r
orld Scientific Monograph Series Mathematics - Vol. 3
Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points
T World Scientific
Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points
World Scientific Monograph Series in Mathematics Eds.
Ron Donagi (University of Pennsylvania), Rafael de la Llave (University of Texas) and Mikhail Shubin (Northeastern University)
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Almgren's Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2 Eds. V. Scheffer and J. E. Taylor
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Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points by G. Gomez, J. Llibre, R. Martinez and C. Simo
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Dynamics and Mission Design Near Libration Points Vol. Ill Advanced Methods for Collinear Points by G. Gomez, A. Jorba, J. Masdemont and C. Simo
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Dynamics and Mission Design Near Libration Points Vol. IV Advanced Methods for Triangular Points by G. Gomez, A. Jorba, J. Masdemont and C. Simo
World Scientific Monograph Series in Mathematics - Vol. 3
Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points
G. Gomez & C. Simo Departament de Matemdtica Aplicada i Analisi Universitat de Barcelona, Spain
J. Llibre & R. Martinez Departament de Matematiques Universitat Autbnoma de Barcelona, Spain
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Library of Congress Cataloging-in-Publication Data Dynamics and mission design near libration points / Gerard G6mez ... [et al.]. p. cm. — (World Scientific monograph series in mathematics ; vol. 3) Includes bibliographical references. Contents: - v. 2. Fundamentals : the case of triangular libration points ISBN 9810242743 (v. 2 : alk. paper) 1. Three-body problem. 2. Lagrangian points. I. G6mez, Gerard. II. World Scientific monograph series in mathematics ; v. 3. QB362.T5 D96 2000 52r.3--dc21
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Preface
It is well-known that the restricted three-body problem has triangular equilibrium points. Those points are linearly stable for values of the mass parameter, /i, below the Routh's critical value (ii. It is also known that in the spatial case it is nonlinearly stable, not for all the initial conditions in a neighborhood of the equilibrium points £4, I/5 but for a set of relatively big measure. The fraction of stable motions tends to 1 when the size of the neighborhood tends to 0, for almost all fi £ (0,^i). This follows from the celebrated Kolmogorov-Arnold-Moser theorem. In fact there are neighborhoods of computable size for which one obtains "practical stability" in the sense that the massless particle remains close to the equilibrium point for a big time interval (some millions of years, say). The question is which part of this stability subsists when the idealized RTBP is substituted by the Earth-Moon system with its real motion and under the very strong influence of the Sun and the milder perturbations due to planets, solar radiation pressure, no spherical shape of the Earth and Moon, etc. According to the literature, what has been done in the problem follows two approaches: a) Numerical simulations of more or less accurate models of the real solar system. Usually the starting point is taken at one of the equilibrium points Li, L5. The results are slightly confusing. Depending on the initial epoch chosen, the orbit escapes in a few months or behaves according to the pattern that we proceed to describe. First the particle spirals from the equilibrium point outwards until it reaches a size of the order of magnitude of the Earth-Moon distance. Then the particle spirals inwards going close again to the equilibrium point. The behavior repeats itself several times or, eventually, escapes after some of these big oscillations, when a closed encounter with one of the primaries is produced. b) Study of periodic or quasi-periodic orbits of some much simpler problem. This can be the bicircular model or a coherent system close to the bicircular one and still periodic or a Hamiltonian system retaining a few leading terms
VI
Preface
of the equations. In this case, the methods of perturbation theory, mainly those based on Lie series, lead to much simpler auxiliary Hamiltonians that can be studied analytically. The other cases can be studied in turn numerically or semianalytically. The results are again confusing: small changes in the approach produce big changes in the size of the periodic orbits or they even disappear. This is a consequence of the lack of convergence of the methods used and of the sensitivity to some resonances. The concrete questions that are studied in this book are: a) Is there some orbit of the real solar system which looks like the periodic orbits of the previous item b) ? That is, are there orbits performing revolutions around L4 covering, eventually a thick strip ? Furthermore we would be pleased if those orbits be quasi-periodic, at least if the motion of the bodies of the solar system is assumed to be quasi-periodic with respect to time. The present knowledge of the motion of the main bodies of the solar system ensures that this assumption can be accepted for moderate time intervals, much larger than the ones for present planned missions. However there is no guarantee that the orbits we look for exist nor they be quasi-periodic. b) If the orbit of a) exists and two particles (spacecrafts) are put close to it, how does the mutual distance and orientation change with time ? As a final conclusion of the work, there is evidence that orbits moving in a somewhat big annulus around L\ and L5 exist, that these orbits have small components out of the plane of the Earth-Moon system, and that they are at most mildly unstable. The mutual distance of two points starting close to these orbits changes by an important factor (at most 1 to 100), and the orientation changes in a regular way, unless some small loops are present in the projection of the relative motion on the (x, y)-plane or this projection comes too close to the origin. In any case we believe that it can be a useful place to locate one or two spacecrafts for scientific purposes because of the nice properties concerning stability. The station keeping necessary to maintain the orbit in its right place can be reduced to an unimportant amount. The contents of this book is the final report of the study contract that was done for the European Space Agency in 1987. This report is reproduced textually with minor modifications: the detected typing or obvious mistakes have been corrected, some tables have been shortened and references, which appeared as preprints in the report, have been updated. The layout of the (scanned) figures has changed slightly, to accommodate to latex requirements. The last page of this preface reproduces the cover page of the report for the European Space Agency showing, in particular, the original title of the study.
Preface
vii
For the ESA's study we also produced software that is not included here, although all its main modules are described in detail in the text. Updates on the state of the art, both concerning theoretical and practical studies, can be found at the end of Volume IV of this collection of works on Dynamics and Mission Design Near Libration Points.
Preface
STUDY ON ORBITS N E A R T H E T R I A N G U L A R LIBRATION P O I N T S IN T H E PERTURBED RESTRICTED THREE-BODY PROBLEM FINAL R E P O R T
ESOC C O N T R A C T NO.: 6139/84/D/JS(SC) ESOC T E C H N I C A L SUPERVISORS: Dr. W. Flury and Dr. J. Rodn'guezCanabal A U T H O R S : G. Gomez \ J. Llibre 2 , R. Martinez 2 , C. Simo 1
2
3
3
Departament de Matematica Aplicada I, ETSEIB, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. Departament de Matematiques, Facultat de Ciencies, Universitat Autonoma de Barcelona, Bellaterra, 08193 Barcelona, Spain. Departament de Matematica Aplicada i Analisi, Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, 08075 Barcelona, Spain.
C O M P A N Y : Fundacio Empresa i Ciencia, Pomaret 21, 08017 Barcelona, Spain EUROPEAN SPACE AGENCY. CONTRACT REPORT The work described in this report is done under ESA contract. Responsibility for the contents resides in the authors that prepared it
Barcelona, February, 1987
Contents
Preface
v
Chapter 1 Bibliographical Survey 1.1 Equations. The Triangular Equilibrium Points and their Stability . . . . 1.2 Numerical Results for the Motion Around L4 and L5 1.3 Analytical Results for the Motion Around L4 and L5 1.3.1 The Models Used 1.4 Miscellaneous Results 1.4.1 Station Keeping at the Triangular Equilibrium Points 1.4.2 Some Other Results Chapter 2 Periodic Orbits of the Bicircular Problem and Their Stability 2.1 Introduction 2.2 The Equations of the Bicircular Problem 2.3 Periodic Orbits with the Period of the Sun 2.4 The Tools: Numerical Continuation of Periodic Orbits and Analysis of Bifurcations 2.4.1 Numerical Continuation of Periodic Orbits for Nonautonomous and Autonomous Equations 2.4.2 Bifurcations of Periodic Orbits: From the Autonomous to the Nonautonomous Periodic System 2.4.3 Bifurcation for Eigenvalues Equal to One 2.5 The Periodic Orbits Obtained by Triplication
1 1 2 6 6 12 12 12
15 15 16 19 21 21 24 26 28
Chapter 3 Numerical Simulations of the Motion in an Extended Neighborhood of the Triangular Libration Points in the E a r t h M o o n System 33 3.1 Introduction 34 ix
x
3.2 3.3
Contents
Simulations of Motion Starting at the Instantaneous Triangular Points at a Given Epoch 35 Simulations of Motion Starting Near the Planar Periodic Orbit of Kolenkiewicz and Carpenter 35
Chapter 4 The Equations of Motion 4.1 Reference Systems 4.2 The Lagrangian 4.3 The Hamiltonian and the Related Expansions 4.4 Some Useful Expansions 4.5 Fourier Analysis: The Relevant Frequencies and the Related Coefficients 4.6 Concrete Expansions of the Hamiltonian and the Functions 4.7 Simplified Normalized Equations. Tests 4.7.1 Tests of the Simplified Normalized Equations
47 47 48 51 52 54 62 65 66
Chapter 5 Periodic Orbits of Some Intermediate Equations 5.1 Equations of Motion for the Computation of Intermediate Periodic Orbits 5.2 Obtaining the Periodic Orbits Around the Triangular Libration Points for the Intermediate Equations 5.3 Results and Comments
71 71 73 74
Chapter 6 Quasi-periodic Solution of the Global Equations: Semianalytic Approach 87 6.1 The Objective 87 6.2 The Algorithm 88 6.3 The Adequate Set of Relevant Frequencies 90 6.4 Avoiding Secular Terms 94 6.5 The Coefficients Related to the Different Frequencies 94 6.6 Determination of the Coefficients of Quasi-periodic Functions Using FFT 95 6.7 Results and Conclusions 103 Chapter 7 Numerical Determination of Suitable Orbits of the Simplified System 107 7.1 The Objective 107 7.2 Description of Two Families of Algorithms. Reduction of the Linearized Equations 108 7.3 Description of the Methods. Comments 112 7.4 Results and Discussion 116 Chapter 8 Relative Motion of Two Nearby Spacecrafts 8.1 The Selection of Orbits for the Two Spacecrafts 8.2 Variations of the Relative Distance and Orientation. Results 8.3 Comments on the Applicability of the Results
121 121 122 135
Contents
xi
Chapter 9 Summary 9.1 Objectives of the Work 9.2 Contribution to the Solution of the Problem 9.3 Conclusions 9.4 Outlook
137 137 138 140 141
Bibliography
143
Chapter 1
Bibliographical Survey
A bibliographical survey of the most important papers about the motion near L\ and £ 5 is presented. This chapter has four sections. In the first one we give the basic definitions and results on the triangular equilibrium points L4 and L$. The second one is devoted to summarize the known results concerning the numerical simulations of motion around the triangular equilibrium points in the Earth-Moon system for different models of motion for the solar system. The third section deals with the motion at some intermediate distance from the equilibrium point, looking for periodic and quasi-periodic orbits. Finally in the last section some miscellaneous results are summarized. We have not reported in this survey all the papers dealing with analytic computation of periodic orbits around the triangular equilibrium points for the restricted three-body problem, because they are not of practical interest in this case.
1.1
Equations. The Triangular Equilibrium Points and their Stability
As a very rough approximation to the real situation, but which shall be useful later on, we start this section summarizing some properties of the triangular equilibrium points of the restricted, circular and elliptic, three-body problem. In a rotating, barycentric, dimensionless coordinate system with the smaller primary on the positive z-axis, the differential equations of motion for the circular three-dimensional restricted problem are:
x-2y
=
ilx,
y + 2i
=
fly,
z =
ttz,
1
2
Bibliographical
Survey
where 2 r\ r\
= =
n 2
2
(x + fi) +y
r2 2
+ z,
2
(x + fi- l) + y2 + z2.
In the above equations, fi denotes the mass of the smaller primary when the total mass of the primaries has been normalized to unity. It is well-known that the above differential system has five equilibrium points. Three of them, denoted by L\, L-i and L3, are collinear with the primaries and the other two are the triangular equilibrium points of coordinates L4 = (1/2 — fi, \/3/2,0), £5 = (1/2 — fi, — -s/3/2,0). Birkhoff (see [40]) showed that the two triangular points are stable, in the linear sense, for values of the mass ratio in the open interval (0,/ii), where fix = 0.038521.... The global (Lyapunov) stability of these points has been studied by several authors: Leontovich [25], Deprit and Deprit [9], Markeev [26], Rvismann [34]. The final conclusion is that, in the planar case, the two equilibrium points are always stable except for two values of the mass ratio for which they are unstable. These two resonant values are: fi% = 0.024293... and n$ = 0.013516.... For the three-dimensional case the same result holds but now leaving aside not only the values fi2 and fi$, but for a fixed fi a set of initial conditions of Lebesgue measure relatively small (see Markeev [27]). For the elliptic restricted three-body problem, the five equilibrium points are also present, with the same coordinates as in the circular case in an adequate frame of pulsating coordinates. Of course now, due to the nonautonomous character of the equations of motion, the analytic computation of the linear stability is not as easy as it is in the circular case. There are also many papers, both analytic and numerical, devoted to this goal (Danby, Benett, Meire, Tschauner, Deprit et al., Giacaglia, Alfriend and Rand, Nayfeh, Kamel, Vinh, Kinoshita,...). The results can be summarized in Figure 1.1, which in the (fi, e)-plane (where e stands for the eccentricity of the primaries) represents the region of linear stability for these three equilibrium points. Finally, it must be said that in the real world the triangular equilibrium points do not exist at all. They must be redefined since the gravitational forces due to the Sun and the planets will disturb the equilibrium forces acting on a body located at the initial equilibrium point.
1.2
Numerical Results for the Motion Around £4 and Z5
Some rough numerical explorations of stability regions near the triangular equilibrium points of the Earth-Moon system were done by McKenzie and Szebehely [29]. The model that they used was the planar and circular restricted problem of three bodies. They computed the regions about L4 and L5 in which a particle, with zero initial velocity, librates about the equilibrium point. The region in the vicinity of
Numerical
0 . 0 2 h1' H-° O.O6 Fig. 1.1
0.08
Stability chart for the equilateral equilibrium points. T h e stable region is dotted ([40]).
iSIS!!MHIi|{iM^!MMi»!HS Fig. 1.2
3
Results for the Motion Around Li and L5
mtnmmi
Regions of stable initial conditions for motions around L4 (left) and L5 (right) ([29]).
the equilibrium points was divided into grids, with a mesh size of 0.005 nondimensional units (1920 km). The trajectories of particles placed at each node point with zero initial velocity were then integrated for 480 time units (1 time unit = 4.7 days). If the orbit did not touch or cross the z-axis, the initial position was considered to be one giving libration. Their results are presented in Figure 1.2. Gyorgyey [18] studied numerically the regions of stability around the triangular equilibrium point L5 in the elliptic planar restricted problem in a similar way to McKenzie and Szebehely. He showed that with increasing eccentricity, the width of these libration regions is decreasing. More realistic models using restricted four-body problems, which shall be detailed, were considered by Tapley and Lewallen [41], and Tapley and Schutz [42], [43] and [44]. The results of [42] were included and improved in [43], so we shall report them
4
Bibliographical
Survey
explicitly in this survey. In the first three papers the authors considered the following model of the solar system. The Earth and the Moon were assumed to move in circular orbits about their mutual mass center. The mass center turns in a circular orbit about the Sun. The Earth-Moon orbit plane was taken with an inclination of 5°9' with respect to the ecliptic. It must be said that when the effects of the Sun are included, the system is nonautonomous and L 4 and L5 are no longer equilibrium points. They are redefined as those points where a particle at rest in a synodical system will have zero acceleration when only the actions of the Earth and the Moon are considered. The first paper [41], studies the motion of a satellite placed at the libration points with zero velocity when the Sun is collinear with the Earth and the Moon, and the Moon is between the Earth and the Sun at the initial epoch. The results obtained for the size of the envelope of motion around L4 are the following: After 8 months 15 months 23 months
Amplitude of x 50 000 km 100 000 km 240 000 km
Amplitude of y 30 000 km 60 000 km 125 000 km
In the z direction they found a near periodic motion (but with increasing amplitude) of period 27.6 days. The amplitude of the motion after 8 months was of 6 000 km. Similar results were obtained for L5. When solar radiation pressure was included, the amplitude of the motion was bigger. The paper includes estimations about the impulse required for forcing a vehicle to remain precisely at L4. For one year 750 m/s are required. Forcing the vehicle to remain at a point near the equilibrium increases the total amount of fuel. For the same model of motion of the primaries as in [41], Feldt and Schulman [13], extended the interval of integration to a larger one. They showed that the envelope of the motion of the spacecraft with the above initial conditions expands up to 270000 km approximately and contracts to some 9 000 km from L 4 . The period of the pulsating motion was about 1 500 days. This was already noticed by Tapley and Schutz in [42]. In [43] Tapley and Schutz showed that after 3 900 days approximately, the spacecraft left a libration-point-centered motion as the result of a near-lunar encounter. The displacement at 5 000 days for this case was over 94 million kilometers from the L4 point, indicating that the spacecraft had escaped from the Earth-Moon system. Then, the problem is to determine the initial angle between the Sun-Earth and Earth-Moon lines so as to minimize the maximum displacement from L4. The same question has already been studied by Wolaver [51] but for a planar model for the motion of the primaries. This study is sensitive to the constant values adopted for the model. In summary, with the choice of constants used by the authors, a libration-point-centered motion continues for a period of at least 8 000 days, and
Numerical Results for the Motion Around L\ and L5
5
the nature of the motion indicates that it may persist for a much longer period. The fourth paper [44] is similar to the above mentioned ones. In it, the equations of motion of the restricted problem of four bodies were numerically integrated using the JPL Ephemeris Tapes DE3 to provide the position of the three primaries: Earth, Moon and Sun. Using initial conditions for the particle which satisfied the elliptic restricted problem of three bodies (this means that, if the effects of the Sun were neglected, the Earth and the Moon would move in elliptic orbits and the particle would have the proper velocity to maintain the equilateral triangle configuration) , the numerical results of the restricted problem of four bodies showed that a particle placed at L5 on Julian Ephemeris Date 2 439 796.735 will follow a libration point centered motion for 2 500 days. The envelope of the particle's motion about L5 expands and contracts with a period of approximately 650 days. Additional computations show that the motion will persist for a period in excess of 5 000 days. Using the same initial epoch, but starting at the geometrical L4 point and with initial velocity determined in the elliptic restricted three-body problem approximation, a near lunar encounter occurs after 579 days. This encounter causes a sudden change of motion. Similar results were obtained when the velocity of the particle was set equal to zero. The Julian Date mentioned was chosen in order to have the Earth's center relatively close (less than one Earth radius) to the Sun-Moon line. In summary, many papers deal with numerical computations about the motion of a particle in a (big) vicinity of L\ or L5 under the influence of the Earth, Moon and Sun (and in some cases, all the solar system [17]). All the simulations seem to confirm the lack of stability of the motion very close to L\ and L5. However, some simulations starting at L4 or L5 have a pulsating character (concerning average distance to the equilibrium point) with a long period. The simulations show a strong sensitivity with respect to the model of the solar system and to the initial epoch as well as to initial conditions. Possible approaches to the Moon after one year are found in several cases. Our own simulations show motions, which shall be explained in detail in Chapter 3, for which the particle is confined to less than 0.8 Earth-Moon distance for 7000 days. For instance, the one starting at the instantaneous L5 point of the EarthMoon system the day 16 000 after 1950.0, when the full solar system is used. The simulations also show that a stable periodic orbit exists for some nearby problem (the periodic orbit found by Schechter, Kolenkiewicz and Carpenter, Kamel and Breakwell, Wiesel, etc.; see the next section) because for some months the motion looks like a quasi-periodic motion around a periodic orbit. However, after a somewhat long period the vicinity of this orbit is left. Starting with the initial conditions given by Wiesel or Kolenkiewicz and Carpenter some evidence of this quasi-periodic motion, for several months, is found. Later, an escape is produced. Results concerning periodic orbits of interest for the problem will be reported in the next section.
6
1.3
Bibliographical
Survey
Analytical Results for the Motion Around L4 and i 5
When we study the motion of a spacecraft near the triangular equilibrium points in the Earth-Moon system, the first question that arises is the adopted model of motion used for the bodies producing the field of forces acting on the infinitesimal one. Some intermediate "mathematical" models, shall be reported here due to its usefulness from a practical point of view. Of course other technical questions, as for example the adopted theory for the motion of the Moon (de Pontecoulant's, HillBrown's,. ..) can be of importance in order to explain quantitative discrepancies between similar results. This is especially true if we take into account the sensitivity of the problem, already displayed in the preceding section, but shall not be explained here in detail. Once the equations have been obtained, the second question deals with the tools to be used to analyze the problem (literal expansions of the equations, perturbation theories,...). Of course these tools are closely related to the kind of searched result, i.e.: the qualitative picture of the phase space near the equilibrium (displaying its periodic orbits), the computation of some particular trajectories, etc. In this section we shall report the results found in the literature concerning these topics.
1.3.1
The Models
Used
It has been conventional in treating this problem to use a very restricted four-body problem, or bicircular problem, in which two masses move in circular orbits about their barycenter, while the barycenter also describes a circular path about the third mass. This model has been widely used in the literature ([46], [38], [40], [51]). Nevertheless its conclusions are not valid for applications, mainly because it does not take into account the indirect influence between the primaries. Mohn and Kevorkian [31] developed the following model: the motion of the three most relevant primaries (Earth, Moon and Sun) is given in terms of asymptotic solutions of the RTBP, for the limiting case where the particle (Moon) remains close to one primary (Earth). They use for this goal both Hill's and de Pontecoulant's lunar theories. As the distances of the Moon and the spacecraft (if it moves near the Lagrangian points) are both 0(y}l3), the variables (including the independent one) are scaled adequately. The final asymptotic equations include the leading terms of the solar perturbation in a dynamically consistent manner. As a final remark we must say that we have not seen any applications of these equations in any other paper. Having in mind the re-computation of the periodic orbits obtained by several other authors, Wiesel [49] produced a model in which the periodic orbits (of the adequate period) known for the genuine RTBP can be analytically continued into periodic orbits of a restricted four-body problem. The major idealization, according
7
Analytical Results for the Motion Around L4 and L$
to the author, is that it neglects the eccentricity of the Sun's orbit. Going from one model to the other can be done roughly in the following steps: (1) Add eccentricity to Moon's orbit (elliptic RTBP). (2) Include the Sun and, optionally, some terms of the lunar motion, to make it coherent. Then, we obtain a restricted four-body problem with the eccentricities of the Moon and the Sun as perturbations. Avoiding the first step, for the moment, in the usual system of units of the RTBP and introducing as a constant the mean motion of the Sun, ns, in an adequate set of coordinates and momenta the Hamiltonian of the problem can be written as: ~{PX + Py +Pz) + VPx - XPy - y 2 rPE
rPM
4/3
—-f ^N(ysm6-xcos6) (1 + ms)2'3
+ rPS
where rpE, rpM and rp$ are the distances from the satellite to the Earth, Moon and Sun respectively, and 9 = (1 — n$)t. The first terms of the above Hamiltonian (outside brackets) correspond to the RTBP. For the motion of the Sun, the author considered a circular one given by:
fl + ms\1/3 xs
=
5— 1 ,
cos l
\
n
~
u
n
s)t,
\ 1/3
ys
=
x 1 +2ms - (— —) '
zs
=
0.
( ~ nsK
sin 1
The motion of the Earth and the Moon can be written, in general, as: rE = -firEM,
rM = (1 - fi)rEM-
Then, for TEM three possibilities are considered (1) Circular motion: TEM = (1,0,0) T . (2) The periodic orbit, r]^'M given by Kolenkiewicz and Carpenter [23], and also reproduced by the author in the paper. (3) The periodic orbit given above plus some Floquet modes: TEM = r^lf + 5 3 V — ~ (LiR(*)
sin
QiM + Lu(t) cos QiM)
i=\
in order to take care of coherence. This orbit is given explicitly in Wiesel [48], and it is good enough so that further analytic refinement becomes unnecessary.
8
Bibliographical
Survey
For the purpose of explicit computation of periodic orbits, some intermediate "mathematical models" are introduced by the author. The general idea is to start at some limit problem, embedded in a two-parameter family of problems, and then continue it, by changing the parameters, till a more realistic one. Two possible choices are displayed in the paper: (1) Writing rEM = (l-e)(-l,0,0)T
+
er%(t),
when e = ms = 0, the RTBP is obtained. If e = 1, ms = actual mass of the Sun, then we get a model equivalent to the second one previously mentioned. In this way resonant orbits of the RTBP can be continued, with some difficulty, across e and ms(2) The second procedure is developed in the following way. Writing the bracket term of the Hamiltonian as: H
T
= ^n2s[(l-3cos2
e)x2+6xysmecose+(l-3sm2
e)y2+z2]+0{ms1/3),
the full Hamiltonian can be written in the form:
K(p, q, e) = hp2x+p2+ p2z) + ypx - xpy - ^ where H'g'jj is equal to H™ deleting the 0(ms
- -JL- + tHfM,
' ) term.
For e = 0 we have again the restricted three-body problem, while for e = 1 we have the asymptotic form (e = l,ms = 0) of the standard system. The above Hamiltonian, K, is known as Hill's limit of the problem. In the paper it is shown that the continuation from the restricted problem to Hill's limit is a far better conditioned process than the direct continuation across the (e, ms)-plane. Once Hill's limit has been reached, the continuation up to the actual mass ms becomes easy and can be done generally in a single step. For this Hill's limit problem Figure 1.3 (first computed by Wiesel [49]) shows the evolution of the yo initial condition for some periodic orbits with the e continuation parameter. The infinitesimal solution at L 4 evolves with e to yield a periodic orbit at point B, which is periodic orbit II of Kolenkiewicz and Carpenter [23]. Constructing their other stable orbit in Hill's limit model yields the orbit at point A. Evolving this orbit with e yields their third (unstable) orbit at point C. The complete evolution of the second family has not been determined. Finally, starting the continuation process across the (e, ms)-plane requires the solution of a bifurcation problem in the infinitesimal e regime near the RTBP. For e small, the Hamiltonian K can be written as K — KRP + ei^pert where KRP is the Hamiltonian of the RTBP and K Pert
- u°o,,n " f l s + W
_,\ 1
Mj
\ (x - n)(l + xp) + yyp _ (x + 1 - /i)(l + xp) + yyp ' [ ((i - /i)2 + y 2 + * 2 ) 3 / 2 ((x + 1 - AO2 + y2 + * 2 ) 3 / 2 .
