit Properties of the Moon TABLE E-7.
Physical Parameters of the Moon. For an extended discussion of lunar proper ties see Eckart [1999] and Heiken, et al., [1991].
Radii: (a) Toward Earth, (b) Along orbit, (c) Toward pole
Mean radius (b + c) / 2 ............................................................... ...................... 1,738.2 km .......................... 1.09 km ...........................0.31 km ...........................0.78 km Semi-diameter at mean d ista n ce .................................................. .......................... 15'32".6 ............ 7.3483 x 1022 kg Mean de nsity.........................................................................................
Surface gravity................................................................................. Surface escape ve lo c ity ................................................................. Extreme range................................................................................. Inclination of orbit to ecliptic oscillating ±9'with period of 173 d , Sidereal period (fixed stars)............................................................. Mean orbital s p e e d ......................................................................... Synodical month (new Moon to new M o o n ).................................. Surface area of Moon at some time visible from Earth.................. Inclination of lunar equator To ecliptic..................................................................................... To orbit.........................................................................................
TABLE E-8.
162.2 cm/s2 - 0.17g .356 400-406 700 km ...................... 5° 8'43".42 .................... 27.321 661 .................... 29.530 588 .................................59% ...................... 1° 32' 32".7 .............................. 6 ° 41'
Orbit of the Moon About the Earth.
Sidereal mean motion of Moon ........................................... . .2.661 699 527 x 10-6 rad S-1
Mean distance of Moon from Earth ..................................... ........................ 384 401 ± 1 km ...................... 60.27 Earth radii ............................ 0.002 570 AU Equatorial horizontal parallax............................................... ................................ 57'02".608 at mean distance ............................................................. .............................. 3,422",608 Mean distance of center of Earth from Earth-Moon barycenter.................... 4.671 x 103 km Mean eccentricity................................................................. .................................. 0.054 90 Mean inclination to e clip tic................................................... .............................. 5.145 394° Mean inclination to lunar equator........................................ ......................................... 6°41' Limits of geocentric declination ........................................... .........................................+29° Period of revolution of n o d e .............. ............................... Period of revolution of perigee............................................ ...................... 8.849 Julian year 1,023 ms-1 _ o.OOO 591 AU/d Mean orbital s p e e d ................................................................... 0.00272 ms-2 = 0.0003 g Mean centripetal acceleration ............................................. Optical libration in longitude (selenocentric displacement) , ............................ +7.883 deg Optical libration in latitude (selenocentric displacement) , . , Saros = 223 lunations = 19 passages of Sun through node =; 6,585 1/3 days Moment of inertia (about rotation axis)................................ ................................ 0.396/1% Gravitational potential te rm ...................................................... ...................... J 2 = 2.05 x 10-4
No, of strong mascons on the near side of the Moon.......... .......... 4 exceeding 80 milligals +107 C (day), 153 C (night) Mean surface temperature................................................... Temperature extremes......................................................... ...................... -233 C, +123 C Moon’s atmospheric density .. ~104 molecules cm-3 (day); 2 x 105 molecules cm-3 (night) No. of maria & craters on lunar surface w/diam. > d ___ 5 x 1010ch20per 106 km2 (din m)
E-5
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
TABLE E-9.
Gravity Field of the Moon.
a = ( C - B ) / A = 0.000 400
C /M H Z= 0.392
/?= (C —A ) ! B - 0.000 628
1= 5,552".7= 1*32'32
Y=
865
( B—A ) / C = 0.0002278
C20 = ~ 0.000 202 7
£30 = - 0.000 006
C32 = + 0.000 004 8
C22 = + 0.000 022 3
C31 = + 0.000 029
S32 = + 0.000 001 7
S31 = + 0.000 004
C^3 = + 0.000 001 8 §33 = - 0.000 001
Phase Law and Visual Magnitude of the Moon A summary of the visual magnitude of the Moon as a function of distance and phase is provided in Section 11.6. (See Table 11-5 for Moon’s phase law.) At the mean distance of the Earth, the visual magnitude of the Moon at opposition is -12.74. However, at first and last quarters, when half of the visible surface of the Moon is illuminated the intensity drops to only 8 % of the full Moon value and the brightness drops by 2.74 magnitudes. For a more extended discussion see Cox [2000].
E.5 Planetary and Natural Satellite Data TABLE E-10. Orbit Data for the Planets. Orbit elements are defined with respect to the mean ecliptic and equinox of J2000.0 (epoch JD 2,451,545.0). Data from Cox [2000] and Setdelmann [1992]. Mean Distance (AU)
Sidereal Period (Julian yrs.)
Planet
Eccentricity
M ercury
0.205 630 69
0.387 098 93
0.240 844 45
Synodic Period (d ) 115.877 5
Venus
0.006 773 23
0.723 321 99
0.615 182 57
583.921 4
Earth
0.016 710 22
1.000 000 11
0.999 978 62
Mean Daily Motion, n (deg)
Orbital Velocity (km/s)
4.092 377 06
47.872 5
1.602 168 74
35.021 4
0.985 647 36
29.785 9
Mars
0.093 412 33
1.523 662 31
1.880 711 05
779.936 1
0.524 071 09
24,130 9
Ju p ite r
0.048 392 66
5.203 363 01
11.856 525 02
398.884
0.083129 44
13.069 7
Saturn
0.054150 60
9.537 070 32
29.423 519 35
378.0919
0.033 497 91
9.672 4
83.747 406 82
Uranus
0.047167 71
19.191 263 93
369.656
0.011 769 04
6.835 2
Neptune
0.008 585 87
30.068 963 48
163.723 204 5
367.486 7
0.006 020 076
5.477 8
P luto
0.248 807 66
39.481 686 77
248.020 8
366.720 7
0.003 973 966
4.74 9
Longitude of Perihelion (deg) 77.456 45 131.532 98 102.947 19 336.040 84
Planet Longitude on Jan. 1.5 2000 (deg) 252.250 84 181.979 73 100.464 35 355.453 32
TABLE E-11. Orbit Data for the Planets.
Planet M ercury Venus Earth Mars
Inclination to Ecliptic (deg) 7.004 87 3.394 71 0.000 05 1.850 61
Longitude of Ascending Node (deg) 48.331 67 76.680 69 -11.260 64 49.578 54
Last Perihelion before 1999 1998 Dec. 2 1998 Sept. 7 1998 Jan. 4 1998 Jan. 7
866
Appendix E
E-5
TABLE E-11. Orbit Data for the Planets, (continued).
Planet Ju p ite r Saturn
Inclination to Ecliptic (deg) 1.305 30 2.484 46
Longitude of Ascending Node (deg) 100.55615 113.715 04
Longitude of Perihelion (deg) 14.753 85 92.431 94
Planet Longitude on Jan. 1.5 2000 (deg) 34.404 38 49.944 32
Uranus Neptune Pluto
0.769 86 1.76917 17.141 75
74.229 68 131.721 69 110.303 47
170.964 24
313.232 18
44.971 35 224.066 76
304.880 03 238.928 81
Last Perihelion before 1999 1987 Jul. 10 1974 Jan. 8 1966 May 20 1876 Sept. 2 1989 Sept. 5
TABLE E-12. Physical Data for the Planets. Data from Cox [2000].
Planet
Mass (102* kg)
Radius (km)
Flattening (geom.)
Mean Density (g/em3)
M ercury
0.330 22
2,439.7
0
5.43
Venus
4.869
6,051.8
0
5.24
Earth
5.9742
6,378.14
0.003 353 64
5.515
Mars
Incl. oJ Equator to Orbit (deg)
J2 ( xlO3)
Sidereal Rotation Period (d) 58.646 225
0
0.027
-243.019 99
177.3
1.082 63 0.997 269 632 3
2345
0.641 91
3,397
0.006 476 3
3.94
1.964
1.02 595 675
Ju p ite r
1898.7
71,492
0.064 874 4
1.33
14.736
0.413 538
25.19 3.12
Saturn
568.51
60,268
0 .097 962 4
0.7
16.298
0.444 009
26.73
Uranus
86.849
25,559
0.022 927 3
1.3
12
-0.718 333
97.86
Neptune
102.44
24,764
0.0171
1.76
3.411
0.671 250
29.58
P luto
0.013
1,195
0
1.1
-6.387 246
119.61
TABLE E-13. Photometric Data for the Planets. So, Cl = Solid, cloud; for lowest visible surface. Data from Cox [2000]. Visual Magnitude
Effective
Temperature
(K)
Visible Surface
-0.2
—
So
-4.22
-230
Cl
-255
So, Cl So
Geometric Albedo
1*1,0)
Mercury
0.106
-0.42
Venus
0.650
-4.40
—
Planet
V0
Earth
0.367
Mars
0.150
-1.52
- 2.01
-212
Jupiter
0.520
-9.40
-2.70
124.4 ±0.3
Cl
Saturn
0.470
- 8.88
0.67t
95.0 ±0.4
Cl
Uranus
0.510
-7.19
5.52
59.1 ±0.3
Cl
0.410
-6.87
7.84
59.3 ±0.8
Cl
variable*
-0.81
15.12
50-70
So
Neptune Pluto
-3.86
* The Pluto visual geometric albedo is variable by 30%. The Pluto color is the combination of the planet and its satellite Charon, t V referts to the Saturn disk only.
£-5
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
867
Notes: 1. The values for the masses include the atmospheres but exclude the satellites. 2. The mean equatorial radii are given. 3. The flattening is the ratio of the difference of the equatorial and polar radii to the equato rial radius. 4. The period of rotation refers to the rotation at the equator with respect to a fixed frame of reference: a negative sign indicates that the rotation is retrograde with respect to the pole that lies to the north of the invariable plane of the solar system. The period is given in days of 86,400 SI seconds. 5. The data on the equator, flattening, period of rotation, and inclination of equator to orbit are based on Davies, et al. [1989]. 6. The geometric albedo is the ratio of the illumination at the Earth from the planet for phase angle zero to the illumination produced by a plane, absolutely white Lambert surface of the same radius as the planet placed at the same position. This is not to be confused with a planet’s bond albedo which is simply the ratio of total reflected light to total incident light. (See Sec. 11.6) 7. V(1,0) is the visual magnitude when the observer is directly between the Sun and the planet and the product of the Sun-planet distance (in AU) is 1. (See Sec. 11.6) 8. Vq *s die mean visual magnitude of the planet when at opposition as viewed from the Earth. Magnitudes for Mercury and Venus are at greatest elongation. (See Sec. 11.6)
TABLE E-14. Natural Satellites: Orbit Data. See Table E-1 for gravitational Data. Orbital Period1
Sat.
#
Satellite Name Moon
Semimajor [R=Retrograde] Axis Orbit (x1Q3 km) Eccentricity (d) EARTH 27.321 661
384.4
0.054 900 489
Orbit Incl. to Motion of Node on Planetary Equator Fixed Plane4 (deg) (deg/yr) 18.28-28.58
19.34S
MARS
I II
Phobos Deimos
0.318 910 23 1.262 440 7
9.378 23.459
I II III IV V VI VII VIII IX X XI XII XIII
Io Europa Ganymede Callisto Amalthea Himalia Bara Pasiphae Sinope Lysithea Carme Ananke Leda
1.769 137 786 3.551181041 7.154 552 96 16.689 018 4 0.498 179 05 250.566 2 259.652 8 735 R 758 R 259.22
0.015 0.000 5
0.9-2.7
422 671 1070 1883
0.004 0.009
0.04 0.47
48.6
0.002
0.21
0.007
0.51
2.63 0.643
181 11480 11737 23500 23700 11720 22600
0.003 0.157 98 0.207 19 0.378 0.275 0.107 0.206 78
0.4 27.63 24.77 145 153 29.02
21200
0.168 7 0.147 62
147 26.07
1
iI [
158.8 6.614
JUPITER
692 R 631 R 238.72
11094
164
12
914.6
868
Appendix E
E-5
TABLE E-14. Natural Satellites: Orbit Data. See Table E-1 for gravitational Data.(Continued) Orbital Period 1
Sat. Satellite # Name XIV Thebe Adrastea XV XVI
Metis
I
Mimas Enceladus Tethys
Semimajor Axis [R=Retrograde] (x 103 km) W) 0.674 5 222 129 0.298 26
Orbit Eccentricity 0.015
Orbit Incl. to Motion of Planetary Node on Equator Fixed Plane4 (deg) (deg/yr) 0.8
0.294 78
128
0.942 421 813 1.370 217 855
185.52
0.020 2
1.53
238.02
0.004 52
0
365 156.25
SATURN II III IV V VI VII Vlll IX X
1.887 802 16 2.736 914 742
294.66
0
1.86
72.2®
Dione Rhea Titan
377.4
0.002 23
0.02
4.517 500 436
527.04 1221.83
0.001
0.35 0.33
30.85s 10.16
Hyperion Iapetus Phoebe
21.276 608 8 79.330 182 5 550.48 R
1481.1 3561.3 12952
0.028 28 0.163 29
0.694 5
151.472
0.007
0.14
0.694 2 2.736 9 1.887 8
151.422 377.4
0.009 0.005
0.34
0.601 9 0.613 0.628 5
294.6S 294.66 137.67
0
0-3
139.353 141.7
0.003 0.004
0
0.575
133.583
Janus XI Epimethus XII Helene XIII Telesto XIV Calypso XV Atlas XVI Prometheus XVII Pandora XVIII Pan
15.945 420 68
1.887 8
0.029 192 0.104
0.521 35
0.43 14.72 1772
0
0
URANUS Ariel 2,52037935 II Umbriel 4.1441772 III Titania 8.7058717 IV Oberon 13.4632389 V Miranda 1.41347925 VI Cordelia 0.335 033 8 VII Ophelia 0.376 400 Vlll Bianca 0.434 579 9 IX Cressida 0.463 569 60 X Desdemona 0.473 699 60 XI Juliet 0.493 065 49 XII Ponia 0.513 195 92 0.558 459 53 XIII Rosalind XIV Belinda 0.623 527 47 XV Puck 0.761 832 87 XVI Caliban 579 R XVII Sycorax 1,289 R I
191.02 266.30
0.003 4 0.005
435.91 583.52 129.39
0.002 2 0.000 8
0.10
0.002 7
4.20
49.77
0.000 26
0.08
53.79
0.009 9 0.000 9
0.10
0.000 4 0.000 13
0.01 0-11
0.000 66
0.07
0.001
0.10
59.17 61.78 62.68 64.35 66.09 69.94
0.30 0.36 0.14
0.19
6.8
3.6 2.0
1.4 19.8 550 419 229 257
0.001
0.30
245 223 203 129
75.26
0.001
0.00
167
86.01
0.001
7169.00
0.082
0.31 139.20
12.214.00
0.509
152.70
81
E-5
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
869
TABLE E-14. Natural Satellites: Orbit Data. See Table E-1 for gravitational Data.(Continued) Orbital Period1
Sat. # 1
II
ill IV V VI VII VIII
Satellite Name Triton Nereid Naiad Thalassa Despina Galatea Larissa Proteus
Semimajor [R=Retrograde] Orbit Axis (x103 km) Eccentricity NEPTUNE 5.876 854 1 R 360.13619
354.76
Orbit IncL to Motion of Planetary Node on Equator Fixed Plane4 (deg/yr) (deg)
0.000016
157.345
0.5232
27.63
0.039
0.7512
5,513.40
0.2943 96 0.311 485 0.334 655
48.23 50.07 52.53
0.0
4.74
0.0
0.21
0.0
0.07
0.428 745 0.554 654 1.122 315
61.95 73.55 117.65
0.0
0.05
0.001 39 0.000 4
0.2
626 551 466 261 143 0.5232
0.55
PLUTO Charon
1
6.38725
19.6
0.001
96.16
TABLE E-15. Natural Satellites: Physical and Photometric Data. Sat. #
Satellite Name
Mass (1 /planet)
Radius (Km)
Sidereal Period1
[i/'O ,o)]
Geo metric Albedo9
+0.21
0.12
Visual Mag.
EARTH Moon
0.012 300 034
1737.4
S
MARS I
Phobos
1.654 X 10“ 8
13.4 x 16.2x9.2
s
+ 11.8
0.07
II
Deimos
3.71 x 10^9
7.5 x 6.1 x 5.2
s
+12.89
0.08
JUPITER I
Io
4.704 1 X10-5
1830x1818.7x1815.3
s
- 1.68
0.63
II
Europa
2.528 0 x1 0 -5
1565
s
-1.41
0.67
III
Ganymede
7.804 6 x 1 0 ^
2634
s
-2.09
0.44
IV
Callisto
5.666 7 x IO- 5
2403
s
-1.05
0.20
V
Amalthea
38 x 10'-10
131 x 73 x 67
s
+7.4
0.07
VI
Himalia
50 x 10-10
85
0.4
+8.14
0.03
VII
Elara
4 x 10-10
40
0.5
+10.07
0.03
VIII
1 x 10-10
18
+10.33
0.1
IX
Pasiphae Sinope
0.4 x 10-1°
14
+ 11.6
0.05
X
Lysithea
0.4 x 10-10
12
+11.7
0.06
XI
Carme
0.5 x 10—10
15
+11.3
0.06
XII
Ananke
0 .2 x 10- 1°
10
+ 12.2
0.06
XIII
Leda
0.03 x 10-10
5
+13.5
0.07
870
Appendix E
E-5
TABLE E-15. Natural Satellites: Physical and Photometric Data. (Continued) S at #
Satellite Name
Mass (1/planet)
Radius (km)
4 x 10-10
55x45
XIV
Thebe
XV
Adrastea
0.1 x IO"™
13x 1 0 x 8
XVI
Metis
0.5 x 10" 10
2 0 x 20
Sidereal Period 1
id ) s
Visual Mag. [ v o ,o)i +9.0
Geo metric Albedo*
+12.4
0.05
+ 10.8
0.05 0.5
0.04
SATURN I
Mimas
II III IV
6 .6 x 10-8
209.1 x196.2x191.4
s
+3.3
Enceladus
1 .0 x 10-7
256.3 x 247.3 x 244.6
s
+2.1
1.0
Tethys
1.10 x IO"*6
209.1 x 196.2 x 191.4
s
+0.6
0.9
Dione
1.95x10-6
560
s
+0.8
0.7
V
Rhea
4.06 x 10-6
764
s
+0.1
0.7
VI
Titan
2.366 7 X 1 0 - 4
2575
s
-1.28
0.22
VII
Hyperion
4.0 x 1 0 - 8
1 8 0 x1 4 0 x 112.5
+4.63
0.3
Vlll
Iapetus
2 .8 x 10-6
718
s
+1.5
(0 .2)2
IX
Phoebe
7 x 10-10
110
0.4
+6.89
0.06
X
Janus
3.385 x 10" 9
97.0 x 95.0 x 77.0
S
+4.4
0.9
XI
Epimethus
9.5 x 10-10
69 x 55 x 55
s
+5-4
0.8
XI!
Helene
18y 16x 15
+8.4
0.7 1.0
XIII
Telesto
15 x 12.5x7.5
+8.9
XIV
Calypso
1 5 .0 x 8 .0 x 8 .0
+9.1
1.0
XV
18.5x17.2x13.5
+8.4
0.8
XVII
Atlas Prom&theus Pandora
XVIII
Pan
XVI
74.0 x 50.0 x 34.0
+6.4
0.5
55.0 x 44.0 x 31.0
+6.4
0.7 0.5
10
In 2000,12 new, small satellites of Saturn were discovered. All are less than 15 km in radius. It is likely that the Cassini spacecraft will discover more.
URANUS I
Ariel
1.56 x 10-s
581.1 x 577.9x577.7
s
+1.45
0.35
II
Umbriel
1.35x10-5
584.7
s
+2.10
0.19
III
Titania
4.06 x 10"5
788.9
s
+1.02
0.28
IV
Oberon
3.47x10-5
761.4
s
+1.23
0.25
V
Miranda
0.08 x 10-5
240.4 x 234.2 x 232.9
s
+3.6
0.27
VI
13 15
+11.4
0.07
VII
Cordelia Ophelia
+11.1
0.07
Vlll
Bianca
21
+10.3
0.07
Cressida Desdemona
31
+9.5
0.07
27
+9.8
0.07
42 54
+8.8
0.07
XII
Juliet Portia
+8.3
0.07
XIII
R o sa lin d
27
+9.8
0.07
IX X XI
£-5
Physical and Orbit Properties of the Sun, Earth, Moon, and Planets
871
TABLE E-1 S. Natural Satellites: Physical and Photometric Data. (Continued) Sat. #
Satellite Name
Mass (1/planet)
Radius (km)
Sidereal Visual Mag. Period1 [ ^ ( 1,0)] (*)
Geo metric Albedo9
33
+9.4
0.07
XV xvt
Belinda Puck Caliban
77
+7.5
0.075
30
0.07
XVII
Sycorax
60
0.07
1
Triton
2.089x 10-4
1353
II
Nereid
2 x 10“ 7
170
+4.0
0.4
III
Naiad Thalassa
29
+ 10.0
0.06
40
+9.1
0.06
Despina Galatea
74
+7.9
0.06
79
+7.6
0.06
VIII
Larissa Proteus
104x89 218 x 208 x 201
+7.3 +5.6
0.06 0.06
1
Charon
+0.9
0.5
XIV
NEPTUNE
IV V VI VII
S
-1.24
0.77
PLUTO 0.125
593
S
Notes: 1. Sidereal periods, except that tropical periods are given for satellites of Saturn. 2. Relative to the ecliptic plane. 3. Referred to the equator of 1950.0. 4. 5. 6. 7. 8. 9.
Rate of decrease (or increase) in the longitude of the ascending node. Rate of increase in the longitude of the apside. On the ecliptic plane. S = synchronous, rotation period same as orbital period. Bright side, 0.5; faint side, 0.05. V (Sun) = -26.75.
References COES A. 1976. U.S. Standard Atmosphere. W ashington, D.C.: Government Printing Office. Cox, Arthur N, ed.. 2000. A lle n ’s A strophysical Quantities (4th ed.). New York: Springer-Verlag. Davies, M.E. et al. 1989. “Report of the IAU/IAG/COSPAR W orking Group on Car tographic Coordinates and Rotational Elements of the Planets and Satellites: 1988,” C elestial M echanics 46, 187—204. Eckart, P. 1999. The Lunar Base Handbook: An Introduction to Lunar Base Design, D evelopm ent, and O perations. NY: McGraw-Hill.
872
Appendix E
E-S
Hedin, Alan E. 1987. “MSIS- 8 6 Thermospheric Model.” J GeophysR, 92, No. A5, pp. 4649-4662. ------- . 1988. “The Atmospheric Model in the Region 90 to 2000 km.” Adv. Space Res., 8 , No. 5-6, pp. (5)9-(5)25, Pergamon Press. ------- . 1991. “Extension of the MSIS Thermosphere Model into the Middle and Lower Atmosphere,” J Geophys R, 96, No. A2, pp. 1159-1172. Heiken, Grant H., David T. Vaniman, and Bevan M. French. 1991. Lunar Sourcebook: A User’s Guide to the Moon. Cambridge: Cambridge University Press. Meeus, Jean. 1998. Astronomical Algorithms (2nd ed.). Richmond, VA: WillmannBell, Inc. Muller, Edith A. and Ainsdt Jappel, eds. 1977. International Astronomical Union Proceedings of the Sixteenth General Assembly, Grenoble, 1976. Dordrecht, The Netherlands: D. Reidel Publishing Co. Nievergelt, Y., and S. Keeler. 2000. “Computing Geodetic Coordinates in Space.” J. Spacecraft. 37:293-296. Seidlemann, P. Kenneth, ed., USNO. 1992. Explanatory Supplement to the Astronom ical Almanac. Mill Valley, CA: University Science Books. U.S. Naval Observatory and H. M. Nautical Almanac Office. 1999. The Astronomical Almanac, 1999. Washington, D.C.: U.S. Government Printing Office.
Appendix F Properties of Orbits About the Moon, Mars, and the Sun Tables F-1, F-2, and F-3 provide fundamental mission data as a function of altitude for orbits around the Moon, Mars, and the Sun respectively. The explanation of each column and the source or formula for the data is given ahead of each table. For any mission which is well defined, these parameters can be easily computed for the exact spacecraft altitude, and the errors estimated, using the formulas provided in the ex planatory material, Chap. 2, or any standard reference. However, the tables here are extremely convenient for preliminary mission design and for determining the impact of variations in basic mission elements. All of the data assumes either a fixed instan taneous altitude or a circular orbit. The rear endpages contain similar, more extensive data for Earth satellites. The Moon and Mars tables provide relevant subsets of the information provided for Earth satellites. The column numbers remain the same for each of these three sets of tables to provide correspondence with the explanations and formulas provided at the front of the endpages. The table of parameters for orbits about the Sun (i.e., interplanetary orbits) is, of course, significantly different in content from the planetary orbit tables. Explanation and formulas for these data are are provided at the front of the table.
Table F-l Lunar (“(”) Satellite Parameters The following table provides a variety of quantitative data for satellites orbiting the Moon. The independent parameter is the distance, r, from the center of the Moon in km. However, the outside column on each page is the altitude, h = r - R where = 1,738 km is the equatorial radius of the Moon. Note that numerical formulas given in the Explanation o f Earth Satellite Param eters will not generally work for a lunar satellite; readers should use the formulas given below or those listed in Chap. 2. • General Changes relative to Earth satellite tables: Re , or R®, (= 6,378.14 km) is replaced by R( ( = 1,738 km) jj® ( = 398,600.4 km3/s2) is replaced by ^ ( = 4,902.798 98 km3/s2) J2,® ( = 1.082 63 x IO-3) is replaced with J2>( (= 0.2027 x IO-3) • Note: Circular orbits, unperturbed by the Earth, have been assumed throughout. Higher altitudes will be more and more affected by pertur bations from the Earth the so data at higher altitudes is less accurate.
873
874
Appendix F
F
Columns 1 through 24 use the same formulas given in the Explanation o f Earth S atel lite P aram eters with the general changes given above. Columns 25 through 40 are omitted because they are not applicable to lunar orbits. Columns 41, 42 and 43 use the same formulas as given in the Explanation o f Earth Satellite P aram eters with the general changes given above. Columns 45, 46 and 47 use the same formulas as given in the Explanation o f Earth S atellite Param eters with the general changes given above. Columns 49 through 53 use the Earth Satellite formulas with the general changes given above. 44.
AV R equired to D e-O rbit (m/s) = the velocity change needed to transform the
assumed circular orbit to an elliptical orbit with apoapse unchanged and periapse of 0 km, i.e. at the lunar surface [Eq. (2-85)]. 48.
Sun Synchronous Inclination (deg) = cos- 1 { 0.985 65° /(- l.5 x 86.400 (fi /r 3) 1/2 x J2i x (R / h)2) } assumes circular orbit with node rotation rate of 0.985 65
deg/day to follow the mean motion of the Earth/Moon system’s rotation about the Sun [Eq. (D-35)].
(#)=■ 1,440IP, where P is from column 52. Note that this is the number of revolutions in a 24 hour time period and not in a sidereal lunar day. {Eq. (D-35)
54.
Revolutions p e r D ay
55.
N ode Spacing (deg) = 360° x ( P I 39.343.1904), where P is from column 52 and
39,343.1904 is the lunar sidereal rotation period in minutes. This is the spacing in longitude between successive ascending or descending nodes for a satellite in a circular orbit [Eq. (D-36)]. 56.
(Rjjd)1 (cos 0 ( 1 - ^ Y 1, where i is the inclination, e the eccentricity (which is set to zero), n is the mean motion (= (p/a3)1/2), a the semimajor axis, J2, the dominant zonal coefficient in the expansion of the Legendre polynomial describing the lunar gravitational poten tial. Note that this is the angle through which the orbit rotates in inertial space in a 24 hour period [Eq. (D-33)].
N ode P recession R ate (deg/day) = -1.5 n J2
Properties of Orbits About the Moon, Mars, and the Sun
F
875
Orbits About the Moon 1
2
3
4
5
6
7
8
Appendix F
876
Orbits About the Moon
532.93
F
F
Properties of Orbits About the Moon, Mars, and the Sun
Orbits About the Moon
877
878
Appendix F
F
Orbits About the Moon 42
43
44
45
46
F
Properties of Orbits About the Moon, Mars, and the Sun
Orbits About the Moon 51
52
53
54
55
879
Appendix F
880
F
Table F-2 Mars ( c f) Satellite Parameters The following table provides a variety of quantitative data for satellites orbiting Mars. The independent parameter in the formulas is the distance, r, from the center of Mars in km. However, the outside column on each page is the altitude, h = r - R^f, where R tf = 3,397 km is the equatorial radius of Mars. Numerical formulas for tne Earth Satellite P aram eters will not generally work for a Mars orbiting satellite; read ers should use the formulas below or those in Chap. 2 . • General Changes relative to Earth satellite tables: Re, or R®, ( = 6,378.14 km) is replaced by R ^ (= 3,397 km) ju^, (= 398,600.4 km3 /s2) is replaced by (= 42,828.3 km3 /s2) J 2 ,® ( = 63 x IO"3) is replaced with J2)Cf (= 1.964 x IO-3) • 17,031 km is the Mars-synchronous altitude at which a Martian satellite orbits the planet in one Mars sidereal day. Columns 1 through 24 use Earth Satellite formulae with the changes noted above. Columns 25 through 40 are omitted as a Mars atmosphere model analogous to the MSIS model is not available. Columns 41, 42 and 43 use Earth Satellite formulas with the changes noted above. Columns 45,46 and 47 use Earth Satellite formulas with the changes noted above. Columns 49, through 53 use Earth Satellite formulas with the changes noted above. 44.
AV Required to D e-O rbit (m/s) = Is the velocity change needed to transform the assumed circular orbit to an elliptical orbit apoapse unchanged and periapse of 0 km, i.e. at the Martian surface [Eqs. (2-85a) and (2-85b)].