Analytical
Results for the Motion Around L4 and L5
0.85
9
0.9 Vo
Fig. 1.3 Evolution with the continuation parameter e (y-axis) of the initial j/o coordinate (i-axis) for some periodic orbits.
where xp and yp are the periodic components of r^M (t). For the obtained results dealing with the analytic study of the Hamiltonian in the vicinity of the triangular points in the Earth-Moon system, standard perturbation techniques have been proven to be unsuccessful because of the occurrence of small divisors in many terms of the assumed series solution. One of the pioneering works in this direction was the one of Breakwell and Pringle [4]. They analyzed a two-dimensional approach of the motion which took into account the dominant nonlinear resonances by examining only the slowly varying terms. In [4] the restricted problem of three bodies is extended to include direct and indirect influence of the Sun on a particle near the L4 point. Perturbations in the displacements from the Lagrange point up to the fourth order, and comparable solar effects, are included in its model. The von Zeipel method was only carried up to the second approximation to remove short period terms. This work was extended and improved by Schechter [35]. The main idea of [35] is to analyze the stability of slow variations around the periodic orbits found to exist in the problem (one of them stable and the other unstable). For the relations giving the Earth-Moon distance and its angular velocity, a classical simplified model (the eccentricity of the Moon is taken equal to zero) was used. Due to the near resonance ui\ w 3o>2 of the coplanar frequencies (uii = ±0.95459, w2 = 0.29791), fourth order terms in the Taylor expansion of the Hamiltonian were retained. The Hamiltonian is split into two parts H = H° + H1, where H° contains the second and third order terms which are linear and quadratic in position and
10
Bibliographical
Survey
momenta, and the perturbation H1 contains the cubic terms in position (H3) and the terms of global order four {Hi). The solution of the linear homogeneous differential equations, governed by H°, consists of a part depending on six constants a*, Pi, i = 1,2,3 and a forced response. The constants ctj, /?$ are taken as independent variables when H1 is included. Two new sets of canonical variables are introduced, one in order to get a slowly varying Hamiltonian and the other to get a time independent one. Let K(a,P) be the final Hamiltonian. If for small a.\, a2, 0:3, there is a stable motion then, in order to determine long-term effects, it is enough to retain only linear terms in the development of K. For this linear expression we have: 0.02425ai + 0.02412a2 + 0.07899a3 - 0.02563ai cos(2/31), which is of Mathieu type and leads to parametric resonance in the a.\ motion. Since 0.02563 > 0.02425, the motion falls into the unstable region of the Mathieu plane, and therefore no motion can exist for which a.\ remains very small. In order to look for periodic solutions, it is seen that the 0:3 variable does not affect very much the a i , 0.2 ones, so an equilibrium is searched for 03 = 0. Using normal coordinates defined by
Qz = (2 ai ) 1/2 sin ft 1 P = (2a i ) 1 /2 cos/3 . J '
. ,1 9 *- '^
two equilibria are found of the form: Q1=Q2= Q1=Q2=
0, 0,
P1 = 0.1093, Pi = 0.1106,
P2 = 0, P 2 = -0.003675,
the first one linearly stable and the second one unstable. These equilibria, when seen as periodic orbits, are quite unrelated to the periodic orbits found by Wiesel and Kolenkiewicz and Carpenter which shall be described later. In a later paper, Kamel and Breakwell [21] outline, without giving any details of its computations, which are in [20], a higher order two-dimensional theory. The small parameter introduced in order to apply perturbation theories, is the ratio, m = 0.074801, of the sidereal month to the sidereal year. The Hamiltonian is expanded in powers of x, y up to 6th degree and re-scaled and split as follows —-r = H0 + \
—rHq,
9=1
where the "unperturbed" Hamiltonian H0 includes all the quadratic terms in x, y, px, py with constant coefficients. After introducing action-angle variables, a perturbation theory based on Lie series [8] with the practical modifications introduced by Kamel [20] is applied in order to skip short periodic terms. The new Hamiltonian is partially given in the paper. Again, looking for the critical points of this
Analytical
Results for the Motion Around L4 and L$
11
Hamiltonian, both the stable and unstable periodic orbits, in this planar situation, are found. The periodic orbits found in [21] are in good agreement (3%) with the ones computed numerically by Kolenkiewicz and Carpenter [22] using an algorithm which shall be now summarized. The standard equations of motion for Moon, Sun and particle are written in a planar model. Then, for the position of the Moon (and also the Sun) a solution with period equal to the synodic period of the Sun in the Earth-Moon system is searched. It is tried f = (1 + a)f0 + 0w, where a — ^ o ° a% cosk9 + ask sinkO, /? = X^o° @k cosk9 + /?| s'mk8, 6 = (UM — ns)t. fo is the position vector in a fixed reference ellipse (usually taken as a circle) and w = ^dfo/dt. Let us suppose first that the motion of the Sun is circular around the Earth. Starting with a = /3 = 0, substitution into the equations of motion and integration gives a new set a,j3. By iteration, they obtain a coherent motion for the Moon, with suitable period and the Sun as stated. Then, the roles of Sun and Moon are changed. The new model of the Moon is kept fixed and one iteration is done for the a,/3 related to the Sun. By iteration of the procedure a periodic solution of the three-body problem is found (in synodical coordinates). Then the same thing is done for the particle, taking r 0 as a vector advanced 60° with respect to the Moon. Two different orbits are found both stable and roughly equal but with the phase changed in 180°. They are larger than the periodic orbit found by Schechter. The shape is that of a 1 : 2 ellipse with semimajor axis « 145 000 km. Another periodic orbit near the Li point describing two loops in one synodic month is found. It agrees with the unstable orbit found by Schechter. The discrepancies between the theoretical results of Schechter and the numerical ones of Kolenkiewicz and Carpenter was one of the motivations of the more accurate theory developed by Kamel and Breakwell previously mentioned ([19], [21]). Finally we shall mention the results of Wiesel concerning periodic solutions. (1) Starting at e = 0, solving first a bifurcation problem, and by continuation of L4 they reach a periodic orbit: the periodic orbit II of Kolenkiewicz and Carpenter. Bifurcations of this family are not analyzed (mainly the resonance 3:1). (2) Starting at Hill's limit problem, another stable orbit is found. It can be continued till the actual mass of the Sun, ms- Further continuation gives another periodic orbit, much smaller and unstable. The author suggests the following extensions: (1) To propagate periodic orbits from RTBP for model (2) and to analyze the stability using model (3). (2) The continuation of periodic orbits bifurcating from short and long periodic orbits near L4.
12
Bibliographical
1.4
Survey
Miscellaneous Results
1.4.1
Station
Keeping
at the Triangular
Equilibrium
Points
De Fillipi [14] in 1978 studied the station keeping at the libration point L 4 for the Earth-Moon system. He considered a model given by a restricted four-body problem in the space in which the Earth and the Moon describe an elliptic orbit around their center of masses, the barycenter of the Earth-Moon moves in an elliptic orbit about the Sun, and the center of the Sun is the center of masses of the full system. In order to control the motion he introduced a periodic gain matrix as the solution of the adequate Riccati's equation. The results that he obtained are: (1) Using the averaged gain matrix, which makes sense due to its periodic character, the computational effort decreases and the results for the control are not worse. The following table displays this comparison for the AV cost: Final Time Optimal System Averaged System
15 0.064286 0.066905
25 0.103100 0.105890
45 0.17738 0.17635
In the above table, 1 time unit is equal to 4.384 mean solar days and 1 velocity unit is equal to 1023 m/s. (2) The cost of the station keeping decreases when the particle is allowed not to be exactly at the equilibrium but in a neighborhood of it, as it should be. (3) Some initial configurations of the primaries, which are computed in the work, minimize the station keeping. The average thrust variation due to different initial configurations is about a 9%. 1.4.2
Some Other
Results
The study of the libration points in the Earth-Moon system has been parallel to the search of natural or artificial bodies near these points. The first observational evidence was anonymous but reported by Kordylewski in 1961. Two "faint cloudlike satellites" in the neighborhood of L 5 were reported by this author. Later, the discovery of such a "cloud" near L 4 was also reported. Simpson, in 1967, confirmed the existence of these clouds as having diameters of the order of 1 to 5 degrees. But subsequent ground based observations: Roosen (1966,1968), Roosen and Wolff (1969), Bruman (1969), Munroe et al. (1975), Roach (1975), Freitas and Valdes (1980) failed to confirm these results. Olszewski [32], in 1971, studied the region of "practical stability" around the triangular points in the elliptic planar restricted problem. Practical stability means
Miscellaneous
Results
13
that there exists a sufficiently small neighborhood of the equilibrium point, where the solutions of the system remain for sufficiently long time. For practical purposes his conclusions are useless. Kunitsyn and Perezhogin [24] in 1978 studied numerically the motion near the triangular points of the circular planar restricted problem by including the solar radiation pressure. These libration points are shown to be Lyapunov stable in the region where the necessary stability conditions are satisfied. They showed that for all the planets of the Solar system the libration points L4 and L5 are Lyapunov stable. Mignard [30] studied the motion near the triangular points of the circular planar restricted problem by including the solar radiation pressure. The triangular equilibrium points disappear. However libration orbits are not completely destroyed.
This page is intentionally left blank
Chapter 2
Periodic Orbits of the Bicircular Problem and Their Stability
In this chapter we look for low period periodic orbits of the bicircular problem. The periods found are equal to the period of revolution of the Sun in the Earth-Moon synodical system and to three times this period. The period must be a multiple of the period of revolution. The bicircular problem is far from the real one. It is even not coherent, that is, the three massive bodies do not satisfy Newton's equations. However it gives us insight concerning the periodic orbits of a system much closer to the real system, as we shall show in Chapter 5. These last periodic orbits will be the basic ones to build up the final quasi-periodic solutions.
2.1
Introduction
The bicircular problem is a very simplified model for the four-body problem that we are considering. In this model we suppose that the Earth and the Moon are revolving in circular orbits around their center of masses, and the Earth-Moon barycenter moves in a circular orbit around the center of masses of the Sun-EarthMoon system. The first thing to state is that this model is not coherent, i.e., the positions of Sun, Earth and Moon, as we have described, do not satisfy Newton's equations (see [40] p. 288). Despite this fact we can consider the equations of the bicircular problem as an approximation to the true equations of motion. We consider it as a test model and in this chapter we study some of its periodic orbits. Using synodical coordinates with respect to the Earth-Moon system, the equations are periodic with period, Ts, equal to the synodic period of the Sun in the EarthMoon system. In Chapter 5 we will see that there is a similar model with periodic equations obtained by keeping, in the general equations of motion, only the terms independent of time and the periodic terms with period T$15
16
Periodic Orbits of the Bicircular Problem and Their
Fig. 2.1
2.2
Stability
Geometry of the bicircular problem.
The Equations of the Bicircular Problem
Let /i be the mass of the Moon, 1 — // the mass of the Earth and m j the mass of the Sun. Let (X, Y, Z) be the coordinates of a particle of infinitesimal mass with respect to the barycenter of the Earth-Moon system, B. Let the distance from the Earth to the Moon be taken as unity. Then, the distance from B to the Sun is asFigure 2.1 displays the relevant geometry. The coordinates of Earth, Moon and Sun with respect to B are given by XE YE ZE
= /icosf, = /usini, = 0 ,
XM YM ZM
= = =
(/x —l)cost, (fi —1) sin t, 0,
Xs Ys Zs
= as cos nst, — assinnst, = 0,
where ns is the mean motion of the Sun, and o | n | = 1. The coordinates (X, Y, Z) are not referred to an inertial frame. However, if (Xi,Yi,Zi) denote inertial coordinates (with respect to the barycenter of the SunEarth-Moon system), we have v Xi
=
Yi
=
Zi
=
v rns X — ascosnst, 1 + ms Y-— assinnst, 1 + ms Z.
The Equations
of the Bicircular
17
Problem
Hence ms
ir
X
*
Y
— Xi 2~ cosn s r, a ss . v> rn = Yi 2~ sinn s r, as
Z
= Zi,
(2.1)
because msasns 1 + ms
ms a2s
For the inertial acceleration we have ^
{X-XE)(1-IM)
=
(X-XM)H
r<J
T
PE
fi
=
' PE M)
-^(1 ~
'PS
(Y-YM)»
mO
Z-
/rt<J
r
PM
(Y-YE)(l-y)
=
(X-Xs)ms
tv*0
z
fi-
« J
' PM
' PS
*•
'
Zms
mO
nfO
rrtii
1
' PM
' PS
PE
(Y-Ys)ms
„o
where rpE, TPM and rps denote the distance from the particle to Earth, Moon and Sun, respectively. We introduce synodical coordinates with respect to the Earth-Moon system by x
=
X cos t + Y sin t,
y
= -Xsint
+ Ycost.
(2.3)
From (2.1), using (2.2) and (2.3), the equations of the bicircular problem are obtained: x — 2y — x —
3—{x - \i) r
PE
•• -
S
TTIQ
r
TYIQ
+ ass'm6) + -^-sinfl,
r
PM
^-cosO, a
PS
j — V ~ ~~3—V- -3-(y r
Z
LI
— (a; - as cos6) r
PM
1 — 11
y + 2x-y =
3—(x - fi + 1) r
PE
(2.4)
a
PS
S
•*•O~ ^Z _ o ^Z _ H±s_ q Z. /fO
' PE
irt<J
ir"J
'PM
'PS
where 9 = (1 - ns)t and rps, rPM and rPs are computed by means of rPE = {x - fi)2 +y2 + z2,r2PM = {x-Li + l ) 2 +y2+z2, r2PS = (x - xs)2 + {y- ys)2 + z2. In the last formula we have xs = ascos9, ys = - a s s i n # , and as can be computed as as
~
if ms and ns are the basic quantities.
yS s
n
18
Periodic Orbits of the Bicircular Problem and Their
Stability
Defining momenta px = x — y, py = y + x, pz = i , equations (2.4) correspond to the Hamiltonian equations with Hamiltonian function: H
1 = 9 (Px + Py + Pz) + VPx ~ xpy 2 -„ - . ---
m
s rPS
P rpE
TpM
rns Triysmff — xcos6). ai
Equations (2.4) have the apparent symmetries Si : 52 :
{x,y,z,x,y,z,t) (x,y,z,x,y,z,t)
(x,-y,z,-x,y,-z,-t), (x,y,-z,x,y,-z,t),
which can be useful in order to find periodic orbits. The variational equations associated to (2.4) are
A =
(2.5)
DF-A,
with A: A(t) and A(0) = I§, where
f
1
DF
\ 1
0 9n
912
921
922
923
\ 931
932
933
913
1 0 2 0 -2 0 0 0 0 0 )
and gij are used for shortness to denote the following functions (we suppose that we are interested in planar orbits, but the vertical stability is also checked): 3(1 -/x) (x - M ) 2
1-H 3n
1
q
~r
' PE
' PE r
n)y
3fi(x + 1 - fi)y
3ms{x - xs){y - ys)
' PM
' PS
921 1
931
922
1-
' PM
PS
3(1 - n){x913
nY
r
PS
PE
= 0, r
923
' PM
3/J,(X + 1 -
+
Zms{x - xs)2
ms
912
fj,
c
PE 932 - 0, 1-/X
+
3(1 -
' PE
r
PM
3/i^2 _ m s ' PM
'PS
3ms(y - ys)2 'PS
ms
933 ' PE
M
M)2/2
' PM
' PS
As it is well-known, the planar and vertical modes are uncoupled for planar periodic orbits. Hence, if A = (ay), i,j — 1,...,6 and A4 = (a^), i,j = 1,2,4,5
Periodic Orbits with the Period of the Sun
19
we have that (2.5) reduces to 0 0
1
3n
312
521
322
0 -2
0 0
d_ ( a33 dt \ a 63
a 36 \ _ / 0 a66 J V 333
1 \ 0 y
1 2 0 , f a33 V a63
a 36 \ ^66 / '
with A(0) = h, i.e., A4(Q) = h, a 33 (0) = a 66 (0) = 1, a 36 (0) = a 63 (0) = 0. In order to compare results with the ones obtained by Wiesel [49] we have adopted the following values: H = 0.012150298,
ms = 328900.12,
ns = 0.07480132816,
all the other parameters being computed from these three magnitudes.
2.3
Periodic Orbits with the Period of the Sun
We look for planar periodic orbits of system (2.4) that, in synodical coordinates, have the same period as the Sun, i.e., Ts = 27r/(l — ns). From now on we write (2.4) with ms replaced by ems, where e is a parameter ranging from 0 to 1, but we do not change the value of as- For e = Owe have the restricted three-body problem, and for e = 1 we get the bicircular problem. One possibility is to start with periodic orbits of the restricted three-body problem. Unfortunately the short periodic orbits of the family Cs around the equilateral points 1/4,5 lying on a big vicinity of £4,5 (mean radius about 1/2 of the EarthMoon distance) have a period shorter than Ts- The long periodic orbits of Ci in the same vicinity have a period slightly bigger than 2>Ts (see section 2.5). Hence, the only simple orbits with the right period, when e = 0, are the equilibrium points £4,5. From now on we only consider the orbits around L5. As the Earth is placed to the right and the Moon to the left in synodical coordinates, the L5 point has positive y coordinate. (We note that this point is called L4 by Wiesel [49], but we have used the notation of Szebehely [40].) As we look for periodic orbits with a fixed period, Ts, the equation to solve is F{x, y, x,y)
=
$ T s (x, y,x,y)
- (x,y, x,y) = 0,
(2.6)
where $t denotes the flow under time t. The monodromy matrix M = D$TS c a n be computed by integration of the variational equations (2.5). Newton's method is used to solve (2.6). This requires that the matrix M—I be regular, i.e., 1 ^ spec(M). We recall that for e = 0 the eigenvalue 1 of M has, at least, multiplicity equal to
20
Periodic Orbits of the Bicircular Problem and Their
Stability
Fig. 2.2 Evolution with the continuation parameter e of the periodic orbits of the RTBP to the bicircular problem.
2. This means that small values of e can produce troubles. We shall return to this point in section 2.3. The first family numerically computed is the one starting at L5 for e = 0. If e is increased, it is easy to do numerical continuation with big steps (of 0.1 units) in e till 0.8. Then, the step should be reduced because det (M — I) reaches a minimum at e RS 0.877. The values of x against e are given, for this family, in Figure 2.2. The orbits in the family are linearly stable even if we consider the vertical stability, i.e., the six eigenvalues are grouped in 3 pairs of complex conjugate values with modulus 1 and different from ± 1 . Comparing with the results of Wiesel [49] we see that the branch emerging from L5 connects with the orbit called B of Figure 4 in [49] instead of connecting with the orbit called A. If now we start with orbit A of [49] when e = 1 we found a nearly periodic orbit for system (2.4). This orbit is again linearly stable. Decreasing e, the orbit can be continued in a natural way till a value of e = ec close to 0.905. For e = ec a saddle-node bifurcation is found. Two of the complex conjugate eigenvalues of the monodromy matrix for the previous family become equal to the unity. Increasing again e an unstable orbit is found. This can be continued till e = 1 with unstable character. It has a reasonable agreement with the orbit called C by Wiesel. The last family is also shown in Figure 2.2. The fact that even qualitatively our results do not agree with those of [49] can be explained because the models are slightly different and in the neighborhood of
The Tools: Numerical
Continuation
of Periodic Orbits and Analysis
of Bifurcations
21
e = 0.9 the matrix M — I is rather close to singular. Small differences in the model can, then, produce a shift from a connection BC to a connection AC. Concerning stability we must add that in both families, ending at A and B, there are points where a couple of eigenvalues related to planar stability become cubic roots of unity. These points are located, in family A, at e = ei « 0.9351784438, and in family B at e = e2 ~ 0.9254693472, and they are denoted by D and E, respectively. The study of the vertical stability shows that all the periodic orbits found are stable concerning the vertical coordinate.
2.4
2.4.1
The Tools: Numerical Continuation of Periodic Orbits and Analysis of Bifurcations Numerical Continuation of Periodic and Autonomous Equations
Orbits for
Nonautonomous
Let us put equation (2.4) in the form *
=
f(v,t,e),
veR6,
(2.7)
where ems is used instead of ms- The function / is Tg-periodic and we look for orbits having a period equal to kTg, k small positive integer. Let $4 denote the flow of (2.7) under time t, starting at t = 0. If the initial condition is v = VQ at t = 0, then we are asking for a solution of the equation G{v0,e)
=
$kTs(vo,e)-vo
= 0.
(2.8)
For a fixed e, equation (2.8) can be solved by Newton's method. Starting at some approximate value VQ ' we define recursively v{0n+1) = vin) - (DV0*kTs(vin),e)
- Id)-H*kTsUn\e)
- «< n) ).
In this formula DVo$kTs(vo >e) denotes the differential of the final point with respect to the initial conditions. This can be obtained either by numerical differentiation or using the variational equations (2.5). We have used the last approach. The implicit function theorem ensures that no troubles should appear in Newton's method unless 1 6 spec(DVo $kTs (vo >e)) (or if there is some eigenvalue rather close to 1). When 1 is an eigenvalue we are faced with a bifurcation. It is also interesting to study the case of an eigenvalue being an mth root of unity. Then the bifurcated orbits have period equal to mTs. This topic will be studied later on in this section. When vo is obtained for a given e, the parameter e can be changed by some amount Ae. As a first approximation we have G{v0 + Av0,e + Ae) ~ DVoG(v0,e)Av0
+
DtG(v0,e)Ae.
22
Periodic Orbits of the Bicircular Problem and Their
Stability
The differential DVoG is the same as that in Newton's method. The differential DeG can again be obtained either by numerical differentiation or by integration of the variational equations, which in this case can be written as
w = Dvf-w
+ DJ,
weR6,
with w(0) = 0 and where w =
Dt$kTs(v0,e).
In the next section we refer also to some periodic orbits for e = 0, i.e., the RTBP. As this system is autonomous, at least one of the eigenvalues of Dv$kTs (v) ls equal to one (in fact at least two due to the Hamiltonian character). This is due to the fact that we can select as initial point any point in the orbit. Hence, in order to get periodic orbits we proceed in the following way. As we are looking for periodic orbits around the triangular points, we use a Poincare section through y = ±\fi/2. Starting at one point (x0,yo = ± - \ / 3 / 2 , z 0 , x 0 , y 0 , i 0 ) we compute the next intersections of the forward flow with the Poincare surface looking for a point coinciding with the initial one. One should be careful concerning the right number of intersections due to the loops which appear in some orbits. Let -P(uJo) be the image of some initial point Wo under the Poincare map. We suppose that the required time is tj. We ask for Q(w0) = P(w0) - W0 = 0. As the energy is preserved, DP(WQ) has an eigenvalue equal to 1. Therefore, we cannot Use directly Newton's method. One possibility is to keep one of the variables fixed and ask only for 4 components of Q(WQ) be equal to 0, the fifth one must be zero due to the conservation of the energy. However this approach has the drawback of the lack of symmetry and, what is worse, one does not know beforehand if the variable kept fixed during the iteration procedure reaches some extremum and hence, cannot be used as local parameter for the family of periodic orbits. Let P be 4 components of P (for instance Xf,Zf,if,Zf). We ask for Q(W0) - P(W0) - (W0)4 = 0, where (u>o)4 contains 4 components of Wo, exactly the same as those we use in P. We note that Q sends a five-dimensional space into a four-dimensional one. The generic Q = 0 is a curve in the five-dimensional space. Given an approximate Wo we refine it to some WQ + Awo such that 0 = Q(W0 + Aw0) « Q{w0) +
DQ(W0)AW0,
with the additional condition that the norm of Awo (with respect to some metric) be minimum.
The Tools: Numerical
Continuation
of Periodic Orbits and Analysis
of Bifurcations
23
The prediction of a point in the Poincare section corresponding to another orbit of the family is done by differentiation of Q(wo) = 0. We have D^0Q • Aw0 = 0. From this (AiiTo)i _ (AtiJo), Ai
Aj
where (AWo)i means the ith component of AWQ and Ai is the minor obtained from Dw0Q when the ith column is suppressed. Finally, using as a convenient parameter the arclength s we can write d(w0)i ds
Ai
= 1,...,5.
7E¥
An Adams-Bashforth method (with increasing order in the first steps) allows to predict the successive points. The differential of the Poincare map can be obtained numerically or using variational equations. In the last approach let M be the 6 x 6 variational matrix obtained when going from wo to P(WQ)- As the time is not fixed and the initial and final values of y are kept fixed we have ( Ax 0 \ 0 Az 0 M Ai0 Ay0
(
x \
( - \ 0
y
z
+
At
X
(2.9)
=
y
V Ai0 J
S
\
\- J
I t=t,
Hence At =
y(tf)
(m 2 i AZQ + m2zAzQ + m2iAx0
+ m2hAy0 + m2&Az0).
The five components left free on the right-hand side of (2.9) should be equal to ( Az 0 \ A^o DP{w0)
A±0 Ayo
V Ai0 J Let DP(w0) = (riij), i,j = l,...,5. n
u
=TOU
h
-m2i-
nu = mi+1,1 - m 2 i - ^ — ,
Then we have, by substitution of At in (2.9)
n\j = m-ij+i
h
- m2,j+i:
riij =mi+1j+i-m2j+i^j—,
2,...,5,
i,j =
2,...,5,
24
Periodic Orbits of the Bicircular Problem and Their
where f1: f2, / 3 , fi, h and f6 denote x(tf), respectively. 2.4.2
y(tf),
z(tf),
Stability
x(tf),
y(tf)
and
Bifurcations of Periodic Orbits: From the Autonomous the Nonautonomous Periodic System
z{tf),
to
Now we turn to the problem of bifurcation. We will consider two different kinds of problems: (1) How a periodic orbit of an autonomous system can be continued when a periodic perturbation is included. (2) How to detect the branches produced when one of the eigenvalues (of a nonautonomous system) becomes equal to the unity. For the first one we consider the system x =
f{x)+eg(x,t),
with g T-periodic with respect to t and x £ E 4 , as a model of the perturbed planar RTBP. Let xo(t) be a periodic solution of x = f(x) with period T. We wish to know if the periodic orbit can be continued for small e and which one is the right initial point (in the autonomous system every point in xo(t) can be taken as the initial point for t — 0). Proceeding to the first order in e we put x = XQ + exi and the equation for X\ is xx{t)
=
D/(x„(t))3;i+ 9 (x 0 (<),().
(2.10)
Let A(t) be the solution of the variational equations A = Df(x0(t))A,
A(0)=ld.
Then, the solution of (2.10) is x1(t)=A(t){x1{0)+yi{t)),
where y\ (t) satisfies yi(t)=A-1ff)
j/i(0) = 0.
The condition for periodicity, up to the first order in e, can be written as x0(0) + ex, (0) = x0(T) + eA(T) [Vl (T) + xx (0)] and from the T-periodicity of xo we obtain (A(T) - Id)Xl(0)
=-A(T)yi(T).
(2.11)
The matrix A(T) has the eigenvalue 1 with multiplicity at least equal to 2. Let us suppose that the other couple of eigenvalues are a ± i/3. Let {ei,e2,e3,e4} be a Jordan basis of A(T), i.e., ei, e 2 belong to the kernel of A(T) and they can be
The Tools: Numerical
Continuation
of Periodic Orbits and Analysis
of Bifurcations
25
chosen orthogonal. We take as e\ the tangent vector to the orbit at the initial point. The vectors e 3 , e 4 satisfy A(T)e3 = ae3 - /3e4, A(T)e4 — fie3 + a e 4 . In this basis A(T) — Id can be expressed as / 0 0 0
7 0 0
V0 0
0 0 a-l
-p
0 0 /3
\
a-l
j
Let ai,... , 0 4 , 6 1 , . . . ,64 be the components of £i(0) and — A(T)yi(T), tively, in the basis {e;}. Then (2.11) can be written as 7^2
=
&i,
0
=
62,
(a - l)a 3 +/?a 4 -/3a 3 + (a - l)a 4
= =
63, 64.
respec-
Hence, what should be done is to require that the second component of —A(T)yi(T), 62, in the basis {e;}, be zero. If this is satisfied then 02, 03, 04 are determined from the other equations. The component a\ remains undefined to the first order. This is clear because a displacement of ea\ in the initial point produces a displacement of ta,\ + 0(c2) in the final one. Therefore, to end the determination of the correction xi, we should use the terms in e2. It is not restrictive to suppose that those terms are zero for t = 0. Instead of equation (2.11) we obtain (again in the {e;} basis) ak
=
[A(T)(yi(T)+Xl(0))]k
+
c lvk(T) + {D{T)Xl(0))k + J^ Rk,ijaiaj
, k = l,...,4,
(2.12)
where the vector v(T) as well as the matrices D{T) and Rk{T), k = 1 , . . . ,4, can be computed by numerical integration from the second order variational equations. From k = 1,3,4 in (2.12) we found only corrections 0(e) for a-i, a 3 , a\. From k = 2, and using the zero order terms in 02, a 3 , 0,4 we obtain a second degree equation for a\. The possible solutions imply potential bifurcated orbits. The proof of the existence of these orbits, if the necessary condition is satisfied, is ended, under generic assumptions, using the implicit function theorem. See the usual techniques in [28]. We return to the condition 62 = 0. In (2.10) we have used a periodic solution, xo(t), of the unperturbed (autonomous) system. But for this system the origin of time can be chosen freely. In an equivalent way g(xo(t),t) can be replaced by g(xo(t),t + s), where s is a time shift ranging from 0 to T. Hence b2 becomes a
26
Periodic Orbits of the Bicircular Problem and Their Stability
function of s. We look for the solutions of b2(s) = 0. The found values of s give possible bifurcated orbits.