48.
Sun Synchronous Inclination (deg) = cos- 1 {0.52407°/ (-1.5 x 86,400 ((j^/r 3) ^ 2 x J 2 d-x (Rq*/ h)2)} assumes circular orbit with node rotation rate of
0.52407 deg/day to match the mean motion of the Sun as seen from an inertial observer at Mars [Eq. (D-34)].
(#)=\ ,477.377 72/P, where P is from column 52. Note that this is revolutions per sidereal Martian day, where the sidereal Martian day is the day relative to the fixed stars which is approximately 24*^ 37™° 22.66sec. [Eq. (D-35)]
54.
Revolutions p e r D ay
55.
N ode Spacing (deg)
56.
N ode P recession Rate (deg/day) = -1.5 n 12,
= 360° x (P / 1,477.377 72), where P is from column 52. This is the spacing in longitude between successive ascending or descending nodes for a satellite in a circular orbit [Eq. (D-36)].
(R^/a ) 2 (cos i) (1 - e 2)~2, where 1is the inclination, e the eccentricity (which is set to zero), n is the mean motion (- (|_iQ’'/
F
Properties of Orbits About the Moon, Mars, and the Sun
Orbits About Mars
881
882
Appendix F
Orbits About Mars
F
Properties of Orbits About the Moon, Mars, and the Sun
17
18
19
20
21
22
23
883
24
884
Appendix F
F
F
Properties of Orbits About the Moon, Mars, and the Sun 49
50
51
52
53
54
55
56
885
886
Appendix F
F
Table F-3 Solar (0) Satellite Parameters The following table provides a variety of quantitative data for spacecraft orbiting the Sun. The independent parameter is the distance, r, from the center of the Sun to the spacecraft in km. The corresponding distances to planets and several asteroids are also listed. Note that numerical formulas given in the Explanation of Earth Satellite P a ram eters on the rear end-pages will not generally work for a solar satellite. You should use the formulas given below or those listed in Chap. 2. Throughout (=398,600.4 knP/s^) has been replaced by (=1.327 245 x IO11 km3/$2) and circu lar, coplanar orbits have been assumed. Columns 1 through 2 use the same formulas given in Explanation o f Earth Satellite P aram eters with the general changes given above.
Columns 3 through 4 use the same formulas given in Explanation o f Earth Satellite Param eters with the general changes given above, except the altitude change is 1,000 km and the plane change is 0.1 deg. Column 5 uses the formulas for a Hohmann transfer given in Chap. 2, Sec. 2.6. Column 6 uses the same formula given in Explanation o f Earth Satellite Param eters with the general changes given above. Column 7 uses equations (2-18), (2-19), and (2-20). Column 8 uses the following equation for the Solar Constant 5 = 1,367 (Rzarth/r)2
(W/m2 )
(F-l)
where R Eart^, is the distance from the center of the Sun to the Earth, r, is the given dis tance of the spacecraft from the center of the Sun. Columns 9 through 10 assume that the spacecraft and the observer on the Earth are in coplanar, circular orbits around the Sun. For spacecraft outside the Earth’s orbit, the maximum distance occurs when the spacecraft is in conjunction, and the minimum dis tance occurs when the spacecraft is in opposition. Refer to Fig. 2-9 for more details, including the distance relationship for spacecraft inside the Earth’s orbit. Columns 11 through 13 assume that the spacecraft and the observer on the Earth are in coplanar, circular orbits around the Sun. The elongation angle is the angle between the Sun and spacecraft with the Earth at the vertex. Quadrature occurs when the elon gation angle is 90 deg. Refer to Fig. 2-9 for more details, including the distance rela tionships for spacecraft inside the Earth’s orbit. Column 14 assumes that the Earth and the spacecraft are in coplanar, circular orbits. The maximum angular velocity of the spacecraft as seen from the Earth is the relative angular velocity of the spacecraft at the point when it is nearest the Earth. Column 15 assumes that the spacecraft and observer on the Earth are in coplanar, cir cular orbits. The synodic rate is simply the absolute value of the difference between the orbit angular velocity of the spacecraft and the orbit angular velocity of the Earth.
F
Properties of Orbits About the Moon, Mars, and the Sun
887
Orbits About the Sun 1
2
Circular
Orbit Angular
Velocity Velocity (km/s) (deg/day)
3 AKReq’d fo r a 1000
km alt.
4
5
W fo r a AV for 0.1 deg Hohmann
6
7 Synodic
Distance from
plane
Transfer
Sidereal
change (m/s) 142
to Earth (km/s) 42.1
(days)
Period
Period (days) 18.77
Center of Sun
81.5
20.2
change (cm/s) 204
40.7 36,4
2.52
25.5
71.1
10.7
142.8
1,80
18,2
63.6
6.58
199.6
33.3
1.37
13.9
58.0
3.46
30.8
1.09
11.0
53.7
1.00
262.4 330.7
27.2
0.747
7.54
47.4
2.626
482.1
1507
180
25.8
0.638
6.44
45.0
4.004
564.6
1034
200
24.6
0.553
5.58
42.9
5.176
651.4
831.5
220
21.0
0.347
3.51
36.7
8.498
1037
563.8
300
18.2
0.225
2.28
31.8
10.926
1597
473.5
400
14.9
0.123
1.24
26.0
12.406 13.377
2934
417.2
600
13.8
0.097 4
0.984
24.0
14.048
3697
405.3
700
12.9
0.0797
0.805
22.5
14.528
4517
397.4
800
12.1
0.0668
0.675
21.2
14.881
391.8
900
11.5
0.0570
0.576
20.1
15.145
5390 6312,
387.7
1,000
9.40
0.0310
0.314
16.4
15.783
11597
377.1
8.1 7.3
0.0202
14.2 12.7
15.953 15.969
17854 24952
372.9 370-7
1,500 2,000
0.0144
0.204 0.146
6.7
0.0110
0.111
11.6
6.2
0.0087
10.7
15.927 15.860
32800 41333
369.4 368.5
3.000 3.500
5.8
0.0071
0.088 0.072
10.1
15.785
50499
367.9
4.000
5.4
0.0060 0.0051 0.0044
0.060 0.052
9.48
15.708
60257
15.632
70574
367.5 367.1
4.500
8.99 8.57
15.558
81421
366.9
5.500
17.9
(106 km)
20
124.4 234,6 440.2
100
80
931.9
120
0.984
436.7
2.500
■ ■
5.2 4.9
0.045
5,000
Body
Appendix F
888
F
Orbits About the Sun 8
9
10
11
12
13
14
15
Distance Max from Distance Distance Max Min Angular from from Distance Distance Earth at Velocity Angle at Center of Solar from from Earth at Greatest Greatest Seen from Synodic Sun Constant Earth Earth Earth Rate Elongation Quadrature Elongation (deg/day) (deg/day) (deg) (10® km) (W/m2) (106 km) (106 km) (106 km) (10* km) 20 76,482 169 129 148 N/A 7.68 1.974 19.2 40 ■; 57-s 60 80 100
19,121 ■3 123 ■' 8,498 4,780 3,059
189 207 . 209 229 249
109
.
144
15.5
22:8.. ;. ■'■■■ ' ■1 ■ ■■i l 23.6 0.953 0.779 32.3 41.9 0.663
r* 89 69 49
137 126 113
N/A N/A N/A
I .1 29
103 89
N/A
1.256
6.14 ililg l 2.89 1.53 0.82
108.2
2,613
257 .
120
2,125
269
0.39
1,561
289
9
52
N/A
53.3 69.4
0.581
140
0.518
0.10
160 180
> -i;3 6 7 '\:’ 1,195 944.2
309 329
10 30
10 30
56
180
100
180
0.469 0.429
0.0945 0.2389
200 220
764.8 632.1
349 369
50 70
50
132
180
0.395
0.3480
70
161
180
0.4330
588.8
377
339.9 u353;1; jJv - 245.4 400 191.2
449
150
■ 7? 150
.172 :.: ■ 260
-189 : ■ 180
0.367 ■0.357. :' 0.288
0.6386
.......?• 0.229
0.7602
;;.-.149J6
227.9 ••::: 300
• 4.14.4...j'.450
SP.2.:'.'- ■ ■ 2 0 3 ^. 549 250
.178.2' 151.1
599
264-.;:,:. 300
;"
....... 250
46.3
. ■
...... 51?..'... 371 386,; ■■■..
:264 300
424
S 500 600 700
122.4
649
850 62.4
749 849
, -r ;i.;-pp;S0'i.= 800 47.8
350 450
350 450
550 .{.-c
550
949
650
650
37.8
1,050
750
750
1,000
30.6
1,150
850
850
1,500 2,000
1,650
1,350
1,350
2,150
1,850
1,850
2,500 4.895 2,650 ::r2;S7t'v'- .,3.712; V ■:C-;3,021l::::: 3.000 3.399 3,150
2,350
2.497
3,350 3,850
3,350
3,500 4.000
1.912
4,498.2;:;
3,650 4,150 .4tg4Q ;■•••
2,850
iS S 477
■' 0.7967
0.208 s
l - S
-
0.8243 0 8629 0.8883
ieo 180
683
180
0.144
785 887
180
L.--J 0i;i32.u-;: 0.129
0.9059
180
0.116
0.9189
988
180
1,493 1,994
2,350 2,496 •^2;72l:M :i ^2 ,8 6 2 ;^: 2,850 2,996 3,850
i g 0.191 0.164
581
..^■,413;...-.:.
•■'1,277;:;;
15.0 > 13.6 7.648
' : =1S° 180
[ i. ,-763.- ■
900 1.42&7;-'.
180
314
:-142;2: '-
!'
N/A
3,497 3,997
180 180 180
0.106
0.9286
0 078 0.075
0.9522
0.058
0.9655
0.047
0.9546 0.9712
7i0.973&:; 180
0.040
0.9747
180 180
0.035 0.031
0.9785
0.9769
1.511 1.224
4,650 5,150
4,350
4,350
4,498
180
0.028
0.9797
5,000
4,850
4,850
4,998
180
0.025
0.9806
5.500
1.011
5,650
5,350
5,350
5,498
180
5,906.4
O0.877
0.023 'V:%<E2V'2
0.9817
4.500
1-575&:
5;9p4.d^ ;v^:180
0.9812
Appendix G Units and Conversion Factors The metric system of units, officially known as the International System o f Units, or SI, is used throughout this book, with the exception that angular measurements are usually expressed in degrees rather than the SI unit of radians. By international agree ment, the fundamental SI units of length, mass, and time are defined as follows (see National Institutes of Standards and Technology, Special Publication 330 [1991]): The m eter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. The kilogram is the mass of the international prototype of the kilogram. The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. Additional base units in the SI system are the ampere for electric current, the kelvin for thermodynamic temperature, the m ole for amount of substance, and the candela for luminous intensity. Taylor [1995] provides an excellent summary of SI units for scientific and technical use. The names o f multiples and submultiples of SI units are formed by application of the following prefixes: Factor by Which Unit is Multiplied 1024
Factor by Which Unit is Multiplied
Prefix
Symbol
yotta zetta
10“ 1 10-2
Prefix
Symbol
deci centi milli
d c rrt
micro
U n
1015
exa peta
Y Z E P
1012
tera
T
10-9
109
giga mega
G M
10-12
nano pico
10—15
femto
103
kilo
102
hecto deka
k h
10r 1S 10-21
da
10-24
1021 1018
10®
101
10“ 3 10-6
atto zepto yocto
P f a z
y
For each quantity listed below, the SI unit and its abbreviation are given in brackets. For convenience in computer use, most conversion factors are given to the greatest available accuracy. Note that some conversions are exact definitions and some (speed of light, astronomical unit) depend on the value of physical constants. . . ” indicates a repeating decimal. AH notes are on the last page of the list.
889
890
Appendix G
G
To convert from
To
Multiply by
Acceleration [meter/second2, m/s2] Gal (galileo) Inch/second2, in/s2 Foot/second2, ft/s2 Free fall (standard), g
m/s2 m/s2 m/s2 m/s2
0.01 0.025 4 0.304 8 9.806 65
E E E E
7L/180 - 0.017 453 292 519 943 295 77
E
Angular Acceleration [radian/second2, rad/s2] Degrees/second2, deg/s2 rad/s2 Revolutions/second2, rev/s2
rad/s2
Revolutions/minute2, rev/min2
rad/s2
Revolutions/minute2 Radians/second2, rad/s2
deg/s2 deg/s2
Revolutions/second2, rev/s2
deg/s2
Angular Area [sr], book also uses deg2 Degree2, deg2 sr Minute2, min2
sr
Second2, s2
sr
Steradian, sr
deg2
Minute2, min2
deg2
Second2, s2
deg2
Steradian, sr
rad2
Notes
2n
= 6.283 185 307 179 586 477 E rc/1,800 = 1.745 329 251 994 329 577 x 1CH5 E 0.1 E 180/it E =57.295 779 513 082 320 88 360 E (7t/180)2 = 3.046 174 197 867 086 x 1
E E E
E E E E
Angular Measure [radian, rad]. This book uses degree (abbreviated “deg”) as the basic unit. Degree, deg rad 71/180 E = 0.017 453 292 519 943 295 77 Minute (of arc), min rad it/10,800 = 2.908 882 086 657 216 x 1(H E Second (of arc), s rad tc/648 000 =4.848 136 811 095 360 x 10~6 E Radian, rad deg = 180/ti = 57.295 779 513 082 320 877 E Minute (of arc), min deg 1/60 E = 0.01666... Second (of arc), s deg 1/3,600 -2 .7 7 7 .. .xlO-4 E
G
Units and Conversion Factors
To convert from
To
Multiply by
r*-
1 O
X
Angular Momentum [kilogram ■meter2/second, kg ■m2/s] kg • m2/s Gram • cm2/second, g ■cm2/s 2.926 396 534 292 x 1(H lbm* inch2/second, lbra ■in2/s kg • m2/s kg • m2/s 9.415 402 418 968 x IO"3 Slug • inch2/second, slug • in2/s kg • m2/s 0.042 140 110 093 80 lbm* foot2/second, lbm - ft2/s kg ■m2/s 0.112 984 829 027 6 Inch • Ibf* second, in • lbf • s kg ■m2/s Slug • foot2/second* slug • ft2/s 1.355 817 948 331 = foot ■lbf *second, ft ■lbf • s
891 Notes
E E D D D D
Angular Velocity [radian/second, rad/s]. This book uses degrees/second as the basic unit. rad/s jr/180 Degrees/second, deg/s = 0.017 453 292 519 943 295 77 E Revolutions/minute, rpm rad/s ti/30 = 0.104719 755 119 659 7746 E Revolutions/second, rev/s 2n rad/s = 6.283 185 307 179 586477 E Revolutions/minute, rpm deg/s 6 E Radians/second, rad/s deg/s 180/tc = 57.295 779 513 082 320 88 E Revolutions/second, rev/s deg/s 360 E Area [meter2, m2] Acre Foot2, ft2 Hectare Inch2, in2 Mile2 (U.S. statute) Yard2, yd2 (Nautical mile)2
m2 m2 m2 m2 m2 m2 m2
4.046 856 422 x IO3 0.092 903 04 1 x IO4 6.451 6 x 1(H 2.589 110 336 x 10* 0.836 127 36 3.429 904 x IO6
E E E E E E E
Density [kilogram/meter3, kg/m3] Gram/centimeter3, g/cm3 Pound mass/inch3, lbm/in3 Pound mass/foot3, lbm/ft3 Slug/ft3
kg/m3 kg/m3 kg/m3 kg/m3
1.0 x IO3 2.767 990471 020 x IO4 16.018 463 373 96 515.378 818 393 2
E D D D
C C
c
10 9.648 70 x IO4 9.649 57 x 104 9.652 19 x IO4 3.335 641 x 10-10
Electric Conductance [siemens, S] Abmho Mho (ft"1)
s s
1 x 109 1
E E
Electric Current [ampere, A] Abampere
A
10
E
Electric Charge [coulomb, C] Abcoulomb Faraday (based on carbon-12) Faraday (chemical) Faraday (physical) Statcoulomb
c C
E NIST NIST NIST NIST
892 To convert from
Gilbert Statampere Electric Field Intensity
Appendix G To
Multiply by
Notes
A
10/471 = 0.795 774 715 459 5 3.335 641 x IO-10
E NIST
A
[volt/meter =kilogram • meter • ampere-1 • second-3, V/m = kg • m - A"1 ■s~3] Electric Potential Difference [volt s watt/ampere s kilogram - meter2 - ampere-1 ■second-3, V =W/A = kg - m2 *A-1 - s-3] Abvolt V lxlO-8 E Statvolt V 299.792 5 NIST Electric Resistance [ohm = volt/ampere = kilogram • meter2 • ampere-”1• second-3, Q = V/A = kg ■m2 • A-2 *s-3] Abohm Q. 1x10-9 E Statohm ft 8.987 552 x IO11 NIST Energy or Torque [joule s newton -meter = kilogram ■meter2/s2, J = N • m = kg ■m2/s2] British thermal unit, Btu (mean) J 1.055 055 852 62x10^ E Calorie (IT), cal J 4.186 8 E Kilocalorie (IT), kcal J E 4.186 8 x IO3 Electron volt, eV 1.602 177 33 x IO-19 J C Erg = gram • cm2/s2 = pole - cm • oersted J 1 x IO-7 E Foot poundal J D 0.042 140 110 093 80 Foot lbf = slug - foot2/s2 J 1.355 817 948 331 4 E Kilowatt hour, kW • hr 3.6 x IO6 J E Ton equivalent of TNT J 4.184 x IO9 E Force [newton s kilogram - meter/second2, N = kg Dyne N Kilogram-force (kgf) N Ounce force (avoirdupois) N Poundal N Pound force (avoirdupois), N lbf = slug ■foot/s2
*m /s2] IX 10-5 9.806 65 0.278 013 850 953 8 0.138 254 954 376 4.448 221 615 260 5
Illuminance [lux = candela *steradian/meter2, lx = cd • sr/m2] Footcandle cd • sr/m2 10.763 910 416 709 70 Phot cd • sr/m2 l x IO4 Length [meter, m] Angstrom, A Astronomical unit (SI) Astronomical unit (radio) Earth equatorial radius, RE Fermi (1 fermi = 1 fin) Foot, ft Inch, in
m m m m m m m
1 x IO-10 1495 978 706 6 x IO11 1.495 978 9 x IO11 6.378 136 49 x IO6 6.378 14 x IO6 1 x 1(F5 0.304 8 0.025 4
E E D E E
E E E AA NIST IERS AQ E E E
Units and Conversion Factors
G To convert from
Light year Micron, |j.m Mil (10-3 inch) Mile (U.S. statute), mi Nautical mile (U.S.), NM Parsec (IAU) Solar radius Yard, yd
To
Multiply by
m m m m m m m m
9.460 730 472 580 8 x IO15
Luminance [candela/meter2 = cd/m2] cd/m2 Footlambert Lambert cd/m2 cd/m2 Stilb
l x 1(H 2.54 x 10-5 1.609 344x103 1.852 x IO3 3.085 677 597 49 x IO16 6.960 00 x 108 0.9144 =3.426 259 099 635 39 (1/Tt) x IO4 =3.183 098 862 x IO3 l x IO4
Magnetic Field Strength, H [ampere turn/meter, A/m] (1/4jc) x 103 Oersted (EMU) A/m = 79.577 471 545 947 667 88
893 Notes
D E E E E D AA E E E E
E,1
Magnetic Flux [weber = volt ■s = kilogram ■meter2 • ampere ^ se c o n d 2, Wb = V • s = kg • m2 ■A- * - s ] E Maxwell (EMU) 1x10-8 wb NIST Wb Unit pole 1.256 6 3 7 x l0 -7 Magnetic Induction, B [tesla s weber/meter2 = kilogram ■ampere-1 ■second”2, T = Wb/m2 s kg - A-1 - s~2] Gamma (EMU) (y) T 1 x IO-9 T Gauss (EMU) lxltH
E,1 E,1
Magnetic Dipole Moment [weber ■meter = kilogram • meter3 ■ampere- 1 ■second-2, Wb ■m s kg - m3 • A--1. g-2] 4ti x IO-10 Pole ■ centimeter (EMU) Wb ■m = 1.256 637 061 435 917 295 x 10-9 e ,1 1 x IO-10 E,1 Wb -m Gauss ■ centimeter3 (Practical) Magnetic Moment [ampere turn ■ meter2 = joule/tesla, A • m2 = J/T] 1x10-3 Abampere • centimeter2(EMU) A • m2 A * m2 Ampere • centimeter2 l x 1(H Mass [kilogram, kg] Y(= 1 Mg) Atomic unit (electron) Atomic mass unit (unified), amu Metric carat Metric ton Ounce mass (avoirdupois), oz Pound mass, lbm (avoirdupois) Slug Short ton (2,000 lbm) Solar mass
kg kg kg kg kg kg kg kg kg kg
1 x IO"9 9.109 389 7 x 10-31 1.660 540 2 x IO-27 2.0 X 1CH l x IO3 0.028 349 23125 0.453 592 37 14.593 902 937 21 907.184 74 1.989 1 x 1030
E, 1 E, 1 E
c c E E E E D E AA
894
To convert from
Appendix G To
Moment of Inertia [kilogram • meter2, kg • m2] Gram • centimeter2, gm ■cm2 kg m~ Pound mass- inch2, lbm • in2 kg m 2 Pound mass- foot2, lbm • ft2 kg m2 Slug • inch2, slug • in2 kg m 2 Inch • pound force- s2, in lbf • s2 kg m 2 Slug - foot2 = ft • lbf • s2 kg m 2
G Multiply by
Notes
l x 10-7 2.926 396 534 292x 10^ 4.214 011009 3 8 0 x l0 -2 9.415 402 418 968 x 10-3 0.112 984 829 027 6 1.355 817 948 331 4
Power [watt - joule/second - kilogram -meter2/second3, W = J/s = kg Foot • pound force/second, ft lbf/s W 1.355 817 948 331 Horsepower (550 ft • lbf/s), hp W 745.699 871 582 3 w Horsepower (electrical), hp 746.0 Solar luminosity w 3.845 x IO2®
E E D D D E
- m 2/s3]
Pressure or Stress [pascal = newton/meter2 = kilogram • meter-1 -second-2, Pa = N/m2 = kg m r1 •s-2] Atmosphere, atm Pa 1.013 25 x IO5 IX 105 Pa Bar Centimeter of mercury (0° C) Pa = 1.333 223 874 145 xlO^ Dyne/centimeter2, dyne/cm2 Pa 0.1 Inch of mercury (32° F) Pa 3.386 388 640 341 x 103 Pound force/foot2, lbf/ft2, psf Pa 47.880258 980 34 Pound force/inch2, lbf/in2, psi Pa 6.894 757 293 168 x 10^ Torr (0° C) Pa (101325/760) =133.322 368 421 052 631
D D
E AQ
E E E E E D D
E
Solid Angle (See Angular Area) Specific Heat Capacity [joule - kilogram-1 • kelvin-1 = meter2 • second2 - kelvin-1, J • kg-1 • K-1 = m2 • s2 • K-1] cal ■g-1 ■K_1 (mean) J • kg-1 ■K-1 4.186 80 x IO3 E Btu • lbm-1 • “F-1 (mean) J • kg-i ■K- l 4.186 80x 103 E Stress (see Pressure) Temperature [kelvin, K] Celsius, ”C Fahrenheit, T Rankine °R Fahrenheit, °F Rankine °R
K K K C C
tK* t c + 273.15 tK = (5/9) (tF + 459.67) tK = (5/9) tR tc = (5/9) (tF - 32.0) tc = (5/9) (tR- 491.67)
E E E
E E
Thermal Conductivity [watt • m eter-1 ■kelvin-1 =skilogram • meter - second-3 • kelvin'1, W • m-1 - K-1 = kg • m ■s-3 - K-1] cal • cm-1 • s-1 ■K-1 (mean) W ■m-1 • K-1 418.68 E Btu • f r 1 ■h r 1 ♦ 'F-1 (mean) W • m-1 • K-1 1.730 734 666 371 39 D Time [second, s] Sidereal day, d* (ref. = T) Ephemeris day, dg
s s
8.616 410 035 2 x IO4 = 23h 56m 4.100 352s 8.64 x IO4
AQ AQ
Units and Conversion Factors
G To convert from
Ephemeris day, de Keplerian period of a satellite in low-Earth orbit
To
Multiply by
d*
1.002 737 795 056 6
min
1.658 669 010 080 x l(Hxa3/2 (a in km)
Keplerian period of a satellite of the Sun Tropical year (ref.= T) Tropical year (ref.- Y) Sidereal year (ref.-fixed stars) Sidereal year (ref.=fixed stars) Calendar year (365 days), yr Julian century Gregorian calendar century
895 Notes
AQ
Table 6-2
s d d
3.652 568 954 757 x IO2 X a 3/2(a i n AU) 3.155 692 597 47x IO7 365.242 198 781 3.155 814 976 320 x IO7 365.256 363 3.153 6 x IO7 36,525 36,524.25
AA AA D AA AA E E E
m/s m/s m/s m/s m/s m/s m/s m/s m/s
5.08 x IO-3 0.025 4 (3.6)“* = 0.277777... 0.304 8 0.447 04 (1852/3600) = 0.514444.., 26.822 4 1.609 344x103 2.997 924 58 x IO8
E E E E E E E E E
de
Torque (see Energy) Velocity [meter/second, m/s] Foot/minute, ft/min Inch/second, ips Kilometer/hour, km/hr Foot/second, fps or ft/s Miles/hour, mph Knot (international) Miles/minute, mi/min Miles/second, mi/s Velocity of Light
Viscosity [pascal • second = kilogram ■meter"1 second-1, Pa ■s = kg • m-1 • s-1] Stoke m2/s 1.0 x IO-4 Foot2 • second, ft2 • s m2/s 0.092 903 04 Pound mass- foot-1 • second-1, lbm ■ft-1 - s-1 Pa ■s 1.488 163 943 570 Pound force- second/foot2, 47.880 258 980 34 lbf • s/ft2 Pa ■s 0.1 Poise Pa • s Poundal second/foot2, poundal s/ft2 Pa *s 1.488 163 943 570 Slug • foot-1 • second-1, slug *ft-1 *s-1 Pa ■s 47.880 258 980 34 Rhe (Pa • s)-*1 10 Volume [meter3, m3] X( \ X = 1 |iL = 1 x lO ^L ) Foot3, ft3 Gallon (U.S. liquid), gal Inch3, in3
m3 m3 m3 m3
1 x IO-9 2.831 684 659 2 x 10-2 3.785 411 784 x IO"3 1.638 706 4 x 10-5
E E D D E D D E E E E E
Appendix G
896 To convertfrom
To
G M ultiply by 1x10-3
Liter, L Ounce (U.S. fluid), oz Pint (U.S. liquid), pt Quart, qt Stere (st) Yard3, yd3
2.957 352 956 25 x IO"5 4.731764 73x 10-4 9.463 529 46 x 1(H 1
0.764 554 857 984
Notes E E E E E E
Notes for the preceding table: AA AQ C D E
Values are those of Astronomical Almanac [Hagen and Boksenberg, 1991]. Values are those of A strophysical Quantities [Cox, 2000]. Values are those of Cohen and Taylor [1986]. Values that are derived from exact quantities, rounded off to 13 significant figures. (Exact) indicates that the conversion is exact by definition of the non-SI unit or that it is obtained from other exact conversions. IERS Numerical standards of the IERS. NIST Values are those of National Institute of Standards and Technology [McCoubrey, 1991], (1) Care should be taken in transforming magnetic units, becausc the dimensionality of mag netic quantities (B, H, etc.) depends on the system of units. Most of the conversions given here are between SI and EMU (electromagnetic). The following equations hold in both sets of units: T B m d
= = = -
m xB = dxH llH IA for a current loop in a plane jam
with the following definitions:
T B H m
= = = =
torque magnetic induction (commonly called “magnetic field”) magnetic field strength or magnetic intensity magnetic moment
I
=
current loop vector normal to the plane of the current loop (in the direction of the angular velocity vector of the current loop about the center of the loop) with magnitude equal to the area of the loop magnetic dipole moment
A =
d = jj
=
magnetic permeability
The permeability of vacuum, jjq, has the following values, by definition: |Hfl = 1 (dimensionless) EMU jiQ s 4tt x IO"-7 N/A2
SI
Therefore, in electromagnetic units in vacuum, magnetic induction and magnetic field strength are equivalent and the magnetic moment and magnetic dipole moment are equivalent. For prac tical purposes o f magnetosiatics, space is a vacuum but the spacecraft itself may have |i * jiq.
G
897
Units and Conversion Factors
Useful Mathematical Constants and Values Constant
Value
71
= 3.141 592 653 589 793 238 462 643
(A)
e
* 2.718 281 828 459 045 235 360 287
(A)
e71
» 23.140 692 632 779 269 006
(A)
lo g 10x
= 0.434 294 4S1 903 251 827 651 128 9 lo g ^
(A)
logex
* 2.302 585 092 994 045 684 017 991 logIC>*
(A)
lo g e7i
= 1.144 729 885 849 400 174 143 427
(A)
(A) are from The Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables [Abramowitz and Stegun, 1970]
References Abramowitz, Milton and Irene A. Stegun, eds. 1970. The H andbook o f M athem atical Functions with Formulas, Graphs, and M athem atical Tables. New York: Dover. Cohen, E. Richard and B.N. Taylor, 1986. CODATA Bulletin No. 63, Nov. New York: Pergamon Press. Cox, Arthur N, ed. 2000. A llen’s Astrophysical Quantities {4th Edition ), New York: Springer-Verlag. Hagen, James B. and A. Boksenberg, eds. 1991. The A stronom ical Almanac. Nautical Almanac Office, U.S. Naval Observatory and H. M. Nautical Almanac Office. 1992. Washington, D.C.: U.S. Government Printing Office. McCarthy, Dennis D., USNO. 1996. “Technical Note 21. ” IERS Conventions. McCoubrey, Arthur O. 1991. Guide fo r the Use o f the International System o f Units (SI). National Institute of Standards and Technology (NIST), Special Publication 811, U.S. Department of Commerce: U.S. Government Printing Office. Seidelmann, Kenneth P., ed. 1992. The Explanatory Supplement to the A stronom ical Almanac. Mill Valley, CA: University Science Books. Taylor, Barry N. 1991. The International System o f Units (SI). National Institute of Standards and Technology (NIST), Special Publication 811, U.S. Department of Commerce: U.S. Government Printing Office.
INDEX
Index
Numerics 1950 C oordinates........................................ 294 2000 C oordinates.................................. 48, 294 3-axis a ttitu d e .............................................. 149 3-axis stabilized sp acecraft............................5 3-burn transfer (= bieUiptic transfer).......... 98 applications o f ........................................ 627 6 am—6 pm o r b it.......................................... 86 621B (mission) navigation system.................................... 201 — A ---AAR (See Area Access Rate) Absolute m agnitude.................................... 655 Absolute stationkeeping (See Stationkeeping) Acceleration units and conversion factors.................... 890 Accelerometers advantages and disadvantages................122 Access area (See also Area access rate) . . . . 470 area formulas for segments o f ................ 429 cutting into equal areas............................ 431 numerical values for Earth orbit................................ inside rear cover numerical values for lunar orbit..............875 numerical values for Mars orbit.............. 881 Access area coordinates...................... 418-419 transformation to geographic and spacecraft coordinates................................ 422^423 acos2 function (spherical geometry)........................................ 791-792 definition o f ............................................ 389 properties of (table)................................ 390 ACR (See Area Coverage Rate) ACS (See Attitude Control System) Active attitude control (See Attitude control) definition o f .................................... 119, 125 Actuator (See Attitude actuator) ADCS (Attitude Determination and Control System; See Attitude Control System) Aeroassist tra je c to ry ............................ 98, 600 Aerodynamic disturbance to rq u e.................................. 128 torque...................................................... 171 Aerodynamic stabilization advantages and disadvantages................ 124 Agrawal, B. N. book b y ...................................................... 32 “Airplane problem” (illustration of full-sky geometry)................................ 389, 391—394 A lbedo.................................................. 558, 568 range of, for Earth.................................... 569 Albedo sensors (See Earth sensors) Allen V arian ce............................................ 830 Allman, M. book by.................................................... 113 Allocation of requirem ents........................ 244 Alpha Centauri (star system nearest the S u n ).......................................... 655-656 relativistic travel to .......................... 661, 663 Altitude as means of classifying orbits.................... 58
899
key parameters in mission requirements.............................. 613-614 of target, impact on mapping and pointing errors.................................................. 254 requirements on for orbit control............ 225 selection of for constellations .................................. 692-694, 725,728 Altitude maintenance definition.................................................. 104 AV for vs. altitude.................................... 599 Altitude plateaus.................................. 692-694 definition.................................................. 684 Ampere.......................................................... 889 Analem m a............................................ 505-508 astronomical............................................ 506 definition o f ............................. 453, 504, 506 equations fo r.................................... 504-505 height and width of (table)...................... 520 Anandakrislinan, S. M. book b y .................................................... 116 Andromeda galaxy travel to .................................................... 663 Angle-angle-angle spherical triangle solutions f o r ............................................ 789 Angle-angle-side spherical triangle solutions f o r ............................................ 785 Angles sum and difference formulas.................... 797 Angle-side-augle spherical triangle............ 787 solutions f o r ............................................ 779 Angle-side-side spherical triangle solutions f o r .................................... 387-388 Angular acceleration units and conversion factors.................... 890 Angular area formulas for.............................. 386, 773-774 on Earth vs. seen from sp a c e .......... 428-430 analysis o f.................................. 429-430 table of formulas................................ 429 units and conversion factors.................... 890 Angular measure units and conversion factors.................... 890 Angular measurements introduction to.................................. 318-325 Angular momentum conversation of as Kepler’s second law , . 43 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic o rb its................ 847 formulas for parabolic orbits.................... 845 role in attitude motion...................... 134—148 units and conversion factors.................... 891 Angular momentum axis vs. rotation axis................................ 135-137 Angular momentum space definition of.............................................. 145 Angular momentum sphere................ 145-147 definition o f ............................................. 146 Angular radius of Mars numerical values for Mars orb it.............. 885 Angular radius of the E arth........................ 418 computation o f ........................ 420, 422—423
900 formulas for circular o rb its.................... 837 formulas for elliptical orbits.................... 843 formulas for hyperbolic orbits................ 850 numerical values for Earth o rb it.............................. inside rear cover Angular radius of the Moon numerical values for lunar orbit.............. 879 Angular rate, seen from E arth formula for circular orbits...................... 460 formula for elliptical o rb its.................... 463 formulas for circular o rb its....................838 formulas for elliptical orbits.................... 844 formulas for hyperbolic orbits................ 851 Angular velocity numerical values for Earth o rb it.............................. inside Tear covcr numerical values for lunar orbit.............. 878 numerical values for Mars orbit..............884 numerical values for solar orbit..............887 units and conversion factors.................... 891 Annular eclipse definition o f ............................................ 563 Annulus area formulas f o r .................... 386, 773-774 Anomalistic year.................................... 53, 859 Anomaly (See also Mean anomaly, Eccentric anomaly. True anom aly)........................ 755 definition o f ............................................ 733 equations for........................................ 50-51 of an orbit.................................................. 49 Antenna (See also Footprint) computation of footprint........ 470-472, 475 coverage equations fo r............................ 475 pattern projected onto the E arth.. . . 434-436 A ntipode...................................................... 296 Antipoint...................................................... 296 Antiproton propulsion for interstellar travel................................ 662 Antisolar p o in t............................................ 296 A phelion........................................................ 46 A poapsis........................................................ 46 Apofocus........................................................ 46 Apogee definition o f .......................................... 8, 46 formulas for velocity a t .......................... 842 radius of for elliptical orbits,.................. 840 Apogee height definition o f .............................................. 46 Apogee motor firings.................................... 12 Apollo 11 mission........................................ 571 use of Kalman filter................................ 164 Apollo 17 mission view of Earth from..................................424 Apollo 8 mission view of Earth from.................................. 569 Apolune.......................................................... 46 “Apparent inclination” ...................... 448, 516 Apparent magnitude (See Apparent visual magnitude) Apparent motion of points on Earth seen from space................................ 440-443 of satellite seen from the Earth . . . . 454—469
Index Apparent solar tim e .................................... 195 Apparent Yisual magnitude (See also Brightness)...................................... 578—582 definition o f ............................................ 655 values for Moon and Planets at opposition.......................................... 580 Apsides (See Line o f upsides) Arc length equations for.................................... 770-771 properties o f ............................................ 298 Arc length m easurem ent.................... 320-321 as part of non-singular data set . . . . 374-376 combining with rotation angle........ 362-368 density for................................................ 328 locus o f............................................ 320, 323 possible solutions w ith............................ 321 singular conditions f o r ............................ 327 solution characteristics............................ 365 uncertainty equations f o r ........................ 330 Arc of a small circle equations for............................................ 771 Arc segment definition o f ............................................ 297 Area (See also Angular area) on Earth vs. seen from sp ace.......... 428—430 analysis of.................................. 429-430 table of formulas................................ 429 units and conversion factors.................... 891 Area Access Rate (A A R )............................ 471 equations for.................................... 475-476 formulas for circular orbits...................... 838 formulas for elliptical orbits.................... 843 formulas for hyperbolic orbits................ 851 numerical values for Earth o rb it............................... inside rear cover numerical values for lunar orbit.............. 875 numerical values for Mars orbit.............. 881 Area Coverage Rate (ACR)........................ 471 average value o f ...................................... 474 computation example.............................. 477 for different instrument types.................. 475 Areal velocity equations for........................................ 50-51 formulas for circular orbits...................... 837 formulas for elliptical orbits.................... 842 formulas for hyperbolic orbits................ 850 formulas for parabolic o rb its.................. 847 Argos m ission.................................................. 6 Argument of perigee definition o f ...................................... 49, 733 effect of small changes in................ 512-513 error bounds............................................. 752 Argument of perihelion................................ 53 Ariane launch vehicle cost/performance table............................ 602 Aries (satellite constellation) summary of.............................................. 672 A rray sensor analysis of measurements................ 353-358 “distortion” in .......................................... 357 field of view of single pixel.................... 355 formulas for field of view................ 353—356
Index geometry for square sensor.............. 305—309 pixel field of view formulas............ 356-357 plane geometry errors.............................. 309 projection onto celestial sphere.............. 354 table of geometry f o r .............................. 358 vs. line scanner to reduce c o s t................ 243 Ascending node definition o f .............................................. 48 rotation of due to J2, equations for. . . . 83-84 Aspects (celestial) definition o f .............................................. 53 Asteroids representative rendezvous mission...................................... 653—654 Astrodynamics (See O rbits)..................37—118 definition o f .............................................. 37 Astrolink (constellation) summaiy of.............................................. 672 Astronomical coordinates..........................438 Astronomical latitude.................................. 861 Astronomical symbols chart o f ...................................................... 55 Astronomical Unit (AU)........................ 44, 578 Astrophysical quantities sources o f ................................................ 767 atan2 function.............................................. 791 Atlas launch vehicle cost/performance table............................ 602 performance for interplanetary flight.................................................. 609 Atmosphere composition vs. altitude.......................... 860 density vs. altitude and fl0.7 index.................................................... 73 effect on terminator definition........ 574—575 layers and transitions.............................. 860 properties vs. altitude.............................. 861 shape of.................................................... 437 Atmospheric d rag .......................................... 68 control of effect on constellations.......... 699 effect on orbit elem ents.......................... 513 effect on satellite d ecay ...................... 68-74 equation for................................................ 68 method for controlling effects of (See also Controlled orbit)................................699 need for margin in AVbudget.................. 601 wide variations in ..........................68, 71-73 Atomic tim e.......................................... 187, 189 characteristics of...................................... 188 Attitude as a coordinate transformation........ 149-150 bibliography o f ................................ 176-178 comparison with orbit.............................. 120 definition o f ........................................ 1, 119 impact of errors on mapping and pointing...................................... 254 methods of specifying...................... 149—152 motion in the absence of control . . . 132—148 motion of natural objects........................ 132 origin of study o f ........................................ 2 sources of requirements...................... 27-32 typical requirements o n .......................... 239
901
Attitude actuators definition o f ..................................... 119, 125 orbilvs. altitude...................................... 120 properties and ty p e s................................ 172 use of........................................................ 128 Attitude and Orbit Systems, combined (See Orbit and attitude systems).......................... 1 Attitude control (See also Attitude Control System)............................................ 167-174 bibliography o f ................................ 176-177 capabilities of alternate approaches........ 168 control laws...................................... 123, 125 control torque.......................................... 144 definition of. .. .'.................................. 3, 119 effects of internal m otion............................ 4 error sources, table o f .............................. 271 passive vs. active...................................... 124 relationship to attitude determination. . . . 275 sources of requirements...................... 27-32 Attitude Control System (ACS; See also Attitude control)............................................ 119-176 actuators (See Attitude actuators) definition o f ................................................. 2 evolution of...................................... 174-176 examples of............................................ 5-23 introduction to.................................. 120-132 sensors (See Attitude sensors) sources of requirements...................... 27-32 Attitude determination (See also Attitude; Attitude control; Attitude Control System) ............................................ 148-167 advantages and disadvantages of alternative methods.............................................. 164 advantages and disadvantages of alternative reference sources........................ 121-122 bibliography o f ........................................ 177 definition o f ................................. 3, 119, 123 definitive vs. real tim e ............................ 123 effect of singularities.............................. 123 examples of............................................ 5-23 process for........................................ 163-165 relationship to attitude control................ 275 sample problems.............................. 165-167 sources of requirements...................... 27—32 state estimation vs. deterministic .. . 163—164 Attitude Determination and Control System (See Attitude Control System) .......... 119-176 Attitude dynamics (See Attitude; Attitude control) Attitude geometry bibliography o f ........................................ 178 Attitude m a trix ............................................ 802 advantages and disadvantages for attitude specification...................................... 151 definition o f ............................................. 150 Attitude measurements........................ 317—376 good vs. bad solutions for spin-stabilized spacecraft.................................. 365—368 Attitude prediction.......................................... 3 Attitude p rop ag atio n.................................. 119 Attitude sensing (See also Attitude determination; Attitude measurements).................... 152—163
902
Index
Attitude sensors (See also Attitude determination; Specific sensors—Earth sensors, Star sensors, Sun sensors. Magnetometers, GPS receiver) .......................................... 152—163 applications............................................ 156 common combinations............................ 121 on-orbit performance history.................. 156 parameters f o r ........................................ 153 tabic of reference sources........................ 122 vs. orbit sensors...................................... .120 Attitude stabilization definition o f ................................................ 4 Attitude system (See Attitude Control System)............................................ 119-176 Attitude transfer system ............................ 131 AU (See Astronomical unit) Autocorrelation function............................ 827 Automation increase of in spacecraft.......................... 179 lack of in spacecraft................................ 180 Autonomous definition o f ............................................ 219 Autonomous distance measurements approximate ran g es................................ 218 methods f o r ............................................ 217 Autonomous navigation sy stem s.................................. 210-218, 594 advantages and disadvantages o f............212 basis f o r .......................................... 215-218 characteristics.......................................... 211 fully autonomous systems.............. 215-218 trades regarding...................................... 242 use in interplanetary missions................ 640 Autonomous orbit control propellant budgets.................................. 224 relative vs. absolute................................ 222 thruster sizing.......................................... 224 Autonomous systems advantages and disadvantages o f ............227 Autoregressive models................................ 827 Autumnal equinox...................................... 304 Auxiliary angle of the hyperbola formulas fo r ............................................ 848 Azimuth coordinate................................................ 295 definition o f ............................................ 304 Azimuth of the ascending node equations for............................................ 771 Azimuthal velocity formulas for elliptical orbits.................... 842 formulas for hyperbolic orbits................ 850 formulas for parabolic orbits..................846 Azimuth-Elevation (Az-El) plot distortion relative to globe p lo t.............. 289
— B— Ballistic coefficient definition o f .............................................. 68 table of representative values.................... 71 Ballistic trajecto ry ........................................ 57 Balloon flights a$ low-cost launch alternative................ 604
Bang-bang c o n tro l...................................... 125 Bankir (constellation) summary of.............................................. 672 “Barbecue mode” ........................................ 362 Barycenter...................................................... 40 definition o f .............................................. 40 Earth-Moon system, location f o r ............ 859 Barycentric Dynamic Time (TD B)............ 200 Bate, R. R., et al. book by.................................................... 113 Battin, R. H. book by.................................................... 113 Beamed power propulsion for interstellar travel................................ 663 Beletsky, V. book by .................................................... 115 Bender, K. book by.................................................... 279 Bernoulli distribution.................................. 812 Bessel functions.............................................. 69 Bias (in measurements) vs. noise.......................................... 261-262 Bias momentum attitude control (See Momentum bias attitude control) Bibliography attitude properties and terminology.............................. 176-178 interstellar exploration............................ 664 orbits................................................ 113-117 requirements definition.................. 279-281 space systems engineering.................. 32-36 spherical trig............................................ 799 time systems, GPS, autonav............ 230-232 Bielliptic transfer (= 3-burn transfer) applications o f ........................................ 627 definition o f .............................................. 98 Binomial distribution.................................. 812 Blackbody radiation.................................... 547 Bode’s L a w .................................................... 53 Body c o n e .................................................... 139 role in dual axis sp ira l............................ 399 Body nutation r a t e .............................. 139, 408 Bond a lb e d o ................................................ 581 Bond, V. R. book by.................................................... 113 Boulet, D. L. book by.................................................... 115 Bousquet, M. book by...................................................... 34 B-plane definition o f ............................................ 648 Brahe, Tycho.................................................. 38 BREM-SAT (mission) clock o n .......................................... 191-192 Brightness, of distant spacecraft and p la n e ts.................................... 578-582 British Summer Time (BST)...................... 182 Broken-plane tra n sfe r................................ 638 Brown, C. D. book by.............................................. 32, 116 Brownian motion (See Random walk) ........................................................ 827 Brumberg, E. V. book by.................................................... 115
Index Brumberg, V. A. book b y .................................................... 115 Bryson, A. E. Jr. book by.................................................... 176 BST (See British Summer Time) Bucy, Richard invention of Kalman filte r...................... 164 Budget allocation and flow-down................ 244-249 definition o f ............................................ 245 error vs. commodity........................ 245—246 items typically budgeted.......................... 245 mapping (See Mapping budget) mathematical process for creating...................... 250-253, 268—279 optimum allocation among components................................ 250-253 pointing (See Pointing budget) political process for creating..................248 timing error.............................................. 273 top le v e l.................................................. 268 Burger, J, J, book b y ...................................................... 32 Bus (spacecraft)........................................ 8, 24 — C —
C3 (See Departure energy) Calendar tim e ...................................... 181-182 Camera projection of field of view onto the Earth.................................... 430—4-37 C andela........................................................ 889 Canters, F. book b y .................................................... 178 Cartesian coordinates transformations between.................. 802—803 transformations to and from spherical.................................... 801—805 Cassini mission orbit sequence.......................................... 651 spacecraft brightness seen from Earth........ 581-582 summary of (table).................................. 633 use of autonomous navigation................ 640 use of gravity a ss ist........................ 638-639 Cauchy distribution .................................... 812 CCD array (See also Array sensor) use in star sensor...................................... 161 Celestial coordinates (See also Inertial coordinates)............................................ 293 1950 coordinates...................................... 294 2000 coordinates...................................... 294 need for date fo r........................................ 53 observability in terms of.......................... 468 True of Date (TO D)................................ 294 Celestial mechanics (See Orbits) definition o f ................................................ 1 Celestial meridian........................................ 194 Celestial pole........................................ 293, 302 Celestial sphere (See also Inertial coordinates) angular measurements as loci on . .. 319—321 area form ulas.................................. 773—774
903
definition o f ............................................. 285 error as probability distribution o n .......... 325 geometry o n .................................... 283-316 projection of rectangular array onto.................... ........................354—358 use for evaluating eclipses...................... 551 Celsat (constellation) summary o f.............................................. 672 Centaur upper stage...................................... 14 Central Limit T h eo rem .............................. 822 CEP (Circular E rror Probability) — 823-824 Ceres (asteroid) gravitational parameters.......................... 855 mission parameters (table)...................... 887 Cesium d o c k s .............................................. 189 CGRO (Compton Gamma Ray Observatory) end-of-life deorbit............................ 760-764 Challenger disaster (1986)............................ 14 Characteristic function (See also Moment generating function)........................ 816, 822 Chaser (in formation flying) definition o f............................................. 528 Chebychev’s inequality............................ .821 Chetty, P. R. K. book b y ...................................................... 33 Cheyenne Mountain debris tracking............................................ 30 China launch vehicles (table)............................ 602 Chobotov, V. A. book b y ............................................ 113, 176 Chord length equations fo r............................................ 771 Churchill, S. E. book b y .............................................. 33, 279 Circle as a conic section................................ 41-42 Circular E rro r Probability (C E P ).. . . 823-824 Circular normal distribution (Rayleigh)........................................ 812, 823 Circular orbits (See also Orbits).............. 40—42 Earth coverage for............................ 470-488 Earth, table of parameters.. .inside rear cover effect of drag o n ........................................ 69 equations, table o f...................................... 51 ground trace o f ................................ 455-461 Mars, table of parameters................ 880-885 Moon, table of parameters.............. 873-879 motion of satellite seen from Earth.. 455-461 orbit and coverage equations.......... 836-839 Sun, table of parameters.................. 886-888 table of station pass equations................ 460 viewed from nearby........................ 314—315 Circular velocity definition o f............................................... 45 numerical values for Earth orbit................................ inside rear cover numerical values for lunar o rb it.............. 878 numerical values for Mars o rb it.............. 884 numerical values for solar o rb it.............. 887 values for Earth, Sun, Moon, Mars............45 Civil orbit........................................................ 82 definition o f............................................. 621
904 Civil tim e...................................................... 182 Clairaut, Alexis C. (mathematician)..........135 Clarke, A rthur C. discovery of G E O .................................... 77 Clock (See also Time) meaning of in relativity.......................... 199 time discontinuities........................ 190-193 Clohessy, w . R..................................... 527-528 Clohessy-Wiltshire equations (See Hill's equations) Cluster definition o f ............................................ 671 CMG (See Control Moment Gyros) Co-elevation coordinate.............................. 295 Coherence (in noise).................................... 827 Co-latitude coordinate................................ 295 Collision avoidance.................... 675,708-718 as long-term threat to constellations............ 674—676, 709—713 definition of parameters in constellation design........................................ 725, 728 rules f o r .................................................. 714 Collision cross section definition o f ............................................ 709 for planetary arrival........................ 648-650 increase with satellite break u p .............. 712 Collision opportunity definition o f ............................................ 709 Collisions, between spacecraft (See also Collision avoidance) consequences o f.............................. 710-711 debris cloud evolution............................ 712 probability o f .................................. 710-711 representative probabilities (table)..........711 Colored n o is e .............................................. 827 Comet Borelly mission to................................................ 632 Comet Temple 1 representative rendezvous mission...................................... 653-654 Comet Wilson-Harrington mission to ................................................ 632 Comets break-up o f.............................................. 710 missions t o .............................................. 632 rendezvous with Giacobini-Zinner........ I l l Committee on Space Research (COSPAR) international designation of satellites by .. 59 Commodity budget definition o f ............................................ 245 Commutation in coordinates systems.................... 154-155 Complete spherical triangle definition o f ............................................ 380 number of solutions f o r .......................... 387 taxonomy o f............................................ 380 Component uncertainty definition o f ............................................ 333 Compton Gamma Ray Observatory (See CGRO) Computers increase of on spacecraft................ 179-180 Conditional probabilities.................... 811-812
Index Configurations, planetary (See Planetary configurations) Conic sections (See also Keplerian orbits) defined...................................................... 40 equations, table o f .................................... 51 orbit properties.......................................... 42 Coning.......................................................... 137 Conjunction, p la n e ta ry .......................... 53-54 Conservation of angular momentum (See Angular momentum) Conservation of energy (See Vis viva equation) Constellation design (See also Constellations, Formations, Orbit design, Earth coverage)........................................ 671—730 alternative patterns.......................... 680-689 build-up.................................. 676, 718-719 collision avoidance.................. 675, 709-718 coverage (See also Earth coverage).................................. 676-680 end-of-life (Set ulsa Disposal o f spacecraft)................................ 676, 723 history of work on (table)................ 681-682 need to break symmetry for collision avoidance.................................. 717—718 outages and replenishment. . . . 676, 720-723 principal issues fo r.......................... 673-676 process table fo r.......................................724 rules for (table)........................................ 728 selection of parameters.................... 690—697 stationkeeping.................................. 697-709 summary of...................................... 723-727 Constellation maintenance (See Stationkeeping)................................ 697-709 definition o f .............................................104 Constellations (See also Constellation design; Orbits;Formations; Relative motion) definition o f ............................................ 671 design o f (See Constellation design) end-of-life........................................ 676, 723 large scale relative motion i n ..........501-510 non-standard patterns...................... 688-689 number of satellites- ................................ 675 orbit control f o r .............................. 220-222 patterns............................................ 680-689 small scale relative motion in.......... 510-518 table of representative.................... 672-673 vs. single satellite.................................... 594 Constraints (as part of requirements; See Requirements).................................. 237—241 definition o f .............................................238 Continuous measurements definition o f .............................................325 error analysis.................................. 334—341 uncertainty formulas........................ 334—340 Continuous thrust form ations.................... 535 Control law (See also Attitude control) definition o f .................................... 123, 125 Control Moment Gyros (CMG) characteristics of...................................... 172 definition o f .............................................127 Control system (See Attitude Control System)............................................ 119—176
905
Index Control torques definition o f ............................................ 144 Controlled o r b i t .......................... 700,702-703 definition o f .................................... 700, 702 Controlled reentry definition o f ............................................ 744 Coordinate systems (See also Celestial coordinates; RPY; Earth-fixed; inertial)............................................ 292-296 cartesian and spherical transformation............................ 801-802 celestial.................................................... 294 common types for space use.................... 293 defining.................................................... 292 Earth-fixed.............................................. 294 geocentric vs. geodetic.................... 861—863 names o f ................................ ..................295 principal components of spherical.......... 304 selecting one to use.................................. 293 transformation between spherical............ 414 transformations between geographic, access area, and spacecraft.................. 422-423 Coordinate transform ations.............. 801—805 Coordinated Universal T im e...................... 1S9 Correlated measurements definition o f .................................... 309, 318 direction accuracy for.............................. 341 error ellipse for........................................ 335 Correlation an g le ................................ 326-332 definition o f ............................................ 326 formal definition...................................... 331 unacceptable values f o r .................. 332-333 Correlation e r r o r s ...................................... 274 Coscon (constellation) summary of.............................................. 672 Cosine detector (Sun sensor type)..............327 COSPAR (See Committee on Space Research) Cost (See also Launch cost, Orbit Cost Function) minimizing via specification process.............................. 251-252, 257 Cost-risk analysis................................ 831-833 Coverage (See Earth coverage) Coverage gap {Figure of M erit).................. 484 Coverage histogram .................................... 478 Crescent E arth photograph from Apollo 1 1 .................... 571 Crescent phase (See Illumination phases) Crew motion effect on spacecraft.......................... 128—129 Critical inclination definition o f .............................................. 88 Critical requirements definition o f ............................................ 240 Cross product equation for direction o f.......................... 772 Cross section (See Collision cross section) Cross-correlation........................................ 827 Crosslinks, satellite advantages and disadvantages for navigation.......................................... 212 characteristics of...................................... 211 definition o f ............................................ 214
Cross-power spectrum (See also Coherence) .................................. .......... 827 .......... 112 Cross-track orbit maintenance C u sp .................................................. .......... 570 — D— d’Alembert, Jean Le Rond (m athem atician)........................ Traite de Dynamique.................. Damping (See Nutation damping) . . . Danby, J. M. A. book b y ........................................ Dark angle definition o f ................................. Darkness of night sky.................................. DATE, software function................ DavidolT, M.