2.4.3
Bifurcation
for Eigenvalues
Equal to One
We restrict ourselves again to the planar bicircular problem. Suppose that a periodic orbit of period T = kTs is available, and that the related monodromy matrix has an eigenvalue equal to one. The case of an eigenvalue equal to an mth root of unity can be studied in the same way considering that the orbit has period mT instead of T. As there are many possible cases, according to the degeneration of the problem, we will present only a couple of them. Due to the Hamiltonian character, the eigenvalue 1 has multiplicity at least equal to 2. Suppose, as in the preceding subsection, that the other eigenvalues are a ± pi. Furthermore, suppose that the Jordan block associated to the eigenvalue 1 is diagonal. This will happen if the monodromy matrix of the periodic orbit has as eigenvalue an mth root of unity and we are considering this orbit traveled m times. The case of nondiagonal Jordan block has been presented essentially in the preceding subsection. Let {e,\,e.i,ez,ei\ be the Jordan basis of the monodromy matrix. We shall use this basis in all the forthcoming analysis. Let Xfc(0), k = 1 , . . . ,4 be the initial conditions for the available periodic orbit and i/ji;(0), A; = 1 , . . . , 4, the corrections to be done when the parameter e is modified by Ae. Expanding the final position $T(X + V, e) in V a n d e we get
$T(x + y,e)
=
$T(s) + - j ^
+
A
e
+ 2
dJ
(
}
+
/ 1 0 0 0 \ ' T,i,j=iai,i}yiVj 0 1 0 0 •y + 0 0 a P \ E i , j = i ai,ijViyj V 0 0 -p a )
^
+ )
d2$T{x) + dedx -AeAyi + . . . , which should be equal to x+y. The partial derivatives can be computed either by the numerical integration of the variational equations or by numerical differentiation. Let
[91,92,93,94)
r
d$T(x)
= —-Q t —•
First we suppose that g\ + g\ / 0. Then, the dominant terms in the preceding
The Tools: Numerical
Continuation
of Periodic Orbits and Analysis
of Bifurcations
27
equations are ffiAe +ai,nj/i+2ai,i22/i2/2+01,222/2 = °> 52Ae +02,112/? + 2a2,i22/i2/2 + 02,222/2 = 0, <73 Ae +aj/ 3 + /?2/4 + a 3 ) n y? + 2a3il22/i2/2 + 03,222/! = 0, p 4 Ae -/?2/3 + c*2/4 + 04,112/? + 2a4ii22/i2/2 + a4,222/| = 0. From these equations it is readily seen that 2/1,2/2 = 0{yfKe), y3,yi = 0(e). The variables 2/3, 2/4 can be obtained from the last two equations when Ae, 2/1, 2/2 are available. To obtain 2/1, 2/2 we put 2/1 = p\/Ae, 2/2 = <7"\/Ae and hence P i + a i , i i p 2 + 2a1:i2pg + ai,2202 2
92 + a2,up
2
+ 2a2,i2pq + a2,22q
=
0,
=
0.
By elimination of the terms independent of p, q and introducing r = q/p, we obtain an equation as c\ + c2r + c3r2 = 0, from which we obtain, generically, two values of r. Substituting q in the first equation we obtain a value of p. At this step the right sign of Ae is also determined. Now we suppose g2 + g2 = 0 and g2 + g\ ^ 0. Let («l,/l2,Al3,«4)
_
2
de2
and Cij
d2$T(x) dedx
»j
The dominant terms are now 4
4
hi(Ae) 2 + ^ c i j j / j - A e + J ^ aiMViVj
=
0,
a2,ij2/i2/j
=
0,
g3Ae + ay3 +/3y4
=
0,
g4Ae -/3y3 + ay4
=
0.
4
4
/i2(Ae)2+^C2j2/jAe+ ^
From this we get that yk = O(Ae) for k = 1 , . . . ,4. The last two equations give 2/3, 2/4 as linear homogeneous functions of Ae . Putting j/i = pAe, 2/2 = ?Ae and replacing yk in the first two equations we found two conies in the variables p, q. The possible intersections of this two conies (generically 0, 2 or 4 points) end the determination of yk to order one in terms of Ae.
28
2.5
Periodic Orbits of the Bicircular Problem and Their
Stability
The Periodic Orbits Obtained by Triplication
First of all we examine the short and long periodic families around L5 for the mass ratio n = 0.012150298 using the tools that we have just described. Letting aside minor variations, the results are the same as the ones obtained in [10] for fi = 0.01215. We summarize them briefly. Data are given in adimensional units. For the short periodic family the limiting period is 6.5826838 and, if we consider the two eigenvalues, Ai, A2, not equal to 1, the limiting value of the trace, Tr = Ai + A2 is -0.764423. When the family grows, the period decreases monotonically. We recall that T5 = 6.791174153. With increasing size of the periodic orbits, the trace decreases till some value —1.67... which is the minimum. Between L 5 and this minimum there is one orbit, L, corresponding to Tr = — 1, which will be found again later. After the minimum, the trace increases monotonically till the value 2. When it reaches the value 2 the orbit becomes symmetrical with respect to the a;-axis, coinciding with a similar orbit in the short periodic family of L4. This orbit, which will be denoted by B45, belongs to the Lyapunov family evolving from L3. Another interesting orbit is found when Tr = 0. We call it B\. For the long periodic family the limiting period is 21.0700687. Increasing the size of the orbit, the period increases till some value close to 26.22 and then decreases again till 25.2943.... For this value the orbit is a quadruplication of B\. We recall that 3Ts = 20.37352246. Hence, there is no orbit either in the short or in the long families which can be continued, to positive values of e, having period mTs, m small positive integer. Now we proceed to the analysis of the orbits D and E of section 2.3, for which one of the eigenvalues is a cubic root of the unity, i.e. Tr = — 1. Using the analysis of bifurcations done in section 2.4.3 we have found a triplication family, i.e. a family of periodic orbits such that for e = t\ (e = £2) consists of the orbit D (E) traveled 3 times. Both families can be continued in the parameter e. First we have considered the branches going up in e till the value e = 1 is reached. The family bifurcated at D (respectively E) has an element with e = 1 which we call F (respectively G). Both orbits have period 3Tg and are mildly unstable. Going down in e, the family bifurcated at D (respectively E) can be continued till some value e3 « 0.010263... (respectively e4 « 0.010138...) when a couple of eigenvalues becomes equal to 1 and a bifurcation of saddle-node type happens. We have not continued these families further. The orbit I (respectively J) which appears in Figure 2.2, is close to the limit orbit. In Table 2.1 we give the initial conditions for t = 0 of several periodic orbits. Table 2.2 contains points in the orbits I and J which belong to y = y/%/2. We remark that the points are rather close. The value of time for which y — y/%/2 is also given. We realize that the difference in the times is roughly 0.5 • Ts- This means that the orbits are very similar but the phase of the Sun is increased roughly
The Periodic Orbits Obtained by
k 1 1 1 1 1 3 3 3 3
e
1 1 1 ei £2
1 1 .010265 .010139
A B C D E F G I J
Xo
yo
-.7189251117 -.0902988768 -.4897494816 -.6437753780 -.2657459249 -.7675885041 .0170293733 -.1805430429 .1988308112
.8167176462 .9476916588 .8705306336 .8326704825 .9206455056 .7798504656 .9909327768 1.0227956815 .9612554737
Triplication
29
xo .0723075174 -.0509735097 .0156873194 .0535767210 -.0318558244 .0693962222 .0015344432 .0556438538 -.0423774283
Vo .2015500525 -.1724139054 -.0001200100 .1229782966 -.1141904296 .2167462379 -.1762336381 .0710693709 -.0883329383
Table 2.1 Initial conditions for t = 0 of several periodic orbits. Orbits A, B, D and E are stable. The largest eigenvalue of the other ones is C: 1.20696, F: 1.62254, G: 1.71030, / : 1.00899, J: 1.00526. The period of the orbit is kTs, where k is given in the table.
I J Table 2.2
X
X
-.281351 -.281141
-.093093 -.093206
y -.123875 -.123965
t 9.201143 5.806122
Point and epoch of intersection of the orbits / and J with y = \ / 3 / 2 .
in half revolution. This is also true for orbits A and B or F and G. As no orbit in the short or long periodic families can be continued to a periodic orbit with period Ts or 3Ts in the perturbed system, we have tried to find an orbit in the RTBP close to the orbits I and J. In this way we have computed an orbit, K, whose characteristics are given in Table 2.3. This orbit belongs to a family of periodic orbits and corresponds to a maximum of the period equal to 20.35848, slightly less than 3Tg. It is easily understood that if the value of e is too small, the training effect of the Sun is unable to produce the periodic orbits with period 3Ts but when we increase e slightly, these orbits are present. The continuation of the family to which K belongs, allows one to detect that the family is started by triplication of one orbit, L, belonging to the short period family. Hence the trace of L is — 1. This orbit is found in the family between the limiting orbit corresponding to L5 and the bifurcation orbit £?i, previously described. Figure 2.3 shows a qualitative summary of some families for e = 0. Figures 2.4 to 2.7 display the orbits A, B, C, F, G, I, J and K, respectively. In Figure 2.5 we have also shown a magnification of the periodic orbit C. It is easily seen that orbits A and B from one part, F and G from another and, finally, orbits I, J and K, are rather close, the main differences being the position of the initial conditions on the orbit.
30
Periodic Orbits of the Bicircular Problem and Their
Stability
T long periodic family
or T/3
T/4
or T/4
triplication family
coordinate
Fig. 2.3 Qualitative representation of the short and long period families of periodic orbits of the RTBP (e = 0).
K L
T 20.35848 6.56317 Table 2.3
X
-.279226 -.154147
y %/3/2 V3/2
X
-.094174 -.197453
y -.125345 -.178815
Period and initial conditions of orbits K and L.
Orbit
A
OrbU B
Fig. 2.4 Periodic orbits A and B of the bicircular problem. Both orbits are quite similar in shape. The most remarkable difference is the initial condition which is marked with a small circle on the orbit.
The Periodic Orbits Obtained by
Triplication
31
Fig. 2.5 Periodic orbit C. It is the dynamical substitute of the equilibrium point L 5 in the bicircular problem. On the right-hand side a magnification of the orbit is displayed.
Fig. 2.6 Periodic orbits F and G of the bicircular problem. Both orbits are quite similar in shape. The most remarkable difference is the initial condition which is marked with a small circle on the orbit.
32
Periodic Orbits of the Bicircular Problem and Their
Stability
Orbit J
Fig. 2.7 Periodic orbits / , J and K of the bicircular problem. As in the preceding figures the main difference between the three orbits is the location of the initial condition.
Chapter 3
Numerical Simulations of the Motion in an Extended Neighborhood of the Triangular Libration Points in the Earth—Moon System
In this chapter we present our own numerical simulations of the motion in an extended neighborhood of the triangular libration points in the Earth-Moon system. We use an analytic model of solar system based in Newcomb's theory for the motion of the planets and in Brown's theory for the motion of the Moon. The system of units used, in all the figures presented, is an adimensional synodical one centered at the triangular equilibrium points. In this system of units, the unit of distance, which is taken as the semiaxis of the Moon, takes the value 384403.678259 km. The unit of velocity, which is taken as the velocity of the Moon with respect to the Earth considered as a system of two bodies, takes the value 1023170.03715 mm/sec. The unit of time is 37569.8724845 sec. In all the figures this system of units is referenced as: adimensional Earth-Moon. The adopted model of motion for the solar system, which in the figures is represented by 16 integers needs some additional explanation. The first 9 integers refer to the planets, the 10th to the Sun and the 11th to the Moon. The remaining 5 integers are not relevant in this context and in all the pictures take the value zero. For all the planets, except for the Earth-Moon barycenter, the following five possibilities are considered for their motion: a) b) c) d) e)
The planet is not taken into account: then it is represented by a zero, Motion according to Newcomb's theory: then it is represented by a one, Circular motion on the ecliptic plane: then it is represented by a two, Elliptic motion on the ecliptic plane: then it is represented by a three, Osculating model: then it is represented by a four.
In the last three models the osculating epoch considered has been January 1 s t year 2000. For the motion of the center of masses of the Earth-Moon barycenter, to the above models we have added the following: f) Osculating with linear variation of eccentricity (5), g) Osculating with linear variation of the argument of the perihelion (6), h) Osculating with the above two variations (7), 33
34
Numerical
Simulations
of the
Motion
i) Newcomb's theory perturbations in longitude, radius vector and latitude due to the effect of the planets, also given by Newcomb (8), j) Same as i) with several periodic perturbations which can be preselected (9). The osculating epoch, as well as the rates of linear variation in f), g) and h) are computed at the same epoch as before. For the motion of the Moon the following models can be used: a) The theory of Brown as given by Escobal (11), b) The same theory deleting the linear variation of the argument of the perihelion of the Earth and using osculating linear expressions for the remaining four arguments (12), c) Same as b) suppressing all the periodic terms (13), d) Same as b) including several preselected periodic terms (14). 3.1
Introduction
In Chapter 1 known results concerning the numerical simulations of motion around the triangular equilibrium points, Earth-Moon system, were summarized, and, as it was pointed out, almost all of them were done using different models of motion for the solar system. All the previously work done seems to confirm the lack of stability for the motion very close to L4 and L5. However, some simulations starting at h\ or L5 also mentioned there, have a pulsating character (concerning the average distance to the equilibrium point) with a long period. The sensitivity of the simulations, with respect to the model of solar system, the initial conditions and the initial epoch were also reported there. In the present chapter we will present our own simulations, using an analytic model of solar system based in Newcomb's theory for the motion of the planets and in Brown's theory for the motion of the Moon. This model will be the one to be used later on by us to get most of the definitive results. The results of this chapter shall be grouped in two sections: one dealing with the motion starting at the instantaneous L4 or L5 equilibrium points at a given epoch, and the other one starting at or near to one of the periodic orbits found by Kolenkiewicz and Carpenter ([22], [23]), which correspond to a planar Earth-SunMoon coherent model. All the graphical output is presented in the adimensional coordinates and units associated to the synodical Earth-Moon reference system. In this reference system the instantaneous equilibrium points are located at L4 = (1/2 — /x, —\/3/2), £5 = (1/2 — ^ , i / 3 / 2 ) , where n stands for the Earth-Moon mass ratio. It must be taken into account that the numerical integration of the equations of motion has, of course, been done in ecliptic coordinates. This fact produces some "anomalies" in the pictures which shall be explained later on.
Simulations
3.2
Starting Near the Planar Periodic
Orbit
35
Simulations of Motion Starting at the Instantaneous Triangular Points at a Given Epoch
Only a sample of results, concerning these kind of simulations, is presented here but it gives evidence of all the main qualitative results that have been obtained. The first two deal with the motion starting at the equilibrium points L4 (Figure 3.1, top) and L5 (Figure 3.1, bottom) at Julian Day 16 000 after 1950.0 (which, from now on shall be indicated as JD50 = 16000 for short). For both simulations the integration has been done for 1000 JD m 3 years, and using a simplified model of solar system. In this model planets are not considered except the Earth, for which an osculating model at 2000.0 has been taken, and the Moon, for which we use an elliptic osculating model at the same epoch as for the Earth. The motion in both cases seems to be confined to a relatively small neighborhood of the equilibria. In Figure 3.2 (top), the motion for the particle is displayed, starting also at L5 and at the same epoch as before JD50 = 16000, but now for a larger time span (until JD50 = 18 250) and using the full solar system with its real (analytic) motion. As it can be seen an escape is produced (approximately at JD50 = 18 250). Let us say that this result is also in good agreement with the previous explorations done by other authors. Although it has not been reproduced here, a similar thing happens for L\. Lastly some explorations have been done, mainly changing the initial epoch in order to have the main primaries coplanar. In them the satellite remains for more than 23 years at a distance, measured from the equilibrium point, less than 0.85 times the mean Earth-Moon one. These explorations have been done starting at £5 and taking the real motion for the solar system except for the Earth, for which the indirect periodic perturbations due to the major planets were not considered. In Figure 3.2 (bottom) this motion has been represented starting at JD50 = 15 619.21 and until JD50 = 24 000. The motion has a pulsating character, mainly when it is seen in the x — y projection (1,2), between the inner part of the Figure and the outer one. This fact can be observed when the plot is produced.
3.3
Simulations of Motion Starting Near the Planar Periodic Orbit of Kolenkiewicz and Carpenter
As it was mentioned in Chapter 1, Kolenkiewicz and Carpenter [22] computed two stable one month periodic orbits, synchronized with the Sun. The main difference between them is that the phase is changed in 180° from one orbit to the other. We have taken as reference orbit the one which is named as II in [22], which has
36
Numerical Simulations
of the
Motion
SIMULATION or NOTION IN THE SOLAR SYSTEM SYSTCT Of UniTSi ASINEHSIOfML EMTH-NOOH ADOPTE© MODCl OF NOT]On FM THE SOlAR SYSTENl • • ' l l • I I I 1 IE I I M I INITIAL AND FINAL EPOCHS OF IHTCfcRAXtONl I C M I . H U I M I 179RI.52133799 INITIAL CONBITIOMS FINAL CONDITIONS V<)> • -t.4ITI4933S«Mfi*«t VF(t) - - f . 3$e*«4SM44S<[»»* Y<2> • -«.ISS«2S4«37R44B+«* YFC2> - -*.934«IC12?IH2D*«* Y<3) • -».ltttlSEtSSM5D-tS YF<3I • • .UL2f442SI4aiD~«l Y(4> • -*.42C3«2326*39f0-lS Y f M I - -A.3322S92S7SS97&-01 V(5> • •.93I33!IS391T1D-1S YFfSl • -•.44t9793243M7a-tl Y<«) • •,292tliLTZ33tlfi-lt VF
«.3IS2D*M
.l3SIB-«t .t7H»**« ,
-V
-.nyiH—
[
».inai-ti
|
t.3iyi
-.H37V»*»
SimjUTIOn OF NOTION IN THE tOLMT SVJTtfl SYsrcfi w w i n s i ABINCNSIOHH EARTH-NOON ADOPTED NOKL Of MTIOH FOA THE SOUR SYSTEHt * » 4 9 t B B O t l l S C » l * » INITIAL AND FINAL EPOCHS OF INTEGRATION) I M M . t l M M M lTOM.BMCBKt INITIAL CONDITIONS FINAL COHWTIONf V d i • -«.4B?S49332»M»+M YK1) - -•.S9«B279t9942D*M Yi2- • • >ISC»*¥4I37144D»«4) YF(«) * t.tSiS4««421SMf*M Vf3) • t.S3*B*23C5K93fi-l« VF(3) • t.2S3321«2«35f 1D-M Y(4> • I.17H3B7473MBB-14 VFH) • 1.13357SB294S430-01 YIO • 9.43I1ZB727M37B-15 Yf<J) - -».l335M4773JMO-«l Yift) • B.IIB2*S7ZKStl*B-lfi YF(«) • I.49145SM33B4I0-O3
Fig. 3.1 Simulation of motion starting at the equilibrium point L4 (top figure) and at L5 (bottom figure) at JD50 = 16 000 during 1000 days.
Simulations
Starting Near the Planar Periodic
1
SIMULATION OF NOTION I N THE M I A * SVSTEN S « T E N OF L H I T S i A&INENSIONAL EA9TM-M0N ADOPTED NODtl OF NOTION FOR THE SOLA* SYSTENt l l l l l t l t l l l t t * * * * I N I T I A L AMD FINAL EPOCHS OF INTEGRATION' I«999.99««9999 19259.9MT9I91 I N I T I A L CONDITIONS FINAL CONDITIONS r<W - -9.49794033Z9«**m9« V F U I > 9.219175IS9S49ID+91 VI81 • 9.9«92549379440»9« V F I i l • 9.17191774971970+91 VI3> | . I N M I N M I M t * H VF(1) • -9.3S37221S1883ID-94 Vt4) • 9 - I T 9 a C 9 4 * 3 t J « 4 D - l 4 VF<4> * . 17423723479929+91 V ( » l - - 9 . 1 E 9 7 2 * 1 7 f i £ l I 3 0 - I 4 VF(S) • -«.I$71|ESS2TI59I>»9I Y(Cl • M H N H M M I I M I VFlCI 9.U143911994C99-96
1
I
37
Orbit
I
•"
•
I
'(1.2)
L. 2 3 7 1 9 * 9 1
[.£•959+91
UltlBD+91
y
L.1543D**/
L.I2E7D/91
L.99WB+9*
L.7IH0+M
|
L4399D**9
U1C31D*W_ -.13»,B»*«1 1
1
I
1
I
1
1
1
" - L.
L22VT9-94
4.U^8fl~-\
UU13&-94
H^^B&JE*
9 . S31,20+99
9.1(4,99*91
1
1
!
— -
j
I . O I I H /
I.34«3^4)4
\
L.4HTD-M
\
\
L~S7SIV-4>4
^*A/
\
/
\
_
^
-.384,99***
|
'(2.3)
~ " _ _
US*t-%y
L.313Zp-M
: . ! 1799-94
L.2310D--+4
^3«3»-94^
L.4C97P-94
/
^ ~ (
*.2SK2D**1
L.U13B/94
^313Z9~9i
-.13^90*91
- . 3 9 4t)D«4M
L3491D-94
L.34919-94
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1
'(1.3)
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(
9.194^0*91
(
9 . 2C<j2D+4>l
uSTS19-94
•.23^!»*«l
9.11^91*91
9.1^79*91
9.7^99+*.
9.1t3jt3*9*
SIMULATION OF NOTION I N THE SOLAR SVSTEft SVSTEn OF UNlTSi ABJNENS10N*t EAATH-NQON •POPTtO ItQDCL OF NOTION FOA THE SOtMt SVSTCNi 9 9 9 9 I N I T I A L AND f | M * t CF-OCHS OF INTEGRATION! J 5 S I 9 . 2 1 f * # 0 9 » 24999.1SS2972S I N I T I A L CONDITIONS FINAL CONDITIONS - • . 4 S ? 8 4 9 3 3 2 A 9 M 0 * A 9 VF I -« . 7944E»999ZISD*9«
• .IU98S4937S440+99 vF ( E i . 4.?ISa4*7749S«CP+99 • 9.19999493192I2D-15 VFO) • 9.S3792SI991949D-91 • .t»7»S6WS??S8D-H VFH> -9.l£€6C11437fiatD-14 VF(S) - 9 . 1 4 9 S S S 7 Z 7 9 9 9 3 D - 1 7 YF CS > •
9.3945C83943479D-91 9.873EECZ99417ID-91 9.15852194923999-91
L.4929S+I
.39^90*99 1
T
1
1
C-7171D-91
j -.9929B***
-.51^30+99
|
-.2fclj7p*99
(
9.52^49-91
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l
-.295,70*99
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|
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rr
9.37LG0+B9
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|
9.68^30*9*
f
9.983/78+1
9.12740*11
Fig. 3.2 Motion starting at the equilibrium point L5. In the top figure time varies between JD50 = 16 000 and JD50 = 18 250. In the bottom figure time varies between JD50 = 15 619.21 and JD50 = 24 000.
38
Numerical Simulations
of the
Motion
the following initial conditions in the adimensional system: x \ I -0.52155 \ y = -1.05137 , z J \ 0.0 /
/ x \ f -0.28400 I y = 0.15178 \ z ) \ 0.0
All the explorations have been done taking an initial epoch at which the Sun, Earth and Moon were in the same vertical plane. This happens, for instance, at JD50 = 17997.574 which has been the epoch adopted. For the first two numerical integrations in Figure 3.3 the behavior of the orbit, starting at the initial conditions displayed above, is shown. The computations are extended till JD50 = 18 360 and 19100, respectively. As it is clear from the second one, an escape is produced. It must be said at this point that the broken line which appears in Figure 3.3 (bottom) {(x,y) projection) is only due to the fact that, though the integration is done in ecliptic coordinates, the figures are represented in adimensional (synodical) ones, which means that some rotation has been introduced producing that kind of facts. For both simulations a simplified model of solar system was used keeping only the Earth in an osculating motion and the Moon in an elliptic osculating one. In Figure 3.4 we repeat the same computation with a displacement in the first component of the velocity equal to 0.01. The other modifications in the coordinates are: Ax = -0.01 for Figure 3.5, Ay = 0.01 for Figure 3.6, and Ay = -0.01 for Figure 3.7. We see completely different behavior among them. Even in Figure 3.4 an escape is found before JD50 = 18 360. In Figure 3.5 (bottom) we can see that the particle remains in the neighborhood of the equilibrium up to the date JD50 = 19100. In other cases, as can be seen in Figure 3.7 (bottom), the particle performs some revolutions around the two primaries, and after that it escapes. Figure 3.8 (top) shows the orbit starting at JD50 = 18 012 and taking the same initial conditions that we used for Figure 3.3 (top). The delay, being equal to 14.4 days, implies that the initial conditions are almost equivalent to the initial conditions of the periodic orbit II of [22]. Final time for Figure 3.8 (top) is JD50 = 21300 and the full solar system is used. For the purpose of clarifying Figure 3.8 (top), in the next three figures (Figures 3.8 (bottom), 3.9) the partial paths from which the first figure is produced are displayed between the epochs 18 012 - 1 9 200,19 200 - 20 700 and 20 700 - 21800, respectively. Good stability properties are obtained. This is in favor of the existence of a set, near orbits I and II of [22], for which some stability is found. In fact it seems that there are quasi-periodic orbits which are confined to a large neighborhood of £4,5. Furthermore, the projections of the orbit, when followed on the graphical screen seem to display motion in a torus for some long period (1 year, say) that is changed to another different torus (bigger or smaller) at the end of this period. The procedure is repeated during the full simulation. It must be noted, looking at the z
Simulations
Starting Near the Planar Periodic
Orbit
39
SIMULATION OF noTion I N T « $OLA« SYSTCR SYSTEfl OF UfllTSl AOMENSIONAl EAUTH-BOON ADOPTED HQDCL OF NOTION FOR THE S O U P svsTiftt
I t 4 I • I t I < 1 IS t M I I INITIAL AND FINAL EPOCHS OF IhTCCXATlONl 17M7.S74AI04M lt3CB.33S0f?f4 INITIAL CONDITIONS FINAL CONDITIONS v m - -O.SZISSMBMMMBD+IO v n t i • -:mnnvnnt*b*t* Y U l - -•.ltS137MBMMD+*l VF(£) - -•.I4CB941432SB7B+** YC3) - -t.H12£12ISa4llD-lS YF(3» - t.2Mt7S?£433tfl0-«2 VI4) • - « . a i 4 » * t M I * M l l * 4 M YF(4t • «.llltC724«S24iO-«l Y d ) > O.tSiTMffCIMMBvM VFtl> - t . 4 « l » m £ f 3 3 4 Q - 0 1 VfC > - #.«1243«|HTn
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Fig. 3.3 Motion starting at the planar periodic orbit around the equilibrium point L5 computed by Kolenkiewicz and Carpenter. The simulation goes from JD50 = 17 997.574 to JD50 = 18 360 for the top figure and from JD50 - 17 997.574 to JD50 = 19 100 in the bottom one.
Numerical Simulations
40
of the
Motion
SIflLLATIQH OF ROTtON IN THE SOLA* SVSTCfl SVSTEK OF UtUTSi At>INtHS!OM»L EHRTH-KOWt AB0PTE9 M K L OF MOTION fOR THE SOW* SYSTEH1 I M t I M I • 1 I M ( I I I IHITIAL AND FINAL EFOCHS OF IriTECAHTIOMi 17997.S74B4H)** 113fi*.i3l1*945 INITIAL CONDITIONS FINAL CONDITIONS V ( t ) • -».t2tS5**t«ltMD»M VFtlt • •.U74$79ISSC»D**1 V(2» • -•.tt5137t4#MMD«tt VFESt - -•.27I87CKS73M0**! V(3t - - t . U l 2 S I » I Z 4 t t 0 - l S VFUi • -f.S2B972M2157*0-U V(4> - - | . 2 7 4 « a i M 0 * t l * D * t l VFUI * -•.17SC79914S7310*** V(5> • f . l $ l 7 M t M # l t W * M VF(5I • -0.4179393951713D»M V(«> > t . l 5 4 S 2 K H 3 » 3 D - l C VF(C> - -t.23»24199e3*ltt-9l
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component, that the departure from the ecliptic increases slowly with time on the average.
Simulations
Starting Near the Planar Periodic
Orbit
41
finuLATiwi OF notion I H THC s o u " SVSTEB SYSTEH OF UNITSI ADtffCHSlOML CAJtTM-ACOtl AfiOTTCB MODEL OF AOTIOM F « t THC SOLA* SVSTEIH i I « t • t I * • I K I t I I f I N I T I A L AND FINAL EPOC** Of INTEGRATION! 17197.S74HIM IIMB.3SB33I99 I N I T I A L CONDITIONS FINAL C 0 M 6 I T I 0 M V I I I • - I . U I S M H H M M I VF(1> • - • . S 8 1 l 3 U l M 0 4 9 D * t t VIII • -•.l»137Mttt«*l>*»l Vfti» • - • . S » 4 » 5 M £ l t t » > W V(JI > -t.U12CI2BS24BlD-15 V F O l • *.3*IH»3tB71*SB-K V M I - - • . 2 B 4 * t * 4 * t M * * J > * « B VF44> ••C3»37tt3Sfif3D-»t YI9I t . l S 1 7 I M t t M t * D * M VT(5) t.497T4«3Zf I B I T B - K Vl«> • B.1C73B579774220-1B Y F C i t • - i . 7 3 l 3 9 i l 7 » 3 2 B O - B 3
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s i f w u t t o n or HOTiON i » THC SOLA* SYSTEM. m m OF UMITSi ADIHCHSI0HAL EMTH-flOOTI ADOPTCft IWBCL OF N.OTIOH FOR THE SOLA* SYSTENi 1 M 1 1 1 t 1 1 1 IS ( • > 1 • I N I T I A L AMP FINAL EPOCHS OF INTECtATIOHt I 7 S S 7 . S 7 4 0 M M 19IBB.SM31131 I N I T I A L CONDITIONS FINAL CONB1T10MS V(l> - - • . S 2 t S S « M « t * t « 0 * * t YFtl> - - * . l t 7 4 M 3 B S t 5 C t D * » Y<*> • - • . l * S 1 3 7 M « t » t « 0 t » l V F I J I • - • . I 1 5 9 1 S 3 M 1 7 4 8 B + A I Y(J) « -t.llltS12f«24f,!0-15 VFI3) • -•.88*4*74B4BlMB-*3 V(4> • - • . 2 9 4 S B * * * « M * * 0 * M V F t l l - - • . 3 3 7 I 0 7 X t t M S I D * « t V(5) * . 1 5 1 7 S * * « 0 « # * B * M V f ( S ) • -t.39«144fi«S*3SSD-01 VIH • B . l f i 7 3 9 S 7 9 7 7 4 2 £ B - l f V F I t l • -•.1SSC35B43B7S2D-01
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42
Numerical Simulations
of the Motion
i
SIHUlAriOlt OF NOTION I N THE S O U * SVtTEN SVSTCfl OF U M T S ! ADIflENSlOMAL C M T t f - M O f l •DOTTED IWDEL OF PWTtOM FOR THE; SOLAR SVSTtfli .{ « M M I I t 1 IS 1 1 M • I « t t J » t AND FINAL EPOCHS OF M T E C M T I O N I 17997.574MOO t l 3 U . t I I 3 N I 4 I N I T I A L CONDITIONS F l M » i CONBITlwtS V U I • -•.S211S*Mt««MO**0 v r i n • -•,43l7IJH*S«ll>«M V(2) • ••.t«S137M***MD*01 * F ( 2 I • - • . t 7 9 * 5 « l T S 7 S l D * M V O I « ••.12*?3*331Ht3D-lE VFOI • • .22SMIBIM2939-K V M 1 • - I . I h M H H H H D H I VFt4> • • .C4522$7iM*Ct&~ll VI5I • t , l 6 t ? S 0 t t M t M D t M VFCSI • - • . 3 2 l S H 2 7 ( 7 7 f l B 6 - i l VtSl • - t . l f » 3 l 5 I I Z I I 7 D - l t V M S ) • - t . 7 l 7 S « 1 7 4 3 B t S O - «
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Fig. 3.6 Motion starting near the planar periodic orbit around the equilibrium point £5 computed by Kolenkiewicz and Carpenter. The second component of the velocity y has been modified by an amount of 0.01. For the top figure the simulation goes from JD50 = 17 997.574 to JD50 = 18 360, and for the bottom one from JD50 = 17 997.574 to JD50 = 19 100.
Simulations
Starting Near the Planar Periodic
Orbit
43
SIMULATION OF NOTION I N THE SOLAR SVSTEft SVSTtB Of UNITSi ADIflEHSIONAl fARTH-NOON ADOPTED HOCEL OF rtOTIOtl FOR THE SOLAR SVSTEAl * • 4 » | « 9 « * t ! $ » • • • • I N I T I A L AND FINAL EPOCHS Of iNTEMATIOHt l?S97.f?4HMI ll3t*.*fif41333 I N I T I A L CONDITIONS FINAL CONDITIONS V
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44
Numerical
Simulations
of the
Motion
SIMULATION OF NOTION I N THC SOLA* svsrcn SvSTEN Or u t t l T $ i ADINENSIOMAL EARTH-NOON A 6 0 » U B NOBEL Or NOTION FOR THE SOLA* $VSTEN i i i i i i i i t i ti # r ' I N I T I A L AMP FINAL EPOCHS OF INTtCRATIOni 11*12.*••*«•»* 2IIM.t2SS7IS» I N I T I A L CONDITIONS FINAL C0H8ITI0HS YC11 • - • . 5 2 l S 5 t * « M » M I > * 0 V F f l J • - * . * » B ? 3 Z C X I 9 3 a D * M V ( 2 ) • - « . i « « i 3 ? » M « M a O * t l YF « . 1 S 1 ? I M * M * R * D * 4 B VF(5> • - • - 1738S18S3B321B-R1 Vi«) • - * . 1 2 7 V M 7 B 1 I 7 « 0 - U VF
SINULATIOtl OF NOTION I « TMC SOLAR SVSTCft SVSTCN OF UHITSt AOIflCHSlOHAL EARTH-NOON ADOPTED NOBEL OF NOTION FOR THC SOLAR $V*T£Ai I 1 1 1 1 1 1 I I 1 tl • • • < I N I T I A L AND FINAL EPOCHS OF INTEGRATION! lttl£.**•!•*•« itU«t.HStU4S I N I T I A L CONOITIOHS FINAL COMMTIOH* -«.St7t«24|9US7D*t» ttli • - t . l « $ l 3 7 t » * * # f l » * A | V F U I • ~*.B7$47t7l4513ID*» Y131 . • . 3 S 9 $ l t S 9 7 t t g * D - l S V F O > - A.SBB17I3S1I3B1D-BI YC41 - - • . 2 l 4 A * O I * M * t D * M YFC4I • -t-13SSR4Rfi»B4B9ff**« V(SI • » . 1 S 1 7 1 » B M « M B * M VF<S> > A . * l 7 » 3 7 1 I t 7 4 2 4 0 t | « VIS) • - f . l 2 7 V 7 4 7 « U 7 2 B £ - U VF(I> • t . 1 1 1 2 M l l t 4 7 3 £ B - S l
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Simulations
Starting Near the Planar Periodic
Orbit
I
SIMULATION Of MOTION IN THE SOLA* EVSTEN SYSTEM OP UNITSl APinEN$IOH*L EAATR-NOON AOOPTEI N H L OF NOTION FOft THE SOLA* SVSTtHt l l l l t l l l l l l l t t * * t INITIAL AHO FINAL EPOCHS OF IHTEOMATlONi l9IB9.M*4tC33 2*7M.4KS1«ES INITIAL CONDITIONS FINAL CONDITIONS V I I ) - *0.?3«im*S23MBtt* VFtl) • -•.47H1223431B10-*! VIA) • -*.7KSKM2igseOD*H VFCK> • •f.lM324B3f9CS3D*tl V(3> • *.1**«3**3?2l*S0-tt VTI)) • • .2M32ECE247UB-tl V(41 - -*.K115*ICStt»B-*C YF<4> • - » . l t * M M S * « 4 M * * * VIS) • 9.|9124ieM4I32D4M VFIII - -•a2S7S977I2323B*»* V
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ilMJUTIOn OF NOTION IN THC SOLA* SVSTCn SYSTtn or imiTft AotnCNSlONAL CARTH-HMN ADOPTED MDCL OF NOTION FOA THE SOLAN SYSTEMi 1 1 1 1 1 1 1 1 1 t 11 • • • • * INITIAL AND FINAL EPOCHS Of IHTCUATIOtfi MC».82I«TI4S 211*1.I2257SI* INITIAL CONDITION! FINAL CONDITIONS V ( l ) - -*-4S*$231C7t7340«** VF(1) - - « . 3»<439447»2»4M V«2I - -*.li3l4Z»41211»4*! VF(2) • -*.7tt2«477US*2B**« Vi3> • *.29SBWt|931930-*l YF(3> - -•.229CI44SI383T9-*! V14) . -*.427*«71313I7»D+M VF«4) • • .22144SSM4293B*** V«S1 - a.UlS93«113M*B»M VFfS) - - * . 1S3BS3IS439I«*M V(Sl • -t.H74S*47M»33ft~*l VF<1> • -*.3ESB»4f1IU84B-II
\i
u4W*BVM/
* . 39-f7»*«l - . I I ^ I » * « I
;
-.
1 H
-.«(«*.. , - .SC^TS***
.D..I
1
I
1
1
1
-.«,...#.
l
'(1.2.
:_4fi22S***^
L.EC7i»tf/ ^7S9«wLI|
^9744B**«V :.1«77D**1 _tl7«C+«I -12IZB**! -.M91D+** , - . I T y f B * * * , 1
1
1
1
1
1
1
1
'(1.3>
1
T
1
1
•.23^30***
(
1
1
* . B^tBtM , 1
*.44*t»«M
'
'<2.3»
L,4S21B-»l^
U4HS»-«1
:
L.34t»H>iJ^JC
-
UMt3y#4/VW L11MB-4&&
L :m D
- * T(yC
\
L.1MUM j \
-
L>34*SB-»VJC
fc"SJ
-
L>22*3B-*I
, \
^7l7K\/ OOvH
"
UIIMB\4/
XlfvA/V ^^^7
L.37MD-u\. t.J*tSD-*f^ ^U*U-*lV
t
i.n*t^*rVKQa&
^333*B^U^06 (
'^^-i_a«W"' -.«Ty?P*M ( - . 1 3 4 , 3 B * * *
t-3»3*o{tT^
4 f ^
i c-44E3fr-*i • , 44*|C&4«* -.121^84*1
i
L.44HM1 -.9^10+M
(
*.l*3|l»+l* ,
-.lBTjTDttl
,
-.•7|*D*M ,
- . M^ifr^** ,
-.4i«a»**«
Fig. 3.9 Pieces of the trajectory displayed in Figure 3.8. In the top figure time varies between JD50 = 19 200 and JD50 = 20 700 and in the bottom one between JD50 = 20 700 and JD50 = 21800.
This page is intentionally left blank
Chapter 4
The Equations of Motion
The equations of the real motion near L 4 or L5 are written. The full solar system and the radiation pressure are considered. The Hamiltonian of the problem is also given. The terms of the Lagrangian and the Hamiltonian which involve Legendre polynomials are expanded as power series in x, y and z with coefficients periodic functions of time (with incommensurable frequencies). A Fourier analysis of such functions, as well as the other functions needed for the equations and the Hamiltonian, is performed. The big size of the region where the motion takes place leads to convergence problems in the expansions. The number of terms to be retained in the expansions of the Hamiltonian and the equations, even with large tolerances, is given. This suggests to write down the equations of motion in a simplified way. The full coefficients which appear in the equations finally used are given. Checks against the full vector field are done. 4.1
Reference Systems
We consider an inertial reference frame with the origin at the center of masses of the solar system and the axes parallel to the ecliptic ones. The equations of motion of a spacecraft in the solar system can be written as £
=
y
GA(RA - R)
A€{S,E,M,Pi,--,Pk}
I
A
~
'
where G is the gravitational constant, and R and RA are the position vectors of the spacecraft and the body of mass A respectively. The summation is taken for the Sun (5), Earth (E), Moon (M) and the planets (Pk). However, the above reference system is not convenient to study the motion of a spacecraft in the vicinity of the libration points L4 or L5 corresponding to the Earth-Moon system. As usual, the libration points are defined as the ones that form an equilateral triangle with the Earth and the Moon. These points are placed at the instantaneous plane of the motion of the Moon around the Earth (see Figure 4.1). 47
48
The Equations
of Motion
TEM
Fig. 4.1
M
Reference systems at the triangular equilibrium points.
We define a normalized reference system centered at the libration points and given by the unitary vectors ei, e
TEM
\TEM\
e3 =
TEM
A
TEM
\rEM ArEM\
e2 = e 3 Aei.
We introduce a modified mass of the Earth, E, to satisfy Kepler's third law K = G(E +
M)=n2MaM,
where UM and OM are the mean motion and the semimajor axis of the Moon around the Earth, respectively. The rest of mass E = E — E will be considered as a perturbation. Then, in this normalized system (x,y,z), we adopt the following units: (1) The unit of mass is chosen such that G(E + M) = 1, (2) The unit of time is defined to have UM = 1, (3) The unit of distance is taken as TEMTherefore, in the L4 case, the coordinates of the Earth and the Moon are {xE,yE,zE) = (-1/2,—\/3/2,0) and (xM,yu,ZM) = ( 1 / 2 , - ^ 3 / 2 , 0 ) respectively. In a similar way, for L 5 we have (XE, VE, ZE) — (~ 1/2, V3/2,0) and (XM,VM, ZM) — (1/2, V5/2,0). For a precise definition of the adopted system see [17].
4.2
The Lagrangian
In the normalized reference system centered at the libration point Li, i = 4 or 5, the Lagrangian (see [17]) can be expanded in Legendre polynomials as: L
-
-lk2{x2+y2
+ z2) + 2kk(xx + yy + zz)
+2k2 [E(xy - yx) + F(yz - zy)} +k2{x2 +y2+ -k2(xEx
z2) + k2(Ax2 + By2 + Cz2 + 2Dxz)}
+ yE) - kk(xEx
+ yEy)
The Lagrangian -k2(AxEx
+ ByEy + DxEz)
+ j ( l - MM + HE) J2
- k2{-EyEx
49
+ ExEy + FyEz)
"•"PnicosE^
n>l
k rs
£i v^y
K
a cos A2 + ^ - V (•?-) 2 f\E,4
Ae{s,M,Pi,...,n}
P„(C0S^!
where a = (x, y, z)T is the position vector of the spacecraft, v stands for the vector v in the normalized system, k = rEM is a scaling factor, and rA-a rEA • a cosAi = — , cosA 2 = . rA a rEA a We denote by v (resp. v) the modulus of vector v (resp. v). The term with fig takes into account the radiation pressure of the Sun. A parameter, S = —a/G, is introduced where .
149 597 871 x 4105 5842 x 86 4002 aa km 3 1000 m modified Julian day
a = 4.56 x 10~ 6 Newton/m 2 is the radiation pressure of the Sun at one A.U., m is the mass (in kg) and a is the effective cross section of the spacecraft to the radiation pressure (in m 2 ). The coefficients A, B, C, D, E and F are functions of the longitude 6 and the latitude 8 of the Moon, as well as its derivatives 6, 5, 6 and 6 defined by A
=
d2cos25 + 52, (cos2 S + R2 sin2 8)62 + (2 + R2)82 + (R + 6 sin S)2 1 + R2 R262-82
D
=
- s i n 8 cos S^vT+R2
R2R2 (1 + R2)2' 9R cos 8 , , Vl + R2
6 cos 8 + 8R
E F
+ (R + 6smS)2 1 + R2
y/T+lP =
0sin8+i
' R +
R2,
where R = -. . 9 cos 8
R2R2 (1 + R2)2'
50
The Equations
of Motion
From the Lagrangian, the equations of motion are easily written as: x-2y-x
= C(l)x + C(2)y + C(3)z + C(4)x + C{5)y + C(7) + -3(1 - fiM + VE)(X ~ XB) Yl
an~2Pn(cosEi)
n>2
,n-2 a"~"
is
x x +TaH( - s) J2 ^+r p «( c o s 5 i) k3' n>2 S r
+ y + 2x-y
K
E
+ (X- XA) ^2 T^+T-Pn(COS A i )
?EA
A€{S,Pu...,Pk,M}
= C(ll)x
an~2^
XEA
MA
i>2
r
A
+ C(12)y + C{13)z + C(U)x + C(15)y + C{16)z + C(17)
+ —(1 -fiM 3
A;
+ HE)(V - VE) ] T a " " 2 P „ ( C O S S I ) n>2
-,n-2
K
x p cos5 +T3t i) k3 s(y - vs) J2 ^+r «( n>2 rS
K 1\ k3
v ^
-W^ + ty-y^Yl ' EA
Ae{S,Pu...,Pk,M}
an~2^ -^+iPn(cosAl)
n>2
r
A
C(21)x + C{22)y + C(23)z + C(25)y + C(26)i + C(27) + ^ ( 1 - HM + VE)z J2 a " ~ 2 p n ( c o s S i ) n>2 +
an~2^ P
tfV§(Z ~ ZS) J2 ^ + T n ( c O s 5 i ) n>2 VS
K_
+
¥
£
fJ-A
A6{S,Pi,...,P fe ,M}
ZEA 3 rf EA
+ (Z - zA) Y, n>2
a " " 2 ^ P COsA ^+T n( ^)
T
A
where C(l) C(3) C(5) C(12) C(16) (7(23) C(ll)
= = = = = = =
2e + e2+A-kk~1, D, 2(e + E), 2e + e2 + £ - M r 1 , 2F, C-JtA;" 1 , -C(2),
C{2) = (7(4) = C{7) = C(13) = C(17) = C(27) = C(14) =
2kk~1E + E, -2kk-\ (kk-1 - A)xE - {2kk~lE + E)yE, 2kk'1F + F, (2kk-1E + E)xE + (kk-1-B)yE, l - £ > i B + {2kk~ F + F)yE, -C(5),
The Hamiltonian
C(15) (7(22) C(26) e 5
= (7(4), = -C(13), = C(4), = (9—1, = B-(l + e)2,
and the Related
Expansions
51
C(21) == C(3), C(25) == -C(16), A
2?
= A-(l + e)2, = £ - ( l + e) 2 .
We have also used Pn(a) = - - r - P „ - 2 ( a ) , where P„ denotes the nth Legendre polynomial. 4.3
The Hamiltonian and the Related Expansions
We start with the expression of the Lagrangian (given in [17]) of the problem when normalized coordinates centered at the instantaneous L45 point of the Earth-Moon system are used. The momenta px, py, pz are introduce through r\ -r
px =
—7ox
= k2x + kkx — k2Ey — kkxE + k2EyE,
TTT- — k2y + kky + k2(Ex - Fz) - kkyE -k2ExE, oy dL, pz = ^ = k2z + khz + k2Fy - k2FyE. dz From these relations it is easy to express x, y, z as functions of the positions and momenta, the coefficients being functions of time. The Hamiltonian is obtained as H = Y^Pi' Qi ~ L, pM-
9 =
where in qi, as well as in x1 + y2 + z2, xx + yy + zz, E(xy — yx) + F(yz — zy), xEx + yEy and —EyEx + ExEy + FyEz. The variables x, y, z are substituted by their expressions in terms of positions and momenta. After a somewhat lengthy computation the following Hamiltonian is obtained: H
=
-k~2(p2x+p2y+p2z)-kk~1(xpx+ypy +F(zpy - ypz) + (kk^XE + l^k2(E2 -k2{EF
- EyE)px
- A)\ x2 + ^k2(E2+F2
+ D)xz
+ zpz) + + (kk^ys
E(ypx-xpy) + ExE)py
- B)j y2 + \^k\F'
+ Fyspz - C)
52
The Equations
+k2(A - E2)xEx
of
Motion
+ k2{B - E2 - F2)yEy
+ k2(EF + D)xEz
+term purely depending on time -Kk~l(l
-HM + n&) Yl
"•"PnicosEi)
*>i
_^-i^^(^y Pn(cos5l ) -Kk-1
Y,
acosA 2
^
1 v ^ / « V n /
T
r A
EA
A€{S,M,Pi,---,Pk}
n > l ^
,. \
r A j
The time-dependent functions which appear as coefficients have been partly computed in the development of the Lagrangian. The only additional functions are k~2,E, k2(E2 - A), k2(E2 + F2 - B), k2(F2 - C), k2(EF + D). The computation of the function A;-1 is also useful.
4.4
Some Useful Expansions
In order to get a quasi-periodic solution of the problem, it is convenient to develop the right-hand side of the equations as series of the following types 2_.a,ijkrXly^zkF(vrt
+ 4>r), or it y ^ a r F ( ^ r ^ + 4>r),
where i) = x,,y or i and F stands for one of the trigonometric functions sine or cosine. A Fourier analysis of the functions that appear in the equations will give the coefficients (aijkr, ar), the frequencies (vr) and the phases (4>r) involved in the dominant terms. We deal separately with the terms of the equations which involve the Legendre polynomials. These terms come through derivation with respect to x, y or z, of the corresponding terms of the Lagrangian. So it is more convenient to develop the summations of the Lagrangian and then to derivate. Using the normalized system of coordinates we have xxE +yyE + zzE cos £1
=
;
arE XXM+WM
cos Mi
=
+
1 /= - — (x + sVoy), Za
ZZM
1 ,
= —(x -
:
^a
Q.TM
Therefore, it is not difficult to see that
n>l
n>l
k=0
V
'
/r
x
sV3y).
53
Some Useful Expansions
^—' m\(n — 2k — m)\ v
m=0
'
*-?
ni!n 2 !n 3 !
2ni+n—2k—m2n2+m d, y
2ns z j
ni+n2+ri3=k
where ( 1 ii A A= =E A= M \ 0 if i4 M ''
_ ff 1 if A 4=E £7 ~ \ \ -- 1 if A A= =M M ' l if
rr _
S
_ f 1 for Li \ - l for L5
_
A routine, EXPEA, gives the coefficients of each monomial xly^zk up to a given order n = i + j + k. This routine has been checked comparing the development with the generating function 1
y/1 — la cos A\ + a2 where A = E or M. It has been seen that to get a difference of the order of 1 0 - 5 , terms up to order 20 are required if the magnitude of a is 0.5. We note that the coefficients in the development of J2 anPn(cos Ar), n>l
for A = E or M are constants and so no Fourier analysis is needed for these terms. For the planets and the Sun, similar expansions are obtained where the coefficients of the monomials are functions of time. We have For the planets
Kk-^A^Tf^-)
P n (cosAi).
For the Sun 1
Kk-\ns+»s)—
/
L
n
\
'E(^ ) rs^VsJ
PnicosSt).
In general, we have the following expansion 1/2]
k
k — ni
n—2k n—2k—l\
S£S!F^^)|£ v f-±\
' fy±\
2
(*±\
3
S zlWa!(„_2Jb - Ix - «a)! (1_\
T2n1+h
U J {rA) {rA) U-J *
2n2+l2y2ns+h
V *>
54
The Equations
of Motion
where fa = n — Ik — fa — fa.
The terms of the expansion can be collected in the form /(9i.92,
£ ?1>92,93
where
The coefficients (c 9ig293 i 1 i 2 i 3 ) and the exponents (qi, q2, q3, fa, fa, fa) are computed up to a given order n = qi + q^ + q% in the routine EXPEAT. The routine is checked using the generating function 1 \Jf\
1 2
- 2CL?A cos Ai + a
rA
Some computations for the Sun, Venus and Jupiter show that order 3 is needed for the Sun, and order 2 is sufficient for the planets in order to have a difference of order 1CT5. A routine, FUN4, has been done to compute the functions at a given day. This routine will be used by program FURI to do the Fourier analysis. We classify the functions in the following way (1) From the Lagrangian: f{qi,q2,
V
8
**M-^)' ^" ("H)' ^" ^(-^)' for the Sun, the planets and the Moon. (3) Other functions that will be useful for the development of the Hamiltonian: k~\ k~2, E, k2(E2 - A), k2{E2 + F2 - B), k2(F2 - C) and k2(EF + D). 4.5
Fourier Analysis: The Relevant Frequencies and the Related Coefficients
First of all the equations of motion as given in section 4.2 are rewritten in the more convenient form displayed below: x - 2y-3x2Ex
- 3(1 -
2fiM)xEyEy
= (7(l)a; + (7(2)?/ + (7(3)* + C(4)± + C{5)y + (7(7) 4- Kk~3
[-3(1 - 2(IM)XEX2
-
3yExy]
Fourier Analysis:
The Relevant Frequencies and the Related Coefficients
+ Kk~3(l
an-2Pn(cosEi)
- nM)(x - xE) Y n>4
+ Kk^fi^x
n 2
- xE) Y
a ~ Tn(cos
E{)
n>2
+ Kk~3fi§(x-xs)Y
r^+iPn(cosSi) r
n>2
S n-2
+ Kk~3[iM{x
- xM) Y
T^+rPnicos
+ Kk~3
y + 2x-3yEy
- 3(1 -
Mi)
V
n>4
M XEA
Y AM Ae{s,plt...,ph}
+ (x-
XA) Y
an~2^ -^+TPn(cosAl)
- - r n>2 A
?EA
2/j,M)xEyEX
C(ll)x
+ C(12)y + C(13)z + C(U)x + C(15)y + C(16)i + C(17) 3
+ Kk-
- 3yEy2]
[-3(1 - 2nM)xExy
+ Kk~3{l
- m)(y
- yE) J2
a^PnicosEi)
n>4
+ Kk-3vE(y
«"" 2 ^n(cos-Ei)
- VE) Y n>2
+ Kk~3n§(y
-ys)Y
T^+rPnicosSx) n>2 'S
an~2 —
3
+ Kk~ nM{y
^+TP«(COS
- VM) Y n>4
+ Kk~3
Y
=
I)
M VEA
^
+ (V-
' EA
Ae{s,Pi,-..,Pk}
z+z
M
V
VA) Y i>2
an~2^
^nTlPnicOsAi)
r
A
C(2l)x + C(22)y + C(23)z + C(25)y + C(26)z + C{27) + Kk~3{\ - nM)z Y
an-2Pn(cosEi)
n>3 3
+ Kk~ fj,Ez
n 2
Y
a ~ Pn(cos
Ei)
n>2
+ Kk~3n-S{z
- zs) Y
an
2 — •^+TPn(cosS1)
>2rS
+ Kk
an~2^ (iMzY-^+rPn(cosM1)
3
n>3
r
M
55
56
The Equations
of Motion
ra-2
Kk~3
M - ^ + (z~ ZA) J2 i+r p »( C0S ^i)
Yl
' EA
Ae{s,pu...,pk}
>2
T
A
where Pn(a) = - —
Pn-2(a),
and C(iy
=
C(l) + (Kk-3-l)(-l
W)
=
C(2) + (Kk-3-1)3(1
+ 3x2E), -2nM)xEyE, 3
0(7) CjU)
= =
C(7)+Kk- [(l-2fiM)xE-fiMXEM} C ( l l ) + (Kk~3 - 1)3(1 2m)xEyE,
g(12)
=
C(12) + ( ^ f c - 3 - l ) ( - l + 32/|),
C(17)
=
C(17) + i r * - 3 y s ,
(7(23)
=
C(23) + (A'A:-3 - 1),
and, in fact, ?M is equal to 1 in the normalized system. Furthermore in the expression of the Hamiltonian given in section 4.3 there appear the following expressions: E2 - A, E2 + F2 - B, F2 - C and EF + D. It turns out that those expressions are identically zero. Indeed, using the definition of the variables, A, B, C, D, E, F we have E2-A
=
(^cos(5 + J2/(6icos(5))2 (6»2cos2<5 + (52)/(6/cos(5)2
- (02 cos2 S + S2) = 0.