, , . 135 ............ 40 .............. 3 .......... 113 .......... 574 549 185
............ 33 Day 181, 189 variations in length o f.................. Day num ber...................................... ..........184 Daylight Savings Time (D S T )........ .......... 182 “Dead-’ satellite 715 consequences of dying in place .. outage due to dying in place........ 720-723 Debris cloud (from satellite break-up) . . . . 710 Debris, in space consequences of collision with................ 711 higher collision cross-section than satellite.............................................. 712 in GEO .................................................... 617 reentry pattern for CGRO................ 761-764 tracking o f.................................................. 30 Decleir, H. book b y .................................................... 178 Declination definition o f ..................................... 304, 802 Deep Space Network (DSN) communications with Pioneer, Voyager.............................................. 632 ground stations f o r ............................ 18, 641 Definitive attitude determ ination.............. 123 Definitive o r b i t .............................................. 61 Degenerate spherical triangles............ 792-795 dealing with.............................................. 795 physical meaning o f ................................ 793 Deimos gravitational parameters.......................... 854 orbit d a ta ................................................. 867 physical and photometric d a ta ................ 869 Delta launch vehicle cost/performance table............................ 602 Orbit Cost Function for.................... 606-607 performance for interplanetary flight.................................................. 609 AV (See also Propellant budget) estimates f o r ............................................ 224 Denmark 0 rsted m ission........................................ 6—9 Density units and conversion factors.................... 891
906 Deorbit, of spacecraft (See also Disposal o f spacecraft) definition o f .............................................. 60 AV for vs. altitu d e.......................... 598-599 equations for............................................ 102 numerical values for Earth o rb it.............................. inside rear cover numerical values for lunar orbit..............878 numerical values for Mars orbit.............. 884 Departure energy (C3) origin of term C3 .................................... 634 role in planetary m issions.............. 634-637 Der-Danchick eq u atio n s............................ 528 Descending node............................................ 48 Despun platform (See also Dual-spin stabilization).................................. 130, 167 Deterministic definition o f ............................................ 261 Deterministic attitude determination definition o f .................................... 123, 163 Deterministic measurements test for good vs. bad........................ 366-367 Deviation of the vertical.............................. 861 Differential spherical trigonometry .. 795-797 Dihedral angles (= rotation angle) definition o f ............................................ 297 Direct orbit (= prograde).............................. 60 Direction calculation of uncertainties i n ................ 332 on Earth vs. from space.................. 426—427 Direction cosine matrix (See Attitude matrix) .................................................... 802 definition o f .................................... 150-151 Direction uncertainty definition o f ............................................ 333 Directional an ten n a.................................... 162 applications o f ........................................ 156 typical parameters.................................. 153 D irectrix........................................................ 42 Discover mission comet exploration.................................... 632 Discoverer I (constellation) summary o f ............................................ 672 Disposal of sp acecraft........................101-103 CGRO example.............................. 760-764 in constellations.............................. 722, 728 in GEO.................................................... 617 need for in a constellation.............. 715-716 part of orbit design.................................. 595 Disposal orbit (See “graveyard orbit") . . . . 592 Distance Autonomous measurements of........ 217-218 from fo cu s................................................ 51 formulas for circular o rb its.............. 836 formulas for elliptical orbits.............. 839 formulas for hyperbolic orbits..........847 formulas for parabolic orbits..........845 in an orbit.................................................. 51 measurements o f ............................ 368-373 to the horizon formulas for circular o rb its.............. 837 formulas for elliptical orbits.............. 843 formulas for hyperbolic orbits .. 850-851
Index Disturbance torques (See also Attitude control) definition o f ............................................ 144 estimating worst case.............................. 171 table o f .................................................... 128 Disturbing potential definition o f .............................................. 63 Divergence in a filter.................................................. 164 DMSP (constellation) summary of.............................................. 672 ‘D onut of position” ............................ 218, 372 Double angle form ulas................................ 798 Double a rc .................................................... 793 Draconic m o n th ............................................ 57 Drag (See Atmospheric drag) Drag coefficient definition o f .............................................. 68 formulas f o r .............................................. 70 table of representative values.................... 71 Draim 4 (constellation) summary of.............................................. 672 Draim, John minimum satellite constellation.............. 682 Drift (in attitude) definition o f ............................................ 145 Drift o r b it...................................................... 99 Drop tower as low-cost launch alternative................ 604 DSN (See Deep Space Network) DSP (constellation) summary of.............................................. 672 DST (See Daylight Savings Time) DT (See Dynamical Time) DTG ('Dynamically Tuned Gyro; See Gyroscope).............................................. 162 Dual spin stabilization capabilities o f ...........................................168 Dual-axis spiral.................................... 394—405 applications o f ........................ 395, 405-416 characteristics of...................................... 402 definition o f ............................................ 394 equations for............................................ 403 equations for relative motion of non-coaltitude satellites............................................ 541 examples of rotating sensor on a spinning spacecraft............................ 405-408 satellite ground tra c k ........ 396,410-114 spacecraft nutation........ .. 408^0 9 geometry o f ............................................ 398 illustration o f .......................................... 395 separation between up and down segments.............................................405 Dual-spin spacecraft definition o f .......................... 5, 18, 130, 167 example of (Galileo)............................ 14—19 Dumping (See Momentum dumping) Duty cy c le.................................................... 474 AV budget (See also Propellant budget)............................................ 596-601 definition o f ............................................ 596 process of creating (table)...................... 597 Dwell time (See also Exposure time)............ 474
Index Dwells (ra d a r).............................................. 582 Dynamical Time (DT).................................. 187 Dynamically Tuned Gyro fDTG; See Gyroscope).............................................. 162 Dyson, Freeman study of interstellar travel........................ 662
---E--Earth angular radius of (See Angular radius o f the Earth) appearance viewed from space........ 424—430 atmosphere of (See Atmophere) atmosphere, effect on satellites (See Atmospheric drag) attitude measurements using (See Earth sensor, Nadir angle measurement, Earthwidth measurement) circular and escape velocities of................ 45 comparison of models of shape.............. 437 geocentric and geodetic coordinates................................ 861-863 geometry viewed from space.......... 418-440 gravitational constant.............................. 835 gravitational parameters.......................... 854 oblateness of (See Oblateness) orbit and coverage equation summary.................................... 835-852 orbit data.................................................. 865 orbit o f ................................................ 52-57 phases (See Illumination phases) phases seen from space (photos) . . . 569, 571 physical and orbit properties.......... 859-863 physical and photometric d a ta ................ 866 rotation rate variations............................ 181 rotational position of (sidereal tim e).......................................... 193-198 satellite data f o r .................. inside rear cover seen from Apollo 8 (photo).................... 569 seen from Apollo 17 (photo).................. 424 shape (numerical values f o r ) .................. 436 spherical vs. oblate model — 418, 421,438 terminator (See Terminator, Illumination phases)...................................... 574—577 thermal balance o f .................................. 568 use in gravity assist trajectory............................ 608, 638—639 E arth central angle as coverage param eter............................ 472 computation o f ........................420, 422—423 numerical values for Earth orbit................................ inside rear cover formula for circular o rb its.............. 460, 837 formula for elliptical orbits..............463, 843 formula for hyperbolic orbits.................. 851 maximum........................................ 456, 476 role in constellation design...................... 690 E arth coverage.................................... 417—498 advantages and disadvantages of different analysis approaches.......................... 488
907
analysis o f ........................................ 469-492 analytic approximations f o r .................... 472 analytic computations fo r................ 479-482 combined role of altitude and elevation angle.......................................... 690-694 constellation figures of m erit.................. 728 coverage patterns............................ 477—479 circular orbit.............................. 477—479 elliptical o rb its.......................... 488-489 G E O .......................................... 490-492 discontinuous variation with altitude...................................... 684-685 effect of spacecraft outage.............. 720-721 equation summary............................ 835-852 excess coverage in a constellation.......... 726 Figures of M e rit.............................. 483—486 for circular o rb its............................ 470—487 for elliptical orbits............................ 488-490 Gaussian statistics not applicable .. . 469—470 importance of swath w idth.............. 690—692 in adjacent orbit planes.................... 676-680 minimum number of satellites for............682 mitigation of spacecraft outage........ 721-723 numerical simulations of.......................... 488 role in constellation design.............. 674-675 role in orbit design.................................. 613 sample analysis................................ 492-497 typical outage in a constellation.............. 720 vs. latitude and inclination...................... 696 Earth departure geometry and constraints................ 645-647 E arth mid-scan an g le.................................. 343 Earth oblateness (See Oblateness) .............. 498 E arth Observing System (See EOS; see also Aqua, EO-1, Terra) E arth sensor.................................. 121, 157-160 advantages and disadvantages................ 122 applications.............................. 156, 341-345 classification of measurement type.. 323-325 definition o f ............................................. 158 full-sky coverage with 2 sensors.. . , 346-347 geometrical gain f o r ........................ 344—345 measurement analysis fo r................ 342—345 measurement equations............................ 343 most common ty p e s................................ 159 on-orbit performance history.................. 156 typical...................................................... 160 typical parameters.................................... 153 use as planet sensor. .................................160 use for orbit and attitude............................ 29 use with star sensing for navigation........ 213 Earth swingby (See also Planetary assist trajectory) use in interplanetary missions.. 100, 653-654 E arth width m easurem ent.......................... 324 Earth-centered inertial coordinates (See Geocentric inertial coordinates) Earth-fixed coordinates.............................. 312 Earth-fixed spacecraft apparent motion of Sun fro m .......... 312-314 Earth-referenced orbit applications and design of................ 612-623
908 Eccentric anomaly definition o f ........................................ 49—50 formulas fo r ............................................ 841 recursive formula fo r................................ 49 Eccentricity formulas for hyperbolic orbits................ 848 Eccentricity vector definition o f ............................................ 107 formulas for elliptical orbits....................840 formulas for hyperbolic orbits................ 848 formulas for parabolic orbits.................. 846 Eccentricity, of E arth’s shape.................... 437 Eccentricity, orbit definition o f ...................................... 42, 733 effect of small changes i n .............. 512—513 for circular orbits.................................... 836 formulas for clliptical orbits.............. ..... 841 formulas for parabolic orbits.................. 846 role in collision avoidance...................... 716 selection of in constellation design........................................ 725, 728 ECCO (constellation) summary o f ............................................ 672 Echo-1 mission ballistic coefficient.................................... 71 Eckart, P. book by.................................................. .. 33 Eclipse.................................................. 563-567 as source of disturbance torque.............. 128 conditions fo r.................................. 564—566 definition o f .................................... 550, 563 duration computation.............. 290-292, 310 for a circular o rb it.......................... 310-311 illustration of ty p e s................................ 564 maximum duration for circular orbits . . . 839 maximum duration for Earth o rb it................................................ inside rear cover maximum duration for clliptical orbits .. 845 maximum duration for hyperbolic o rb its................................................ 851 numerical values for Earth orbit numerical values for lunar orbit.............. 879 numerical values for Mars orbit..............885 of the M oon............................................ 550 of the satellite.......................................... 563 of the Sun................................................ 563 terminology dependant on observer........ 563 Ecliptic as reference for interplanetary o rb its............................................ 52-53 definition o f .............................................. 47 Ecliptic coordinates.................................... 293 Effective h o riz o n ........................................ 457 definition o f ............................................ 418 Einstein, Albert (See also Relativity) the twin paradox...................................... 661 theory of relativity.................................... 39 Elbert, Bruce R, book by...................................................... 33 Electric charge units and conversion factors.................... 891
Index Electric conductance units and conversion factors.................... 891 Electric current units and conversion factors.................... 891 Electric field intensity units and conversion factors.................... 892 Electric potential difference units and conversion factors.................... 892 Electric propulsion (See also Solar electric propulsion) applications o f ........................................ 628 transfer orbit equations........................ 94—96 use by New Millenium............................ 640 use in interplanetary m issions........ 652-653 Elcctric resistance units and conversion factors.................... 892 Electromagnet (See Magnetic torquer) Electrostatic propulsion.............................. 652 Elements, of an orbit (See Orbit elements; Keplerian elements) effects of small changes in .............. 512-513 Elevation (in Az-El coordinate system)............................................ 295, 304 Elevation angle computation o f ........................ 420, 422—423 definition o f ............................................ 420 formula For circular o rb its...................... 460 formula for elliptical o rb its.................... 463 mitigation of satellite outage.................. 721 selection of in constellation design........................................ 725, 728 Ellipse as a conic section................................ 41—42 defined...................................................... 40 terminology........................................ 41-42 Ellipso (constellation).......................... 622, 687 illustration................................................ 687 summary of.............................................. 672 Elliptical orbits (See also Eccentric orbits; O rbits).................. 40-42, 46, 449-452, 462 Earth coverage f o r .......................... 462-490 effect of drag o n ........................................ 69 equations, table o f .................................... 51 geometry o f .............................................. 41 ground station coverage.................. 462—463 ground trace o f ................................ 449—452 motion of satellite seen from Earth.......................................... 462-463 orbit and coverage equations.......... 839-845 table of station pass equations................ 463 terminology.............................................. 46 use in constellations........................ 673, 687 EUipticity of the E arth (See Flattening factor).........................................................64 Elongation (astronomical)............................ 54 Emitted radiation (See Infrared radiation)................................................ 568 End-of-life (See Disposal o f spacecraft). . . . 118 in a constellation...................................... 723 Energy (See also Specific energy) conversation of (See Vis viva equation) formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840
Index formulas for hyperbolic o rb its................ 847 units and conversion factors.................... 892 Energy ellip so id .................................. 145-147 definition o f ............................................ 146 Ensemble of systems................................................ 824 Entry corridor in planetary missions.............................. 649 Environment importance of in constellation design........................................ 674—675 models o f ................................................ 274 role in orbit design.......................... 613-614 Ephemeris definition o f .............................................. 38 lunar and planetary.................................. 104 of satellite, definition o f ..........................104 Ephemeris time (E T ) .................................. 1S7 characteristics of...................................... 188 history of.................................................. 200 Epoch definition o f .............................................. 38 of an orbit.................................................. 48 Epsilon Eridani (nearby Sun-like sta r).. . . 655 Equation of tim e.................................. 195, 506 Equator (in a coordinate system ).............. 302 Equatorial plane definition o f .............................................. 47 Equilateral right triangle definition o f ............................................ 301 Equinoxes ('See also Vernal Equinox).......... 304 definition o f .............................................. 53 Equipotential surface.................................. 861 E rror analysis...................................... 807-823 bias vs. noise............................................ 262 definition of sources................................ 258 introduction to analysis o f .............. 25 S—268 mathematical process for “summing” ................................ 258-268 of angular measurements................ 325—353 optimum allocation among components................................ 250-253 sources of for pointing and mapping . . . . 251 E rror bounds................................................ 752 E rror budget definition o f ............................................ 246 examples of geopositioning.................................. 247 mapping.................... 253-256, 276-277 pointing...................... 256-257, 276-279 pointing (simplified).......................... 246 for reference frames................................ 272 mathematical process for “summing” ................................ 258-268 optimum allocation among components................................ 250-253 pointing error budget for Space Telescope.......................................... 246 process for creating (tables)............ 260-261 E rro r ellipse definition o f ............................................ 334 for nonorthogonal measurement?............ 335
909
for orthogonal measurements.................. 335 probability interpretation........ 339, 823-824 E rror equation.. ........................................ 258 E rror parallelogram ............................ 326, 333 ERS-1 ballistic coefficient.................................... 71 ESA (See European Space Agency) E-Sat (constellation) summary o f.............................................. 672 Escape velocity definition o f ............................................... 45 equation for................................................ 45 formulas for circular orbits...................... 837 formulas for elliptical orbits.................... 842 formulas for hyperbolic o rb its................ 850 major solar system bodies................ 853-855 numerical values for Earth orbit................................ inside rear cover numerical values for lunar o rb it.............. 878 numerical values for Mars orb it.............. 884 values for Earth, Sun, Moon, Mars............ 45 Escobal, P. R. book b y ............................................ 115, 117 ET (See Ephemeris Time) Euler angle............................................ 398-400 definition o f..................................... 135, 399 Euler axis.............................................. 398-400 application to ground track analysis......................446—447, 459—461 co-latitude of formulas for circular orbits .................837 formulas for elliptical orbits.............. 842 formulas for hyperbolic o rb its..........850 definition o f ..................................... 135, 399 role in ground track analysis............ 410-414 table of values for satellite ground track. .412 use in better approximation of ground tra c k .................................................. 413 Euler axis/angle advantages and disadvantages for attitude specification...................................... 151 Euler rotation rate formulas for circular orbits...................... 837 formulas for elliptical orbits.................... 843 formulas for hyperbolic o rb its................ 850 Euler symmetric param eters (See quaternion) advantages and disadvantages for attitude specification.......................................151 Euler, Leonhard (Swiss m athem atician).............................. 135, 399 Euler’s theorem............................................ 399 Europe time zones................................................ 182 European Space Agency (ESA) ISS participation.................................. 19—21 launch vehicles table................................ 602 Excel (software) use in date calculations............................ 186 Excess coverage, in constellation................ 726 Expectation, of a fu n c tio n .................. 819-820 Explorer I (space m ission).......................... 148 Explorer-11 ballistic coefficient.................................... 71
910
Index
Explorer-17 ballistic coefficient.................................... 71 Exposure time (See also Dwell tim e)..........474 Extended Kalman filter.............................. 164
— F— fl0.7 index definition o f .............................................. 72 effect on drag and satellite decay........ 72-75 historical daily v alu es............................ 857 historical monthly values........................ 858 Failure, of a satellite in a constellation motion o f ................................................ 715 outage due t o .................................. 720-723 ways to mitigate.............................. 721—723 FAISAT (constellation) summary o f ............................................ 672 Farrell, J. A. book by.................................................... 230 Field of view direction, shapes, and area projected onto the Earth.......................................... 426-430 instantaneous.......................................... 470 projection onto the Earth’s surface...................................... 430-437 Figures of Merit (FoMs) for coverage.................................... 483—486 Filtering (See also Kalman filter; Attitude determination; Orbit determination} .................................123, 164 effect of singular measurements o n ........ 329 errors i n .................................................. 215 First Point of Aries (See Vernal Equinox) First quarter Earth........................................................ 571 M oon...................................................... 580 First quarter phase (See Illumination phases) “Fish” spherical triangle............................ 382 illustrations o f ........................................ 381 inside vs. outside (illustration)................ 381 taxonomy o f............................................ 380 FIX, software function................................ 185 Flat spin definition o f ............................................ 124 Explorer 1.......................... - ....................148 Flattening factor (= E lliptidty).......... 859, 861 definition o f .............................................. 63 of the Earth.............................................. 437 Fleeter, R. book by...................................................... 33 Flight path angle definition o f .............................................. 50 formulas for circular o rb its....................836 formulas for elliptical orbits.................... 840 formulas for hyperbolic orbits................848 formulas for parabolic orbits..................846 use in determining orbit elem ents.......... 107 Flow-down of requirements...................................... 244 FLTSATCOM (constellation) summary o f ............................................ 672
Flux density.................................................. 578 Flyby (See also Planetary assist trajectory)................................................ 600 Focus, of a conic section............................ .. 40 Football nutation o f .............................................. 132 F oo tp rin t...................................................... 473 area formula............................................ 471 average overlap...................................... 474 size computations............................ 472-474 Force “living” vs. “dead” .................................... 40 units and conversion factors.................... 892 Foreshortening near horizon as seen from space.............. ............................424—426 Formations of satellites continuous-thrust............................ 534-536 definition o f .................................... 518, 671 relative motion in ............................ 519-527 Fortes cue, P. book b y ...................................................... 34 Forward, Robert interstellar bibliographies........................ 664 invention of statite.................................... 91 study of interstellar missions.................. 654 Fourier transform s...................................... 816 FOV (see Field o f view) Franklin, G. F. et al. book by.................................................... 176 Free return trajecto ry ................................ 626 French, J. R. book by...................................................... 34 Frozen orbit applications o f .................................. 76, 622 characteristics and applications o f .......... 615 definition o f .............................................. 90 use o f ...................................................... 622 Full E arth photographed from Apollo 8 .................. 569 Full sky coverage by rotating sensor on spinning spacecraft.................................. 405-408 Full sky spherical geometry................ 377-416 introduction t o ................................ 378—394 simpler than traditional spherical geometry............................................ 388 Full sky spherical triangles (= complete spherical triangles) complete solutions f o r .................... 779-795 definition o f ............................................ 779 introduction t o ................................ 378-394 Functional requirements (See also Requirements).................................. 237—241 definition o f .............................................238 in specifications...................................... 241 Fusion rockets for interstellar travel................................ 663
— G— G&C (See Guidance and Control) Galileo Galilei.......................................... 38, 40
Index Galileo mission to J u p ite r ...................... 14—19 orbit............................................................ 17 orbit and attitude system............................ 19 satellite functional block diagram ............ 18 spacecraft.................................................. 15 summary of (table).................................. 633 timeline................................................ 13, 16 use of gravity a ss ist........................ 638—639 GalileoSat (constellation) summary o f.............................................. 672 Gamma Ray Observatory (See CGRO) Gander (constellation) summary of.............................................. 672 GAS container (Spacc Shuttle) as low-cost launch alternative................ 604 Gas Jets (See Thrusters).............................. 126 Gauss's equation............................................ 49 Gauss’s Form ula.......................................... 775 Gaussian statistics not applicable for Earth coverage... 469-470 GCI (See Geocentric Inertial Coordinates) GDOP (See Geometric Dilution o f Precision) General Theory of Relativity...................... 199 GEO (See Geosynchronous orbit) Geocentric coordinates................................861 Geocentric Inertial Coordinates (GCI) .. . 293 Geocentric latitude.............................. 438, 861 Geodetic coordinates.................................. 861 Geodetic latitude.................................. 438, 861 Geographic coordinates (Latitude, Longitude) transformation to access area and spacecraft coordinates................................ 422-423 Geographic latitude (See also Geodetic latitude) 438,861 Geoid (See also Mean sea level)..................437 Geometric alb ed o ........................................ 5S1 Geometric Dilution of Precision (G D O P).................................................. 205 Geometrical gain (See Measurement density) definition o f ............................................ 330 Geometrical horizon (See also Horizon) . . . 457 definition o f ............................................ 418 Geometry (See also Attitude geometry; Mission geomery) bibliography o f ........................................ 178 direction, shape, and area on Earth as seen from space.......................................... 426—430 on the celestial sphere...................... 283—316 introduction t o .......................... 284—296 Sensor FOV projected onto Earth.,. 430-437 Geopositioning accuracy model (See also Pointing).................................................... 28 Geopotential model.................................. 63-67 definition o f .............................................. 64 Geostationary orbit (See Geosynchronous orbit) Geosynchronous orbit (GEO).................................. 76-79, 616-617 apparent daily motion in............................ 77 applications o f ................................ 616—617 characteristics and applications o f ..........615 constellations i n ...................................... 683 definition o f ...................................... 4 ,9 ,5 8 discovery o f .............................................. 77 Earth coverage f o r .......................... 490-492
911
example mission (Instelsat).................. 9—13 motion of pole due to Sun, Moon.............. 78 motion of satellites in LEO from . . . 542-543 motion of, as seen from L E O .......... 544-545 Orbit Cost Function for (table)................ 607 radiation environment.............................. 614 recommended methods for handling perturbations...................................... 699 satellite distribution i n ............................ 616 satellite ground trace........................ 453-454 shape of as seen from E a rth ............ 462—464 stationkeeping.......................................... 104 use of................................................ 616-617 viewed from Earth surface.............. 462-465 Geosynchronous ring appearance of from E arth........................ 464 Geosynchronous transfer orbit (G T O )........ 10 GGS (constellation) summary o f.............................................. 672 GHA (See Greenwich Hour Angle) Gibbons phase (See Illumination phases) Gibbs vector advantages and disadvantages for attitude specification ....................................... 151 Global geom etry.................................. 283—316 definition o f ,............................................ 302 Global Positioning System (See GPS) Globalstar (constellation) summary o f.............................................. 672 Globe plot advangates and disadvantages........ 285—289 definition o f............................................. 285 lack of distortion relative to az-el........................................ 286-289 GLONASS (Russian Global Navigation Satellite System )...................... 23,208-209 comparison to GPS.................................. 209 summary o f.............................................. 672 GMT (see Greenwich Mean Time) Goddard Earth Model 10b (GEM lOb)........ 64 GOES (constellation) summary o f.............................................. 672 Gonetz (constellation) summary o f.............................................. 672 Gordon, G. D. book b y .......................................................34 GPS (Global Positioning System) . . . . 201-209 advantages vs. disadvantages for navigation.......................................... 212 as orbit/attitude system.......................... 3, 29 characteristics o f...................................... 211 comparison to GLONASS...................... 209 computation of excess coverage.............. 726 coverage as a function of altitude............ 204 difference with BREM-SAT clock.......... 192 inverse GPS.............................................. 214 navigation message.......................... 206—207 orbit parameters...................................... 202 required satellites in view........................ 208 summary o f.............................................. 672 use in satellite navigation........................ 179 use on ISS.................................................. 22
912 GPS receiver applications............................................ 156 classification of measurement ty p e ........ 323 on-orbit performance history.................. 156 typical parameters.................................. 153 use for attitude determination................ 163 use for orbit and attitude...................... 3,29 GPS time characteristics o f .................................... 188 definition o f ............................................ 190 transfer error budgets.............................. 190 Graceful degradation in constellation design............................ 594 Graveyard orbits (See also Disposal o rb it)........................................................ 12 in GEO .................................................... 617 role in orbit design.................................. 614 Gravitational constant (#1 accuracy o f........................................ 44,835 major solar system bodies.............. 853-855 table of values.......................................... 44 Gravitational m ass........................................ 39 Gravity-assist trajectory (See Planetary assist trajectory) Gravity-gradient disturbance torque.................................. 128 role in formation flying.......................... 535 torque computation................... ............ 171 Gravity-gradient stabilization............ 133-134 advantages and disadvantages................ 124 capabilities o f.......................................... 168 definition o f ........................................ 5, 124 example of (0rsted)................................ 6-9 hidden cost of.......................................... 167 of M oon.......................................... 132, 134 Grazing angle (see also Elevation angle) definition o f ............................................ 420 Great circle as a degenerate triangle.......................... 794 definition o f ............................................ 297 equations for.................................... 770—773 Great circle arc ............................................ 297 Greater a n g le ...................................... 382, 385 Green, R. book by.................................................... 178 Greenberg, J. S. book by.................................................... 280 “Greenland effect” (on a m a p )..................303 Greenwich hour angle (GHA; See also Greenwich sidereal time)................196—197 Greenwich Mean Time (GM T).................. 182 Greenwich m e rid ia n .................................. 181 definition o f ............................................ 304 Greenwich Sidereal Time (G S T ).............. 467 definition o f ............................................ 196 determining............................................ 197 Gregorian ca le n d ar.................................... 184 Griffin, M. book by...................................................... 34 GRO (Gamma Ray Observatory; see CGRO) Ground station coverage of by circular o rb its........ 455-461 coverage of by elliptical orbits........ 462—463
Index definition o f ............................................ 454 table of coverage formulas.............. 460, 463 Ground station pass computation of parameters fo r........ 454—469 Ground trace (See Ground track) Ground tr a c k ...................................... 409-412 better approximation for.......................... 413 circular LEO orbits.......................... 444-449 definition o f ............................................ 443 elliptical orbits................................ 449-452 equations for............................................ 411 geosynchronous orbits.................... 453—454 instantaneous rotation axis of (table). . . . 412 supersynchronous orbits.................. 454—455 use for coverage analysis................ 477—478 Ground Tracking advantages vs. disadvantages for navigation.......................................... 212 Ground-based systems................................ 227 GST (See Greenwich sidereal time) Guidance and Control (G&C; See Attitude control) definition o f ............................................ 103 Guidance, Navigation, and Control (GN&C; See Attitude control) definition o f .................................... 103, 120 Gurzadyan, G. A. book by.................................................... 115 Gyroscope advantages and disadvantages................ 122 applications o f ........................................ 156 characteristics of...................................... 162 classification of measurement ty p e ........ 323 types o f.................................................... 162 typical parameters.................................. 153
—H— H-2 launch vehicle cost/performance table............................ 602 HA (See Hour angle) Haiti coverage simulation o f .................... 492—497 Half angle form ulas.................................... 798 Halo o r b i t.................................................... 624 Hankey, W .L. book by.................................................... 115 HDOP (See Horizontal Dilution o f Precision) HEAO-2 ballistic coefficient.................................... 71 Heel, of footprint (See also Footprint)........ 473 Heliocentric arrival velocity in planetary missions.............................. 649 Heliocentric flight path angle in planetary missions.............................. 649 Heliopause.................................................... 631 Helv^jian, H. book by...................................................... 34 Hemisphere definition o f ............................................ 299 Hemisphere function (spherical geometry) definition o f .................................... 389, 422 properties of (table)................................ 390
Index Hemispherical Resonator Gyro (HRG; See Gyroscope).............................................. 162 Hertzfeld, H, R. book b y .................................................... 280 High-energy transfer applications o f ........................................ 628 Higher order harmonics (in geopotential) control of effect on constellations.......... 699 effect on orbit elem ents.......................... 513 method for controlling effects o f ............699 Hill, George.................................................. 527 Hill's equations.................................... 527-533 definition o f ............................................ 519 vs. Der-Danchick.................................... 528 Hofmann-Wellenhof, et al. book by.................................................... 231 Hohmann transfer orbit..........................92-95 applications o f ........................................ 628 definition.................................................... 92 diagram of.................................................. 92 AV for vs. altitude.................................... 598 equations for.............................................. 93 ground trace and swath width of . . . 450—451 numerical values for solar orb it.............. 887 Hohmann, W alter.......................................... 92 Horizon effective.................................................. 457 Horizon sensor (See Earth sensor).............. 158 Horizontal definition o f ............................................ 426 Horizontal Dilution of Position (H D O P).................................................. 208 Horizontal plane role in satellite relative motion................ 509 Hour Angle (HA) definition o f .................................... 194, 468 Housekeeping as source of orbit, attitude, and timing requirements........................................ 25 definition of functions.............................. 24 HRG (Hemispherical Resonator Gyro; See Gyroscope)........................................ 162 Hubble Space Telescope (HST; See Space Telescope) Hughes, P. C. book by.................................................... 176 Huygens probe (Cassini m ission)..............651 Huygens, C hristian........................................ 40 Hyperbola as a conic section................................ 41^42 definition.................................................... 42 terminology.......................................... 41-42 Hyperbolic anomaly formulas f o r ............................................ 849 Hyperbolic excess velocity..........................634 definition o f ............................................ 609 formulas f o r ............................................ 850 role in planetary missions........................ 645 Hyperbolic orbits (See Orbits)................ 41^42 equations, table of...................................... 51 orbit and coverage equation............ 847—852 Hyperbolic velocity definition o f .............................................. 45
913
—I— IAA (See Instantaneous Access Area) IAU (See International Astronomical Union) ICO (constellation) summary o f.............................................. 672 IFOG (Interferometric Fiber Optic GyrO; See Gyroscope).............................................. 162 Dluminance units and conversion factors.................... 892 Illumination sources o f ................................................ 547 Illumination phases (See also Lighting, Terminator)...................... 550-554, 567-577 brightness as a function o f .............. 578-582 brightness table for Moon........................ 580 computation of, on an arbitrary spacecraft face............................ 554—558 dependence on subsatellite point (fig) . . . 572 diagram o f................................................ 570 equations fo r .................................... 576-577 photos o f .......................................... 569, 571 progression of in an orbit............................ 552-554, 570-572 Imarsat (constellation) summary o f.............................................. 672 Impact parameter in planetary m issions...................... 647-650 Impact-plane definition o f ............................................. 648 Improper spherical triangle........................ 385 Inclination.................................................... 734 apparent vs. re a l.............................. 448, 516 definition o f ....................................... 47, 733 cffcct of small changes in ................ 512-513 effect of solar/lunar perturbations on . . . . 517 equations fo r............................................ 771 error bounds............................................ 753 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic o rb its................ 848 formulas for parabolic orbits.................... 846 impact on coverage vs. latitude................ 696 relative............................................ 501-503 role in collision avoidance.............. 716-718 selection of in constellation design........................ 695-697,725, 728 Incrossing (Earth sensor)............................ 158 Independent Junctions........................ 811-812 Independent measurements........................ 309 Indiction (historical).................................... 183 Inertial coordinates (See also Celestial coordinates) need for date f o r .................................. 48, 53 relative motion o f .................................... 284 Inertial nutation rate definition o f ..................................... 139, 408 Inertial sensors (See also Gyroscope; Accelerometer) advantages and disadvantages................ 122 definition o f ............................................. 120 use for autonomous navigation................ 217 Inertial space................................................ 293
914 Inertially fixed definition o f ............................................ 312 Inferior conjunction...................................... 54 Inferior p la n e t.............................................. 53 Infinitesimal trian g les........................ 796-797 Information content of measurements................................ 319-321 Instantaneous Access Area (IAA).............. 471 equations for............................................ 475 formulas for circular o rb its.................... 837 formulas for elliptical orbits.................... 843 formulas for hyperbolic orbits................ 851 numerical values for Earth o rb it.............................. inside rear cover numerical values for lunar orbit.............. 875 numerical values for Mars orbit.............. 881 Instantaneous area coverage ra te .............. 474 Instantaneous coverage area (See Field o f view, Footprint) definition o f ............................................ 471 Instantaneous rotation axis {See Euler a xis).............................................. 399 definition o f .................................... 135, 446 Instrument definition................................................ 470 projection of field of view onto the Earth.................................... 430^t37 INT, software function................................ 185 Intelsat (constellation).................................... 9 seen from Space Shuttle (photo)............499 summary o f ............................................ 672 Intelsat (organization).................................... 9 Intercos-16 ballistic coefficient.................................... 71 Interference, R F .................................. 585-587 Interferometric Fiber Optic Gyro (TFOG; See Gyroscope)........................ 162 International Astronomical Union (IAU) adoption of astronomical constants........ 767 definition of Modified Julian Date.......... 186 role in defining time................................ 199 International Atomic Time fTAI; See Atomic Time) International designation for satellites........ 59 International Space Station (ISS).......... 19-23 assembly sequence.............................. 20-21 drawing o f ................................................ 19 first tourist................................................ 19 functional block diagram.................... 23—24 orbit and attitude system .......................... 22 potential for satellite refurbishment........ 595 International System of Units (S I)............889 Interplanetary launch opportunities........ .5 7 Interplanetary missions definition o f ................................................ 5 Orbit Cost Function for (table).............. 607 summary tab le........................................ 633 Interplanetary orbits alternative propulsion techniques -- 651-654 design of.......................................... 630—654 Earth departure................................ 645-647 guidance, navigation, and control .. 640-642 launch issu es.................................. 642-644
Index motion in as seen from Earth.......... 465-469 observability times.................................. 468 overview of interplanetary transfer...................................... 632-640 planetary arrival and insertion........ 647—651 transfer times f o r .................................... 642 Interplanetary probe definition o f .............................................. 58 Intersatellite navigation.............................. 214 Intersatellite visibility................................ 507 Interstellar exploration...................... 654—664 bibliographies.......................................... 664 getting to the s ta rs .......................... 662-664 initiated by Pioneer and Voyager............ 631 introduction t o ............ ................ 654-657 relativistic space travel.................... 657-662 the twin paradox...................................... 661 Interstellar p ro b e ...................................... 5, 58 Interstellar ranyet for interstellar travel................................ 663 In-track orbit maintenance (See also Stationkeeping)........................................ 112 Inverse angle spherical tria n g le ........ 382, 385 Inverse G P S ................................................ 214 Inverse side spherical triangle............ 383, 385 INX (constellation) summary of.............................................. 672 Ion thruster propulsion.............................. 652 Iridium (constellation) computation of excess coverage.............. 726 summary of.............................................. 672 Iridium fla re ................................................ 548 Irregular spherical triangle........................ 385 illustrations of.......................................... 381 taxonomy o f ............................................ 380 ISEE-C (m ission)........................................ I l l orbit design.............................................. 590 iSky (constellation) summary o f ............................................. 672 Isotropic definition o f ............................................ 548 ISS (See International Space Station) — J— J - l launch vehicle cost/performance table............................ 602 J 2 (oblateness term in geopotential expansion; See also Oblateness)............................ 65-67 control of effect on constellations..........699 effect on orbit elements.......................... 513 method for controlling effects o f ............ 699 node rotation for circular orbits.............. 838 node rotation for elliptical orbits............ 838 perigee rotation due t o ............................ 844 role in Molniya orbits.......................... 87—88 role in Sun synchronous orbits............ 83—85 J am m ing .............................................. 585—587 geometric susceptibility.................. 586-587 Japan ISS participation.................................. 19-21 launch vehicles (table)............................ 602 time zones................................................ 182
915
Index JD (See Julian Date) Joint d e n sity ................................................ 811 Joint distribution........................................ 811 Julian calendar............................................ 182 Julian Date (J D ).................................. 182-187 history o f ................................................. 183 tables for...................... ....................183-184 Julian perio d ................................................ 183 Jujakins, J. L. book by .................................................... 176 Jupiter (See also Galileo mission) gravitational parameters.......................... 854 Io (m oon).................................................. 16 mission parameters (table)...................... 887 motion relative to background stars........ 467 orbit data.................................................. 865 physical and photometric d a ta ................ 866 repeating ground track orbits f o r ............621 repeating ground track orbits for moons o f............................................ 621 satellite d a ta ............................................ 869 satellite gravitational parameters............854 summary of missions to (table).............. 633 Sun synchronous orbits for...................... 618 transit of moon (photo)............................ 559 use in gravity assist trajectory (See also Planetary assist trajectory} ---- 637-640
—
K—
Kalman filtering definition o f ............................................ 123 divergence i n .......................................... 164 history...................................................... 164 Kalman, Rudolph invention of Kalman filtering.................. 164 Kane,L. book by .................................................... 177 Kaplan, E. D. book b y .................................................... 231 Kaplan, M. H. book b y .................................................... 177 Kay, W. D. book by.................................................... 280 Kelvin (unit of tem perature)...................... 889 Kepler, Jo h a n n e s.................................... 38—40 laws of planetary motion (See Kepler’s Laws of Planetary Motion) Kepler’s equation.......................................... 49 Kepler’s Laws of Planetary Motion defined...................................................... 39 first law................................................ 39-43 second la w .......................................... 39, 43 third la w ........................................ 39, 43-44 Keplerian elements (See also Orbit elements).............................................. 45-51 definition o f .............................................. 45 determination of from position and velocity.............................. 106-108 determination of position and velocity fro m ............................ 108-109 Keplerian o rb its ...................................... 38-52
definition o f ............................................... 38 elements and terminology.................... 45—47 equation summary............................ 835—852 Kilogram definition of. . ........................................ 889 Kosmos launch vehicle cost/performance tab le............................ 602 — L —
L-5 Society formation o f ............................................ 625 Labunsky, A.V., et al* book b y .................................................... 115 Lagrange point orbits.............................. 76, 88 about other planets.................................. 626 applications of.......................................... 625 definition o f......................................... 58, 88 equations fo r.............................................. 90 parallax of as seen from E a rth ................ 469 physics o f .................................................. 89 Lagrange points........................................ 58, 89 Lagrange, Joseph L. (m athem atician).............................. 88, 135 Lagrange’s planetary equations definition o f ............................................... 63 Landmark tracking...................................... 214 advantages and disadvantages for navigation.......................................... 212 Landsat-1 ballistic coefficient.................................... 71 Large spherical triangle...................... 382, 385 Larson, W. J. books b y ............ 35-36, 117, 231, 281-282 Laser illumination of surfaces.................. 582—585 Last quarter Earth photo from space............................ 571 Latitude definition o f ............................................. 304 Launch cost as element of orbit design........ 595-596 cost estimating process.................... 609-612 cost vs. on-orbit m ass...................... 601—612 effect of delay on interplanetary m issions............................................ 646 interplanetary missions.................... 642-647 low-cost alternatives (table).................... 604 options.............................................. 601-605 options for constellations........................ 675 orbit cost function............................ 605-609 role in orbit design.................................. 613 Launch azimuth definition o f............................................. 738 Launch energy formulas for hyperbolic o rb its................ 850 Launch on dem and...................................... 723 Launch period definition o f............................................. 735 Launch vehicles Orbit Cost Function for.................... 606-607 selection o f ...................................... 601-605
916 table o f ..................................................602 vs.on-board propulsion.........................629 Launch window........................................ 645 definition o f .......................................... 735 interplanetary missions.........................645 Mars launch opportunities, table o f..........56 Law of Cosines for angles.............................................. 775 for plane triangles.................................392 for sides................................................775 Law of Large Numbers.............................821 Law of Sines..............................................775 LDEF mission ballistic coefficient.................................. 71 Legendre, Adrian-Marie (mathematician).................................. 135 Leibnitz, Gottfried...................................... 40 Leick, A. book by..................................................231 Length units and conversion factors...................892 LEO (See Low-Earth Orbit) Leo-One (constellation) summary o f .......................................... 672 LEOSAT (constellation) summary o f .......................................... 672 Lesser angle definition o f .................................. 382, 385 Levin, E. M. book by.................................................. 115 Libration points (See Lagrange points) Libration, of the M o o n..................... 134, 864 Lifetimes, of satellites due to orbit decay ...................................... 71-75 graph o f ..................................................75 Light as absolute upper limit of velocity......... 657 Light y e a r..................................................550 Lighting conditions...................................... 550-558 conditions for satellite and target... 547-588 diagram o f ............................................ 551 effects of................................................555 introduction t o ............................... 550-554 looking at Earth from space........... 567-578 LightSats use for constellations.............................594 Limit cycle.................................................. 125 Line of apsides (See also Major axis; Apogee; Perigee)
definition o f ............................................46 equations................................................87 rotation of due to ................................ 87 Line of nodes definition o f ............................................ 48 equations for rotation of due to J2 . . .. 83-84 illustration........................................ 46—47 Line scanner vs. array sensor to reduce cost............... 243 Lit horizon................................................568 Local Mean Time computing at spacecraft subsatellite point..................................................86
Index Local reference sensor use for autonomous navigation.............. 217 Local Sidereal Time (L S T )................ 196-197 Local tangent coordinates.......................... 293 Local v e rtic a l.............................................. 861 Local Vertical/Local Horizontal (LVLH) . 293 Loci, on the celestial sphere as representation of measurement .. 319-321 Logarithms invention o f ............................................ 304 Logsdon, T. book by.................................... 113, 116, 231 Long a r c .............................................. 382, 385 Long Duration Exposure Facility (See LDEF) Long March launch vehicle cost/performance table............................ 602 Longitude definition o f ............................................ 304 Finding from time and R A .............. 197-198 in solar system vs. Earth or celestial........ 53 Longitude of perihelion................................ 53 Longitude of the ascending node............ 48, 53 Longitude of the satellite.............................. 52 Longitude shift per o rb it............................ 445 Loral (See Space systems Loral) Lorentz contraction............................ 658—659 Low E arth Orbit (LEO) defined by Van Allen belts.......................... 4 definition o f .......................................... 4, 58 motion in, seen from G EO .............. 542-543 motion of GEO satellites from ........ 544-545 Orbit Cost Function for (table)................ 607 radiation environment............................ 614 Low thrust tra n s fe r................................ 92-96 vs. high thrust.......................................... 629 LST (See Local sidereal time) Luminance units and conversion factors.................... 893 Luminosity class (stars).............................. 655 Lunar cycle.................................................. 183 Lunar eclipse (See Eclipse).......................... 550 Lune area formulas f o r .................................... 773 as a degenerate triangle.......................... 794 definition o f ............................ 299, 383, 385 illustration o f .......................................... 384 LVLH (See Local Vertical/Local Horizontal)
—M— Madonna, R. G. book by.................................................... 113 Magellan (constellation) summary of...............................................672 Magellan spacecraft radar map of V en u s................................ 583 Magellenic clouds travel to.................................................... 663 Magnetic disturbance to r q u e .................... 128 Magnetic field measurements accuracy with respect to magnetometer axes............................ 348 torque computation.................................. 171
Index Magnetic stabilization (See also Magnetic torquers)
advantages and disadvantages............... 124 Magnetic torquer (electromagnet) advantages and disadvantages o f ........... 126 characteristics of.................................... 172 use o f ....................................................127 Magnetic units units and conversion factors................... 893 varying dimensionality...........................896 Magnetometer.................................... 121,153 advantages and disadvantages............... 122 applications o f ...................................... 156 classification of measurement ty p e ........323 measurement analysis.............................348 on-orbit performance history................. 156 typical parameters.................................. 153 use for orbit and attitude...........................29 Magnitude (brightness measurement) as measure of distance................... 217—218 definition o f .......................................... 578 of spacecraft and planets............... 578—582 MagSat (space mission).............................150 Main sequence (star classification)........... 655 Major axis (See also Semimajor axis)............40 Manned Maneuvering Unit use of cold gas thrusters.........................170 Manned spaceflight (See also International Space Station, Space Shuttle)
Mars mission........................................ 627 MANS (See Microcosm Autonomous Navigation System)
Mapping definition o f .................................... 25, 250 errors in formulas f o r .................................... 254 sources............................................251 examples of error computations . . . . 262-266 typical error budget tree.........................247 Mapping budget creating.......................................... 268-279 error sources, tables o f ................... 268, 273 introduction t o ...............................250-258 representative................. 253—256, 276—277 Maral, G. book by....................................................34 Marathon (constellation) summary of. .........................................672 Margin definition o f .......................................... 245 Marginal density........................................ 811 Marginal distribution.........................810—811 Mark, H. book by..................................................114 Mars circular and escape velocities of............... 45 gravitational constant.............................835 gravitational parameters.........................854 manned flight to .................................... 627 mision parameters (table).......................887 Orbit Cost Function for (table)............... 607 orbit data................................................865 physical and photometric d ata............... 866
917
repeating ground track orbits f o r ............ 621 repeating ground track orbits for Moons o f............................................ 621 satellite d a ta .................................... 867, 869 satellite gravitational parameters............ 854 satellite parameters (numerical table).......................................... 880-885 summary of missions to (table)................ 633 Sun synchronous orbits fo r...................... 618 table of launch opportunities...................... 56 Mars central angle numerical values for Mars orb it.............. 882 Mars Global Surveyor (mission) use of aero-assist trajectory........................ 98 Mars Pathfinder mission summary of (table).................................. 633 M ascons.......................................................... 67 Mass increase in for moving objects........ 658-659 inertial vs. gravitational............................ 39 units and conversion factors.................... 893 Mass expulsion source of unintended disturbance .............128 MatLab use in date calculations............................ 186 Maximum coverage gap (Figure of Merit) .. .............................................. 484 Maximum E arth central angle.................... 418 Maximum gap . . ........................................ 484 Maximum uncertainty................................ 333 Mean angular rate (See Mean motion)........ 118 Mean anomaly definition o f ............................................... 49 equations fo r.............................................. 51 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 841 formulas for hyperbolic o rb its................ 849 formulas for parabolic orbits.................... 846 Mean coverage g ap ...................................... 484 Mean gap d u ratio n ...................................... 470 Mean motion definition o f ....................................... 50, 450 equations fo r.............................................. 51 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 841 formulas for hyperbolic o rb its................ 849 formulas for parabolic orbits.................... 846 Mean orbit elements.......................................52 Mean response time (Figure of M erit)...................................................... 484 Mean sea level...................................... 437, 861 Mean solar time (See also Solar tim e )........ 195 Mean Sun...................................................... 195 definition o f ............................................. 506 Mean uncertainty equation for quantized measurement . . . . 334 Measurement bias (See Bias) Measurement density (Geometrical g ain )........................................ 275,326-332 definition o f ..................................... 328, 330 formal definition...................................... 331 interpretation o f....................................... 330 unacceptable values for.................... 332—333
918 Measurements (See also Attitude measurements, Position measurements).................. 317—376 arc length (See Arc length measurement) as loci on the celestial sphere.......... 319-321 classification of sensor types..................323 error in as probability density on the celestial sphere........................................ 319-321 error probabilities.................................... 339 evaluation of uncertainty................ 325-353 good vs. bad measurement sets . . . . 373-376 tests for...................................... 366-367 horizon crossing geometry...................... 342 independent............................................ 326 information content o f ....................318-321 importance o f .................................... 318 math models of sensor and environment.............................. 278-279 rotation angle (See Rotation angle measurement) rotation angle vs. arc length............ 322-323 types of............................................ 321-324 illustration o f .................................... 322 loci corresponding to ........................ 323 uncertainty evaluation.................... 325—341 Medium-Earth Orbit (M E O ).................. 4, 58 MEO (See Medium-Earth Orbit) M ercator Projection.................................... 303 Mercator, G erhardus.................................. 303 Mercury gravitational parameters.......................... 853 mission parameters (table)...................... 887 orbit d a ta ............................ ....................865 physical data. ...........................................866 repeating ground track orbits fo r ............ 621 Merges, R. P. book by.................................................... 280 Meridian (in a coordinate system; See also Celestial meridian).......................... 295, 304 definition o f ............................................ 461 Meteor shower cause o f .................................................. 710 Meteorite s w a rm ........................................ 710 Meter definition o f ............................................ 889 Metric prefixes............................................ 889 Metric units (= International System of Units, S I ) .................................................. 889-897 Meyer, R. X. book by...................................................... 34 Michelson-Morley experim ent..................661 Microcosm Autonomous Navigation System (MANS).................................................. 213 advantages and disadvantages for navigation.......................................... 212 characteristics o f .................................... 211 Microgravity.................................................. 22 role in formation flying.......................... 535 Microwave background.............................. 550 Middle Angle Law (MAL).......................... 792 Middle Side Law (M S L )............................ 792 Mid-scan rotation angle.............................. 324 Milani, A., et al. book by.................................................... 116
Index Miniaturization application to spacecraft.......................... 594
Minimizing cost example........................................ 831-833 Minor axis definition o f ............................................ 40 illustration............................... , , ............ 46 Mission Geometry (See Geometry) bibliography o f.......................................178 Mission operations (See operations) Mission orbit definition................................................ 60 Missions (See also specific missions) representative examples....................... 5-23 MJD (Sec Modified Julian Date) /i-mesons time dilation in decay.............................658 Modified Julian Date (MJD)..................... 186 Modified launch mode use to reduce launch cost....................... 608 Mole........................................................... 889 Molniya (constellation) summary of............................................ 672 Molniya o rb it.................................. 76, 86-88 about other planets.................................620 applications o f ...............................619-620 characteristics and applications o f..........615 definition o f ..................................... 88, 687 ground trace and swath width of . .. 450-451 parameters o f .......................................... 88 properties.............................................. 619 use o f ............................................619—620 Moment generating function (See also Characteristic function) ......................... 822 Moments of a function....................... 819-820 computing higher moments............ 822-823 Moments of inertia (See also Principal moments o f inertia)
units and conversion factors................... 894 Momentum bias definition o f .......................................... 127 Momentum bias control system (attitude control approach) capabilities o f........................................ 16S definition o f .................................. 130, 169 Momentum dumping......................... 127,130 Momentum unloading (See Momentum dumping) Momentum wheel alternative configurations....................... 169 definition o f .......................................... 127 Monte Carlo simulation approach to error analysis (process table)................................................ 261 use in error analysis...............................819 Month alternative definitions o f..........................57 M oon......................................................... 873 as navigation source (MANS)............... 213 as navigation source (space sextant) . . . . 213 cause of lengthening of day ................... 181 circular and escape velocities of............... 45 definitions of orbit period......................... 57 eclipse of (See E clipse) .........................550
Index cffcct on GEO satellite orbit...................... 78 equation for effect on orbits...................... 67 gravitational constant.............................. 835 gravitational parameters.......................... 854 gravity field o f ........................................ 865 gravity-gradient stabilization o f . . . . 132, 134 libration o f .............................................. 134 motion relative to background stars........ 466 Orbit Cost Function for (table)................ 607 orbit data.................................................. 867 orbit o f ................................................ 52-57 orbit perturbations due to (See solar-lunar perturbations, gravity gradient) phase law and visual magnitude.............. 865 phases (See Illumination phases) physical and orbit properties.......... 864—865 physical and photometric d a ta ................ 869 potential for water o n .............................. 134 repeating ground track orbits f o r ............621 role in Lagrange point o rb its.............. 88-90 satellite parameters (numerical table).......................................... 873-879 source of tides.......................................... 125 terminator (See Terminator, Illumination phases)...................................... 574-577 terminator radius......................................575 use as a stable platform............................ 126 use for gravity assist................................ 627 use for large plane change...................... 627 Moon central angle numerical values for lunar orbit.............. 876 Moore, R. C. book by ...................................................... 35 Morgan, W. L. book b y ...................................................... 34 Moving average............................................ 827 autoregressive.......................................... 828 M-SAT (constellation) summary of.............................................. 672 MSSP (Multi-Satellite System Program; constellation).......................................... 689 summary of.............................................. 672 11(See Gravitational constant)...................... 118 Multi-Satellite System Program (See MSSP)
—N— Nadir definition o f .................... 292, 296, 418, 568 Nadir a n g le .....................................................295 computation o f ........................ 420, 422-423 definition o f ............................ 326, 363, 420 formula for circular o rb its...................... 460 formula for elliptical orbits...................... 463 maximum................................................ 456 numerical values for Earth orbit................................ inside rear cover numerical values for lunar orbit.............. 877 numerical values for Mars orbit.............. 883 Nadir v e c to r................................................ 418 Napier, J o h n .......................................... 304, 776
919
Napier’s Analogies.....................................775 Napier’s Rules.................................... 300, 776 NASA satellite designation................................... 59 use of disposal guidelines for CGRO........................................ 760-764 National Bureau of Standards................... 767 National Space Society formation o f .......................................... 625 Natural satellites orbit data................................................ 867 physical and photometric d ata............... 869 Navigation (See also Orbit determination) advantages and disadvantages of alternative methods............................................ 212 attitude requirements for......................... 225 definition o f....................................... 3, 103 historical use of spherical geometry........284 requirements on for orbit control............225 rhumb line for terrestrial......................... 303 Navigation, guidance and control.......................................... 640-642 NavStar (See GPS)........................................ 201 NEAR mission summary of (table).................................633 use of autonomous navigation............... 640 Neptune gravitational parameters......................... 855 mission parameters (table)..................... 887 orbit data................................................ 865 physical and photometric d ata............... 866 repeating ground track orbits f o r ........... 621 satellite d a ta .................................. 869, 871 satellite gravitational parameters............855 Sun synchronous orbits for..................... 618 New Millennium Program (NMP)............. 105 asteroid tracking in .................................210 use of autonomous navigation............... 640 use of solar electric propulsion............... 640 Newton, Isa a c ........................................ 38-40 Philosophiae Naturalis Principia Mathematica........................................ 38 second law.......................................... 38-39
Nicogossian, A. E., et al. book b y .................................................... 34 NMP (See New Millennium Program) Nodal vector formulas for circular orbits..................... 836 formulas for elliptical orbits................... 840 formulas for hyperbolic orbits............... 848 formulas for parabolic orbits................... 846 Node definition o f............................................. 48 rotation of due to J2 circular orbit.....................................838 elliptical orbit...................................844 equations for.................................83-84 numerical values for Earth orbit.........................inside rear cover numerical values for lunar orbit........ 879 numerical values for Mars orbit........ 885 spacing of, in constellations................... 697
920
Index
Node spacing................................................ 445 formulas for circular o rb its.................... 839 formulas for elliptical orbits.................... 845 numerical values for Earth o rb it........................ - - inside rear cover numerical values for lunar orbit.............. 879 numerical values for Mars orbit.............. 885 Nodical m o n th .............................................. 57 Noise vs. b ia s.................................................... 262 Non-inertial formations.............................. 535 Non-spherical mass distribution effects o f ............................................. 63-67 Noon'midnight o r b i t .................................... 86 Normal distribution (See Gaussian distribution) Normal spherical trian g les........................ 774 Norstar 1 (constellation) summary o f ............................................ 672 ‘‘Notch” spherical tria n g le ........................ 382 illustrations o f ........................................ 381 taxonomy o f............................................ 380 Noton, M. book by.................................................... 116 Nuclear electric propulsion for interstellar travel................................ 662 Nuclear pulse propulsion for interstellar travel................................ 662 Null in rotation angle measurement................ 360 N u ta tio n .............................. 137-143,408-409 definition o f ............................................ 137 Nutation a n g le ............................................ 138 Nutation d a m p in g ...................................... 138 definition o f ............................................ 148
—o — “O” (symbolfor “o f order").......................... 50 O ’Neill, G erard space colonization.................................. 625 Oblate (shape).............................................. 130 Oblate spacecraft moments of inertia and rotation rates f o r .................................... 408-409 Oblate spheroid natural attitude motion of........................ 143 Oblateness, of the Earth (See also J2) ............................................ 418, 421,438 control of effect on constellations..........699 effect on geopotential.......................... 63-67 effect on orbit elements.................. 513-516 method for controlling effects o f ............ 699 models o f ........................................ 437—440 Oblique full sky spherical triangles . . 779-792 solutions for angle-angle-angle.............................. 789 angle-angle-side................................ 785 angle-side-angle................................ 787 side-angle-side.................................. 784 side-side-angle.................................. 782 side-side-side.................................... 780 Oblique spherical triangle.......................... 305 definition o f ............................................ 779
Observability computing for interplanetary spacecraft.......................................... 468 Observation e rro rs...................................... 274 O ccultation.......................................... 558-563 conditions fo r.................................. 560-562 definition o f .............................................558 OCF (See Orbit Cost Function) Odyssey (constellation) summary of.............................................. 673 Off ground track a n g le .............................. 441 Olbers' P aradox .......................................... 549 solution.................................................... 550 Onboard orbit control (See autonomous orbit control) Onboard processing (See also Computers, spacecraft).............................................. 594 Onboard systems advantages and disadvantages o f ............ 227 On-orbit spares definition o f ............................................ 723 use o f ...................................................... 626 Operational requirements (See also Requirements).................................. 237-241 definition o f ............................................ 238 Opposition (planetary configuration)... 53-54 definition o f ............................................ 579 Mars, table o f ............................................ 56 Optical autonomous navigation (See also Autonomous navigation)........................ 216 ORBCOMM (constellation)...................... 703 summary of.............................................. 673 O rbit and attitude systems, com bined........29 combined sensors...................................... 29 definition o f ................................................ 1 examples o f .......................................... 5-23 need for a systems approach................ 27-32 orbit vs, attitude characteristics.............. 120 representative combined systems.............. 31 trade between orbit and attitude................ 28 Orbit control (See also stationkeeping) autonomous.................................... 219-226 definition o f ................................ 3, 103, 700 error sources, table o f.............................. 271 reasons for needing.................................. 110 sources of requirements...................... 27-32 O rbit Cost Function (O C F)................ 605-609 definition o f ............................................ 605 representative values of (tab le).............. 607 Orbit decay dependence on solar cy cle.......................... 8 Orbit design (See also Constellation design)............................................ 589-669 available on-orbit mass.................... 601-612 bibliography o f................................ 116-117 collision avoidance.......................... 708—719 AV budget........................................ 596-601 interplanetary orbits........................ 630-654 interstellar travel.............................. 654—664 launch cost estim ates...................... 601-612 of Earth-referenced orbits................ 612-623 of space-referenced o rb its.............. 623-626 of transfer orbits.............................. 626-630
Index process f o r ...................................... 590-596 summary ta b le .................................. 591 selection of constellation parameters.................................. 690—697 O rbit determination (See also Navigation) ...................................... 104-105 basic techniques...................................... 106 common measurement sets...................... 105 definition o f ........................................ 3, 103 definitive, definition of............................ 104 error sources, table o f.............................. 270 from position and velocity.............. 106—108 of position and velocity..................108-109 real-time.................................................. 104 sources of requirements...................... 27—32 Orbit elements (See also Keplerian elements) definition o f .............................................. 45 determination of from position and velocity.............................. 106-108 determination of position and velocity from ...........................108-109 effects of small changes in .............. 512-513 mean vs. osculating.................................. 52 need for date fo r ........................................ 48 poorly defined in many eases....................52 Orbit insertion in planetary missions...................... 647-650 O rbit maintenance definition o f ........................................ 4, 700 sources of requirements...................... 27-32 stationkeeping in Earth o rb it..........697-709 O rbit maneuvers (See also Transfer orbits, Planetary fly-bys, Hohmann transfer, 3-bum transfer) ............................................ 91—103 satellite rephasing in a constellation.............................. 721-722 O rbit number (designation o f)...................... 60 Orbit perturbations (See also Specific perturbations, i.e., Solar-lunar, Atmospheric drag, Solar radiation pressure).......... 61—74 cyclic vs. secular........................................ 38 definition o f ........................................ 38, 61 effect on constellation structure — 675, 698 in L E O ...................................................... 61 summary of................................................ 61 treatment in constellation design............728 Orbit plane change (See Plane change maneuver) O rbit prediction .............................................. 3 O rbit propagation................................ 2-3, 104 O rbit pumping definition o f ............................................ 651 use by Cassini mission............................ 651 O rbit shaping in planetary missions.............................. 647 O rbit transfer (See Transfer orbits)........ 92-96 definition o f ........................................ 60, 92 high thrust vs. low th ru st........................ 629 interplanetary transfer times.................... 642 Orbit, attitude, and timing systems... 179-233 architecture.......................... ............226-230 representative example............................ 229 typical components.................................. 230
921
Orbits (See also Constellations; Specific orbits, i.e., Geosynchronous, Lagrange point, Sun synchronous, Frozen, Repeating Ground Track, Molniya, Hohmann transfer, Planetary a ssist)................................................ 37—118 appearance viewed from nearby___314-315 bibliography.................................... 113-117 classification of by altitude........................ 58 comparison with attitude.................. 119—120 decay due to d rag ................................ 73-75 definition o f ..................................... 1, 37, 59 design of ("See Orbit design; Constellation design) determ. of pos. and velocity from Orbit elem ents.................................... 108-109 determination of from position and velocity.................................106-108 Earth, numerical parameter tables.............................. inside rear cover Earth-referenced........................ 60, 612-623 epoch o f..................................................... 48 equation summary............................ 835—852 equations for impact of errors on mapping and pointing.............................................. 254 error boundaries...................................... 734 first interstellar spacecraft................ 654-664 history o f.............................................. 38-39 interplanetary.................................. 630-654 maintenance and control.................. 110-112 maneuvers.......................................... 91-103 Mars, numerical parameter tables. . . 880-885 methods for reducing AV required for plane change.................................................. 98 Moon, numerical parameter tables.......................................... 873-879 natural satellite d a ta ........................ 867-869 of Moon and planets............................ 52-57 of spacecraft........................................ 57-60 orientation of planetary........................ 47-48 orientation within the p la n e ................ 48-49 origin of study.............................................. 2 parking............................................ 626-630 perturbations (See also Orbit perturbations)................................ 61-74 planetary d a ta.................................. 865-866 principle types of (table).......................... 592 properties and terminology................ 37—118 satellite motion as seen from E arth.......................................... 454-469 selection of (See Orbit design) size and shape............................................ 46 space-referenced........................ 60, 623-626 summary of solar system data.......... 853-871 Sun, numerical parameter tables. . . . 886—888 terminology for.................................... 57-60 transfer............................................ 626—630 typical requirements on............................ 239 Orb view (constellation) summary o f.............................................. 673 Orion (interstellar travel project).................................................... 662
922
Index
0 rsted (science m ission)............................ 6-9 mission timeline.......................................... 7 orbit and attitude system ............................ 8 spacecraft.................................................... 7 Orthogonal measurements error ellipse for........................................ 335 Orthographic projection (See Globe p lo t)........................................................ 285 Oscar-1 ballistic coefficient.................................... 71 Osculating elements...................................... 52 Osculating ellipse........................................ 533 OSO-7,-8 ballistic coefficient.................................... 71 Outages (See Spacecraft outage) in constellation........................................ 720 Out-of-Ecliptic maneuver use in interplanetary missions........ 653-654 Overlap area, between a great circle and a small cirde formula fo r.............................................. 774
—P— Pallas (asteroid) gravitational parameters.......................... 855 Parabola as a conic section................................ 41-42 definition o f .............................................. 42 terminology........................................ 41-42 Parabolic anomaly formulas fo r............................................ 846 Parabolic flight (aircraft) as low-cost launch alternative................ 604 Parabolic orbits (See also O rbits).......... 41-42 equations for.................................... 845-847 Equations, table o f .................................... 51 Parabolic velocity (See Escape velocity) . . . . 45 formulas for parabolic orbits.................. 846 Parallax ................................................ 217, 370 in satellite positions................................ 469 Parallels (in spherical coordinates)..........295 Parking orbit applications and design o f .............. 626-630 definition o f ............................................ 626 Parkinson, B. W. book by.......... ..........................................231 Partial eclipse of the S u n ............................ 563 Pass, ground station (See Ground station pass) Passive stabilization (See Stabilization; Attitude control) definition o f .................................... 119, 123 Patched conic approximation definition o f ............................................ 630 Payload advantages and disadvantages as source of attitude information.......................... 122 as source of orbit, attitude requirements .. 26 definition o f .................................. 8, 24, 121 orientation control error sources (table).. 270 orientation determination error sources (table).................................. 270 pdf (probability density function).............. 808
PDOP (See Position Dilution o f Precision) Pegasus launch vehicle cost/performance table............................ 602 Pegasus-3 ballistic coefficient.................................... 71 Pentriad (constellation) summary of.............................. . 673 P en u m b ra............................................ 550, 563 Penumbral eclip se...................................... 564 Percent coverage (Figure of M erit)............ 483 Performance requirem ents........................ 241 P e riap sis........................................................ 46 P eriastron...................................................... 46 Pericynthiane................................................ 46 Perifocal coordinate system........................ 108 Perifocal distance definition o f ...............................................46 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic orbits................ 848 formulas for parabolic o rb its.................. 846 Perifocus........................................................ 46 Perigee drfinitioa o f .......................................... 7, 46 equations for...............................................87 rotation of due to J2 .......................... 87, 844 velocity at formulas for elliptical orbits.............. 842 formulas for hyperbolic orbits.......... 849 formulas for parabolic orb its............ 846 Perigee h eight................................................ 46 P erih elio n................................................ 46, 53 Perihelion passage time o f ...................................................... 53 Perijove.......................................................... 46 Perilune.......................................................... 46 Period effect of small changes in................ 512-513 for hyperbolic orbits................................ 849 formula for circular o rb its.............. 460, 837 formula for elliptical o rb its............ 463, 842 major solar system b odies.............. 853-855 numerical values for Earth o rb it.............................. inside rear cover numerical values for lunar orbit.............. 879 numerical values for Mars orbit.............. 885 Perturbations, orbit (See Orbit perturbations) ............................................ 38 handling of in orbit control.............. 220—221 methods for controlling.................. 698-699 Phase an g le.......................................... 548, 579 Phase difference (in a Walker constellation).......................................... 685 Phase law definition o f ............................................ 579 for the Moon (table)................................ 580 Phase shift, in orbit (See Drift orbit) role in collision avoidance.............. 716-718 Phases (of Moon, Earth; See Illumination phases) Phasing lo o p s .............................................. 732 Phobos gravitational parameters.......................... 854 orbit data.................................................. 867
Index Photon rocket equations for............................................ 662 Photosphere.................................................. 856 Physical constants sources o f ................................................ 767 Pioneer mission first interstellar spacccraft...................... 654 leaving solar system........................ 631-632 summary of (table).................................. 633 use of gravity a ss ist................................ 637 Pipper (horizon sen so r)...................... 159—160 definition o f ............................................ 158 Pisacane, V. L. book b y ...................................................... 35 Pitch axis definition o f ............................................ 151 Pitch, Roll, Yaw coordinates (See Roll, Pitch, Yaw coordinates) Pixel geometry of in array sensor............ 356—358 Planar scanner full sky Earth coverage with two sensors................................ 346-347 geometry in L E O .................................... 346 Plane change maneuver application of alternative mechanisms . . . 628 AV for vs. altitude.................................... 599 in interplanetary transfer.......................... 638 numerical values for Earth orbit................................ inside rear cover numerical values for lunar orbit.............. 878 numerical values for Mars orbit.............. 884 numerical values for solar o rb it.............. 887 Plane change maneuver (orbits; See also Planetary-assist trajectory)................ 96—99 methods for reducing A v f o r .................... 98 Plane geometry as approximate solution to spherical geometry problems............................................ 392 errors in array computations.................... 309 use for array computations.............. 306-307 Plane triangle number of solutions f o r .......................... 387 Planet sensor modifications to Earth sensor to create .. 160 Planetary arrival geometry, orbit insertion, orbit shaping.................................. 647-651 Planetary assist trajectory (See also Flyby)................................................ 98-101 applications o f ................ 627-628, 637, 640 energy gain.............................................. 600 powered vs. passive swing-by................640 use to reduce launch c o s t........................608 Planetary configurations.............................. 53 Planetary missions summary ta b le ........................................ 633 Planets computing observability tim es........468—469 geometry viewed from space (See Earth, Earth coverage) gravitational d a ta ............................ 853-854 Molniya orbits about................................ 620 orbital d a ta ...................................... 865-866
923
52-57 orbits o f ............................................................................. physical and photometric data ........ 866 repeating ground track orbits for ........ 621 Sun synchronous orbits fo r............ 618 Plume im pingem ent.......................... ........ 128 Pluto flyby mission.................................. ........ 653 gravitational parameters................ 855 mission parameters (table) ...................... 887 orbit data ......................................................................... 865 physical and photometric data — ...............866 621 repeating ground track orbits for satellite d a ta .................................... . 869, 871 satellite gravitational parameters . . ........ 855 Pocha, J. J. .