For the other expressions the checks are carried out in a similar way. The coefficients C(j), C(j) and the ones coming from the effect of Sun and planets are computed by the routine FUN4. We introduce the following notation for further use: 50(1) 50(2) 50(3) 50(4) 50(5)
= = = = =
Kk~3 - 1, 0(1), 0(2), 0(3), 0(4),
50(6) = 0(5), 50(7) = C(7) 50(8) 0(11), 50(9) = 0(12), 50(10) = 0(13),
50(11) = 0(16), 50(12) = 0(17), 50(13) = 0(23),
for terms and coefficients coming from the noncircular motion of the Moon, and we expand the term K k - ' f l s - T t - ) r
s „>i
\rsJ
PnicOsS,),
Fourier Analysis:
The Relevant Frequencies and the Related Coefficients
57
accounting for part of the contribution of the Sun to the Hamiltonian as constant +SOS(l)x +SOS(6)xz
+ SOS(2)y + SOS(3)z + SOS(4)x2 2
+ SOS(7)y
+ 2
+ SOS(8)yz + SOS{9)z
SOS(5)xy + SOS(10)x3
....
The contributions of the planets can be expanded in a similar way. Then the contribution of the Sun to the equations is of the following form: 1 s t equation: 2 n d equation: 3
rd
equation:
5 0 5 ( 1 ) + 2SOS(4)x + S0S{5)y
+ S0S(6)z
5 0 5 ( 2 ) + 5 0 5 ( 5 ) z + 2SOS(7)y +
+
3SOS(10)x2
S0S(8)z
5 0 5 ( 3 ) + 505(6):r + SOS(8)y + 2SOS(9)z + ... ,
where 505(1)
=
505(1)
-Kk~3fiS^, r
ES
505(2)
3
SOS{2)-Kk- ns^§1,
=
r
ES
505(3)
=
505(3)
3
-Kk~ ns^-, r
ES
which is quite useful because of the cancellations (i.e. SOS(j) , j — 1,2,3 are quite large but SOS(j), j = 1,2,3, are rather small). A Fourier analysis of the functions SO(i), i = 1 , . . . , 13 and SOS(i), i = 1 , . . . , 9 has been performed. Other functions SOS(i), i > 10 have also been computed but the fact that in the denominator appears at least r^s makes them negligible. The Fourier analysis is carried out in two steps. First we have performed several FFT in order to identify the relevant frequencies. Some of them appear both in the SO(i) functions and in the SOS(i) ones. The frequencies retained are, in principle, the ones such that the peak of the modulus of the discrete Fourier Transform is greater than 10~ 4 . The unit frequency has been taken equal to the frequency of the mean longitude of the Moon. Then the frequencies have been checked against linear combinations of the four more important terms appearing in the motion of the Moon (the motion of the EarthMoon barycenter perihelion has been neglected). Let v be one of the obtained frequencies. We look for expressions of the type: v = miv (1) + n 2 z/ 2 ) + n 3 ^ ( 3 ) +
niV^\
where VW
= frequency of the mean longitude of the Moon = 1,
1/(2)
=
frequency of the mean longitude of lunar perigee,
3
z/ >
=
frequency of the mean longitude of the ascending node of the Moon,
VW
=
frequency of the mean elongation of the Sun.
58
The Equations of Motion
The FFT has been sidereal revolutions of centuries since 1900.0 appear also the phases
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
carried out using 2 16 points in a time interval equal to 2 1 3 the Moon centered at the epoch r = 1 — 0.5/36525 Julian . The results obtained are given in Table 4.1 where there at the epoch.
Frequency VJ 0.1326984719029936D+0 0.7840427350868880D+0 0.8463731326263370D+0 0.8588467495676956D+0 0.9167412069898816D+0 0.9251959855191924D+0 0.9915452214706892D+0 0.1004018838412048D+1 0.1016492455353406D+1 0.1066349235951497D+1 0.1717693499135391D+1 0.1775587956557577D+1 0.1850391971038385D+1 0.1983090442941378D+1 0.2709238720606080D+1 0.2841937192509074D+1 0.2974635664412068D+1 0.3700783942076770D+1 0.3833482413979763D+1 0.3845956030921122D+1 0.3978654502824115D+1
Phase 4>j 0.5820858844887022D+0 0.3056181124678582D+1 0.2388511629531575D+1 0.1659447913253431D+1 0.3638267009167057D+1 0.5092083509087575D+1 0.2241533797742133D+1 0.1512470081463988D+1 0.7834063651858438D+0 0.8448005863172090D+0 0.3318895826506862D+1 0.5297714922420715D+1 0.3900981710995564D+1 0.4483067595484266D+1 0.5560429624248995D+1 0.6142515508737697D+1 0.4414160860468126D+0 0.1518778114811541D+1 0.2100863999300243D+1 0.1371800283021871D+1 0.1953886167510574D+1
m 2 -2 -1 -1 0 0 1 1 1 2 -2 -1 0 2 -1 1 3 0 2 2 4
n2 -2 1 0 1 -1 0 -1 0 1 -1 2 0 0 -2 1 -1 -3 0 -2 -1 -3
"3
714
0 0 1 0 0 0 0 -1 -2 0 0 0 0 0 0 0 0 0 0 -1 -1
-2 3 2 2 1 1 0 0 0 -1 4 3 2 0 4 2 0 4 2 2 0
Table 4.1 Main frequencies appearing in the Fourier analysis and its identification as a linear combination of the four fundamental ones.
The normalized time is taken with origin at the epoch r and such that the sidereal period of the Moon is equal to 27r. Hence, if JD50 is the Julian date counted from 1950.0 (and then r = 18 262 JD50) we have the relation TN = ( J D 5 0 - 1 8 2 6 2 ) n M , where TN (sidereal) When by means
stands for the normalized time and UM for the mean motion of the Moon in rad/day (note UM = v^ letting aside the units). the frequencies are known it is possible to obtain the related coefficients of the Fourier integral 1
-J
rto+T
e-^f(t)dt,
where the time interval T is a multiple of the period associated to the frequency. The integral has been discretized using the trapezoidal rule. Several time intervals and steps have been used in the numerical integration to allow for interval checks.
Fourier Analysis:
The Relevant Frequencies and the Related Coefficients
59
The results for the functions SO(i), i = 1 , . . . , 13 are given in Tables 4.2 and 4.3, and those of the functions SOS(i), i = 1 , . . . , 9 in Table 4.4. We remark that in fact we have only retained terms in SO(i) giving an amplitude bigger than 5 • 10~ 4 . This is done to avoid an unmanageable number of terms. This leads to the conclusion that all the planets can be skipped from now on, as well as the effects of the solar radiation pressure and the spherical terms coming from the Earth and the Moon. Tables 4.2, 4.3 and 4.4 give the results for the point £4. We remark that the frequencies 6, 13 and 18 are multiples of the frequency of the mean elongation of the Sun, the one which appeared in the bicircular problem. Even keeping only the terms with absolute value of the modulus greater than 5 • 10~ 4 the following number of terms is found: - Terms coming from the noncircular motion of the Moon around the Earth: 142, among them 27 with the synodical period of the Sun, - Terms coming from the Sun: 34, among them 15 with the synodical period of the Sun. Those 27+15 periodic terms are the ones which have been used to find the periodic solutions of the intermediate equations which shall be presented in the next chapter.
t
+
M
II
o
B
o'
CO
•a
35
CD
O o
to
a;
h-'
h-»
o o
h-» 4^
h-'
00
h—* h-'
o
h-»
h-1
o
o
o
o
o
o
i—'
00
o
h- l
O
CO
as
O
1—*
1 o o 4^ o
to
-«j oi
4*. to
i—1 ^•i
Ol
as
o to o
h-!
-J
as
Ol
4*.
to
1—1
to to to to to
Ol
t— l
o o
as
l—»
Cn
O
O
\—L
to as
o
IO
h-'
h-*
-v|
t—I
o o
H-* Ol
Cn
O
O
4^
•—'
Ol
(—1
Ol
t—*
Ol
1
o o CO
h->
1
^1
Ol
as
Ol
l
CO
~4
h-i
1 Ol
o
o o o as to o
to to o
1—»
Ol
h
O
O
4
O
O
O
ts
*
a
h-*
CO
1 1 o a
as
O
Cn
Ol
1 o o o h-1 o o o o to o -a o o o o o
1 1 1 1 to 1 o to o 00 CO o o -a 0 0 -a 0 0 O l 4^ h-
to 1 1 t—' 1 *. 4^ to o o o o - 4 H-» o 0 0 o o 0 0 C O as O l to 0 0 as t—» to to to o
t—' i—'
CO
IO
1 1 to 1 to o o o o to o o 4^ o 0 0 h-» C n C O to as to as to o
h-1
(—' 1—'
as as as as as as as as as
4^ to to
OS
as
I-1
h->
to to
1 o o o o o o o o o o o
-a to o
I—1
o to o o o o o o o o 00 o o o
O to o
Ol
h-» Cn
H-* l—i ~J OS
1—>
OS
a> as
o
CD
h-1
o to
O
as
h-»
-~4
1—>
o to as 1—> o o to o 1 to to o o 0 0 0 0 to I—
h->
oo
to
CO
h-1
to to to to to to to to to to
1
o
to
I-" O C O O to C O
t—*
to to to
4^ to
W
OS Cn
CO
Fourier Analysis:
i 7
7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 Table 4.3
3
4 13 16 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16
Au
The Relevant Frequencies and the Related Coefficients
• 103
Bn • 103
0.5
0.8
-4.1 -0.5
-7.1 -0.8
6.0 2.2 0.4
0.0 -0.3
39.9
-0.8
0.3
1.2 0.0
-1.0 208.4 -0.8 -0.2
0.0 33.7 17.0
1.0 5.3 1.4 10.7
3.6 0.7 71.9
0.5 -1.9 370.5 -1.4 -0.4
1.1 0.8 51.6 30.3
1.6 8.5
1.7
-0.6
0.0 1.0 2.3 8.3 0.0 0.2 0.9 0.0 0.0 -0.4
3.0 0.1 2.2 0.0 -0.6 -0.1
1.7 0.0 3.4 0.0 0.0 0.0 0.0
i
3
9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13
17 18 19 3 8 20 21 3 8 20 21 4 7 11 13 16 18 19 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17
Mi • 103
Bn • 103
2.4 1.0 1.4 0.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.4
-0.7
3.9 1.2 0.0 0.0 0.0 0.0 0.9 0.5 0.6
-1.4
2.0 0.6 -0.5 -0.1
-7.1 -0.8
0.2 4.1 0.4
0.6 0.7
-0.1 -0.2
-6.4 -2.2 -0.4 -39.5 -0.3
0.0 0.2
1.1 -218.3
0.8 0.2 -0.8 -53.5 -20.8 -1.7 -8.8 -1.8
-1.6
0.0 -1.2
0.0 0.4 0.0 -1.0 -3.4
0.0 0.0 0.0 0.0 0.0
Coefficients of the Fourier expansion of SO(i) (continuation of Table 4.2).
61
62
The Equations
i
1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 Table 4.4
J Aij • 103 0 1.4 4 -0.5 12 0.5 4.1 13 16 0.5 0.1 18 0 2.5 4 0.8 12 0.2 -7.2 13 16 -0.8 18 -0.1 0.6 3 8 -0.6 0 1.4 4 -0.5 4.1 13
Bij
• 10 3
0.0 0.8 0.2 -7.1 -0.8 -0.1
0.0 0.5 -0.5 -4.1 -0.5 -0.1
0.4 -0.4
0.0 0.0 0.0
of Motion
i
4 4 5 5 5 5 5 6 6 7 7 7 7 7 8 8 9
J 16 18 4 12 13 16 18 3 8 0 4 13 16 18 3 8 0
Coefficients of the Fourier expansion of SOS(i)
Aj
• 10 3
0.5 0.1 0.0 0.5 0.0 0.0 0.0 0.0 0.0 1.4 0.5 -4.1 -0.5 -0.1
0.7 -0.7 -2.8
B^
• 10 3
0.0 0.0 0.9 -0.1 -8.3 -0.9 -0.1
0.7 -0.7
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
= ^ • Aij cos(vjt + 4>j) + Bij sm{yjt +
4.6
Concrete Expansions of the Hamiltonian and the Functions
As mentioned in section 4.4 the expansion of the terms (l-2acosAi
+a2)-l>2,
with A either Earth or Moon, is carried out by a suitable routine. However at this point we introduce a piece of information that in fact, will only be available in section 5.3 of the next chapter. This is the real size of the periodic orbits obtained from the previous equations when only the frequencies 6, 13 and 18 retained. These orbits have as period the synodical period of the Sun in the Earth-Moon system (or a multiple of it). They can be considered as perturbations of the orbits of the bicircular problem. But the change in size is big, leading to orbits of the types A, B, F and G (see Chapter 2) far away from Lii5. The terms mentioned at the beginning of this section come purely from the RTBP. We analyze now the number of terms to retain in the expressions of the Hamiltonian and in the equations. The worst case corresponds obviously to the equations, because a term like xk in the Hamiltonian produces a term kxk~1 in the equations and, hence, it'is multiplied in modulus by k/\x\. We have selected 11 points which are close to the periodic orbit of type A of the intermediate equations. As this orbit is planar we have selected a small uniform
Concrete Expansions
of the Hamiltonian
and the
Functions
63
value of z (z = 0.1) in the computations but the contribution of z is not significant. The selected points have normalized coordinates as follows: (-0.8,0.1), (0.4,-0.3), (-0.6,-0.1).
(-0.6,0.3), (0.2,-0.4),
(-0.4,0.4), (0..-0.4),
(0.0,0.4), (-0.2,-0.3),
(0.2,0.2), (-0.4,-0.2),
In the Lagrangian or in the Hamiltonian, there appears the function 1 - V-M (l + x + sVSy + a2)1/2
(1-x
RA£ + sVSy +
a2)1/2
(4.1)
with s = 1 for L 4 , s = - 1 for L 5 and a2 — x2 + y2 + z2. To have some idea about the suitable order to be reached in the expansions we have computed: i) The value of (4.1) directly, ii) The value of (4.1) expanded in powers of the coordinates up to a certain order, iii) The same as in ii) but skipping those terms whose absolute value is less than a given threshold. The objective is to obtain small differences i)—iii) but keeping as few terms as possible. The study has been done for the 11 points above mentioned and looking for the worse case. We show a sample of results in Table 4.5.
Order 25
30
35 40 Table 4.5
Threshold 2.0 xlO" 4 1.2 xlO" 4 1.0 x l 0 ~ 4 0.8 xlO" 4 0.6 xlO" 4 2.0 xlO" 4 1.2 xlO" 4 1.0 x l O - 4 0.8 x l 0 ~ 4 0.6 xlO" 4 1.0 xlO" 4 1.0 xlO" 4
i)-iii) 19.5 xlO" 4 6.4 xlO" 4 5.0 xlO" 4 6.0 xlO" 4 8.0 xlO" 4 18.0 xlO" 4 5.0 x l O - 4 2.7 x l O - 4 5.9 xlO" 4 3.6 x l O - 4 3.0 xlO" 4 2.7 xlO" 4
Number of terms to retain 314 347 363 392 422 372 422 446 484 521 510 556
Results of the different truncations of the expansions of function (4.1).
If we are satisfied with errors less than or equal to 5 • 1 0 - 4 one should use at least 363 terms in the expansion of the Hamiltonian. This is much worse in the equations.
64
The Equations
Order 40
35
30 32 34 36 38 Table 4.6
Threshold 1.0 x 10" 4 1.2 x 10- 4 1.5 x 10" 4 2.0 x 10" 4 3.0 x 10- 4 4.0 x 10~ 4 5.0 x 10- 4 6.0 x 10" 4 8.0 x 10" 4 10.0 x 10- 4 2.0 x 10" 4 3.0 x 10" 4 4.0 x 10~ 4 5.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4
of Motion
Maximum error 1.7 x 10~ 3 1.9 x 10" 3 1.7 x 10" 3 1.8 x 10~ 3 2.4 x 10~ 3 2.9 x 10" 3 3.4 x 10~ 3 5.7 x 10" 3 5.0 x 10" 3 6.6 x l O - 3 3.1 x 10~ 3 1.7 x 10~ 3 3.6 x 10~ 3 3.4 x 10" 3 6.2 x 10~ 3 7.5 x 10" 3 3.2 x 10" 3 2.2 x 10" 3 3.0 x 10" 3
Total number of terms to retain 1428 1375 1320 1233 1140 1086 1033 1000 943 887 1037 962 918 879 776 853 925 994 1070
Bounds of the errors of the expanded equations.
We have expanded the equations in powers of x, y, z. Then we have compared the direct evaluation of the derivatives of (4.1) against the evaluation of the expansion keeping only the terms greater than some threshold in absolute value. The maximum of the differences at the 11 preselected points is displayed. We remark that the terms retained in the expansions are not the same for the 11 points. So we count the terms that have been used at least in one of the points and give the total number of terms to be used. We merely note that the expansions up to order 40 (the maximum order allowed in our program) require 6390 terms. Some unavoidable error is due to that truncation at order 40. We present some results in Table 4.6. Hence one of the most appealing parameters are: order 35, threshold 3.0 x 10~ 4 giving errors as small as 1.7 x 10~ 3 and requiring 962 terms. Even taking these "excellent" parameters the number of terms in the equations becomes unmanageable. As a conclusion, the approach which uses the expansions of the Hamiltonian or of the equations is not feasible. Hence we should look for other forms of the equations which are more useful for the applications. Furthermore we wish to comment the results of Kamel in [19]. There, a Lie series method is used to study the motion near 1*4,5 in the Earth-Moon system, taking into account some of the terms of the lunar and solar motion. The perturbation method is used to eliminate short and long
Simplified Normalized Equations.
Tests
65
periodic terms of the Hamiltonian. At the end, a formally integrable Hamiltonian is found and the author looks for fixed points. Those will give rise, going back in the transformation, to quasi-periodic solutions. It is found that there are fixed points, which appear if the theory is carried out to a given order, but not if the order is increased. This can be due to the big amount of terms required to have convergence. We will also refer to convergence problems in Chapters 6 and 7. 4.7
Simplified Normalized Equations. Tests
Prom what has been said in section 4.6 it is concluded that a simpler form should be used for the equations. In this new form we retain the terms called SO(i), SOS(i) in section 4.5 and the terms coming from the RTBP are left as they are (with respect to the point L4 or L5 taken as origin and multiplied by the factor Kk~3). Let us introduce some auxiliary functions P(i), i = 1, • • •, 20 which account for the terms appearing in SO(i) and SOS(i) according to the following definition: P(l) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) P(ll) P(12) P(13) P(14) P(15) P(16) P(17) P(18) P(19) P(20)
= = = = = = = = = = = = = = = = = =
5 0 5 ( 1 ) + 50(7), 1 + 2 5 0 5 ( 4 ) + 0.2550(1)+ 50(2), 5 0 5 ( 5 ) + 50(3) + a 50(1), 5 0 5 ( 6 ) + 50(4), 50(5), 2 + 50(6), 1 + 50(1), 5 0 5 ( 2 ) + 50(12), 5 0 5 ( 5 ) + 50(8) + a 50(1), 1 + 2505(7)+ 50(9)-1.2550(1), 5 0 5 ( 8 ) + 50(10), -2-50(6), 50(5), 50(11), SOS(3) + yESO(lQ)-xESO(4), 5 0 5 ( 6 ) + 50(4), 505(8)-50(10), 2 5 0 5 ( 9 ) + 5 0 ( 1 3 ) + 50(1), -50(11), S0(5),
where a stands for expressed as
Q^MXEVE-
With these definitions the functions P(i) can be
m
m
P(i) = Aito + ^2 Ai,j cos6j + ^2 Bid sindj, j=i
j=i
66
The Equations
of Motion
where Oj = Ujtn + j, Vj, fa being the frequency and phase as given in section 4.5 and tn the normalized time. The different values of i, j , Aij, Bij, are listed in Tables 4.7, 4.8 and 4.9. With this notation the equations of motion written in a simplified form, to be used from now on are (we shall denote them by S.E.):
=
P(7)
X-XE •—^
(1 -
X + XE 3 MM -
MM)
' PE
.
.
xE{l
-
2/J.M)
' PM
+P(1) + P(2)x + P(3)y + P(4)z + P(5)± + P(6)y, V
=
P(7)
y-VE,, •—^ U ' PE
—
. VM) 1
y-yE 3 MM PM
+P(8) + P(9)x + P(10)y + P(ll)z =
^(7)
--o—(l-MM) 1 PE
-
VE
+ P(12)x + P(13)y + P(14)i,
-fJ-M ' PM
+P(15) + P(16)x + P(17)y + P(18)z + P(19)y + P(20)z, where rps, TPM, denote the distances from the particle to the Earth and Moon, respectively, given by rpE = (x - xE)2 + (y- yEf We recall (xE,yB) 4.7.1
+ z2,
rPM = (x + xEf
= (-1/2,--y/3/2) for L4 and {xE,yE)
Tests of the Simplified
Normalized
+ (y -
VE?
+ z2.
= ( - 1 / 2 , ^ 3 / 2 ) for L5.
Equations
To be sure that the simplified system is meaningful we have written a test program which computes the vector field on the particle in ecliptic coordinates and then performs the transformation to the normalized ones as described in section 4.1. On the other hand we can evaluate directly the simplified vector field (already in normalized coordinates). Then we can select either a given point in the neighborhood of L4 in the phase space (introducing the position and velocity) or some points from the periodic orbits of the intermediate system, to be found in the next chapter. The relevant angle in these orbits is the mean elongation of the Sun which is expressed, as stated before, as 6 = 0.9251959855 • TN + 5.0920835091. We recall that TN = (JD50 - 18 262)/n M , JD50 = Julian Date since 1950.0 and nM mean motion of the Moon longitude. The biggest difference between the two vector fields in normalized coordinates, both in absolute terms and relative to the modulus of the vector field, is of the order 10~ 3 . The vector fields have been computed on points along the periodic orbit called A, following the notation of Chapter 5. Hence it seems reasonable to believe that the orbits found using the simplified equations can be easily modified to satisfy the complete system of equations.
Simplified Normalized Equations.
i 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 5 5 5 5 5 5 5
j 0 4 12 13 16 18 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 18 20 2 4 5 6 7 10 12
Ajj 0.140000e-2 0.000000e+0 0.500000e-3 0.000000e+0 0.000000e+0 0.100000e-3 0.100757e+l 0.142500e-2 0.275000e-3 0.314750e-l 0.250000e-3 -0.800000e-3 0.165000e+0 -0.650000e-3 -0.150000e-3 0.800000e-3 0.300000e-3 0.266500e-l 0.134500e-l 0.200000e-3 0.425000e-2 0.107500e-2 0.100000e-2 0.100000e-2 0.424867e-4 0.451547e-4 -0.802668e-4 -0.280141e-4 0.464887e-4 0.140448e-4 0.137926e-4 -0.394663e-4 0.535112e-4 0.992977e-3 -0.169897e-4 0.147495e-4 -0.140448e-4 -0.237352e-4 0.568839e-5 0.000000e+0 0.000000e+0 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0 -0.200000e-3 0.600000e-3 0.190000e-2 Table 4.7
Bij 0.000000e+0 0.160000e-2 0.200000e-3 -0.142000e-l -0.160000e-2 -0.100000e-3 0.000000e+0 -0.500000e-4 0.132500e-2 0.000000e+0 0.950000e-3 0.000000e+0 -0.275000e-3 0.000000e+0 0.800000e-3 0.000000e+0 0.122500e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.535112e-4 0.521771e-4 0.180000e-2 -0.675560e-4 0.000000e+0 0.180266e-3 0.000000e+0 -0.140448e-4 -0.154845e-3 -0.165000e-l 0.000000e+0 -0.200000e-3 -0.180000e-2 0.000000e+0 -0.100000e-3 -0.100000e-2 -0.100000e-3 -0.169000e-l -0.100O00e-3 0.500000e-3 -0.107900e+0 0.100000e-3 -0.400000e-3
i 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9
j 13 14 15 16 17 0 2 4 5 6 7 9 10 11 12 13 14 15 16 17 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 0 4 7 11 12 13 16 18 19 0 1 2 4
Tests
Mi 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.200000e+l 0.300000e-3 0.382000e-l 0.300000e-3 -0.110000e-2 0.217600e+0 -0.800000e-3 -0.200000e-3 0.500000e-3 0.600000e-3 0.429000e-l 0.148000e-l 0.100000e-2 0.530000e-2 0.100000e-2 0.100470e+l 0.170000e-2 0.300000e-3 0.315000e-l 0.200000e-3 -0.800000e-3 0.164400e+0 -0.600000e-3 -0.200000e-3 0.400000e-3 0.266000e-l 0.134000e-l 0.800000e-3 0.420000e-2 0.110000e-2 0.250000e-2 0.170000e-2 0.500000e-3 0.600000e-3 0.200000e-3 -0.143000e-l -0.160000e-2 0.500000e-3 0.700000e-3 0.424867e-4 0.451547e-4 0.197331e-4 -0.280141e-4
67
^±L -0.295000e-l -0.880000e-2 -0.700000e-3 -0.380000e-2 -0.700000e-3 0.000000e+0 0.150000e-2 0.000000e+0 0.120000e-2 0.000000e+0 -0.400000e-3 0.000000e+0 0.100000e-2 0.000000e+0 0.280000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.200000e-3 0.130000e-2 0.000000e+0 0.100000e-2 0.000000e+0 -0.300000e-3 0.000000e+0 0.800000e-3 0.170000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.100000e-3 0.200000e-3 -0.500000e-3 0.000000e+0 -0.100000e-3 -0.200000e-3 -0.200000e-3 0.000000e+0 -0.464887e-4 0.521771e-4 0.100000e-3
Cosine and sine coefficients of the functions P(i), i = 1 , . . . , 20.