.35, 117
Pointing definition o f ........................... .............. 26, 250 errors formulas for.............................. ........ 254 sources.................................... ........251 Pointing budget creating.......................................... 268-279 error sources, tables o f .................. 268,273 representative.................. 256—257, 276-279 812 PoissOn distrib u tio n .......................... “Pork chop plot” ................................ ........635 Position Dilution of Position (PDOP)............................................ 208 Position measurements........................ 317-376 Position, of spacecraft
and velocity, determination of from orbit elem ents ....................................................... . . 108-109 and velocity, determination of orbit elements 106-108 Power ..............894 units and conversion factors Power spectral density.................. ..............827 Powered swingby in gravity assist trajectories......................640 Precession definition o f............................... ..............144 Precession of the equinoxes definition o f ............................... 53, 294, 304 effect on inertial coordinates. . . ................48 numerical value........................ ..............859 Pre-emphasis (in attitude control) ..............125 Pressure units and conversion factors — ..............894 Primary (in orbital mechanics) . . .................40 Primary axis (in dual axis spiral) ..............394 Prime m eridian............................................302 136, 408 Principal axis.................................. Principal moments of in e rtia ......................136 Principal of Equivalence in re la tiv ity .................................... .......................... 199 Principle of Relativity ...................................................661 Probability introduction to ................................................... 808-815 Probability density........................ 333-334 Probability density function........................ 808 Probability distribution................ 325, 808 Prograde o rb it................................................ 48
924
Index
Project 21 (constellation) summary o f ............................................ 673 Projection sensor fields-of-view onto the Earth.................................... 430-437 Prolate (shape)............................................ 130 Prolate spacecraft moments of inertia and rotation rates f o r ............................................ 408 Prolate spheroid natural motion o f ............................ 140-143 Propagated o r b i t .......................................... 61 Propagation (See Attitude propagation; Orbit propagation) Propellant slo sh .......................................... 128 Proper distance............................................ 658 Proper le n g th .............................................. 659 Proper m a s s ................................................ 658 Proper spherical triangle.................... 382, 385 more complex than complete triangles .. 388 number of solutions f o r .......................... 387 Proper tim e.......................................... 657, 659 Propulsion alternatives for planetary missions.................................... 651-654 for interstellar travel........................ 662-663 on-board spacecraft to reduce launch cost........................................ 60S Proton launch vehicle cost/performance table............................ 602 use for ISS assembly.......................... 19-21 Proxima Centauri (star nearest the S u n ).......................................... 655-656 Prussing J.E., B.A. Conway book by.................................................... 114 Przemieniecki, J. S. book by.................................................... 280
— Q— Q uadrantal spherical triangle....................385 definition o f ............................ 301, 305, 776 formulas fo r ............................................ 778 Quadrantal three-quarter spherical trian g le.................................. 383 Q uadrature.................................................... 54 Quantized m easurem ents.......... 325, 333-334 equations........................................ 333-334 Q uarter phase (See Illumination phases) Q uaternion.................................................. 150 advantages and disadvantages for attitude specification...................................... 151
— R— RA (See Right Ascension) RAAN (See Right Ascension o f the Ascending
Node)
Radar
coverage equations fo r............................ 475 illumination of surfaces.................. 582—585 jamming.......................................... 585-587 power dependence on distance and beam size.................................. 583-585
R adar ranging definition o f .............................................369 Radial velocity formulas for elliptical orbits.................... 842 formulas for hyperbolic orbits................ 850 formulas for parabolic orbits.................. 847 Radian (angular measure).......................... 300 Radiation belts (See Van Allen radiation belts) Random definition o f ............................ ................261 Random processes...................................... 826 Random variable (= variate)...................... 809 addition o f ...................................... 815-819 functions of........ *............................813, 818 history o f ................................................ 809 Random walk (Brownian m o tio n )............ 827 Range (See Distance) Range measurement (See Measurements, Error analysis) Range to surface numerical value for Earth o rb it.............................. inside rear cover numerical value for Mars orbit................ 883 numerical values for lunar orbit.............. 877 Range to targ et............................................ 457 RapidEye (constellation) summary o f ............................................. 673 Rayleigh distribution (Circular n o rm a l).......................................... 812, 823 R-Bar rendezvous ap p ro a ch ...................... 534 Reaction wheels advantages and disadvantages for attitude control................................................ 126 characteristics of...................................... 172 definition o f .................................... 127, 170 vs. momentum w heels............................ 170 Real-time attitude determ ination.............. 123 Rechtin, E. book by.................................................... 280 Reduction formulas (in trigonom etry). . . . 798 Re-entry (See De-orbit o f spacecraft; Disposal of spacecraft) Reference frame error budget for........................................ 272 errors i n ...................................................274 Reference launch energy formulas for hyperbolic orbits................ 850 Reference o rb it...............................................61 Regan, F. J. book by.................................................... 116 Regression of the no d es................................ 83 Regular spherical triangle.................. 382, 385 illustrations of.......................................... 381 number of solutions f o r .......................... 387 solution equations for.............................. 391 taxonomy o f .............................................380 Relative inclination between orbits with the same inclination.................................. 678-679 definition o f ............................................ 501 equations for.................................... 502—503 Relative motion.................................... 499-546 causes of small scale m otion.................. 511 equations for............................................ 541
Index in continuous thrust formations---- 534-536 in formations.................................... 518—534 of co-altitude satellites.................... 500-536 large scale.................................. 500-510 small sc a le ................................ 509—525 of nearly co-altitude satellites.................... 526-527,537-538 of non-co-altitude satellites............ 536-545 resulting from orbit variations (table)512—513 summary of results.......................... 510, 523 vs. apparent m otion................................ 509 Relative phase definition o f ............................................ 501 equations for.................................... 502—503 Relative stationkeeping (See Stationkeeping)........................................ 220 Relativistic effects absolute upper limit of velocity of lig h t.............................................. 657 increase in mass of moving objects........................................ 658-659 lack of absolute simultaneity..........658-659 Lorentz contraction.......................... 658-659 time dilation.................................... 657-660 Relativistic rocket equation definition o f ............................................ 660 Relativistic ti m e .................................. 198-201 characteristics of...................................... 188 typical effects.......................................... 200 Relativity, theory of applicability to space travel............657—662 effect on Mercury...................................... 62 effect on time system s.................... 198—201 equations for.................................... 659—662 summary of effects.................................. 200 Rendezvous.......................................... 527-534 equations for.................................... 528-530 representative trajectories................ 531-534 thrust-free........................................ 533-534 Repeating ground track o r b it..........76, 79—82 applications o f ................................ 620—621 around other planets................................ 621 characteristics and applications o f .......... 615 equations for...................................... 81, 620 representative, table o f .............................. 82 use o f .............................................. 620-621 Replacement, of satellites (See Satellite replacement) Requirements basic ty p e s.............................................. 238 budget fo r................................................ 244 critical issues for development of............237 Earth-referenced orbits............................ 6 13 examples of good vs. bad................ 242—243 from payload vs. housekeeping.............. 238 need to trade o n .............................. 235—236 orbit, attitude, and timing system examples............................................ 239 orbit-related............................................ 593 process for defining........................ 235-281 source of for orbit and attitude............ 24—27
925
space-referenced orbits............................ 623 trading o n ................................................ 235 transfer orb its.......................................... 626 typical reasons for not trading o n ............ 240 validation/verification of definition o f....................................... 241 iterative process for............................ 248 methods of.......................................... 237 Resource 21 (constellation) summary o f.............................................. 673 Response Time (Figure of Merit) . . . . 484—485 Rest length (= proper length)...................... 659 Restricted three-body p ro b lem .................... 88 Retrieval of spacecraft consideration in orbit design.................... 595 Retrograde o rb it...................................... 48, 60 Revolution vs. rotation................................................ 60 Revolutions per day formulas for circular orbits...................... 838 formulas for elliptical Orbits.................... 844 numerical values for Earth orbit................................ inside rear cover numerical values for lunar o rb it.............. 879 numerical values for Mars o rb it.............. 885 Reynolds, G. H. book b y .................................................... 280 RF interference ....................................585—587 Rhumb li n e .................................................. 303 Right Ascension (RA) definition o f ............................... 48,196, 304 relation to longitude........................ 197-198 Right Ascension of the Ascending Node definition o f ....................................... 48, 733 effect of small changes in ................ 512-513 equations.............................................. 83-84 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic o rb its................ 848 formulas for parabolic orbits.................... 846 rotation of due to J2.............................. 83-84 Right spherical tria n g le .............................. 776 definition o f..................................... 300, 304 formulas for.............................................. 777 Right three-quarter spherical triangle.................................................... 383 Ring (See Armulus)........................................ 773 Ring Laser Gyro i'RLG; See Gyroscope) . . . 162 RLG (Ring Laser Gyro; See Gyroscope) . . . 162 Rocket equation definition o f .............................................596 relativistic version.................................... 660 traditional................................................ 596 Rocket technology not useful for interstellar exploration___656 Roll a x is........................................................ 150 Roll, Pitch, and Yaw (RPY) coordinates (See also Local horizontal coordinates) advantages and disadvantages for attitude specification...................................... 151 definition o f ............................................. 150 do not com m ute.............................. 154—155 Root Sum Square (See R SS)........................ 821
926 Rotating reference frame cffcct on apparent inclination.................. 448 Rotation angle formulas fo r............................................ 770 formulas for segments o f ................ 773-774 Rotation angle m easu rem en t............ 352-368 as part of non-singular data s et . . . . 374—376 combining with arc length.............. 362—368 definition o f ............................................ 297 illustration o f .......................................... 352 locus of.................................................... 323 properties o f............................................ 298 solution characteristics............................ 365 Type 1...................................... 322, 353-356 Type I vs. Type II............................ 322, 352 Type I I .................................... 322, 359-362 plane geometry equivalent................ 359 uncertainty measurements f o r ................ 330 Rotation vectors addition o f .............................................. 400 Rotation, of the Earth (See Earth).............. 181 Roy, A.E. book by.................................................... 114 Royal Greenwich Observatory.................. 182 RPY coordinates (See Roll, Pitch and Yaw coordinates) ............................................ 293 RSS (Root Sum S q u a re )............................ 821 of independent errors...................... 259-260 treatment of errors.................................. 246 Rubidium clock use on GPS.............................................. 201 Russia (See also Soviet Union) ISS participation.................................. 19—21 launch vehicles (See also Proton, Soyuz)................................................ 602 navigation system (See GLONASS) use of Molniya orbits.............................. 619
S-80 (constellation) summary o f ............................................ 673 Sarsfield, L. book by...................................................... 35 Satellite (See also Orbits) consequences of dying in place..............715 definition o f .................. ............................ 58 ground trace.................................... 443^-54 international designation f o r .................... 59 motion as seen from Earth.............. 454—469 NASA designations f o r ............................ 59 observability times far from Earth..........468 orbital mechanics definition o f ................ 40 outage due to dying in p la c e .......... 720-723 position within orbit............................ 49-52 Satellite collisions (See Collisions) Satellite crosslinks (See Crosslinks)............214 Satellite failure (See Failure) Satellite ground track (See Ground track) equations for............................................ 411 Satellite navigation (See Navigation) Satellite relative motion (See Relative motion)
Index Satellite replacement in constellations.............................. 720-723 Satellite tether (See Tether) Satellite viewing conditions for Sun, planet, observer........................ 563 Saturn Cassini orbit sequence............................ 651 gravitational parameters.......................... 854 mission parameters (table)...................... 887 orbit data.................................................. 865 physical and photometric d a ta ................ 866 repeating ground track orbits f o r ............ 621 satellite d a ta .................................... 868, 870 satellite gravitational parameters............ 854 summary of missions to (table).............. 633 Sun synchronous orbits for...................... 618 Saturn launch vehicle cost/performance table............................ 602 SBIRS (constellation) summary of...............................................673 Scale height definition o f .............................................. 68 Scaliger, Joseph origin of Julian Date................................ 183 Scanner (type of E arth sensor).......... 159-160 Scanning sensor coverage equations fo r............................ 475 Second (unit of time) definition o f .................................... 187, 889 relativistic definition...................... 198-199 Second eccentricity...................................... 437 Secondary (in orbital mechanics)................ 40 Secondary axis (in dual axis s p ira l)..........394 Secondary payload as low-cost launch alternative................ 604 Sectoral harmonics........................................ 64 Seidelmann, P. K. book by............................................ 114, 231 Semiautonomous navigation.............. 214-215 definition o f ............................................ 210 types o f.................................................... 214 Semim^jor a x is............................................ 734 definition o f ...................................... 40, 732 effect of small changes in................ 512-513 error bound.............................................. 751 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic orbits................ 848 formulas for parabolic o rb its.................. 846 Semiminor axis definition o f ...............................................40 formulas for elliptical orbits.................... 841 formulas for hyperbolic orbits................ 848 formulas for parabolic o rb its.................. 846 Sem iparam eter...................................... 41, 109 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic orbits................ 848 formulas for parabolic o rb its.................. 846 Sensors (See also Attitude determination; Specific sensors—Earth sensors, Star sensors, Sun sensors, Magnetometers, GPS receiver).................................................. 120
Index hardware parameters................................ 153 models, definition o f ............................... 274 un-orbit performance.............................. 156 orbit vs. attitude...................................... 120 projection of field of view onto the Earth.......................................... 430-437 rotating on a spinning spacecraft . . . 405-408 Series expansions trig functions............................................ 798 SFOF (See Space Flight Operations Facility) Shadow direction of as seen from space . . . . 573—574 Shadow cone computations for.............................. 564—565 definition o f .................................... 550, 563 Shishko, R. book b y .................................................... 280 Short arc definition o f .................................... 382, 385 Shuttle Tether E xperim ent........................ 555 SI (metric u n its ).................................. 889-897 Side-angle-side spherical triangle solution fo r.............................. 389-391, 784 Sidereal day definition o f ...................................... 77, 453 length o f .................................................. 193 Sidereal m o n th .............................................. 57 Sidereal period numerical values for solar orbit.............. 887 Sidereal tim e ................................ 187,193-198 characteristics of...................................... 188 definition o f .................................... 193,196 relation to longitude........................ 193-198 vs. solar time............................................ 193 Sidereal y e a r.................................................. 53 Side-side-angle spherical triangle solutions f o r ............................................ 782 Side-side-side spherical triangle solutions f o r ............................................ 780 Sidi, M. J. book by.................................................... 177 Simpson, j . A. book by .................................................... 280 Simultaneity in relativity.............................................. 199 lack of in relativity theory.............. 658-659 Single-axis spiral (= small circle) analysis.................................................... 397 definition o f ............................................ 395 Single-string.................................................... 8 definition o f ................................................ 8 Singularities dealing with in spherical tr ig .......... 792-795 physical meaning o f ................................ 793 Singularities, measurement avoidance of by sensor placement.................................. 346—347 definition o f ............................................ 327 effect on attitude sensing........................ 123 in arc length, rotation angle measurements............................ 363-366 in Earth width measurement.................... 345 in error analysis.............................. 266-267
927
resolution of by using other measurement ty pes.......................... 364-365, 373-376 tests fo r ............................................ 366—367 Sky darkness of at night.................................. 549 Skybridge (constellation) summary o f.............................................. 673 SkyLab attitude m otion................................ 12S-129 ballistic coefficient.................................... 71 use of CMGs............................................ 127 Slit sensor configurations.......................................... 349 measurement analysis of.................. 349-352 straight vs. curved slits.................... 351-352 Slosh (See Propellant slosh).......................... 128 Small circle area formulas for.............................. 386, 773 definition o f ............................................. 299 equations fo r............................................ 770 overlap area formula........................ 386, 774 Small spherical tria n g le...................... 382, 385 Software impact on changing spacecraft design . . . 180 on-orbit upgradability.............................. 180 Solar constant numerical values for solar o rb it.............. 888 Solar cycle historical.................................................. 183 seen in fl 0.7 index.................................... 72 Solar eclipse (See Eclipse).................... 563-567 geometry of.............................................. 564 Solar dcctric propulsion use by New Millennium.......................... 640 use in interplanetary missions.......... 652—653 Solar flux (See Sun, f l 0.7 index) Solar radiation intensity...................... 556—558 Solar radiation pressure balancing o f ............................................. 129 control of effect on constellations............699 disturbance torque.................................... 128 effect on orbit............................................ 67 effect on orbit elements............................ 513 method for controlling effects o f ............ 699 torque computation.................................. 171 Solar sail........................................................ 600 Solar system travel outside of (See Interstellar exploration) Solar tim e.............................................. 193-195 characteristics o f...................................... 188 Solar/lunar perturbations control of effect on constellations............ 699 effect on inclination................................ 517 effect on orbit elements............................ 513 method for controlling effects o f ............ 699 Solar-photonic assist.................................... 653 Solar-sailing use in interplanetary missions.................. 653 Solid Angle (See Angular Area) units and conversion factors.................... 894 Soop, E. M. book b y .................................................... 116
928
Sounding rockets as low-cost launch alternative................604 definition o f .............................................. 57 Soviet Union (See also Russia) launch vehicles (table)............................ 602 use of elliptical orbit constellations.................................... 687 use of Molniya orbits........................ 88,619 Soyuz launch vehicle cost/performance table............................ 602 use for ISS assembly.......................... 19—21 Space colonization...................................... 625 “Space cone” ................................................ 139 role in dual axis sp iral............................ 399 Space debris (See Debris in space) Space Flight Operations Facility (SFOF) control of interplanetary missions..........642 Space missions (See also Specific missions) representative examples........................ 5-23 types of........................................................ 6 Space navigation (See Navigation) Space Sextant.............................................. 213 advantages and disadvantages for navigation.......................................... 212 characteristics o f .................................... 211 Space Shuttle (Orbiter) Challenger disaster.................................... 14 cost estimates.......................................... 602 cost/performance table............................ 602 GAS container on.................................... 604 ISS assembly...................................... 19-21 retrieval of spacccraft.............................. 595 use for boost of Space Telescope.................................. 606-608 view of INTELSAT satellite from..........499 Space Station (See International Space Station) Space Systems Engineering (See Systems engineering; Orbit and attitude systems) Space Systems Loral (organization)............10 Space Telescope (Hubble Space Telescope, HST) ballistic coefficient.................................... 71 boost by Space Shuttle.................... 606-608 control system ........................................ 129 simplified pointing error budget . . . . . . . 246 use of Moon as future platform f o r ........ 126 Space tourism ................................................ 19 Space Tracking Network debris tracking.......................................... 30 Space-based orbit, attitude, and timing sy stem s.......................................... 179—230 changes in, over tim e...................... 179—180 Space-based rad ar {'See ra d a r).......... 582-5S7 Spacecraft consequences of dying in space..............715 disposal of (See Disposal o f spacecraft) effect of shape on drag coefficient. .. . 70-71 orbit terminology f o r .......................... 57-60 outage due to dying in space.......... 720-723 Spacecraft bus (See Bus) Spacecraft coordinates transformation to access area and geographic coordinates................................ 422-423
Index Spacecraft dynamics (See Attitude dynamics, Attitude control) definition o f ................................................ 1 Spacecraft eclipse (See Eclipse).......... 563-567 Spacecraft elevation angle (See Elevation angle) Spacecraft failure (See Failure) Spacecraft ground track (See Ground track) Spacecraft lighting (See Lighting) . . . . 550-554 Spacecraft longitude.................................... 503 Spacecraft nutation (See Nutation) Spacecraft outage (See also Failure) in constellations.............................. 720-723 Spacecraft position vector definition o f ............................................ 418 Spacecraft-centered celestial sphere definition o f ............................................ 288 Earth geometry projected onto........ 418—440 Space-referenced orbit applications and design o f .............. 623—626 requirements fo r...................................... 623 Spaceway (constellation) summary of...............................................673 Special relativity (See Relativity) Specialized o rb its .................................... 74-91 applications o f ........................................ 593 causes o f.................................................... 76 Earth-referenced, applications and design o f.................................... 614-622 space referenced, applications and design o f.................................... 624-625 transfer, applications and design o f.................................... 627-628 Specific energy definition o f .......... ...................................40 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 840 formulas for hyperbolic orbits................ 847 formulas for parabolic o rb its.................. 845 Specific heat capacity units and conversion factors.................... 894 Specific impulse (Isp ).................................. 596 for solar electric propulsion.................... 652 Specific launch c o s t............................ 601-603 definition o f .................................... 601, 610 table of values.......................................... 602 Specifications...................................... 241-244 definition o f .................................... 236, 241 typical documents.................................... 244 Spectral type (stars) definition o f ............................................ 655 Specular reflection bright spot on Earth d is k ........................ 569 from Earth seen from space.................... 424 Sphere surface area o f ........................................ 773 Sphere of influence...................................... 630 Spherical area (See Angular area) Spherical coordinate sy stem ...................... 801 definition o f ............................................ 302 transformations between...................... 414-416, 803-805
Index Spherical excess definition o f .................................... 300, 384 Spherical figures area form ulas.................................. 773-774 Spherical geometry (See also Global geometry; Spherical
trigonometry) . .283-316, 377-416, 769-800 advantages and disadvantages vs. vectors........................ 377-378, 391-394 area form ulas.......................... 386,773-774 basic formulas.................................. 296—302 bibliography............................................ 799 coordinate system definition.................... 302 definition o f ............................................ 302 differences from plane geometry . . . 299-302 Napier’s R u les................................ 776-778 vs. vectors........................................ 287-289 Spherical polygon area formulas for...................... 384-386, 773 Spherical triangle 3 vertices define 8 shapes........................ 381 area formulas for...................................... 773 complete.................................................. 380 definition o f ............................................ 297 differential relations between p a rts .......................................... 795-796 general rules.................................... 774-775 methods of specifying.............................. 387 table of defiedtions.................................. 385 taxonomy o f ............................................ 380 types o f .................................... 301,382-383 Spherical trigonometry bibliography............................................ 799 classical vs. full s k y ........................ 379-380 definition o f ............................................ 302 degenerate triangles........................ 792-795 formulas f o r .................................... 769-800 general rules.................................... 774-776 oblique full-sky triangles................ 779-792 right and quandrantal triangles........ 776—778 Spilker, J. J. book b y .................................................... 231 Spin stabilization advantages and disadvantages................ 124 capabilities o f .......................................... 168 definition o f ............................................ 124 of plan ets................................................ 132 physics o f ........................................ 134-148 Spin stabilized spacecraft definition o f ........................................ 5, 312 good vs. bad attitude solutions........ 365-368 solar radiation intensity o n ...................... 557 Sputnik (First man-made satellite)................ 2 SSP (See Subsatellite Point).......................... 568 ST (See Sidereal time) Stabilization (See also Attitude control; Spin; Dual spin; Gravity-gradient) passive methods o f ............................ . 124 Stable d istrib u tio n ...................................... 822 Star sensor.................................................... 160 advantages and disadvantages................ 122 applications o f ........................................ 156
929
classification of measurement type.......... 323 on-orbit performance history.................. 156 typical parameters.................................... 153 use for orbit and attitude............................ 29 use with Earth sensor for navigation........ 213 “Star” spherical triangle illustrations Of.......................................... 381 taxonomy o f ............................................ 380 Stark, J. book b y .......................................................34 Starnet (constellation) summary o f.............................................. 673 Stars nearest to the Sun (tab le)........................ 655 State estimation (See Filtering) .................... 164 State v e c to r.....................................................37 Stationkeeping (See also Orbit control) ....................111-112, 675, 697-708 absolute.................................................... 700 absolute vs. relative.................. 222, 700-703 box build-up............................................ 706 box definition.................................. 725, 728 cross-track........................................ 707-708 definition o f............................... 698, 707 need for.............................................. 707 definition o f ..................................... 103, 697 east-west.................................................. 706 in G E O .................................................... 617 in-track............................................ 703—707 definition o f....................................... 698 illustration.......................................... 705 simulation results.............................. 707 north-south.............................................. 706 relative.................................................... 700 relative to a near-by satellite............ 534—536 relative vs. absolute.................................. I l l Stationkeeping box build up of................................................ 706 definition of in constellation design........................................ 725, 728 Statistical error analysis...................... 807-833 Statistical measurements test for good vs. b a d ........................ 366-367 Statistical solutions accuracy tests f o r ............................ 366-367 Statistics not applicable for Earth coverage. . . 469—470 Statite................................................ 76, 91, 624 Stellar refraction characteristics o f...................................... 211 use for navigation............................ 212—214 Steradian (unit of angular area) definition o f ..................................... 300, 773 value o f .................................................... 773 “Stiffness” (angular m om entum )........ 18,130 Stochastic process........................................ 826 Storage orbit (See also Parking orbit) applications and design o f ............... 626-630 definition o f ............................................. 626 Streets of coverage...................................... 683 constellation pattern........................ 684—685 definition o f ..................................... 677, 685
930 Stress units and conversion factors.................... 894 Suborbital vehicles as low-cost launch alternative................ 604 definition o f .............................................. 57 Subsatellite point (SSP)...................... 418, 568 Subsolar point.............................................. 568 Summer Time (Europe).............................. 182 Sun circular and escape velocities o f .............. 45 eclipse of (See Eclipse).................. 563-567 effect on GEO satellite o rb it.................... 78 equation for effect on orbits...................... 67 gravitational constant.............................. 835 gravitational parameters.......................... 853 missions to via Jupiter.................... 638-639 motion of in spacecraft s k y ............ 290-292 orbit perturbations due to (See Solar-lunar perturbations, Solar radiation pressure) physical properties.......................... 856-858 role in Lagrange point orbits.............. 88-89 satellite parameters (numerical ta b le )........................................ 886-888 solar radio flux (fl0.7)........................ 72-73 historical daily v alu es...................... 857 historical monthly values.................. 858 summary of missions to (table).............. 633 sunspot cy cles................................ 856-858 Sun angle constraints for circular orbits.................. 839 constraints for elliptical orbits................ 845 constraints for hyperbolic o rb its............ 851 definition o f .................... 326, 363, 569, 572 on an arbitrary spacecraft face........ 312-314 computation geometry . . . . 313, 554-558 on Earth as seen from S pace.................. 572 Sun sensor.................................... 121,156-157 advantages and disadvantages................ 122 analysis of in plane geom etry................ 307 analysis of in spherical geometry............308 analysis of with vectors.......................... 307 applications............................................ 156 classification of measurement ty p e ........ 323 construction............................................ 306 field of view on the celestial sphere........ 308 most common types................................ 157 typical parameters.................................. 153 use for orbit and attitude.......................... 29 V-slit measurement analysis.......... 349-352 Sundial equation of time applied t o ....................506 Sunspots cycles.............................................. 856-858 maxima and minima................................ 856 Sun-synchronous orbit...................... 76, 82-86 applications o f ................................ 617-618 characteristics and applications o f.......... 615 definition o f .............................................. 84 for other planets...................................... 618 inclination for circular orbits.................. 838 inclination for Earth orbit... inside rear cover
Index inclination for elliptical orbits................ 844 inclination for lunar orbit........................ 878 inclination for Mars orbit........................ 884 local mean time for.................................... 86 motion of Sun relative to orbit plane........ 85 Orbit Cost Function for (table)................ 607 poor reasons for choosing...................... 242 representative.......................................... 618 table of inclinations.................................. 85 to eliminate second gimbal...................... 618 use o f .............................................. 617-618 Superior conjunction.................................... 54 Superior planet.............................................. 53 Supersynchronous orbit................................ 58 definition o f ................................................ 5 motion of satellite seen from Earth.......................................... 465^169 observability tim es.................................. 468 Surface brightness definition o f .............................................548 independence on distance........................ 548 source of Olber’s paradox...................... 549 Survivability role in orbit design.................................. 613 Swath definition o f ............................................ 443 Swath width as a function of altitude, elevation angle (table).......................................691 characteristic maximum w idths.............. 692 computation for elliptical orbits. . . . 450-451 definition o f .................................... 443, 476 selection of in constellations.......... 690-692 Swing-by (See Planetary assist trajectory) Symbols standard astronomical................................ 55 Symmetry in constellation design............................ 728 Synchronization (See also Timing) typical requirements o n .......................... 239 Synchronous altitude for any planet, formula.............................. 79 Synchronous orbits (See also Geosynchronous orbits) for Moon, Sun, planets, table of................ 80 Synodic m o n th .............................................. 57 Synodic p e rio d ........................................ 54—56 definition o f ...................................... 54, 537 Earth satellites relative to the Space Station (table)................................................ 538 numerical values for solar orb it.............. 887 relevance to interplanetary transfer........ 538 table of for planets.................................... 56 Synodic rate definition o f ............................................ 537 numerical values for lunar orbit.............. 878 Synodical m onth.......................................... 864 Synthetic aperture radar coverage equations fo r............................ 475 System requirements definition o f ............................................ 236 System specifications (See Specifications)
Index System trades basic ty p e s.............................................. 240 potentially good vs. b ad .......................... 244 Systematic erro r (See Bias) Systfeme International, SI (units)........ 889-897 time unit (sec).......................................... 188 Systems approach (See Orbit and attitude systems)................................................ 27-32 Bibliography o f ................................... 32-36 Systems engineering (See also Orbit and attitude systems) need for................................................ 26—32 relation to requirements definition.......... 235 Syzygy............................................................ 54 Szebehely, V. G. book by.................................................... 114
— T— Taff, L. G. book by.................................................... 178 TAI (See Atomic time; International Atomic Time) Target (See also Ground station) definition o f .................................... 422, 454 Motion of satellite as seen f rom. . . . 454—469 Target s p a c e ................................................ 823 Tau Ceti (nearby Sun-like star)..................655 Taurus launch vehicle cost/performance table............................ 602 performance for interplanetary flight. . . . 609 use for solar sail launch.......................... 653 Taylor, J. R. book by .................................................... 281 TDB (See Barycentric Dynamic Time) TDOP (See Time Dilution of Precision) TDRS (Tracking and Data Relay Satellite)............................................ 32, 129 advantages and disadvantages for navigation............................ ..............212 summary of................................ ..............673 TDT (See Terrestrial Dynamic Time) TechSat 21 (constellation) summary o f.............................................. 673 Teles, J. book by.................................................... 117 Temperature units and conversion factors.................... 894 Terminator appearance seen from space.. . 424, 574—577 dark angle corrections for................ 574—575 definition o f .................... 158, 548, 552, 568 equations for computing visibility.......... 576 radius of on Earth, M oon........................575 Terrestrial Dynamic Time (T D T).................................... 187, 200 Terrestrial Time (TT).......................... 190, 199 history o f ................................................. 187 Tesserai harmonics definition o f .............................................. 64 Tether visibility from Earth................................ 555 Thermal analysis of input on spacecraft face.............. 554-558
931
Thermal balance of the Earth.............................................. 568 Thermal conductivity units and conversion factors.................... 894 Thermal radiation (See Infrared radiation).......................................... .. 568 definition o f ............................................. 547 Third body interactions................................ 67 Third quarter Earth (See Last Quarter E arth)...................................................... 571 Third quarter phase (See Illumination phases) Three-axis attitude definition o f ............................................. 149 Three-axis stabilized spacecraft definition o f ,................................................ 5 Thruster advantages and disadvantages for attitude control................................................ 126 characteristics o f...................................... 172 sizing for Orbit control............................ 224 use for attitude control............................ 168 Thrust-free rendezvous...................... 533-534 Tidal forces (see Gravity-gradient forces) Tides source o f .................................................. 125 Timation (navigation sy stem ).................... 201 Time (See also Second)........................ 180-201 absolute vs. calendar................................ 181 budget allocation exam ple...................... 249 clock errors.............................................. 181 common systems o f ................................ 188 definition time intervals.......................... 181 discontinuities on spacecraft............ 190-193 errors in relative vs. absolute measurement of.................................. 242 history of measurement systems.............. 187 relation to longitude........................ 197—198 units and conversion factors.................... 894 Time average gap (Figure of M erit)...................................................... 484 Time dilation (relativity)............ 198, 657-660 Time Dilution of Precision (TDOP)............ 208 Time in v ie w ................................................ 458 formula for circular orbits........................ 460 formula for elliptical orbits...................... 463 maximum duration for circular orbits.. . . 838 maximum duration for elliptical orbits.................................................. 843 maximum duration for hyperbolic orbits.................................................. 851 numerical values for Earth orbit................................ inside rear cover numerical values for lunar o rb it.............. 876 numerical values for Mars o rb it.............. 882 Time zones.................................................... 181 Timeline allocation o f ............................................ 249 Timing typical requirements on............................ 239 Timing budget creating budget f o r .......................... 268-279 error sources, table.................................. 273 errors........................................................ 275
932 discussion of...................................... 276 impact on mapping and pointing erro rs.......................................... 254 sources, table o f ................................ 273 Timing systems combination with orbit and attitude...................................... 226—230 sources of requirements...................... 28-32 Titan launch vehicle cost/performance table............................ 602 performance for interplanetary flight.. . . 609 Tito, Dennis (space to u rist).......................... 19 ToD (See True o f Date coordinates) Toe, of footprint (See also Footprint)........ 473 “Tombstone plot” (= coverage histogram )...................... 478-479, 489, 497 TOPEX m ission............................................ 27 Torque definition o f ............................................ 144 response to ...................................... 143-148 units and conversion factors.................... 892 Torque equilibrium attitude IS S ............................................................ 22 Torque-free m o tio n ............................ 134—143 Total eclipse of the satellite.......................................... 563 of the Sun................................................ 563 Total response time definition o f ............................................ 485 Tourism, sp ace.............................................. 19 Tracking and Data Relay Satellite (See TDRS) Trades, system-level basic ty p es.............................................. 238 potentially good vs. bad.......................... 242 Trading on req u irem ents.......................... 235 TraitS de Dynamique (See d ’Alembert, Jean) Trajectory (See Orbits) definition o f ........................................ 37, 59 Transfer orbit (See also Planetary fly-bys, Hohmann transfer, 3-burn transfer) . . 92—96 applications and design o f .............. 626—630 definition o f ................................ 60, 92, 626 AV for vs. altitude.................................. 598 Transformations rectangular to spherical coordinates........ 801 T ransit.................................................. 558—563 calculating limits for elliptical orbits. . . . 562 conditions fo r.................................. 560-562 conditions on Earth-centered celestial sphere................................................ 561 Transit (navigation satellite system).................................................... 201 Transit time formulas for circular o rb its.................... 839 formulas for elliptical orbits.................... 845 formulas for hyperbolic orbits................ 852 Transit, of the meridian definition o f ............................................ 466 Triangulation (See also Parallax).............. 370 Trig functions common relations among................ 797-799 Tropical y e a r .............................................. 859 definition o f .............................................. 53
Index True anomaly definition o f ........................................ 49—50 formulas for circular orbits...................... 836 formulas for elliptical orbits.................... 841 formulas for hyperbolic orbits................ 849 formulas for parabolic o rb its.................. 846 rate of change for circular o rb its............ 836 rate of change for elliptical o rb its.......... 841 rate of change for hyperbolic orbits........ 849 rate of change for parabolic orbits.......... 846 series expansion f o r .................................. 50 True horizon................................................ 418 True of Date coordinates............................ 294 definition o f ...............................................48 TRUNC (software fu n ctio n )...................... 185 Tsiklon launch vehicle cost/performance table............................ 602 TT (See Terrestrial Time) Tumbling of small natural bodies............................ 132 Turn angle (of a hyperbola).......................... 42 formulas f o r ............................................ 848 in hyperbolic transfer.............................. 640 in planetary assist trajectory............ 100—103 equations for...................................... 101 table of values for plauets.................. 101 Twilight o r b it................................................ 86 Twin p arad o x.............................................. 661 Type 1 arc segment...................................... 793 Type 2 arc segment...................................... 793 Type-I missions, in interplanetary tra n sfe r.................................................. G37 Type-II missions, in interplanetary tra n sfe r.................................................. 637
—u— 224 Ullage.................................................. . Ulysses mission summary of (table).................................. 633 use of gravity a ssist........................ 638—639 U m b ra .................................................. 550, 563 Uncorrclatcd measurem ents.............. 309, 326 Uniform distribution.................................. 812 Unit vector advantages and disadvantages vs. spherical trig.............. 377—378, 391—394 equivalent to points on celestial sphere........................................ 287-288 solution to spherical triangle problem...................................... 392-393 transformations to spherical coordinates........................................ 296 United States (See also individual missions and launch vehicles)
TSSparticipation.................................. 19—21 time zones................................................ 182 Units and conversion factors.............. 889-897 Universal time (U T )............ 182,187,189, 467 characteristics of...................................... 188 Universe age o f ...................................................... 550 expansion o f ............................................ 550
Index Unloading, momentum (See Momentum dumping) Unraodeled definition o f ............................................ 261 Uranus gravitational parameters.......................... 855 mission parameters (table)...................... 887 orbit data.................................................. 865 physical and photometric d a ta ................ 866 repeating ground track orbits f o r ............ 621 satellite d a ta .................................... 868, 870 satellite gravitational parameters............ 855 Sun synchronous orbits for...................... 618 UT (See Universal Time) UTC (See Coordinated Universal Time)
— V— V infinity (See Hyperbolic excess velocity) Validation of requirements (See Requirements, validation of) Vallado, D, A. book b y .................................................... T14 Van Allen radiation belts role in defining LEO.............................. 4, 58 role in orbit design.................................. 614 Vanguard-2 ballistic coefficient.................................... 71 Variate (See Random variable)....................809 VDOP (See Vertical Dilution of Precision) Vector magnetometer measurement uncertainties...................... 348 Vectors (See also Unit vectors) advantages and disadvantages vs. spherical trig.............. 377-378, 391-394 vs. spherical geometry.................... 287—289 V elocity........................................................ 108 addition of in relativity............................ 660 and position, determination of from orbit elements.................................... 108—109 and position, determination of orbit elements fro m .......................................... 106-108 equations for.............................................. 51 formulas for circular orbits...................... 837 formulas for elliptical orbits.................... 842 formulas for hyperbolic orbits................ 849 formulas for parabolic o rb its.................. 846 in an orb it.................................................. 51 major solar system bo d ies.............. 853—855 numerical values for Earth orbit................................ inside rear cover numerical values for lunar orbit.............. 878 numerical values for Mars orbit.............. 884 numerical values for solar orb it.............. 887 units and conversion factors.................... 895 Velocity of escape (See Escape velocity) Velocity of light as absolute upper limit for velocity.............................................. 657 Venus gravitational parameters.......................... 853
933
mission parameters (table)...................... 887 orbit data.................................................. 865 physical and photometric d a ta ................ 866 radar map o f ............................................ 583 repeating ground track orbits f o r ............ 621 use in gravity assist trajectory............................ 608, 638-639 Verification of requirements (See Requirements, validation of) Vernal Equinox.................................... 196, 294 definition o f ....................................... 48, 304 precession of the equinoxes.................... 294 Vertical definition o f ............................................. 426 Vertical Dilution of Precision...................... 208 Vesta (asteroid) gravitational parameters.......................... 855 mission parameters (tabic)...................... 887 Viewing and lighting conditions for satellite and target .. . 547—588 Viewing Area cutting into equal p arts............................ 431 Viking (Mars missions) ballistic coefficient.................................... 71 loss o f ...................................................... 225 summary of (table).................................. 633 Vinti, J. P. book b y .................................................... 114 Virgo (constellation) summary o f.............................................. 673 Vis viva energy . ...........................................634 Vis viva equation................................ 40, 44-45 Viscosity units and conversion factors.................... 895 Visibility (See Viewing and Lighting) Visible light sensors (See also Horizon sensor)...................................................... 568 Visual magnitude (See also Brightness) definition of. .............................................578 of spacecraft and planets.................. 578—582 Vitasat (constellation) summary o f.............................................. 673 Volume units and conversion factors.................... 895 Voyager mission leaving solar system ........................ 631-632 summary of (table).................................. 633 use of optical navigation.......................... 640 V-Slit sensor.......................................... 349-352 analysis o f ................................................ 351 definition o f ............................................. 349 straight vs. curved slits............................ 352
—w — Walker 5 (constellation) summary o f.............................................. 673 Walker constellations.......................... 685-686 definition o f............................................. 682 example of................................................ 686 Walker Delta Pattern........................ 685-686 Walker, John work on constellations.................... 682, 685
Index
934 Wallops Islands coverage simulation o f.................... 492-497 Weightlessness (See also Microgravity) definition o f .................................... . .. 39 Wertz, J. R. books by .................... 35,117, 177, 231,281 Wheels (attitude actuator) on spacecraft.................................. . . .. 127 127 types of............................................ use for attitude control.................... 169 S27 White noise.......................................... Wie, B. book by............................................ 177 Wiesel, W. E. book by............................................ 114, 177 Williamson, M. book by............................................ 36, 281 Wiltshire, R. S...................................... 527-528 Wookcock, G. 36 book by............................................ — Y— Yaw axis definition o f .................................... 151 Yaw, Pitch, Roll coordinates (See Roll, Pitch, Yaw coordinates)............................ . . . . 151 Youdan, K. 176 book by............................................ — z — Zarchan, P. book by............................................ . . . . Zee, C. book by............................................ Zenit launch vehicle cost/performance table.................... . . . . Zenith definition o f .................... 194, 296, 302, Zenith a n g le ........................................ definition o f .................................... . . . . Zero g (see also Microgravity)
116 116 602 418 295 468
39 definition of............................... Zero momentum attitude control. . . . . . . . 169 capabilities o f.................................. . . . . 168 definition o f .................................... . .. 130 Zonal harmonics definition o f .................................... ........64 Zulu Time (Z ).............................................. 182
EARTH SATELLITE PARAMETERS
Equations are available throughout the book for calculating all of the basic mission parameters for circular Earth orbits. Here they are evaluated numerically to provide a quick reference, a check on computations, and an estimate of the sensitivity of the relevant parameters to altitude.
Explanation of Earth Satellite Parameters The following table provides a variety of quantitative data for Earth-orbiting satel lites. Limitations, formulas, and text references are given below. The independent parameter in the formulas is the distance, r, from the center of the Earth in km. The outer column on each table page is the altitude, h = r where = 6,378.14 km is the equatorial radius of the Earth. 1. Instantaneous Access Area fo r a 0 deg Elevation Angle or the Full Geometric Horizon (106km2). All the area that an instrument or antenna could potentially see at any instant if it were scanned through its normal range of orientations for which the spacecraft elevation is above 0 deg [Eq. (D-29)]. 2. Instantaneous Area Access fo r a 5 deg Minimum Elevation Angle (IO6 km2) = same as col. 1 but with elevation of 5 deg. 3. Instantaneous Area Access fo r a 10 deg Minimum Elevation Angle (IO6 km2) = same as col. 1 but with elevation of 10 deg. 4. Instantaneous Area Access fo r a 20 deg Minimum Elevation Angle (IO6 km2) = same as col. 1 but with elevation of 20 deg. 5. Area Access Rate fo r an Elevation o f 0 deg (IO3 km2/s) = the rate at which new land is coming into the spacecraft’s access area [Eq. (D-30)]. 6. Area Access Rate fo r an Elevation Limit o f 5 deg (103 km2/s) = same as col. 5 with an elevation of 5 deg. 7. Area Access Rate fo r an Elevation Limit o f 10 deg (IO3 km2/s) = same as col. 5 with an elevation of 10 deg. 8. Area Access Rate fo r an Elevation Limit o f 20 deg (IO3 km2/s) = same as col. 5 with an elevation of 20 deg. 9. Maximum Time in View fo r a Satellite Visible to a Minimum Elevation Angle o f 0 deg (min) = JP/lm^ t /180 deg, where P is from col. 67 and Xmax is from col. 13. Assumes a circular orbit over a nonrotating Earth [Eq. (D-31)]. 10. Maximum Time in View fo r a Satellite Visible to a Minimum Elevation Angle o f 5 deg (min) = same as col. 9 with for 5 deg taken from col. 14. 11. Maximum Time in View fo r a Satellite Visible to a Minimum Elevation Angle o f 10 deg (min) = same as col. 9 with Xniax for 10 deg taken from col. 15. 12. Maximum Time in View fo r a Satellite Visible to a Minimum Elevation Angle of 20 deg (min) = same as col. 9 with Xmax for 20 deg taken from col. 16. 13. Earth Central Angle fo r a Satellite at 0 deg Elevation (deg) = Maximum Earth Central Angle = acos(/?E / r). Alternatively, Maximum Earth Central Angle is = 90 - p, where p is from col. 49 [Eqs. (9-2), and (D-28)].
Explanation of Earth Satellite Parameters 14. Earth Central Angle fo r a Satellite at 5 deg Elevation (deg) = 90-6-7), where rj = asin (cos£ sinp), p is from col. 54, and £ = 5 deg [Eqs. (9-4), (9-5) and (9-6)]. 15. Earth Central Angle fo r a Satellite at 10 deg Elevation (deg) = same as col. 14 but with £ = 10 deg. 16. Earth Central Angle fo r a Satellite at 20 deg Elevation (deg) = same as col. 14 but with £ = 2 0 deg. 17. Maximum Range to Horizon = Range to a satellite at 0 deg elevation (km) = (r2 - /?E 2)1/2, where /?E = 6,378.14 km is the equatorial radius of the Earth. 18. Range to a Satellite at 5 deg Elevation (km) —Maximum Range fo r Satellites with a Minimum Elevation Angle o f 5 deg (km) = R^ (sinA / sinry), where /?£ = 6,378.14 km is the equatorial radius of the Earth, A is from col. 14, T] = 90 deg - A - s, and £ = 5 deg [Eq. (9-7)]. 19. Range to a Satellite at 10 deg Elevation (km) = same as col. 18 with A from col. 15 and e= 10 deg. 20. Range to a Satellite at 20 deg Elevation (km) = same as col. 18 with A from col. 16 and e = 20 deg. 21. Maximum Nadir Angle fo r a Satellite at 0 deg Elevation Angle (deg) = Max. Nadir Angle fo r Any Point on the Earth = Earth Angular Radius = asin (J?E/ r), where flE = 6,378.14 km is the equatorial radius of the Earth [Eq. (9-2)]. 22. Nadir Angle fo r a Satellite at 5 deg Elevation Angle (deg) = Maximum Nadir An gle fo r Points on the Ground with a Minimum Elevation Angle o f 5 deg = 90 deg - f - A, is the Earth central angle from col. 14 [Eq. (9-6)]. 23. Nadir Angle fo r a Satellite at 10 deg Elevation Angle (deg) = same as col. 22 with £ = 1 0 deg. 24. Nadir Angle for a Satellite at 20 deg Elevation Angle (deg) = same as col. 22 with £ = 20 deg. 25. Atmospheric Scale Height (km) = RT / Mg, where R is the molar gas constant, T is the temperature, M is the mean molecular weight, and g is the gravitational ac celeration [inside front cover]. 26. Minimum Atmospheric Density (kg/m3), from MS IS atmospheric model [Hedin*t*, 1987, 1988, and 1991]. The solar flux value, F10.7, was chosen such that 10% of all measured data are less than this minimum (65.8 x IO-22 W m-2 Hz-1). See Sec. 8.1.3. The MSIS model is limited to the region between 90 and 2,000 km. Below 150 km and above 600 km the error increases because less data have been used. All data have been averaged across the Earth with a 30 deg step size in lon gitude and 20 deg steps in latitude (-80 deg, to +80 deg). This over-represents the * Hedin, Alan E., 1987- “MSIS-86 Thermospheric Model,” J. Geophys. Res., 92, No. A5, pp. 4649-4662. f Hedin, Alan E-, 1988. “The Atmospheric Model In The Region 90 to 2,000 km,” Adv. Space Res., 8, No. 5-6, pp. (5)9-(5)25, Pergamon Press. * Hedin, Alan E., 1991. “Extension of the MSIS Thermosphere Model into the Middle and Lower Atmosphere,” J. Geophys. Res., 96, No. A2, pp. 1159-1172.
Explanation of Earth Satellite Parameters Earth’s polar regions; however, satellites spend a larger fraction of their time at high latitudes. The solar hour angle was adapted to the individual location on the Earth with UT = 12.00 Noon. 27. Mean Atmospheric Density (kg/m3) = same as col. 26 but with a mean F I0.7 value of 118.7x IO-22 W m - 2 H z-'. 28. Maximum Atmospheric Density = same as col. 26 hut with a F10.7 value of 189.0 x IO-22 W n r 2 Hz-1. This is the F I0.7 value such that 10% of all measured values are above it. 29. Minimum AV to Maintain Altitude at Solar Minimum (m/s per year) = n (CpA/m) x prv/P, where p is from col. 26, v is from col. 56, P is from col. 67 expressed in years, and the ballistic coefficient, m!CDA, is assumed to be 50 kg/m2. AV esti mates are not meaningful above 1,500 km [Eq. (2-36)]. 30. Maximum AV to Maintain Altitude at Solar Maximum (m/s per year) = same as col. 29 with p from col. 28 and the ballistic coefficient, m/CpA, assumed to be 50 kg/m2. 31. Minimum AV to Maintain Altitude at Solar Minimum (m/s per year) = same as col. 29 with p from col. 26 and the ballistic coefficient, m/Q)A, assumed to be 200 kg/m2. 32. Maximum AV to Maintain Altitude at Solar Maximum (m/s per year) = same as col. 29 with p from col. 28 and the ballistic coefficient, m/Q)A, is assumed to be 200 kg/m2. 33. Orbit Decay Rate at Solar Minimum (km/year) = ~2k (Cq A /m) p r2IP, where p is from col. 26, P is from col. 67 (expressed in years), and the ballistic coefficient, m!CDA, is assumed to be 50 kg/m2. Orbit decay rates are not meaningful above 1,500 km [Eq. (2-34)]. 34. Orbit Decay Rate at Solar Maximum (km/year) = same as col. 33, with p from col. 28 and the ballistic coefficient, m/Q> A, assumed to be 50 kg/m2. 35. Orbit Decay Rate at Solar Minimum (km/year) = same as col. 33 with p from co l 26, and the ballistic coefficient, m/CDA> assumed to be 200 kg/m2. 36. Orbit Decay Rate at Solar Maximum (km/ycar) = same as col. 33, with p from col. 28 and the ballistic coefficient, ra/Q) A, assumed to be 200 kg/m2. 37. Estimated Orbit Lifetime at Solar Minimum (days) = Data was produced using the software package SatLife. Ballistic coefficient, mlCDA, assumed to be 50 kg/m2. 38. Estimated Orbit Lifetime at Solar Maximum (days) = same as col. 37 with the bal listic coefficient, m/Q)A, assumed to be 50 kg/m2, 39. Estimated Orbit Lifetime at Solar Minimum (days) = same as col. 37 with the bal listic coefficient, mlCDA, assumed to be 200 kg/m2. 40. Estimated Orbit Lifetime at Solar Maximum (days) = same as col. 37 with the bal listic coefficient, m/CDA, assumed to be 200 kg/m2. 41. Euler Axis Co-Latitude (deg) = atan ((Oorbit sin i / (ft)^, + (Qorbit cos /)), where toorbi{ is the orbital angular velocity from col. 57, 0)day is the Earth’s angular
Explanation of Earth Satellite Parameters velocity on its axis - 0.250696 deg/min, and i is the assumed inclintion of 0 deg for this column. [Table 8-8 and Eq. (8-23a)] 42.-47. Euler Axis Co-Latitude (deg) = same as col. 41 with value of the inclination at the top of the column. 48. Inertial Rotation Rate (deg/min) = Orbit angular rate = same as col. 57. Data is repeated here for convenient comparison with columns 50-55. 49. Euler Rotation Rate (deg/min) = Angular rate of rotation about the Euler axis = oyorbit sin i / sin EACL, where coort)il is the orbital angular velocity from col. 57, EACL is the Euler Axis Co-Latitude from col. 41, and / is the assumed inclination of 0 deg for this column. [Table 8-9] 50.-55. Euler Rotation Rate (deg/min) = same as col. 49 with value of the inclination at the top of the column and Euler Axis Co-Latitude from the corresponding column 42^17. 56. Circular Velocity (km/s) = (juE/r) M = 631.348 l r
[Eq. (2-8)].