68
The Equations i
3
Ajj
Jjjj
9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12
5 6 7 9 10 12 13 14 15 16 17 18 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 3 8 20 21 0 2 4 5 6 7 9 10 11 12 13 14 15 16
0.464887e-4 0.140448e-4 0.137926e-4 -0.394663e-4 0.535112e-4 -0.702240e-5 -0.169897e-4 0.147495e-4 -0.140448e-4 -0.237352e-4 0.568839e-5 0.000000e+0 0.100762e+l 0.147500e-2 0.325000e-3 0.335250e-l 0.250000e-3 -0.900000e-3 0.165000e+0 -0.650000e-3 -0.150000e-3 0.110000e-2 0.300000e-3 0.101500e-l 0.135500e-l 0.600000e-3 0.225000e-2 0.102500e-2 0.800000e-3 0.140000e-2 0.130000e-2 -0.140000e-2 0.390000e-2 0.120000e-2 -0.200000e+l -0.300000e-3 -0.382000e-l -0.300000e-3 0.110000e-2 -0.217600e+0 0.800000e-3 0.200000e-3 -0.500000e-3 -0.600000e-3 -0.429000e-l -0.148000e-l -0.100000e-2 -0.530000e-2
-0.675560e-4 0.000000e+0 -0.219733e-3 0.000000e+0 -0.140448e-4 0.451547e-4 0.000000e+0 0.000000e+0 0.200000e-3 0.000000e+0 0.000000e+0 -0.100000e-3 0.000000e+0 -0.150000e-3 0.137500e-2 0.100000e-3 0.950000e-3 0.000000e+0 -0.225000e-3 -0.100000e-3 0.700000e-3 0.000000e+0 0.127500e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.150000e-2 0.000000e+0 -0.120000e-2 0.000000e+0 0.400000e-3 0.000000e+0 -0.100000e-2 0.000000e+0 -0.280000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0
of Motion i
12 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 15 15 15 15 16 17 17 17 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 20 20 20 20
Table 4.8 Cosine and sine coefficients of the P(i), Table 4.7.)
j
Ajj
r>ij
17 2 4 5 6 7 10 12 13 14 15 16 17 3 8 20 21 3 8 20 21 20 3 20 21 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 3 8 20 21 2 4 5 6
-0.100000e-2 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0 -0.200000e-3 0.600000e-3 0.190000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.803847e-4 0.621778e-5 -0.337749e-2 -0.103923e-2 0.000000e+0 0.100000e-3 -0.390000e-2 -0.120000e-2 -0.730000e-2 -0.500000e-3 -0.100000e-3 -0.800000e-2 -0.100000e-3 0.300000e-3 -0.539000e-l 0.200000e-3 0.000000e+0 -0.400000e-3 -0.269000e-l -0.740000e-2 -0.900000e-3 -0.460000e-2 -0.700000e-3 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0
0.000000e+0 -0.100000e-3 -0.169000e-l -0.100000e-3 0.500000e-3 -0.107900e+0 0.100000e-3 -0.400000e-3 -0.295000e-l -0.880000e-2 -0.700000e-3 -0.380000e-2 -0.700000e-3 0.140000e-2 -0.140000e-2 0.200000e-2 0.600000e-3 0.500000e-4 -0.500000e-4 -0.500000e-3 0.000000e+0 -0.100000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.300000e-3 0.000000e+0 -0.200000e-3 0.000000e+0 0.100000e-3 0.000000e+0 -0.200000e-3 -0.170000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.140000e-2 0.140000e-2 -0.200000e-2 -0.600000e-3 -0.100000e-3 -0.169000e-l -0.100000e-3 0.500000e-3
i = 1 , . . . , 2 0 functions.
(Continuation of
Simplified Normalized
l
J
•'^•ij
20 20 20 20
7 10 12 13
-0.200000e-3 0.600000e-3 0.190000e-2 0.000000e+0
Equations.
ij
-0.107900e+0 0.100000e-3 -0.400000e-3 -0.295000e-l
20 20 20 20
Table 4.9 Cosine and sine coefficients of the P(i), Table 4.8.)
Tests
69
J
~^ij
14 15 16 17
0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0
i = 1 , . . . , 2 0 functions.
ij
-0.880000e- -2 -0.700000e- -3 -0.380000e- -2 -0.700000e- -3 (Continuation of
This page is intentionally left blank
Chapter 5
Periodic Orbits of Some Intermediate Equations
The aim of this chapter is first to describe the model closer to the simplified equations that still has periodic orbits trained by the Sun, i.e., whose period is the one of the Sun in the Earth-Moon system. Second, to determine those periodic orbits and relate them to those of the bicircular problem. A continuation method has been used.
5.1
Equations of Motion for the Computation of Intermediate Periodic Orbits
We look for equations of motion such that they are as close as possible to the simplified ones but still have periodic solutions around L4 or L§ in a suitable region (not too far from the libration point). What we propose is merely to keep only those terms in the simplified equations of Chapter 4, section 4.7, which are either independent of time or have the same period of the solar elongation. This means to retain in the time-dependent functions P(i) the independent term and the frequencies number 6, 13 and 18. Before going into the details of the procedure we consider worthy a short discussion. Keeping those periodic terms, the passage to the simplified equations is done adding an important number of terms. Furthermore, some of those terms are much more important than the retained periodic ones. For instance, looking at the table giving the functions SO{i) (Chapter 4, section 4.5) there are only 4 frequencies with coefficients larger than 1 0 - 2 . They correspond to frequencies number 4, 7, 13 and 14. It is easily checked that frequency number 14 is the double of frequency number 7, and that frequency number 4 is the difference between frequencies numbers 13 and 7. Hence, the really important frequency besides the retained ones (6, 13, 18) is number 7. The related coefficients are of the order of 1 0 - 1 . As it is natural, this frequency is the mean motion of the mean anomaly of the Moon and the coefficients are closely related to the lunar eccentricity. After realizing this fact it looks natural to consider the following approach: 71
72
Periodic Orbits of Some Intermediate
Equations
a) To consider as intermediate equations the simplified ones keeping only the periodic terms whose period is the one of the mean anomaly of the Moon, that is, the terms with frequencies number 7, 14 and 17. b) These equations should be rather close to the planar elliptic RTBP. c) To look for periodic orbits of the planar elliptic RTBP. Their period should be the anomalistic period of the Moon. d) When those orbits are available, we consider the perturbations coming from the previously skipped terms. As they are relatively unimportant, the effect should be small. However, the program just described has some difficulties, which make clear that the terms to retain are those with period equal to the lunar synodical one: (1) To obtain a periodic orbit of the planar elliptic RTBP the natural process is: (a) To obtain, for the selected value of /i, a 27r-periodic orbit. As we look for orbits around L4 it should belong to the short period family. (b) When this orbit is available to perform natural continuation with respect to the eccentricity of the primaries. (2) As it is well-known (see, for instance [10]) for the Earth-Moon system, considered as a planar circular RTBP, the short periodic family contains one orbit with period 2n. But this orbit is placed between the quadruplication orbit (for which the short and long periodic families meet) and the orbit labeled B45, which belongs to the short periodic families associated to L4 and L5 (and which is also an orbit of the Lyapunov family around L3). This orbit is extremely large for our purposes. (3) Furthermore, looking at the simulations done in Chapter 3 using the full solar system or a nice approximation to it, the "natural" period of the orbits, obtained by dividing the time interval between the number of revolutions around L4 or L5 is placed between 29 and 30 days. This is close to the synodical period, roughly 29 d 12 /l 44 m . What precedes makes reasonable our election of the intermediate equations. However we wish also to relate them to the RTBP and to the bicircular problem. Hence we write down the intermediate equations (I.E.) as follows: q
=
acceleration of the RTBP +pi x terms coming from the noncircular motion of the Moon +P2 x terms coming from the Sun + (1 — P2)
x
terms coming from the Sun in the bicircular problem,
The Periodic Orbits Around the Triangular
Points
73
where q stands for one of the coordinates (x, y, z), and pi and p2 are continuation parameters ranging from 0 to 1. Then for p1 — p2 = 0, we obtain the bicircular problem, for pi — P2 = 1, we obtain the intermediate equations. The experience has shown that it is better to use 2 parameters rather than using a single one p. At this point we have already introduced all the systems of equations to be used in this work. The logical sequence for increasing difficulty is:
RTBP —> Bicircular Problem —> Intermediate Equations —>
—> Simplified Equations —> Real Equations.
5.2
Obtaining the Periodic Orbits Around the Triangular Libration Points for the Intermediate Equations
We have set up a program, TRIANG, for the computation of those periodic orbits. The leading idea is to start with the periodic orbits of the bicircular problem of types A, B, C, F and G and to perform continuation till the I.E. First of all we remark that, at least with the tolerance which we are using in the coefficients of SO(i), SOS(i), the change of variables (x,y,z,x,y,z,t)
—>
(x,-y,z,-x,y,-z,~t),
leaves the equations invariant. Hence, we only need to work with periodic orbits around L4, the ones around L5 being obtained by the symmetry. The program starts at some initial conditions. After one period, which is equal to k times the synodical period of the Moon, with k — 1 for orbits A, B and C and k = 3 for orbits F and G, a point is obtained. We require that this point coincides with the initial one in the phase space. The difficulties mentioned in the bicircular problem appear also here. That is, the system is a small periodic perturbation of an autonomous Hamiltonian system. Hence, at least two of the eigenvalues of the monodromy matrix D$T are close to 1. As Newton's method requires the inversion of D$T ~ I, this produces some problems unless the initial point, in Newton's method, is rather close to the solution. This means that even using some linear or quadratic extrapolation, when several orbits are available for different values of the active parameter pi, i = 1 or 2 the parameter step should be small (even less than 10- 2 ).
74
Periodic Orbits of Some Intermediate Equations periodic orbit of the intermediate equations (1.1)
(0.0) periodic orbit of the bicircular problem
(1.0)
Fig. 5.1 Evolution in the parameter space from the bicircular problem to the intermediate equations.
For the evaluation of the vector field, the coefficients SO(i) are multiplied by Pi, the SOS(i) by pi and the terms ms ,
as cos 6)
" -r 3 — \x
ps
m s
1
,
• a\
--3— {y + a s sinfj)
ms cos 8, a s ms al suit
r
PS
ms 3 'PS
^">
expressing the difference between the bicircular problem and the RTBP (see Chapter 2, section 2.2), are multiplied by (1 — P2). The variational equations are integrated simultaneously. This allows to apply Newton's method looking for the initial conditions and also to know the eigenvalues of the monodromy matrix, reflecting the stability properties of the orbit. The program also computes, optionally, the Fourier analysis of the orbit found. It is useful, for requirements of Chapter 7, to know the point of the orbit and the corresponding value of 9, when the projection of the orbit in the (x, y)-plane meets the segment joining L\ with the Earth. The periodic orbits for the I.E. have been found using different paths on the parameter plane (pi, p^). Small steps have to be used, especially with p\, due to the bad condition number of the linearized problem. However, the results obtained, to be presented in section 5.3, are independent of the followed path. The diagram in Figure 5.1 shows the paths in the parameter space.
5.3
Results and Comments
The orbits obtained by continuation of orbits A, B, C, E and F of the bicircular problem are given in the following pages as well as the corresponding plots. In the plots the origin is placed at L4 and the dots in the lower part correspond to the positions of Earth (left) and Moon (right). For comparison purposes we present also in the same plot the bicircular and intermediate periodic orbits of type A (note the
Results and
Comments
75
change in the sign of x: now the Moon is on the left and the Earth on the right). As it is natural we denote the periodic orbits by the same letters used in the bicircular case. At the light of the results, several comments should be done: a) Orbits A, B, F and G have increased the size in a significant way. For orbits F and G the maximum distance to L4 is close to the Earth-Moon distance. This increase has already been mentioned in Chapter 4, section 4.6, and produces problems of convergence in the expansions. In the plots each division corresponds to 0.1 normalized units, except in the magnification of orbit C where each division means 0.0025 units. b) The orbit C is quite small and for the I.E. looks like a periodic path traveled twice. c) The plots of orbits A and B and of F and G are almost indistinguishable except in what concerns the initial point (marked with a dot on the orbit). They are almost the same orbits but with a time delay of roughly 1/2 synodical period. d) The Fourier analysis of orbits A and B shows again the almost coincidence of both orbits with a suitable time shift. It can also be said, with the level of error in the equations, that the first 4 or 5 harmonics are enough to represent the orbits accurately. For these orbits the main harmonic is the first one giving as result that the shape is a distorted ellipse. For the orbit C the two first harmonics are enough and the dominant one is the second in agreement with b). e) The stability of the periodic orbits is quite close to that of the bicircular problem. The five orbits are now unstable but the instability of all of them (especially in what makes reference to orbits A and B) is not significant. The values of the modulus of the dominant eigenvalues of the monodromy matrix, scaled to the synodical period Ts, are 1.0011 for A and B, 1.1666 for C, 1.8012 for F and 1.8140 for G. Even for the worse case, G, to have an amplification of the errors by a factor of 1000 requires near 12 synodical periods (roughly 1 year).
76
Periodic Orbits of Some Intermediate
PERIODIC ORBIT
A
INITIAL CONDITIONS 0.3446348364766992 0.1131305748645592
Equations
AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O. -5.5680350668244804E-02 -0.3702379416714056
CONTINUATION PARAMETERS
1.000000000000
1.000000000000
PERIOD = 6.791193871917804 EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.8042860126098644, 0.5960657272228537) (-0.8042860126098644, ••0.5960657272228537) ( 0.9744318402714565, 0.2198111634920723) ( 0.9744318402714565,-•0.2198111634920723) ( 0.8315322727454194, 0.5554764435894994) ( 0.8315322727454194,-•0.5554764435894994)
Orbit A
Fig. 5.2
Periodic orbit A of the intermediate equations. LP. denotes the initial point for 0 = 0.
Results and
Comments
77
P.O. of the intermediate equations
* M
Fig. 5.3 Periodic orbit A of the bicircular and the intermediate equations. LP. denotes the initial point for 0 = 0.
78
Periodic Orbits of Some Intermediate Equations VARIABLE X COEFFICIENT OF COSINE
0 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-0 .1820373734425437D+00 0 .5473727418033956D+00 -0 .2612533555332382D-01 0 .3596130222497096D-02 0..2344569225597054D-02 -0,.4575103314521871D-03 -0 .1283596091750251D-03 0 .8115733431706914D-04 -0 .6006697781577388D-05 -0 .7998225804570571D-05 0..2902385293734465D-05 0..2671358749239722D-06 -0..4466534961041022D-06 0..8072182940984220D-07 0..3628311894839477D-07 -0..2085549226832394D-07 0..9006514180829629D-09 VARIABLE Y COEFFICIENT OF COSINE
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.3074625593793142D-01 -0.8108614861426111D-01 -0.1353602163256157D-01 0.9323372828908902D-02 -0.6696404904180181D-03 -0.6781769862557938D-03 0.2243659953483073D-03 0.1784110670955425D-04 -0.2865629165719000D-04 0.5521312671056611D-05 0.2034387741816418D-05 -0.1259385321574008D-05 0.6153485844562423D-07 0.1461539099788795D-06 -0.4661939029472928D-07 -0.6300820342498891D-08
16
0.8235503779228720D-08
COEFFICIENT OF SINE
0.1021536482035979D+00 0.1969261999112519D-01 -0.8006804264988614D-02 0.7371063032005168D-03 0.6051812116823743D-03 -0.2131976043251629D-03 -0.1378901338908217D-04 0.2723903977540330D-04 -0.5546731166285052D-05 -0.1915504731193367D-05 0.1225612300955243D-05 -0.6720753096896170D-07 -0.1407865417703950D-06 0.4621568855238718D-07 0.5862414195848801D-08 -0.8079707750053004D-08
COEFFICIENT OF SINE
-0.3279162252541900D+00 -0.3144486918061280D-01 -0.5644561891475041D-02 0.2620597115032109D-02 -0.5087362805546621D-03 -0.1465742159558731D-03 0.8538414503175204D-04 -0.5310073093758154D-05 -0.8427090278480657D-05 0.2955041510378497D-05 0.2963664177241574D-06 -0.4614450552628417D-06 0.8033867877525882D-07 0.3795713981675681D-07 -0.2116564775316567D-07 0.8161916230497732D-09
Table 5.1 Fourier coefficients of th x and y coordinates of orbit A.
Results and
PERIODIC ORBIT
B
INITIAL CONDITIONS
Comments
79
AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O.
-0.7582100971505726 -3.6463329908348591E-02
8.9087947614105868E-02 0.2717079806443636
CONTINUATION PARAMETERS
1.000000000000
PERIOD
1.000000000000
6.791193871917804
EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.8083524564940744, 0.5905803225873873) •0.5905803225873873) (-0.8083524564940744, • ( 0.9744672713141360, 0.2195441596004288) ( 0.9744672713141360,- 0.2195441596004288) ( 0.8314962415092812, 0.5555303775275825) ( 0.8314962415092812,-•0.5555303775275825)
Orbit B
Fig. 5.4
Periodic orbit B of the intermediate equations. LP. denotes the initial point for 8 = 0.
80
Periodic Orbits of Some Intermediate
VARIABLE X COEFFICIENT OF COSINE
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-0,.1829095046378868D+00 -0..5485886673345659D+00 -0,.2578065911946599D-01 -0,.3542603775514775D-02 0,. 2368594561928400D-02 0,.4520705863907640D-03 -0,. 1326625497700241D-03 -0,.8151340436942062D-04 -0..5460716124152157D-05 0,.8209793896144044D-05 0,.2881079098107230D-05 -0..3069937620521880D-06 -0,.4546225570443204D-06 -0..7676823204406415D-07 0..3859534953274974D-07 0., 2093160189132245D-07 0.. 5623238525211474D-09 VARIABLE X COEFFICIENT OF COSINE
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.3083494565531459D-01 0.8113074533596854D-01 -0.1375549583216786D-01 -0.9339186468253033D-02 -0.6443154890681684D-03 0.6885485971883162D-03 0.2234431872573132D-03 -0.1962622749851291D-04 -0.2901338989277210D-04 -0.5364937796184402D-05 0.2135853165996285D-05 0.1265255429078654D-05 0.4712719912378712D-07 -0.1510518557543697D-06 -0.4581665863319156D-07 0.7269941138508557D-08 0.8405390861468149D-08
Table 5.2
Equations
COEFFICIENT OF SINE
-0.1020528523635009D+00 0.1983618130474269D-01 0.8036535305076123D-02 0.7159895565795747D-03 -0.6149682407757302D-03 -0.2125403078480607D-03 0.1544898101060362D-04 0.2760222564002016D-04 0.5402964622344150D-05 -0.2014544666400152D-05 -0.1232061712601718D-05 -0.5335713298378076D-07 0.1456208343812417D-06 0.4547668119814228D-07 -0.6811210312841061D-08 -0.8251535600915437D-08
COEFFICIENT OF SINE
0.3284296996209018D+00 -0.3177683546152209D-01 0.5733817517258685D-02 0.2633667402750067D-02 0.5022993202481919D-03 -0.1512532488421611D-03 -0.8568456295413692D-04 -0.4721802148857870D-05 0.8641196629111997D-05 0.2930565983724559D-05 -0.3375223421981380D-06 -0.4694078630035761D-06 -0.7618065057130223D-07 0.4031175252572723D-07 0.2122615936322471D-07 0.4695181133461478D-09
Fourier coefficients of th x and y coordinates of orbit B.
Results and
PERIODIC ORBIT
C
INITIAL CONDITIONS
Comments
81
AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O.
-4.5395552497062299E-03 1.3665787366715615E-02 2.9866888241593641E-02 -3.8408486363138898E-03 CONTINUATION PARAMETERS PERIOD =
1.000000000000
1.000000000000
6.791193871917804
EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.4707137878243862, 0.8819740228870461) (-0.4707137878243862,- 0.8819740228870461) ( 1.166584855221183, 0.0000000000000000) ( 0.8576749916489031, 0.0000000000000000) ( 0.8491731641193959, 0.5281145115781721) ( 0.8491731641193959,- 0.5281145115781721)
Orbil C (magnification) Orbit C
- 1
1
1
Fig. 5.5
1
1
1
1
1
1
<*
Periodic orbit C of the intermediate equations. LP. denotes the initial point for 8 = 0.
82
Periodic Orbits of Some Intermediate VARIABLE
Equations
X
COEFFICIENT OF COSINE
COEFFICIENT OF SINE
0
-0 .2179102134061469D-02 -0 .2107116018230248D-03 2 -0 .2137452148380368D-02 3 -0,. 3259771617048419D-06 4 -0..1193378643053741D-04 5 -0,.2597770517753912D-08 6 -0,.2631941310461543D-07 7 -0,.1428383302411770D-11 8 -0..6840371710204618D-09 1
VARIABLE Y COEFFICIENT OF COSINE 0..1287565922122336D-03 0..1677729964962324D-03 2 0..1330085725319693D-01 3 -0..2711233945969800D-05 4 0..7083741223480185D-04 5 -0,.3888973709361927D-07 0.,3131832474273168D-06 6 7 -0.,2490711093250972D-09 8 0..3069055314824465D-09
0.. 1395328423750097D-03 0.. 1585335344079418D-01 -0..2769446234382116D--05 0..1107273425686815D--03 -0..4502402335281610D--07 0., 1790448102527333D-06 -0.. 1920932024624830D-09 -0..5834563869003051D--10
COEFFICIENT OF SINE
0 1
Table 5.3
0..6908350144302873D--04 -0..2157688908730818D--02 0..2050304600591596D--05 0..2196109390574436D--04 -0,.2118121938244490D--08 0., 1537361445238035D-06 -0..6094382076330908D--10 -0..2001528419400221D--09
Fourier coefficients of the x and y coordinates of orbit C.
Results and
PERIODIC ORBIT
F
INITIAL CONDITIONS 0.4100071795043306 0.1137594676750601
Comments
83
AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O. -0.1028862236602286 -0.3961422724835140
CONTINUATION PARAMETERS
1.000000000000
1.000000000000
PERIOD = 20.37358161575341 EIGENVALUES FOR THE STABILITY OF THE P.O. ( 5.843986862671638, 0.0000000000000000) ( 0.1724795897350981, 0.0000000000000000) ( 0.7605877747758888, 0.6431180099343742) ( 0.7605877747758888,-0.6431180099343742) (-0.2139898585913298, 0.9768358820293541) (-0.2139898585913298,-0.9768358820293541)
Orbit F
^y
J initial point
J for 0 = 0
E
Fig. 5.6
M
Periodic orbit F of the intermediate equations.
84
Periodic Orbits of Some Intermediate
PERIODIC ORBIT
G
INITIAL CONDITIONS
AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O.
-0.9589832398888400 9.8372900887546655E-02
0.1274822373224818 0.3249701100444401
CONTINUATION PARAMETERS
1.000000000000
PERIOD =
Equations
1.000000000000
20.37358161575341
EIGENVALUES FOR THE STABILITY OF THE P.O. ( 5.969270559081630, 0.0000000000000000) ( 0.7604233537556916, 0.6431852520302338) ( 0.7604233537556916,-0.6431852520302338) ( 0.1688874183109395, 0.0000000000000000) (-0.2146137627679631, 0.9766989980698135) (-0.2146137627679631,-0.9766989980698135)
Orbit
initial poi, for 0 = 0
Fig. 5.7
Periodic orbit G of the intermediate equations.
G
Results and
Comments
85
In Chapters 6 and 7, we look for orbits of the simplified equations, S.E., revolving around L4 or L5 during a given time interval. The S.E. can be considered as the intermediate ones plus a perturbation. So, initial conditions of the periodic orbits A, B, C, F, or G do not correspond to initial conditions of periodic orbits for the S.E. Nevertheless we can hope that near them, there exist orbits of the S.E. with a behavior similar to the periodic ones at least during some interval of time. The methods described in Chapters 6 and 7, to compute such orbits for the S.E., require initial conditions of a periodic orbit of the I.E. (we call it "basic periodic orbit"). These initial conditions will be modified to get a suitable orbit of the S.E. We note that the basic periodic orbit plays a role similar to the halo orbits of the RTBP in [37]. We think that only orbits A and B are good candidates as basic periodic orbits to obtain the desired orbits of the S.E. We remark that orbits F and G are too big. On the other hand, one of their loops is too close to the equilibrium point. The small size of orbit C makes it not significant in front of the perturbations to be added to the I.E. to reach the S.E. Hence, from now on we concentrate our efforts on orbit A around L4. We remark that, the orbits C, F and G can be investigated using the same approach.
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Chapter 6
Quasi-periodic Solution of the Global Equations: Semianalytic Approach
The purpose of this chapter is the exposition of an iterative method for the computation of quasi-periodic solutions of the equations of motion. The algorithm proposed is close to Picard's method for solving the initial value problem. In it, the right-hand side of the differential equations shall be replaced by suitable trigonometric approximations of trivial integration. The algorithm is presented in a more general framework, of nonautonomous differential equations, than the one which corresponds to our real situation. Some of the sections of the chapter are devoted to explain the tools which have been used to solve the technical difficulties which appear in the development and application of the method.
6.1
The Objective
The objective of the method proposed is to compute a quasi-periodic solution "close" to some of the periodic orbits computed in the preceding chapter for the intermediate equations. The quasi-periodic orbit to be computed shall be of the form m
x{t) = A0 + 2_. M c o s vit + Bi sini>it, 1=1
where vi, I = 1,2, ...,m will be a preselected set of fixed and known relevant frequencies. These relevant frequencies will be linear combinations of a small set of fundamental ones related to the motion of the celestial bodies, as it must be. To them we shall pay attention in section 6.3. We note the very small unstable character which exhibit the periodic orbits of the intermediate equations, as well as the pulsating character, around the triangular points, of the true orbits computed in the simulations for the real solar system during a finite time interval (see Chapter 3). This gives us some hope that these kinds of orbits, to which we are looking for, shall exist in the real situation for all time. 87
88
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
The method developed is an iterative one that remembers Picard's algorithm for the solution of an initial value problem. The main difference between them is that we shall not keep any initial condition fixed for which the solution must go through, of course now convergence cannot be ensured, as in Picard's theorem [2], because the iteration function shall be changed at each step of the algorithm. Another difference between them is that in our method, once the numerical coefficients Ai, Bi have been computed, the approximate solution remains valid for all t, while in Picard's method we are computing the solution for values of t within a certain time interval.
6.2
The Algorithm
Assume that the nonautonomous differential equations of our problem look like x =
f(t,x,x,e),
where e is a continuation parameter that in the real situation will take the value 1. The quasi-periodic solution shall be looked "nearby" to a periodic solution of the differential equations, for the value zero of the parameter. For this last value of the parameter it is assumed that, although the system is still nonautonomous, the time-dependent terms are periodic. In the problem in which we are interested, we have adopted as these equations those named as intermediate equations, in which the only period that appears is the synodical period of the Sun. So, in this situation, it makes sense to assume that at least one periodic solution has been computed. For the value of the parameter e = 1 we will denote, for short, the system of differential equations by x = f(t,x,x),
(6.1)
and for e = 0 we will denote the periodic solution computed by xo(t). As it has been said, the quasi-periodic solution, that we are looking for, looks like 771
x(t) = Ao + 2_] Aicosvrt
+ Bi
smu
rt>
i=i
where vi, I = 1,2,..., m is a preselected set of fixed and known frequencies. The known periodic solution, Xo(t), is considered as the first approximation to the final solution in our algorithm. After inserting it in the right-hand side of (6.1), a Fourier analysis is done of the function f(t, xo,xo). It is assumed that the main frequencies appearing in this analysis belong, all of them, to our preselected set {v{\. If this is not the case, then the set of frequencies must be enlarged. This point is not foreseen to be done automatically. Two exceptions must be considered. If the frequency is too large and with a small, or not,
The
Algorithm
89
amplitude associated to it, it will produce rapidly oscillating terms in the solution of small amplitude, so it can be skipped. If a small frequency, UJ, appears in the analysis of the function / , after the integration, the corresponding coefficients will be divided by w2. As these coefficients have been computed with a certain degree of accuracy, it may happen that the final figure is not meaningful, so these frequencies are skipped too. At this point it must be said that if we allow an independent term in the Fourier development of f(t, xo,x0), it will produce a secular term after the integration of the trigonometric sum, which is, of course, the next step in the algorithm. This has been avoided by adding the adequate "integration constants" to x0{t) in order to fulfill the purpose. Once the integration has been done we get our next approximation, xi(t), to the desired solution. This procedure completes one loop of the iterative algorithm, which is schemed in the following diagram
nth approximation to the solution: xn(t)
Fourier analysis of
f(t,xn(t),xn(t)) Integration of the equations when their right-hand sides are replaced by suitable trigonometric approximations. The result, xn+i(t), plays the role of xn (t) in the next step
Fig. 6.1
Diagram of the iterative algorithm.
Of course this loop is stopped when two consecutive approximations are close enough. The role of the parameter e, introduced in the differential equations, is relevant in two senses: - The first one is that, taking xo(t) as the first approximation to the solution, we can have a small rate of convergence to it, or, in the worst of the cases, we cannot have convergence at all. Introducing intermediate models of the differential system, via the continuation parameter e, and varying slowly
90
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
the parameter, these two facts can be avoided. - The second one, in certain sense related to the first, is the following. Assume, as it is the case in our real situation, that we can compute numerically a quasi-periodic solution, during a certain time interval and for a given value of e greater than zero. After performing its Fourier analysis, this solution can be taken as the starting point, xo(t), in the algorithm, which shall be much closer to the desired one that the periodic solution computed for e = 0. The procedure that has been explained has four important critical points which are: (i) The definition of the adequate set of relevant frequencies Vi, i = 1,2,..., m. (ii) The adequate selection of the integration constants of xn(t) in order to avoid secular terms in the solution. (iii) The adequate identification of the frequencies appearing in the Fourier developments of f(t,xn(t),xn(t)) according to the preestablished set. (iv) The good computation of the coefficients appearing in the Fourier developments, when these coefficients are firstly, computed by an FFT routine and after are shifted to those corresponding to the adequate set of frequencies. These questions have been solved in the way that shall be discussed in detail in the following sections. A modular program has been developed which performs all the above mentioned questions. The program tries to minimize memory and CPU time and the only external routine that it uses is the Fast Fourier Transform in Sine and Cosine (FFTSC) of the IMSL library. The flow chart of the program mainly follows the one which has been displayed for the algorithm. The results obtained shall be discussed in the last section of this chapter.
6.3
The Adequate Set of Relevant Frequencies
For the selection of the relevant frequencies which shall be considered in the solution, an examination of the equations of motion is needed. As it has been explained in Chapter 5, the equations in a normalized reference system, centered at one of the triangular equilibrium points, can be written in a compact form as: x
=
x—
P(7) L
XE
,-. (1 - n)x
r
x — g
r
'PE PE
XE
fj, - xE(l - 2/i)
PM ' PM
J
+P(1) + P{2)x + P{3)y + P{4)z + P(5)x + P(6)y, P(7)
y-VE,-. " Ir 3 PE
t1
_
, y-VE W ~ ^ 3r /* - VE PM
The Adequate Set of Relevant
+P(8) + P(9)x + P(10)y + P(ll)z =
Frequencies
91
+ P(12)± + P(13)y + P(14)i,
P(7) +P(15) + P(16)z + P(17)j/ + P(18)^ + P(19)2/ + P(20)i,
where {xE,yE) = ( - 1 / 2 , - V 3 / 2 ) for L 4 and ( - 1 / 2 , 7 3 / 2 ) for L 5 , and where the time-dependent functions P(i), i = 1,2,..., 20, have already been given and studied in Chapter 4. This set of equations has been taken as the definitive one, corresponding to e = l. In the Fourier analysis of the functions P(i) it was detected that only 21 frequencies appear, after keeping only those terms with absolute value of the coefficient greater than 5 • 1 0 - 4 . These 21 frequencies are linear combinations of the four fundamental ones, which correspond to the following angles: 61 = 02 = #3 = 04 =
mean mean mean mean
longitude of the Moon, longitude of the lunar perigee, longitude of the ascending node of the Moon, elongation of the Sun.
In fact, of the above mentioned 21 frequencies, only three of them shall be considered as more relevant, in the sense that the coefficients associated to them, or their multiples, are greater than 5 • 10~ 3 . These three frequencies are related to the angles: 04
=
-81+e2
+ 284,
fa = 64, fa
=
01-02-
Note that the second one corresponds to the frequency of the periodic orbit computed for the intermediate equations. Related to these three frequencies, the greatest coefficients that appear in the Fourier analysis are: 0.072, 0.6, 0.370 respectively. These quantities shall be taken as the weights associated to the three angles fa, fa, fa. We have looked for all the combinations of these angles giving weights greater than or equal to the established bound of 5 • 10~ 4 . The frequencies, and the corresponding phases, are given in Tables 6.1 and 6.2. On the right-hand side of Tables appear four integer numbers which are the coefficients of the linear combination of the fundamental frequencies giving the obtained ones (i.e.: ni0i + 7i202 + "303 + rn9\). The above given set of frequencies is the one which has been adopted as {i>i, i = 1,2,..., 61}. Related to these frequencies, the question that arises is its identification with those that appear in the FFT of f(t,xn(t),xn(t)).
92
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
In the process of identification some care must be taken with respect to the effect termed as leakage. This effect is inherent in the discrete Fourier transform because of the required time domain truncation (see [5]). Recall that even for a periodic function, its truncation at other than a multiple of the period results in a sharp discontinuity in the time domain, or equivalently results in side lobes, such as those of sin(:r)/a;, in the frequency domain. These side lobes produce spurious frequency components which are known as leakage. To reduce leakage it is necessary to employ a time domain truncation function with side lobes characteristics much smaller in magnitude than those of sin(:r)/a;.
Frequency 8.326763869672732 7.467917120105036 7.401567884153539 6.675419606488837 6.609070370537340 6.542721134585843 6.476371898634347 5.882922092872638 5.816572856921141 5.750223620969645 5.683874385018148 5.617525149066651 5.551175913115154 5.484826677163658 4.957726107353446 4.891376871401949 4.825027635450452 4.758678399498956 4.692329163547459 4.625979927595962 4.559630691644465 4.032530121834254 3.978654502824115 3.966180885882757 3.899831649931260 3.845956030921122 3.833482413979763 3.767133178028266 3.700783942076770
Phase 1.846454431531298 0.187006518277412 3.037556229623082 1.960194200858354 4.810743912204023 1.378108316370106 4.228658027714866 3.733381883439295 0.300746287605377 3.151295998951047 6.001845710295807 2.569210114461890 5.419759825807560 1.987124229972733 4.924483681531079 1.491848085697162 4.342397797042831 0.909762201208914 3.760311912553674 0.327676316719757 3.178226028065427 6.115585479622863 1.953886167510574 2.682949883788946 5.533499595134615 1.371800283021871 2.100863999300243 4.951413710645913 1.518778114811541
Table 6.1
n3
"i
n2
0 1 0 3 2 1 0 5 4 3 2 1 0 -1 5 4 3 2 1 0 -1 -5 4 4 3 2 2 -1 0
9 0 0 7 0 -1 8 0 0 4 0 -3 -2 5 0 6 0 -1 7 0 0 1 0 -5 2 0 -4 3 0 -3 4 0 -2 5 0 -1 6 0 0 7 0 1 0 0 -5 1 0 -4 2 0 -3 3 0 -2 4 0 -1 5 0 0 6 0 1 1 0 5 0 -3 -1 0 0 -4 1 0 -3 2 -1 -1 2 0 -2 0 -3 1 4 0 0
The relevant frequencies.
rii
The Adequate Set of Relevant
Frequencies
Frequency Phase 3.634434706125273 4.369327826157211 3.040984900363564 3.874051681881184 2.974635664412068 0.441416086046812 2.908286428460571 3.291965797392482 2.841937192509074 6.142515508737697 2.775587956557577 2.709879912903780 2.709238720606080 5.560429624248995 2.642889484654584 2.127794028415078 2.115788914844372 5.065153479972968 2.049439678892875 1.632517884139051 1.983090442941378 4.483067595484226 1.916741206989882 1.050431999650122 1.850391971038365 3.900981710995564 1.784042735086888 0.468346115161646 1.775587956557577 5.297714922420715 1.717693499135391 3.318895826506862 1.124243693373683 2.823619682230835 1.066349235951497 0.844800586317209 1.057894457422186 5.674169393576277 1.016492455353406 0.783406365185843 1.004018838412048 1.512470081463988 0.991545221470689 2.241533797742133 0.925195985519192 5.092083509087575 0.916741206989881 3.638267009167057 0.858846749567695 1.659447913253431 0.846373132626337 2.388511629531575 0.792497513616198 4.509997624599100 0.784042735086888 3.056181124678582 0.726148277664702 1.077362028764729 0.199047707854490 4.014721480322619 0.132698471902993 0.582085884488702 0.066349235951496 3.432635595834144 Table 6.2
ni
-1 -4 3 2 1 0 -1 2 -4 -3 2 -1 0 -1 -1 -2 -3 2 -2 1 1 1 0 0 -1 -1 -2 -2 -3 -3 2 -1
93
Tl2
n3
ri4
1 5 0 1 4 0 0 -3 0 1 -2 0 2 -1 0 3 0 0 4 1 0 5 -2 0 2 4 0 1 0 3 0 -2 0 1 0 -1 2 0 0 3 1 0 3 0 0 4 2 0 2 3 0 -1 0 -1 1 2 0 0 1 -2 0 0 -1 0 -1 0 1 0 0 1 -1 0 2 1 0 2 1 0 2 3 0 1 3 0 4 3 0 3 0 3 -2 0 -2 1 0 1
The relevant frequencies. (Cont. of Table 6.1.)
The truncation function that has been used is Hanning's one: 1 2
"• w W = -^ —
1 2
T: COS
2nt N
——,
where N is the number of points used in the FFT analysis. This function, as is well-known, exhibits the desired characteristics and has given us good results. The only disappointing question is that the memory requirements are enlarged
94
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
due to the fact that the FFT must be used twice, first for the functions properly and after for the functions multiplied by H(t). When the frequencies given by the FFT are known, then they are identified, by a minimum distance criteria, with those that appear in the relevant set. As a final question we shall mention that the Fourier analysis has been done using 2 13 = 8192 equally spaced distributed points during 450 revolutions of the Moon. The epochs at which the points are taken are centered at JD50 = 18262.
6.4
Avoiding Secular Terms
Avoiding secular terms in the "solutions" xn(t) is possible due to the freedom that we have in choosing the integration constants of xn-.i(t). The problem is in fact reduced to the computation of the zeros of a system of three nonlinear equations depending on three variables. The variables correspond to the integration constants, one for each component of the solution, and the equations correspond to the independent terms of the Fourier series of each of the three functions that appear as right-hand side in the equations of motion. As the independent terms are the mean values of the functions which define the differential system, we can skip for their computation all the periodic contributions that appear in them. These periodic parts are the most CPU time consuming ones, due to the large number of trigonometric functions evaluations that must be done for their computation. As algorithm for the computation of the zeros we have selected Newton's method. The computation of the partial derivatives has been done by numerical differentiation using a centered difference formula. The results obtained are good in the sense that they are fast and that no more than 4-5 iterations are needed to cancel the constant terms with an adopted tolerance established at 10~ 8 .
6.5
The Coefficients Related to the Different Frequencies
The question to which this section is devoted is the following: Assume that we have a real quasi-periodic function from which we know beforehand the frequencies that appear in it. Its discrete Fourier transform shall give us the coefficients of the cosine and sine terms at equally spaced frequencies that shall not be, in general, equal to those ones that we know that appear. After having filtered those spurious frequencies due to leakage, we want to recover the right coefficients of the cosine and sine terms, when the function is represented in terms of the known set of frequencies instead of those given by an FFT algorithm. There are several ways to solve this question: The conceptually most elemental is a direct computation of the coefficients using some minimization technique. This is not possible with our computational media due to the large size of the linear
Determination
of the Coefficients
of Quasi-periodic
Functions
Using FFT
95
system that, in general, must be solved. Another way is its computation argument by argument, using a trapezoidal rule for the integration and starting at an epoch such that the related argument is zero. The integration interval is then taken equal to an integer multiple of the period related to the analyzed frequency. The large number of frequencies, 61, that appear in our situation does not allow, also by CPU time reasons, this way of computation. The third one follows from a careful study of the coefficients computed by the FFT method. This has been the option adopted and it is explained in the following section.
6.6
Determination of the Coefficients of Quasi-periodic Functions Using F F T
We are faced with the following problem: Given a function f(t) which is supposed to be of the form m
f(t)
=
m
A0 + ^2Aicos(pit)
+
J2Bism{uit),
i=i
i=i
where the frequencies, vi, I = 1,2,... ,m are known, we try to determine the coefficients AQ, AI, Bi, I = 1 , . . . , m. We shall expose in the sequel the FFT approach. We recall the formulae giving the discrete Fourier transform. Let ax,...,ajv be a set of real values. They shall be identified later as the values of the function f(t) at equally spaced points t0,..., ijv-i. Then, we obtain coefficients c\,..., cN/2+i, s2,. . . , SJV/2) N being supposed even, through:
Ck+1 =
S v^
AT 1^
aj+1 cos
2irjk
~W
=
,
°''''' I '
3=0
2 ^v1 Sk+l
=
a
2-Kik lsin
Jsrz2 i+ -N->
k=
l,...,N/2-l,
j=0
where 5 = 1 for k = 0, N/2 and 2 otherwise. The initial values a i , . . . , a j v are recovered from the coefficients c 1 ; . . . , cN/2+i, s 2 , . . . , sN/2 by means of 2irjk
E cjt+icos—— + fc=i
^-^ 2^
2-KJk Sfc+isin—_.
*=i
The given function, / , contains terms with frequencies which are, in general, unrelated to the frequencies k/N which appear in the Fourier transform. Therefore, the first question is to find the values of c^+i, s^+i in the cases aj+\ = 1, a,j+i = COS(WJ) and a,j+i = sin(wj).
96
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
To carry out the computations a preliminary result is useful. Let M
1 eaiNa
=
_ ^
1
Then cos(iVa) — 1 + i sin(Na) cos a — 1 + i sin a cos((N — l ) a ) - cos(Na) — cos a + 1 4 sin2 (a/2) sin((./V — l ) a ) — sin(./Va) + sin a + 4 sin2 (a/2) '
M
and hence Re(M) Im(Af)
1+
sin(^i)a^
=
-
=
1 / c o s f - c ov s (22 ^ ) a ' - ' —-^ "~ 2 1 sh
We return to the computations, presenting separately the three cases. C a s e I: Oj+i = 1.
c\
=
c fc+1
=
1, - ^ c o s - ^
AT
r
= — £(e*»
j=o
+
e - * )
j'=o 027rifc
27V
Sk+i
e ^ - l
+
e~~i$~ — 1
2 ^ . 2njk , 1 ..TV 1 n -jj } _ ; sin - ^ - = 0, fc = l , . . . , — - 1 . 2 AT JV
=
i=o
Case II: Oj+i = cos(wj). We denote 2-Kk/N by ZA JV-l Ck+l
=
JV
^
cos(wj) cos(j/j)
J=0 JV-l
= I7v£( e ^ + e "^)(e ^ j + e - ^ j ) JV-l
=
— V
(ei{u+l/)j
4/V ^—' V j=o
+ conj. +ei^-u)j
+ conj. ) /
Determination
of the Coefficients of Quasi-periodic -ei(u+v)N ei(u+v)
AN
_ i
ei(w-v)N
+ conj. +
_ I
Functions
Using FFT _ i
ei(ui-v)
con].
_ I
2+^(y^y))+^(^
_5_ AN
sin (I ( - + ¥ ) )
97
("-¥))
sin(I(^-^))
* = o,...,f. If k = 0 or AT/2, the second and third terms in the square brackets are equal. N-l
Sk+1
=
AT J2 C0S (^') s i n ( I / i) 3=0 N-l
efc,i + e iuij ei j e ivi = —-YI ( ~ )( " ~) 2Ni 3=0
ei(u+v)N
1 2Ni
_ ^
ei{u+v)
,i(-u+u)N
— conj. + J '
_ I
ei(-u+v)
cos(i(^+^))-cos(^i(g;
1 2N
conj.
_ \ +
^))
Bin ( I (a,+ 2 ^ ) )
cos ( I ( - „ + 2 S * ) ) - C O B (3^=1 ( - a , + 3 ^ ) )
+
,fc=l,...,y-l.
Bin(|(-W+^))
Case III: a,+i = s i n ^ j ) . N-l
Ck+l
=
~M Yl sin ( w i) c o s (^) 3=0 •
AAfi 6 ANi AN
+
N-l
1^^
. e-™i)(eW
+
e-W)
3=0
ei(w+v)N ei(ui+v)
_ i _ J
ei{w-u)N
con
^ ^ J" j -+i
_ j
..v,...,/. gifa — v) _ 7j
con
-
J-
COB ( 1 ( ^ + 3 ^ ) ) - C O B (2^=1 ( a i + 2 ^ ) ) Bin ( ! ( „ + * * ) )
cos (I (w - ^ ) ) - c
0
s(2^1(c-^))'
sin(I(--^))
,
*=0,...,y.
As in case II, the two fractions inside the square brackets are equal if k = 0
98
Quasi-periodic
or k =
Solution of the Global Equations:
Semianalytic
Approach
N/2. N-l
Sk+i
=
-T7 Yl
sin
(wi)
sm
(^i)
i=0 N-l j
)(eivj
2TV f-r v
e~ivj)
-
j=0
1 27V 1 27V
; i(w-i/)iv
_
j
gi^+^JV _
+ conj. —
e i(w-j/) _ ^
s i n ( ^ i ( g , - ^ ) ) s i n ( I ( w _ ¥ ) )
ei(u+v)
x
_ J
-
conj.
sin ( 3 ^ ( 0 , + ^ ) ) s m ( I ( w + 2 ^ ) ) * - l
-
1.
It is readily checked t h a t in case II, if w = 2irm/N, m = 0 , . . . , TV/2 we have Cfc+i = 1 for A; = m and c^+i = 0 otherwise, and Sk+i = 0 for any k. In a similar way, in case III all the Ck+i are zero and Sk+i — 1 for k = m, Sk+i = 0 otherwise. To get t h e values for k = m some limiting procedure is required t o avoid indeterminacy. To discuss the dependency of Ck+i, sjt+i with respect t o u> let us introduce e through to — v = (27r/7V)e, where v = 2irk/N as before. Hence e = 1 when the difference between the frequencies equals the step in the frequency space under the discrete Fourier transform. If we skip the terms related to sums of frequencies in the expressions of c^+i, Sfc+i and approximate sin(7re/TV) by ire/N (both things are allowable if TV is big enough) and suppose e not too big (at most some units) we obtain: Case II. sin 2ixe Cfc+l
2we '
1 — cos 2ire Sk+l —
2TTC
'
Case III. 1 — cos 2we Cjfc+l
2?re
'
sin 2ire Sk+l
-
2ne
It is enough to discuss two of the functions
si 00 =
sin 2ne 2ne
sin 7re cos 7re 7re
,
. . 1 — cos 2-7re 52(e) = 2ire
T h e modulus 93(e) = ((91(e))2+
(g2(e))2)1/2
=
sin7re
is also useful. Figure 6.2 shows the behavior of
sin 2 7re
Determination
of the Coefficients of Quasi-periodic
Fig. 6.2
Functions
Using FFT
99
Behavior of the functions gi(e), i = 1,2,3.
It is clear that given u in the range [0,7r], there is some Vk of the form lirk/N, k = 0 , . . . ,N/2, such that the related e satisfies |e| < 1/2. For that e the related matrix sin ire / cos 7re — sin ire ire \ sin7re cos7re is invertible with inverse
Mr1
=
ire sin7re
cos 7re sin ne - sin ire cos we
Later on we shall need the supremum norm of M e and M e l. For IIM.- i | and |e| < 1/2, we have the bound
IIM,"1!
<3'
(the maximum is attained roughly at e = ±0.384). Also in that range ||MC" Concerning the norm of Mt we have 34
=
HMeHoo = sm7re 7TC
> 1.
cos7re + sm7re
The graph of this function is given in Figure 6.3 (it is obviously symmetric with respect to the y-axis). The successive maxima in each interval of the form [n,n+ 1] are point P value V point Q
0.19 1.308
1.34 0.283 0.83
2.36 0.162 1.77
3.36 0.114 2.76
4.37 0.088 3.74
5.37 0.072 4.73
100
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
1.4 1.2
0.8 0.6 0.4 V 0.2
0
0.5
Q
P
1
1.5
2
2.5
3
E
Fig. 6.3
Behavior of the function 94(e).
In the table we give also the values of the point Q (see Figure 6.3). For instance, for |e| > 2.76 we have HM^oo < 0.114. The k + 1 maximum is less than (1 + V2)/(2nk). Now we proceed to solve our initial question: To find the coefficients AQ, A[, BI, I = 1 , . . . , m. Let a,j+i = f(tj), j = 0 , . . . , N/2. For the sake of definiteness we assume to = 0. We suppose that the frequencies i/[, I = 1 , . . . , m are given taking as unit some given frequency (in our application the unit frequency will correspond to the frequency of the mean longitude of the Moon). Suppose also that the time span, £JV — t0, is equal to n\ times the period associated to the unit frequency. Hence, in our units, the frequencies associated to the FFT range from 0 to N/(2ni) with step 1/ni. Taking k/N as the frequency associated to ck+i, Sk+i we see that the frequencies range from 0 to 1/2 with step 1/iV. Hence to express 1/1 in the units inherent to the FFT, to obtain the generic value w, we put wj = vitw/N. Let K(I) be the closest integer to ni v\, i.e. K(l) = [n\ vi +0.5]. The frequency related to K(I), in the same units used for ui, is K{l)/ni = F;. We put also w; = VitN/N = 2irk/N. Then the passage from A0, Ai, Bi, I = l , . . . , m to c\, ck^+i, sk(i)+i, I = 1 , . . . ,m is given by f(l) I/ 11 Jo,i ff1) un /1,1 nu f(3) Ji,i
\
J0,2 /(I) J 1,2
(2)
AD
/i,i
(1)
u
f
/2,1
J2,l
0
f(l)
f(2)
u
Jm,l
Jm,l
u
Jm,l
Jm,l
0
f
AD
f(2) Jo,i /(2) Ji,i /W
A3) Jl,2 J2,2
/(I)
Jm,2 A3)
f(l) J0,m
Ai) Jl,m A3) Jl,m f(l) J2,m
Ai) J m,m A3)
f{2) J0,m y(2)
\ )
(
A
"
Jl,m
Ai
Jl,m
Bx A2
f(2) J2,m
A2) Jm,m
M
\
(
^
s
k(l) + l
c
k(2)+l
=
Am
j \Bm
Cl
Cfc(l) + 1
Ck(m) + 1
)
\
Cfc(m) + 1 /
(6.2)
Determination
of the Coefficients
of Quasi-periodic
Functions
Using FFT
101
The form factors f\") are computed as given in cases II and III, i.e. f{1)
-
1 2N
1+
COS {\u)j)
sin ( ^ " i ) sin [\UJ)
f(2)
_
1
Jo,j
-
2iV
in ( i W i ) sin
( 1 )
=
1 2iV ^
sin ( | ( W J +
/ •
f (2)
1
=
- COS ( S ^ f a J j )
/COS {\{Uj
+ Ui)) - COS (^~(0Jj
2iV ^ cos
+ A3)
1
=
sin (\(UJ - u7j))
WJ))
+Ujj))
sinJl^+Wi)) ( | K - ~ ^ ) ) - ^ ( ^ ( a j j - uJi)) \ sin {\{UJ -tot))
/ c o s ( | ( ^ j + a^)) - cos {^Y^juJj + coj))
COS {\{Uj
- UJi)) - COS (^f^-{bJj
- UJi)) \
s i n d ^ - -oJi)) /(4)
=
1 ^sin(^K-aJ,)) 2./V \
) '
sin(^i(^+^))' sin
sin (\(u>j — u>i))
(|(WJ
+Wj))
The matrix of form factors in the linear system would be the identity matrix if wi = wi for / = 1 , . . . ,m. This is not the case but it retains some of the good properties. We skip the first row because we can obtain A0 from c\ when Ai, Bi, I = 1 , . . . , m are available. If we define e,j by means of _ _ i - u t -
u
2TT
jftij,
we know \eij\ < 1/2. The matrix is then well-behaved if e^ is big enough for i ^ j . The linear system can be solved by the block Jacobi's iterative method. We state without proof the Lemma: Let A x = b, A e £ ( E n « , E n 9 ) , and assume that A has a block form / An
with Aij e C(W,W).
A12
A21
A22
\ Anl
An2
Let x =
(xux2
Am A2n
\
, x n ) T , XJ E Rq. Then, the block Jacobi's
102
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
method x
k = Akk ( h ~ XI Akixi
J'
k = 1
'' • •
is convergent provided
maxJI^H^H^IKl.
Q
*=
In our case A^ is (essentially) the matrix M~l with norm bounded by 5/3. Using the properties of 34(e), we can select the right values oi N, ni to obtain convergence. It is clear that we should avoid too close frequencies Uj, w;. Let pij defined as
Then a sufficient condition to ensure convergence of the block Jacobi's method in our problem is given by T h e o r e m : The block Jacobi's method for the solution of (6.2) is convergent provided \Pij\ > 2, for i ? j
and
V ( | ^ | - l)"1 < W
6
5
* « 1.5615, ( + v 2) :
for all i,j = l,...,m.
D
The proof follows immediately from the lemma and the properties of 34(e). As the initial frequencies are given, this means that ni should not be too small. We recall that n\/N should not be too large to be able to detect large v\ if any. Finally we should be aware that the noise in the expression of f(t) (due to skipped frequencies, errors in the computations, etc.) will produce some error in the determination of Ao, A[, Bi, I = 1 , . . . , m. An auxiliary program has been implemented to test the effectiveness and robustness of the method proposed. It has been useful to give us some idea of the values of the parameters to be taken. The program is divided into three main steps: - In the first one it performs the computation of the function to be FFT analyzed at the number of points indicated. This function is defined as a finite trigonometric sum with two kinds of noises allowed. The first one is a uniformly distributed error, of given amplitudes, generated by a random number generator. The second one allows the introduction of biases in some of the frequencies which define the function. This is the most CPU time consuming part of the program. - In the second step a FFT of the data computed is performed. Working with 8192 points, less than one minute is necessary to complete this step.