57. Orbit Angular Velocity (deg/minute) = 360/P = 2.170 415 x 106r -3/2, where P is from col. 67. This is the angular velocity with respect to the center of the Earth for a circular orbit. (See col. 62 for angular rate with respect to ground stations) [Eq. (9-65)]. 58. Escape Velocity (km/s) = (
2
= 892.8611 r~M = ( 2 ) ^ x vcirc [Eq. (2-7)].
59. AV Required to De-Orhit (m/s) = the velocity change needed to transform the as sumed circular orbit to an elliptical orbit with an unchanged apogee and a perigee of 50 km [Eqs. (2-85a), and (2-85b)] 60. Plane Change AV ((km/s)/deg) = 2 vcirc sin (0.5 deg), where vcirc is from col. 56. Assumes circular orbit and linear sine function; [Eq. (2-78)] 61. AV Required fo r a i km Altitude Change (m/s) = assumes a Hohmann Transfer with rB - r A = 1 km; [Eqs. (2-69), and (2-70)]. 62. Maximum Angular Rate /is Seen from a Ground Station (deg/s) = 2nr/hP: where h = r - /?E is the altitude and P is from col. 67. This is the angular rate as seen from the surface of a non-rotating Earth of a satellite in a circular orbit passing directly overhead. (See col. 57 for the angular velocity as seen from the center of the Earth.) [Eq. (9-106)]. 63. Sun Synchronous Inclination (deg) = acos (-4.773 48 x IO-15 r 7/2); assumes cir cular orbit with node rotation rate of 0.9856 deg/day to follow the mean motion of the Sun. Above 6,000 km altitude there are no Sun synchronous circular orbits [Eq. (D-34)]. 64. Angular Radius o f the Earth (deg) = asin (R g / r ), where RE = 6,378.14 km is the
equatorial radius of the Earth [Eqs. (9-2)]. 65. One Degree Field o f View Mapped onto the Earth’s Surface at Nadir from Altitude h (km) = The length on the Earth’s curved surface of a 1 deg arc projected at nadir from this altitude. Note: This data is very nonlinear [Eqs. (9-4), (9-5), and (9-6)].
Explanation of Earth Satellite Parameters 66. Range to Horizon (km) = same as col. 17 = (r2 - RE 2)1/2, where R e = 6,378.14 km is the equatorial radius of the Earth. For the range to points other than the true horizon (i.e., £ * 0 deg) use columns 18, 19, and 20 [Eq. (9-7)]. 67. Period (min) = 1.658 669 x IO-4 r3/2 = (1/60) x 2% (r3//i)1/2. Assumes a circular orbit, r is m easured in km, and n = 398,600.5 km 3/s2. Note that period is the same for an eccentric orbit with semimajor axis = r; [Eq. (2-4b)]. 68. Revolutions per Day (#) = 1,436.07/P, where P is from col. 67. Note that this is revolutions per sidereal day, where the sidereal day is the day relative to the fixed stars which is approximately 4 minutes shorter than the solar day of 1,440 min utes. [Eq. (D-35)] 69. Maximum Eclipse (minutes) = (p/180 deg)P, where p is from col. 64 and P is from col. 67. This is the maximum eclipse for a circular orbit. Eclipses at this al titude in an eccentric orbit can be longer. [Eq. (D-37)] 70. Node Spacing (deg) - 360 deg x ( P 1 1,436.07), where P is from col. 67. This is the spacing in longitude between successive ascending or descending nodes for a satellite in a circular orbit [Eq. (D-36)]. 71. Node Precession Rate (deg/day) = -2.06474 X 1014 r ~7/2 cos i = -1.5 n ^ (Re la)2 (cos i) (1 —e2)-2, where i is the inclination, e the eccentricity (which is set to zero), n is the mean motion (= (p/a3)y z ), a the semimajor axis, and J2 the domi nant zonal coefficient in the expansion of the Legendre polynomial describing the geopotential. This is the angle through which the orbit rotates in inertial space in a 24 hour period. Assumes a circular orbit; r is in km in the first expression [Eq. (D-33)],
Earth Satellite Parameters 1
2
3
4
5
0.00
5 deg elevation (lOfikm2) 0.00
3.95
10 deg elevation (lOGkm2)
7
8
A R E A A C C E S S R ATE
IN S T A N T A N E O U S A C C ESS A R E A Odeg elevation (IC^km2)
6
Odeg 5 deg 10 deg 20 deg 20 deg elevation elevation elevation elevation elevation (10®krn2) (103km2/s) (1 (PkrrPfs) (103km2/s) (lOSkrrvVs)
Alt. (km)
0.00
0.00
0.00
0.00
0.00
0.67
0.21
17.24 . . . 10.71
7.15
3.96
1.31
0.44
20.76
14.01
9.86 ■
5.69
150
2.06
0.73
23.56
16-73
12.20
7 28
200
0.00
0 .
too
5.17 ' .......2.89 ” .......1.08
25.90 *
19.04
14.27
8.76
250
11.48
6.48
3.77
1.48
27.89
21.06
16.11
10.12
300
13.30
7.81
4.70
1.92
29.63
22.83
17.76
11.38
350
9.64
2.39
'.I 15.0$ 10.50
12.56
2.89.
32.S0
20.02
13.G6
450 ■
3.42
33.71
27.10
21.86
14.68
500
8.66
3.97
34.78
28.25
22.99
15.63
550
14.54
9.69
4.53
35.75
29.30
24.03
16.52
600
15.88
10.72
5.12
36.62
30.25
24.99
17.36
650
31.12
25.87
18.58
11.85
20.29
13.20
21.98 23.64
6.64
24.41
5.72 26.89 ■-^28:49^: ''I';: 1^:85:^;
■ ■12.81
6,33
38.11
31.91.
26.68
' 18.87
13.85
6.05
■ ■38:75
32.63
27.43
, 19jB6
800 '
30.06
21.15
14.90
7.58
39.33
33.29
28.12
20.20
31.61
22.44
15.94
8.21
39.85
33.90
28.76
20.80
900
33.14
23.73
16.98
8.86
40.32
34.45
29.35
21.37
950
'M .6A
2&0Q '
. 34.96 . 42.29
31.18
: -23.-14
61.02 ”
47.97
.....37.51
71.98
57.81
46.14
28.96
42.71
81.77
66.70
54.05
34.86
74.76
61.29
:':^4ilv89:^ 48.67
28.12
90-52 105.74 '
82.10 ftft 70
112.32
94.93
123.90 133.74
16.14
43.12
38.10
' " " '22.6 9 " " 43.43
39.07
45.52.'
. . if. m
. 36.90 ■
m
850
k
. 24,10 ■
' 32.06 33.50 * ™4’ 93
25.69 27.59
2,000
38.89
35.19
28.39
2,500
41.47
38.10
34.78
28.52
3,000
! 39.95
36.98
28.22
3,500
38.33 •Sft fifl
32.98
27.68
4.000 •
50.31
i:''.:-**-'siV :V:
31.86
26.97
4,500
79.63
54.77
35.05
32.92
3070
26.18 ' ....5,000
105.78
89.62
62.81
31.97
30.22
23.37
24.49
6,000
115.06
98.23
69.84
29.18
27.73
26.16
22.80
7,000
142.22
123.10
105.71
149.59
130.11
112.27
:: 156.06 179.35
136.29
118.07
86.35'
22.57
158.65
139.22
104.35
15.69
81.46
25.48
24.13
.. 23^47
.22.30 ;;
193.80
172.65
152.56
115.89
11.63
15.21 11.34
194.23
173.06
152.95
116.24
11.52
11.23
203.65
182.23
161.73
123.91
;3EIS1
134.31
6.03
134.93
5.86
210.79'
: 189.19
216.20
194.49
216.94
195.21
^^68li42;^: 174.22
8.85 "
7.29
See Front of Table for Formulas and Sources.
8,000
19.72 .
9,000
20.66
18.36
10,000
13.25
15,000
10.97
10.04
20,000
10.87
9.95
20,184
"8.60'
7.92
25,000'
$.33 5.76
21.20
... 14.63...
.6.96 .
.
5.62
|S|S3
30,000
!i:U;$SRi ;'!:i
35,000.
..... 5.23
35,786
Earth Satellite Parameters
See Front of Table for Formulas and Sources.
Earth Satellite Parameters
See Front of Table for Formulas and Sources.
E arth Satellite P aram eters
25
26
27
28
29
30
ATMOSPHERIC DENSITY Ait (km)
Atm. Scale Ht. (km)
0
8.4
Minimum (kg/m3)
Mean (kg/m3)
31
32
AV TO MAINTAIN ALTITUDE
Maximum (kg/m3)
Solar Min Solar Max Solar Min Solar Max 50 kg/ m2 50 kg/ m2 200 kg/ m2 200 kg/ m2 (m/syyr (m/s)/yr (m/syyr (m/s)/yr 2,37x1013
1.2
1.2
1.2
4,61x10-7
4.79x10“7
5.10x10-7
25.5
1.65x10-3
1.81x10^' 2-04x10-9
3.17x104
3.94x10+
37. C
1,78x10"10 2.53x10-10 3.52x10-1° 3.40xi(P
6.72X1CP
... 250
' ' 44.8
3.35x10-11 6.24x10"11 1.06x10-1° 6.36x102
2.02x103
8:51X102 UaA*y<wxjw.Uxw 1.59x102
300
50.3
8,19x1 CT12 1.95x10-11 3.96x10-11 1.54x102
7.47X102
3.86x101
1.87x102
350
54.8
2.34x10-12 6.98x10"12 1.66x10-11 4,37x101
3.11x102
1.09x101
7.78x101
400
58.2
7.32x1 Q-->3 2.72x10~’ z
1.30x101
1.40x102
3.40x100
■■m i 500
61.3
2.47x10-13 1.13x10~15 3.61x10-12 4.55x100
6.66x10“*
1.14x10°
64.5
8^98x10“14 4.89x10-13 1.80x10^12 1.64x10°
3.29x10*
4.11X10-1
’ 550...
68.7
3.63x10-14 2.21x10-13 9.25x10-13 6.59x10-1
1^68x101
1.65x10-1
4,20x10®
600
74,8
1.68x1CH4 1.04x10"13 4.89x10-13 3.03x10-1
8.81x10°
7.58x10-2
2.20x10°
650
84.4
9.14x10-15 5.15x10-14 2.64x10-13 1.64x10-1
4.73x10°
4.09x10-2
1.18x10°
; 700
99.3
5.74xl6“ i5 2.72X10-14 1.47x1Crl3 1.02x10-1
2.61x10°
121 ■ 3.99x10 - if 1.55x10-14 8.37X10 -‘<+ 7 . 0 4 * ^
1:48x10°-
3 .^ x 1 ( r i
.,* 1 0 *
■ § m m
150
''f
2.37x1013
5.92x1012
9.90X106
2.24x10s
5.92x1012
■■..(..■ifi'■■;,!»!i|wii;;ij 5.04x102
850
188
2.'96x 10-15 9.63x10-15 4.39x10-*4 5.19x10-2 2.28x10-15 6.47x10-15 3.00x10-14 3.97x10-2
5.23x10-1
9.94x10-3
1.31x10-1
900
226
1.80x10-15 4.66x10"15 1.91X10*14 3,11x10-2
3,30x10-1
7.78x10-®
8.25x10-2
1.44x10-15 3.54x1 CHS 1.27x10-14 2.48x10-2
2.18x10-1
6.19x10-3
5.45x10-2
800
950 t.000-
263
;
' 2,000
m .™ 408
’ 8 2 9 ....
2,500
1,220
3.000
1,590
3.500 ' z m s p t 4.000 2,180 4.500
2,430 ..
5.000
2,690
6,000
3,200
7.000
3,750
6.84x10-15 1.99x1!*!
•1.17x10-15
2.59x10“ '5 7.69x10-3
—
‘ iVi:if'.i■>.!v‘ i.:.■iv it*!:: —
Y>Y"AS <Wi'':. •.'f<'n'\<"< ..... ■: '
—
rx ::“‘0'v
—
__
8.000
'rrri. \ : *:
—
III
— -
1^95x10"2
—
—
i
i
i
Y,~.krd±
■■ r-t-.;'1
—
i g
15,000
9,600
20,000
14,600
—
—
—
—
20,184
14,600
—
—
—
—
— ■— ■
i i i S I
®
35.000 ■ 1 1 ® ! : ’ 35,786 37,300
■i: :j:
p
S
-I
; ■ i-!:!
■.'■■■•■: : —
—
See Front of Table for Formulas and Sources.
—
—
—
—
—
—
IS B P 3 m m m
j:1!;.
i:-: —
—
,:i' ! ! ‘V !‘! r'.A:-..i-.ti.ii-.". "
l f
25.000 30.000
—
.r.'i -VJ-.Ji • -.1. \ t 'm •J.::i - V::^;e«i'*■'i "• '
w.,wsr:-,..\.-,rr,.r.' B
_
i
9.000 10.000
I lS lE
H S ll:;
— ■
l i l i :
1.92x10-3
2.30x10 -1® 5.21x10-1S 1.22x10--5 3.68x10-3 — — — —
iiii
:v;i i —
i
::i.!
: —
:i
Earth Satellite Parameters 33
34
35
36
37
38
39
40
ORBIT DECAY RATE ESTIMATED ORBIT LIFETIME Solar Min Solar Max Solar Min Solar Max Solar Min Solar Max Solar Min Solar Max 50 kg/ m2 50 kg/ m2 200 kg/ m2 200 kg/ m2 50 kg/m2 50 kg/m2 200 kg/ m2 200 kg/ m2 (kmTyr) (km/yr) (km/yr) (km/yr) (days) (days) (days) (days) 3.82x1013 3.82x1013 9.55x1012 9.55x1012
0.00
0.00
I H S f I l S S S I I P S 6:" l ! S S ! f f f p f ? ! 1 5.30x10* 6.58xt04 M i s S S illi'lS ..... § £ ■ £ . 5.75x10s
0.00
0.00
0.06 tl$l i
0.06
Alt (km) 0
0.48 ■5.99 yj-lilk
i l l
1.09x103
3.45x103
2.72X102
7.99x102
10.06 '
3.82
40.21
3.6 14.98
2.67x102
1.29x103
6.67x101
2.95x102
49.9
11.0
196.7
49.2
7.64x101
5.44X102
1.91x101
1.23X102
195.6
30.9
2.40x101
2.43x102
6.01x10°
5.50x101
552.2
8.12x10°
1.19X102
2.03x1 go ■ 2.60*101
2.97x100
5.95x101
7 4?x10~1
1.28*101 ’
1.20x10°
3.07x101
3.01x10-1
6.53x10°
1,638
801
5,470
4,775
550"
5.60x10-1
1.63x101
1,40x10-1
3.41X10°
2,580
3,430
14,100
13,400
600
3.05X10-1 8.83x10°
1.83X10°
5,560
4,550
28,500
27,900
650
4 92x10°
7.64X10-2 4.81x10-2
12,800
53,400
^52,700 ■
2.82x1 (jt>
3.36x10-2
5.67x10-1
|^ 3 0 0 :;|!:
98,500
41,000
175,200
6.12x10-2 6.49x10-1 1.53x10-2 1.26x10-1 4.92x10-2 4.33x10"1 1.23x10-2 8.26x10-2
76,600 " 76,200 127,000 128,000 21,1000 210,000
307,400 521,000 853,000
306,700 520,000
4:ddki^2':;; 3.03x10-1 9.99x10-3 5.70xiqr2
341.000
1,361,000
1,362,000
1.92x10“ ’
1.00x10°' 42,000 .
1.93X10-2 1.99x10-1
.
340,000
140.3 £ 346.9
1,497
iy
7 2 4
250 300 350
.
5 i S K ; :; w M B :
3.310
I
■ iv M H rim .j!.:;;,;
; 97,700
111 L-' Th'i'J Id!;!.: :'1 N ‘
174,200
852,000
2§K S 850 900 950
1,700,000 1,700,000 6,800,000
1.62x1
4.B10.000 4.810,000 19,250,000 19,250,000
^ ■ ^ S O O ': ':
' 2,000
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
2,500
—
—
—
—
—
—
—
—
3,000
_ ,. . . . . .
.24,400
615.9 1024.5 2.S77
ite :* :
3.30x10-1.
1.66x10° 7.74x10"2 1.02x10°
S lte i := n
.
i
U '.i'ii; ■i';
,,k,
"
. " .jh
1 1 1 1 1 1
—
—
_ I# :!
L-sJ:
—
;
■—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
— i; iTr.....
— ■
'E ' ' 8 :S ;' —
—
—
li' ■ — —
_
— —
—
— — —
—
— — —
S S il5 — ■■■ — — — .........
:: —
:v —
—
— —
i S S S i!,:j
—' —
—
See Front of Table for Formulas and Sources.
—
— —
—
— —
— — —
■■S —
—
■ — —
7,000
—-
•'y1
— —
”5,000 6,000
—
# '
; !ib|ocfi|
15.000 20,000 20,184
25,000 ~
.'ilW Pp:—
35,786
Earth Satellite Parameters
See Front of Table for Formulas and Sources.
Earth Satellite Parameters
48
49
52
51
50
53
54
55
Inc = 120.0
Inc = 150.0
Inc 180.0
4.480
EULER ROTATION RATE (deg/min)
Inertial Rot. Rate (deg/min)
Inc = 0.0
4.261
Inc = 30.0
Inc = 90.0
4.046
4.141
4.268
4.392
:: 3.948- ’
' 4:043- '
4.170'
g $ 2 9 3 ;|
3.884
3.900
3.995
4.123
4.246
3.817
3.053
3.949
4.076 .
4.010 ■■ 3.912
4.068
Inc = 60.0
Alt
4.512
0
4.413
,';io o
4.334'
4.366
V iS b - Y
4.287
4.319
3.771 ‘
3.807
3^903
4.030
4.152
4.241
4.273
3.977
3.726
3.702
3.858
3.985
4.108
4.196
4.228
300
3.933
3.682
3.718
3.814
3.941
4.064
4.153
4.183
350
3.770
3.897
4.021
3.855
3.978
4.066
4.097
3.813
3.936
. 4.024.
■ 4.056 “ 4 .014”
4.022
'
V; ;a839’;:T'
;3.639'x' : 'S-674.:;::.: 3.554
. 3;632 3.590
250
4.140 S S ;!!
3.764
... 3.513....
3.549
3.645.... ....3.772....
3.895
3.983
3.723
3.473
3.508
3.605
3.732
3.855
3.942
3.974
600
3.684
3.433
3.469
3.565
3.692
3.815
3.903
3.934
650
3.526 '
/"';."3i64S 3.356
!
. 3.488'
3-392.
3.532
. .. .3.318 . ......-*i 3.281
3.317
i i 3.413
3.496
3.245
3.281
3.460
3.209
' :|1 0 $ ; V
S
3.653
3.776
3.864
'3.615
3.738
3.826
3.857
3701
'
550
3.820
3.541
3.664
3.751
3.783
850
3.377
3.505
3.627
3.715
3.746
900
3.245
3.342
3.469
3.592
3.679
3.711
3.210
3.306
' 3.4-34
3.043 ■
3.140
3.267
2.890
2.986
3.114 ... 2 041
3 644 3.390 ' ruijfi.iiin.jiStMi ij|i•i 3.237
!!:i!-" ‘.sv.l!".J! '!,'!
3.477
950 f
1,000
P ^ fi
ilS ! S !
3.355
'1,500
2.964
3.050
3.081
2,000
2.830
2.580
2.616
2.714
2.595
2.344
2.381
2.479
2.607
2.729
2.814
2.845
2,500
2.390
2.139
2.176
2.275
2.403
2.525
2.610
2.641
3,000
2.346
2.431
2.461
O S S S ''■
2.068
2.189
2.273
2.304
4,000
: ^ il|.8 b iij: 1.677
1.929'
2.050
2.164
4.500 5,000
1.998 ■!' 1.-S02 i :.gie2: 1.913 V
' 1.700
.... 1788....
1.538
1.576
1.806
1.926
2.009
2.039
1.576
1.325
1.365
1.467
1.596
1.715
1.797
1.827
6,000
1.403
1.152
1.192
1.296
1.425
1.543
1.625
1.653
7,000
1.284
1.401
■ 8,000
1.259
^ 7 0.087 0.823
"
ji:
1.1 G5
': 0.936
1.065
1.481
J:: 1.510
■;i.3 s i
1.389
1.259
'1.286
9,000 10,000
0.694
0.444
0.493
0.609
0.738
0.848
0.920
0.945
15,000
0.507
0.256
0.315
0.439
0.565
0.668
0.734
0.757
20,000
0.501
0.251
0.311
0.434
0.663
0.729
0.752
: 0.313
0,401:.^!
:;^:0ip62,:.:;;: ::9;p07:^ f^o fi:3 2 ;*:{
0.251
0.561 V!:;'0^0;.
0.390
0.000
0.130
i.v d Sfed.o!:;; 0.469
,:!i,:o .s 4 $ v l . 0.564
§ 0 ;3 8 f:ih ! ....0.251......
0.355
See Front of Table for Formulas and Sources.
0.434
20,184 25,000
0.484
30,000;
0.509
35,000
0.000
35,786
Earth Satellite Parameters 56
57
58
59
60
61
62
63
AV Req’d fo r a 1 km Alt Chg (m/s)
Max Ang Rate from Gnd Stn (deg/s)
Sun Syn chronous Inclination (deg)
VELOCITY-RELATED PARAMETERS Orbit Circular Angular Velocity velocity (km/s) (deg/min) 4.261 7.905
Alt (Km) 0 100
7.W4
Escape Velocity (km/s)
AV Req’d to Deorbit (m/s)
Plane Change AV (m/s)/deg
11.180
—
137.97
0.62
—
95.68
4.163
.11,093'
■15.2
136.90
0.61
449
96.00
0 60
2.98
96.16
150
7.814
4.115
■11051
-30.2
■136.38
200
7.784
4.068
-45.0
135.86
250
7.755
4.022
11.009 .. 10.967
-59.6
135.35
0.58
1.78
300
7.726
3.977
10.926
-74.0
134.84
0.58
1.48
96.67
350
7.697
3.933
10.885
-88.3
134.34
0.57
1.26
96,85
400
7.669
3.889
10.845
' -102.3
• 133.84
0.57
450 ■
7640
3.847
10.805'
—116.2
7.613
3.805
10.766
-129.8
'!
2.23
.
■
1.10
0.66
0.07
;; 132.86
0.55
0.87
1 3 3 .3 6 '
96.33 ' 96.501' '
f s W W 'j § ;J S '; i
550
7.585
3.764
10.727
' -1 4 a 3
132.38
0.55
0.79
97.59
eoo
7.558
3.723
10.688
-156.7
131.91
0.54
0.72
97.79
650
7.531
3,684
10.650
-169.8
131,44
0.54
0.66
97.99
700. •;;P:75iiP;i:!
7.504
3.645
10.613
-182.8
130.97
0.53
0.61
7.478
3 606
10.575
■ -195.6-
130.51
0,52
7.452
3.569
10.538
130.06
0.52
850
7.426
3.532
10.502
-208.3 \hm 'iI.mmm,,;:;,w.tf/.i't.u. -220.8
129.61
0.51
0.50
900
7.400
3.496
10.466
-233.1
129.16
0.51
0.47
99.03
950
7.375
3.460
10.430 1
-245.3
128.72
0.50
0.44
99.25
‘ 7.35G
3.425
10.395
-257.4
128.28
3,258
10.223
-315.4'
126.16
0.47
3.104
10.059
■ -370.1
■ 124.14
0.45
0.27'
. 1,500
M
98.82
0.42V ’ ipo;66 ' 101.96
2,000
6.898
2.830
9.755
-470.2
120.38
0.41
0.20
104.89
2,500
6.701
2.595
9.476
-559.6
116.94
0.38
0.15
108.35
6.519
2.390
9.220
-639.8
113.78
0.35
0.12
112.41
6.352
2.211
0.32
o !io
3,000
... 4^5Q(j
6.197
8.764
-777.0
6.053 .
8.561
-836.0
' ,r,i..108.16 i'i:.V : i^| 1:05^65;.;:::: -
0.30
0.09
0:28
0.08
5,000
5.919
1.788
8.370
-889.5
103,30
0.26
...... 0.07
6,000
5.675
1.576
8.025
-982.8
99.04
0.23
0.05
—
7,000
5.458
1.403
7.719
-1,060.8
95.27
0.20
0.04
—
8,000
5.265
1.259
7.446
' 9,000
10,000
0.04
■ 91.89
...
:
I
: l 4.933
1:035;
:!■!
15,000
4.318
0.694
6.107
-1,381.9
75.36
0.10
..... 0.02
20,000
3.887
0.507
5.497
-1,453.8
67.85
0.07
20,184
3.874
0.501
5.478
-1.455.5
67.61
0.07
0.01 0.01
5.040
-1,485.7 ~
•0.390
25,000 30,000
54.17
: : 4.389 3.075
0.251
4.348
0.06
I
-1,493.2
See Front of Table for Formulas and Sources.
53.66
0,01 0.04
— — —
°-01
57.77 '
4^681
35,000. 35,786
138.59
0.00.....
i l l . ® W :s. p i —
Earth Satellite Parameters
64
65
66
67
68
69
70
71
Node Spacing (deg)
Node Precession / Day (deg/day)
GENERAL PARAMETERS Angular Radius of the Earth (deg)
1 deg FOV on Earth’s Surface (km)
Range to Horizon (km)
90.00
0.00
0
84.49
42.24
Revo lutions per Day <#) 17.00
21.18
-9.96 cos /
0
79.92
1.75
1,134
86.48
38.40
16.61
21.68
-9.44 cos i
100
Period (min)
Max Eclipse (min)
Alt (km)
77.69
2.62
1,391
87.49
37.76
16.41
21.93
-9.19 cos i
150
75.84
3.49
1,610
88.49
37.28
16.23
22.18
-8.94 cos i
200
74.21
4.36
1,803
89.50
36.90
16.04
22.44
-8.71 cos /
250
72.76
5.24
1,979
90.52
36.59
15.86
22.69
-8.48 cos i
300
71.44
6.11
2,142
91.54
36.33
15.69
22.95
-8.26 cos i
350
70.22
6.98
2,294
92.56
36.11
15.51
23.20
-8.05 COS /
400
69.08
7.85
2,438
93.59
35.92
15.34
23.46
-7.85 COS /
450
68.02
8.73
2,575
94.62
35.75
15.18
23.72
-7.65 COS /
500
67.02
9.60
2,705
95.65
35.61
15.01
23.98
-7.46 cos /
550
66.07
10.47
2,831
96.69
35.49
14.85
24.24
-7.27 cos /
600
65.16
11.34
2,952
97.73
35.38
14.69
24.50
-7.09 cos /
650
64.30
12.22
3,069
98.77
35.29
14.54
24.76
-6.92 cos /
700
63.48
13.09
3,183
99.82
35.20
14.39
25.02
-6.75 COS (
750
62.69
13.96
3,293
100.87
35.13
14.24
25.29
-6.59 cos /
800
61.93
14.84
3,401
101.93
35.07
14.09
25.55
-6.43 cos /
850
61.20
15.71
3,506
102.99
35.02
13.94
25.82
-6.28 cos /'
900
60.50
16.58
3,608
104.05
34.97
13.80
26.08
-6.13 cos i
950
59.82
17.45
3,709
105.12
34.94
13.66
26.35
-5.98 COS /
1,000
56.73
21.82
4,184
110.51
34.83
13.00
27.70
-5.33 cos /
1,250
54.06
26.18
4,624
115.98
34.83
12.38
29.08
-4.76 COS /
1,500
49.58
34.91
5,433
127.20
35.03
11.29
31.89
-3.84 cos i
2,000
45.92
43.64
6,176
138.75
35.40
,10.35
34.78
-3.13 cos /
2,500
42.85
52.36
6,875
150.64
35.86
9.53
37.76
-2.58 cos /
3,000
40.22
61.09
7,543
162.84
36.38
8.82
40.82
-2.16 cos /
3,500
37.92
69.82
8,187
175.36
36.94
8.19
43.96
-1.81 cos /'
4,000
35.90
78.54
8,812
188.19
37.53
7.63
47.18
-1.54 cos i
4,500
34.09
87.27
9,422
201.31
38.13
7.13
50.47
-1.31 cos /
5,000
31.02
104.73
10,608
228.42
39.36
6.29
57.26
-0.98 cos i
6,000
28.47
122.18
11,760
256.66
40.60
560
64.34
-0.75 cos i
7,000 8,000
26.33
139.64
12,886
285.97
41.84
5.02
71.69
-0.58 cos i
24.50
157.10
13,993
316.31
43.06
4.54
79.29
-0.46 cos /'
9,000
22.92
174.55
15,085
347.66
44.27
4.13
87.15
-0.37 cos /
10,000
17.36
261.85
20,405
518.46
50.00
2.77
129.97
-0.14 cos /
15,000
13.99
349.16
25,595
710.60
55.24
2.02
178.14
-0.07 cos /
20,000
13.89
352.37
25,785
718.05
55.42
2.00
180.00
-0.07 cos /'
20,184
11.73
436.49
30,723
921.94
60.07
1.56
231.11
-0.04 cos i
25,000
10.10
523.85
35,815
1,150.85
64.56
1.25
288.50
-0.02 cos /'
30,000
8.87
611.24
40,884
1,396.10
68.77
1.03
349.98
-0.01 cos /
35,000
8.70
624.98
41,679
1,436.07
69.41
1.00
360.00
-0.01 cos /
35,786
See Front of Table for Formulas and Sources.