Results and
Conclusions
103
- Finally the refinement of the coefficients computed in the previous step is performed, in order to adjust them to the adequate frequencies. This is done using Jacobi's block method for the solution of the linear system that appears in the algorithm. With a tolerance of 10~ 4 for the computation of the coefficients the number of iterations is far from the value 15 which is given as upper bound. This last part is done in a few seconds. Some results are given in Tables 6.3 and 6.4.
6.7
Results and Conclusions
A program has been developed which carries out all the above mentioned computations. Several things must be said about it and on the results obtained. The first one is the large memory requirements. In order to perform the Fast Fourier Analysis, mainly 26 functions must be evaluated and stored at the number of points that we have adopted. This number is 2 13 = 8192 and is the smallest one which can be taken in order to have the precision required in the results. These 26 functions correspond: 20 of them to the functions P(i), which appear in the equations of motion, and the other 6 to the position and momenta of the spacecraft. This means roughly 1.7 Mb of memory. In practice more than 2.5 Mb are required. For an operating system working with virtual memory the price paid for this quantity is the large CPU time needed due to the also large number of page faults required to manage the data. On the other hand, remember that all these functions are trigonometric sums, in sine and cosine, depending on 61 frequencies. If we assume that 1 0 - 4 seconds are required to perform a trigonometric evaluation, then the total number of seconds required for one step, of the main loop in the algorithm, is roughly of 100 seconds. In practice, and taking into account that to the above quantities we must add: the evaluations of Hanning's function, the time required for the FFT computations and the time required for the refinement of the coefficients computed, for one loop more than 2 minutes are needed. This figure should not be critical if only a few steps where needed to go from the periodic orbit, x0(t), to the quasi-periodic desired solution. This has not been the case and it is at this point where the parameter e plays an important role. Due to convergence problems e must vary very slowly in order to fulfill our goal. In fact these small variations of e are seen in the next chapter, devoted to the numerical computation of quasi-periodic solutions using a parallel shooting algorithm in which, in some cases, only steps of 0.001 where allowed. A very fast machine, or, in the better of the cases an array processor, should be the adequate tool to perform the above computations. On the other hand, the program has been tested in the following way. Starting with the periodic solution, xo(t), numerically computed, it has been recovered for the value e = 0 of the parameter in the differential equation, which in this case only
104
Quasi-periodic
Solution of the Global Equations:
Semianalytic
Approach
depends on 21 frequencies. This is not an immediate result because the first thing that it is done is the addition of adequate constants, to the periodic orbit, in order to have no constant terms in the Fourier developments of f(t,xo(t),io(t)).
Results and
Conclusions
105
ASSIGNED VALUES OF THE FREQUENCIES 1 0.100000000000 2 1.500000000000 3 3.000000000000 4 4.000000000000 5 5.500000000000 6 8.000000000000 POWER OF 2 IN THE FFT = 12 REVOLUTIONS UNIT FREQUENCY = 450 TOLERANCE ACCEPTED FOR THE COMPUTED COEFFICIENTS = 1.0000000000E-04 AMPLITUDE UNIFORM ERRORS = 0.2000000000 ITERATIONS NEEDED = 1 NUMBER OF THE FREQ. RIGHT COEFFICIENT COMPUTED COEFFICIENT 1.000002303121793 1.00000000000 0 1.999472692426594 2.00000000000 1 2.998743261251667 3.00000000000 1 4.002945077310253 4.00000000000 2 5.002936413578079 5.00000000000 2 5.999524245633162 6.00000000000 3 7.000017748852902 7.00000000000 3 8.001621320051628 8.00000000000 4 9.005207876667423 9.00000000000 4 10.0000000000 5 -2.6790233731987183E-15 11.0000000000 5 3.5304831780709426E-15 12.0000000000 6 -8.9332607928651565E-16 13.0000000000 6 2.3727133675703948E-15 ASSIGNED VALUES OF THE FREQUENCIES 1 0.100000000000 2 1.500000000000 3 3.000000000000 4 4.000000000000 5 5.500000000000 6 8.000000000000 POWER OF 2 IN THE FFT = 13 REVOLUTIONS UNIT FREQUENCY = 450 TOLERANCE ACCEPTED FOR THE COMPUTED COEFFICIENTS = 1.000OO000OOE-04 AMPLITUDE UNIFORM ERRORS = 0.5000000000 ITERATIONS NEEDED = 1 NUMBER OF THE FREQ. RIGHT COEFFICIENT COMPUTED COEFFICIENT 0 1.00000000000 0.995565781284678 1 2.00000000000 1.990133131744210 1 3.00000000000 3.001785093565403 2 4.00000000000 3.996384805474221 2 5.00000000000 4.992675230978226 3 6.00000000000 6.000145060896423 3 7.00000000000 7.007416327319529 4 8.00000000000 8.001198043611448 4 9.00000000000 9.004732237392236 5 10.0000000000 9.993052889946045 5 11.0000000000 10.99825638707591 6 12.0000000000 11.99740534144984 6 13.0000000000 13.01258430421064 Table 6.3
Tests of the algori thm proposed in section 6.6.
106
Quasi-periodic Solution of the Global Equations:
Semianalytic
Approach
ASSIGNED VALUES OF THE FREQUENCIES 1 0.100000000000 2 1.500000000000 3 3.500000000000 4 3.505000000000 5 5.500000000000 6 8.000000000000 POWER OF 2 IN THE FFT = 13 REVOLUTIONS UNIT FREQUENCY = 450 TOLERANCE ACCEPTED FOR THE COMPUTED COEFFICIENTS = 1.0000000000E-04 AMPLITUDE UNIFORM ERRORS = 0.3000000000 ITERATIONS NEEDED = 2 NUMBER OF THE FREQ. RIGHT COEFFICIENT COMPUTED COEFFICIENT 1.00000000000 0.997340455219723 0 2.00000000000 1.994081853008091 1 3.00000000000 3.001071012407096 1 4.00000000000 3.997833149455256 2 4.995604352731482 5.00000000000 2 6.00000000000 5.997144853020941 3 7.00000000000 7.000737222585756 3 8.00000000000 7.993378184096384 4 9.000086284403584 9.00000000000 4 9.995831447032455 10.0000000000 5 10.99895498097890 11.0000000000 5 11.99844344144705 12.0000000000 6 13.00755081974184 13.0000000000 6 ASSIGNED VALUES OF THE FREQUENCIES 1 0.100000000000 2 1.500000000000 3 3.500000000000 4 3.505000000000 5 5.500000000000 6 8.000000000000 POWER OF 2 IN THE FFT = 13 REVOLUTIONS UNIT FREQUENCY = 450 TOLERANCE ACCEPTED FOR THE COMPUTED COEFFICIENTS = 1.0000000000E-04 AMPLITUDE UNIFORM ERRORS = 1.0000000000 ITERATIONS NEEDED = 2 NUMBER OF THE FREQ. RIGHT COEFFICIENT COMPUTED COEFFICIENT 0.991134850732429 1.00000000000 0 1.980272843360290 2.00000000000 1 3.003570041357003 3.00000000000 1 3.992777164850604 4.00000000000 2 4.985347842438488 5.00000000000 2 5.990482843402317 6.00000000000 3 7.002457408620150 7.00000000000 3 7.977927280322015 4 8.00000000000 9.000287614677178 4 9.00000000000 9.986104823440083 5 10.0000000000 5 10.99651660326429 11.0000000000 11.99481147148802 6 12.0000000000 13.02516939914143 6 13.0000000000 Table 6.4
Continuation of Table 6.3.
Chapter 7
Numerical Determination of Suitable Orbits of the Simplified System
A numerical method, based on the parallel shooting idea, is proposed to compute an orbit of the simplified equations which performs a given number of revolutions around the point L4. A suitable parameter is introduced to change continuously from the intermediate equations up to the simplified ones. The orbits obtained using this procedure, as well as their stability, are discussed.
7.1
The Objective
The purpose of this chapter is to obtain an orbit of the simplified model which makes a given number of revolutions around the point L4, without loops. We refer to these orbits as quasi-periodic orbits (q.p.o.) despite the fact that we have no guarantee they are quasi-periodic. To obtain these orbits, a parallel shooting (P.S.) method ([39]) with suitable free variables and final conditions has been used so that the initial requirements are satisfied. The P.S. method has been implemented together with a continuation method with respect to a suitable parameter e ([39]). For e = 0, the system of equations reduces to the intermediate ones, for which we have some periodic orbits (see Chapter 5). For e = 1 we obtain the simplified equations. The perturbations for e = 1 with respect to the periodic orbit are not small, especially the perturbations due to the eccentricity of the Moon. So, if the numerical integration of the complete equations starts with the initial conditions corresponding to a periodic orbit, the orbit escapes from a neighborhood of the £4 point after a short time span. This behavior can be seen in Figure 7.1 for the periodic orbit A. Therefore, the numerical method consists of starting with a periodic orbit for e = 0 and compute a q.p.o. for values of e between 0 and 1, taking a suitable step on e that allows to follow the family of q.p.o. The number of revolutions required for a q.p.o. should correspond to a time interval for which the relevant angles involved in Moon's theory (mean longitude, 107
108
Numerical
Determination
of Suitable Orbits of the Simplified
System
Fig. 7.1 Integration of the complete equations starting at the periodic orbit A of the intermediate equations. A fast escape is produced.
argument of the pericenter, argument of the ascending node of the Moon and elongation of the Sun) after this time, were approximately congruent to the initial values modulus 2ir. That means an interval of time of roughly 19 years. Taking into account that one year corresponds to 12 revolutions of the Sun's elongation, which is the frequency associated to the periodic orbit of the intermediate system, we obtain a total amount of 228 synodic revolutions of the Sun with respect to the Earth-Moon system. The development of the program stated before for 228 revolutions, requires a large amount of computing time. So we restrict in the computations the number of revolutions to 12 and 24, approximately 1 and 2 years respectively. We believe this is sufficient to illustrate clearly the proposed method. We use for the computations the simplified equations instead of the total solar system. This is due to the time required for the numerical integration at least for our analytic implementation of the complete model of the solar system ([12]). We want to remark that the problem of finding a q.p.o., as it is considered in this section, is not a well-posed problem. The reason is that we only ask for an orbit that performs a given number of revolutions around L\ and nothing else. However we have no guarantee that such an orbit could exist and in the case it exists, we have no evidence that it is unique. So the problem can be ill-conditioned. 7.2
Description of Two Families of Algorithms. Reduction of the Linearized Equations
The different approaches based on the P.S. method that can be used are explained in this section. Due to the freedom we have to choose variables and conditions in the implementation of the P.S. method, we can consider two families of methods: - Family (a), for which the number of free variables coincides with the number
Description
of Two Families of Algorithms.
Fig. 7.2
Reduction
of the Linearized Equations
109
Definition of the point Y{.
of equations, - Family (b), such that the number of free variables is greater than the number of equations. First, we give the main ideas of the methods of family (a). Let us write the vector field of the current equations as vector field
=
(1 — e) x intermediate equations + e x simplified equations,
(7.1)
where e is a parameter, e £ [0,1]. From now on x, y, z, x, y, z, stand for the normalized coordinates. We take S
=
{(x,y,z,x,y,z)\y
= V3x, x < 0 j ,
as a surface of section. A point V o n S will be represented by Y = (p,z,x,y,z), where p is the distance from the point (x,y) to L\. Let ti be a given time, Yi — (pi, Zi,±i,yi, Zi)T £ S and
=
We define Yi = y>(tj;tj,Yj) where U
=
min {t > ti\ip(t;ti,Yi)
£?:} ,
if it exists (see Figure 7.2). We put Xi = (ti, Yi). We look for a solution, ip(t;t0,Yo) =
i = 0,...,n,
(7.2)
110
Numerical Determination
of Suitable Orbits of the Simplified
System
in such a way that the function *(t)
=
te[n,ti+1],
for i = 0 , . . . ,n — 1, is continuous, and solution of (7.1). This yields the following 6n conditions: Xi
— Xi+i,
(7.3)
i = 0, . . . , n - l ,
for the 6n + 6 unknown components (7.2), where Xi = (Ij, Yj). We assume that 6 —fc,0
n(x0,xn)
(7.4)
= o, i = i,...,k.
These conditions will be chosen to prevent the orbit to move away from a given region. That is, in them we introduce a convenient feedback. Altogether, (7.3) and (7.4) represent a system of equations of the form (
X0-X1
\
X\ — X2
= 0,
G(X)
(7.5)
\ rk(X0-Xn) J Xn)T. It can be solved iteratively using Newton's method:
where X = (X0,Xi,...,
xU+i)
=
xW-iDGixWfi^GiXU)),
(7.6)
for j = 0 , 1 , . . . , provided an initial guess X(°) is known. The Jacobian matrix DG(X^) has the following especial structure
DG{X)
( Gi -I 0 0 G2 -I
... ...
0 \ 0
0
G„ 0
-I B J
= \
where Gi,i = 2,...,n, matrices defined by
A
0
are 6 x 6 matrices, G\, A and B are 6 xfc,fcxfcandfcx 6 d
= Dx^pCi-i),
* = l,2,...,n,
A
=
DXo(r(X0,Xn),
B
=
DXn(r(X0,Xn),
Description
of Two Families of Algorithms.
Reduction
of the Linearized Equations
111
and / stands for the identity matrix. For each j > 0, (7.6) can be written in the following way DG(X(i))D
S(XU)),
=
where D =X^+^-X^ = (du ... ,dn+1)T, and S(X^) = -G(X^) = (su...,sn+1)T. This linear system can be reduced taking into account that it is equivalent to G\di - d2
=
si,
G2d2 - d3
=
s2, (7.7)
Adi+Bdn+i
=
sn+i.
It is easy to express all dj successively in terms of d\: d2
=
G\d\ — s\
dn+i
=
GnGn-i
• • • Gidi — 2_^ I 1 1 ^ M
s
j-
From the last equation of (7.7) we get a k x k linear system for the unknown vector d\: Qd!
=
b,
(7.8)
where Q
=
A +BGnGn-i
(
• • • Gi, m I n
\
8 G n <) i 3 = 1 \l=i+l J
E
The system (7.8) is solved by means of Gauss elimination. Once d\ is known, we recover d2,..., dn+i recurrently from (7.7). The methods of the family (b) are obtained when none of the components of X0 is fixed, and the number of final conditions (7.4) is less than 6. Then, (7.3) and (7.4) form a nonlinear system of 6n + 6 variables, and Newton's method cannot be applied as before. Nevertheless, given a vector X^ near the solution of (7.5), we can approximate the system of equations by flG(I(i))(I(i+1)-I<J'») = -G(X^). This system can be reduced to (7.8) using the same argument as before. We note that G\, A and B are 6 x 6, k x 6, and k x 6, matrices respectively. So, Q is a k x 6
112
Numerical Determination
of Suitable Orbits of the Simplified
System
matrix and the system (7.8) will be solved by a least squares method as we explain in what follows. We require to minimize
\dlMdu where M is a given symmetric weight matrix, with the condition (7.8). We consider the function $
=
+ \T(Qd1
)-d\Md1
-b),
where A (Lagrange multiplier) is given by d\M + XTQ Therefore, d\ = —M~1QT\.
=
0.
By substituting d\ in (7.8) we get -QM~1QTX-b
= 0.*
Then A =
-{QM~1QT)-lb,
and so d!
=
M~lQT{QM-1QT)-lb.
As before, it is easy to recover di,- • •, d n +i from (7.7). Usually, the weight matrix M will be taken as the identity. 7.3
Description of the Methods. Comments
We have some freedom to choose the final conditions (7.4) to define a concrete P.S. method. Also for the (a) family of methods, we need to fix some initial values. The goal of this section is to summarize some variants of the P.S. method that have been implemented. Comments on each method are also included. The methods 1, 2, 3, 4, 5, 6 and 7 given in the following table belong to family (a), and methods 8 and 9 to family (b). In Table 7.1 r* means a fixed value. In methods 3, 4, and 5, r* can be taken as n x p where p is the period of the basic periodic orbit. Conditions pn — p0 = 0 and zn — ZQ = 0 appear in every method. The first one is imposed to avoid the natural behavior of the orbit to go very near to the L4 point as some previous simulations showed. As it will be explained later, this is only partially achieved. The condition zn — ZQ = 0, is imposed to have small contribution in the z component. We recall that the basic periodic orbit is a planar one. Experience shows that conditions which involve the third component give no trouble in the determination of the q.p.o.
Description
of the Methods.
113
Comments
Final equations
Method
Initial variables
variables
1
to, Po, z0, io
xo = iP, 2/o = 2/p
Pn ~ PO = 0,
Zn - Z0 = 0,
xn - xo = 0, Pn~ PO = 0, Xn - Xo = 0,
yn - 2/0 = 0, Z„ - 2^0 = 0, i n - -ZO = 0,
*n - io - T* = 0, ^n - 2^0 = 0, tn-to-T* = 0, 2TI - -20 = 0,
pn- Po = 0, i n - io = 0, pn - po = 0, Xn — XQ = 0,
*n - to - T* = 0, 2 n - 2 0 = 0, Vn - 2/0 = 0,
Pn~ P0 = 0, Xn - ±o = 0, i „ - i 0 = 0, Z„ - Z0 = 0,
Fixed
2
to, po, zo, zo
xo = iP, 2/o = Vp
3
to, Po, zo, zo
±o = xp, 2/o = VP
4
to, Po, zo, 2/0
±o = xp, z0 = 0
5
to, Po, zo, xo, 2/o, z0
6
Po, ZQ, z0
t0 = T*, ±o = Xp,
Pn~ P0 = 0,
i« - io = 0,
zo, x0, 2/o
2/o = 2/P to = T*, po = pP,
7
XQ
8 9
to, Po, Xo, 2/o, to, Po, io, 2/o, Table 7.1
z
o, zo z o, z0
— Xp
Pn - PO = 0,
Zn - Z0 - 0,
in - io = 0, Pn-
PO = 0,
2„ - Z0 = 0,
i« - io = 0, tn-to-T* = 0, 2n - 20 = 0,
p„ - /9o = 0, i„ - i0 = 0
Boundary conditions for the different parallel shooting procedures.
Preliminary explorations indicate that method 1 is not adequate to our problem because the components x, y are uncoupled from the component z. This gives rise to an ill conditioning of the system (7.8) even for the determination of the periodic orbit for e = 0. To avoid this difficulty it is necessary to include the same number of conditions as initial variables which involve the z component. This requirement is satisfied by the other methods in Table 7.1. The main difficulty of methods belonging to family (a), is the variation of the determinant of the matrix Q in (7.8), which for values of e between 0 and 1, goes to zero. The passage of the determinant through 0 corresponds to a "turning point" of the family of q.p.o. This behavior is common to all methods of family (a). By construction, methods of family (b) do not present this kind of difficulty. However, they have the disadvantage that the step in e to advance in the continuation of the q.p.o., must change in different regions of e. In fact, for some ranges of e, the step must be taken less than 10~ 3 . This makes the procedure very slow. The methods shown in the Table have been used at different steps of the continuation from a periodic orbit for e = 0 up to a q.p.o. for e = 1. We wish to give a view on the difficulties that appear in the implementation as well as the criteria that lead us to the different methods. To this purpose we will
114
Numerical Determination
Fig. 7.3
of Suitable Orbits of the Simplified
System
When method 3 is applied, the thickness of the orbit increases as e does.
explain some of the numerical computations that we have done. As it was mentioned before, method 1 is not adequate to our problem, so we skip it. Method 2 allows to get a q.p.o. for e = 1 if the number of revolutions is small, for example 1 or 4. The situation changes when the number of revolutions is increased. For example, if n = 12, the determinant of matrix Q is equal to 0 for a value of e in the range (0.12,0.13). This is mainly due to the variation of time. This consideration gives rise to method 3. Method 3 has been implemented taking the parameter r* equal to n x p where p is the period of the basic periodic orbit. In this way, it is possible to get a q.p.o. for n = 24 and values of e in the range [0,0.508]. As we can see from Figure 7.3, the region on the (:c,2/)-plane which contains the orbit is a strip that grows up in amplitude when e increases. It seems not possible to avoid this kind of spiraling for larger values of the parameter. It is not possible to continue the family of q.p.o. for values of e > 0.508 using method 3. The reason is that the family has a turning point as can be seen in Figure 7.4, taking to as a function of e. Using the solutions obtained for e = 0.504, 0.506 and 0.508, we extrapolate to get an initial approximation for the orbit corresponding t o e = 0.506 on the upper branch. This second branch has been computed going back in e up to the value e = 0. It is a surprising thing that the new orbit obtained for e — 0, is not a periodic orbit but an orbit which makes 24 revolutions around L4 spending the same time as the periodic one. That orbit can even be continued for negative values of e. To overpass e = 0.508 we modify method 3. It seems that the condition imposed on the time becomes excessively rigid when e increases. So, we remove that condition and fix the initial time equal to the initial time corresponding to the orbit for e = 0.508, that is, we use method 6 with r* = 3.1054339. In this way, we can obtain a q.p.o. for values of e in the range [0.508,0.62], but the method fails for e > 0.62. At this point several checks have been done as to continue using method 3 again, and method 7 where r* was taken as the corresponding value for the orbit at e = 0.62.
Description
of the Methods.
0.508 Fig. 7.4
115
Comments
e
At € = 0.508 a turning point appears.
All these attempts were unsuccessful because they do not allow to get a q.p.o. even for e = 0.63. Method 8 can be applied to continue the orbit corresponding to e = 0.62 up to e = 0.628. This method allows some relaxation of the interval of time tn — to. Starting at e = 0.629 and using method 9, it is possible to obtain a q.p.o. up to e = 0.7 approximately. For values of e greater than 0.7 the time step in e becomes excessively small. We want to remark that methods 4 and 5 have been applied with r* = n x p, starting at e = 0 for a different number of revolutions. In all the cases that have been studied, the family of q.p.o. presents a turning point of the same pattern as the one explained before. We can also say that this kind of point has no relation with the vertical component because in the planar problem it has been observed the same behavior. As a conclusion, it seems convenient to use methods of the family (b) because the conditions imposed in all the family (a) seem to be too strong for this problem. Concerning the methods (b), our experience has shown that neither method 8 nor method 9 can be used alone. The reason is that method 9 has one condition too strong, that is, it fixes the difference between initial and final times which is slightly artificial. On the other hand, if this condition is suppressed as in method 8, it seems that there is excessive freedom as it would happen in an autonomous problem. Therefore, the strategy that we have followed (see next section) is to use both of them, changing from one to the other when it is necessary, that is, when the step in the parameter becomes too small.
116
7.4
Numerical Determination
of Suitable Orbits of the Simplified
System
Results and Discussion
The implementation of P.S. methods is slightly different for the two families of methods. It has been observed that methods of family (b) are more efficient than the ones of family (a) at least for this problem. So, in this aspect, we refer only to family (b). A program PSL45 has been implemented to compute q.p.o. of system (7.1) by methods 8 and 9. The step on e in the continuation method is modified automatically by the program depending on the number of iterations necessary to get the q.p.o. The same program can be used to draw a given solution. We discuss in this section the results obtained by program PSL45. The quasiperiodic orbits we present here have been obtained starting with the periodic orbit A. The main results are divided into two groups of computations. One of them concerns q.p.o. of 12 revolutions, while orbits for 24 revolutions form the second group of results. The two families of q.p.o. display almost the same behavior. The beginning of the families from e = 0 up to e = 0.5 approximately, is computed by method 9 without trouble. This means that a step on e , Ae, of the order of 0.05 is sufficient to get the q.p.o. After the value e = 0.5, the Ae required decreases quickly. This decreasing is more pronounced when the number of revolutions is bigger. That means, whereas for n = 12, Ae is of the order of 0.01, for n = 24, it becomes less than 0.001. When this happens, we use method 8 to relax the interval of time. In what follows, we comment each group of q.p.o. Quasi-periodic orbits of 12 revolutions have been computed for values e £ [0,0.82] by method 9. Method 8 allows to follow the family from e = 0.82 up to e = 0.98. Finally, method 9 has been used again to compute the solutions up to e = 1. The value r* adopted is that corresponding to the q.p.o. of e = 0.98, that is, 81.437548. Figure 7.5 shows the projection on the (x,y)-plane of the q.p.o. of 12 revolutions for e = 0.25, 0.5, 0.75 and 1. The plot on the (x, z)-plane of the solution for e = 1 is given in Figure 7.6 where the window has been taken as [—1,1] and [—0.1,0.1]. A nice behavior of the third component is observed. The corresponding initial conditions (t0, Po, zo, io, Vo, io), the interval of time At = tn — t0 and the final conditions (xn,yn) for n = 12, are given in Table 7.2. Also the eigenvalues of the variational equations after 12 revolutions are included in Table 7.2. The evolution of the eigenvalues is very regular when e 6 [0,0.5]. From e = 0 up to e = 0.3, all of them are complex with modulus approximately equal to 1. For e = 0.4 there appear two real eigenvalues with modulus greater and less than 1 respectively. From e = 0.5 on, they change from real to complex and vice versa in small ranges of e. A sample of this behavior is shown in Table 7.3. When method 8 is applied, four eigenvalues remain on the real line for e 6 [0.83,0.96]. In this range, the greatest modulus decreases from 40 to 1 approx-
Results and
Discussion
117
Fig. 7.5 From left to right and from top to bottom, (x, y) projections of the q.p.o. of 12 revolutions for e = 0.25, 0.5, 0.75 and 1 respectively.
imately. After e = 0.98 all the eigenvalues become complex changing again for e = l. Concerning the second set of results, that is, for 24 revolutions, orbits for e G [0,0.72] have been computed using method 9. The orbit in Figure 7.7 for e = 0.3 belongs to this range. For e > 0.72 we try two ways. The first one consists of following using method 9. Figure 7.8 shows the orbit obtained for e = 0.75. We
Fig. 7.6 ( i , z) projection of the last orbit of Figure 7.5 (e : 1). The size of the displayed window is [-1,1] x [-0.1,0.1].
118
Numerical Determination
to Po zo xo yo zo At Xn
yn
X
Table 7.2
e = 0.25 3.16863186 0.27894180 0.00186329 -0.57035092 0.19101187 -0.00353491 81.49432646 -0.55681660 0.19825060 -0.8605±0.3358i 0.8078±0.6844i 0.7210±0.6769i
of Suitable Orbits of the Simplified
e = 0.50 3.17677778 0.27023059 0.00432479 -0.56557670 0.16462492 -0.00794562 81.49432646 -0.52301208 0.20856667 4.7565 0.2437 -0.8906±0.8842i 0.7350±0.6506z
e = 0.75 3.19056777 0.23118090 0.00824644 -0.48965628 0.12376409 -0.01252960 81.49432646 -0.43559649 0.20647764 -106.3709 -1.2813 -0.5821 -0.0113 0.7958±0.5592i
System
e = 1.00 3.18268735 0.21126947 0.01859491 -0.44781851 0.11988688 -0.01998493 81.43754800 -0.38283159 0.18143498 -1.9351 -1.4629 -0.4367±0.3312i 0.9045±0.3223i
Initial conditions, final conditions and eigenvalues of the q.p.o. of 12 revolutions.
e = 0.775 -53.5933 -1.2697 -0.6110 -0.0214 0.8108 ± 0 . 5 3 5 6 ; Table 7.3
e = 0.79 -0.6013 -0.2287 - 2 . 1 2 2 3 ± 1.4007i 0.8192 ± 0.5215i
e= -0.3443 -0.5364 0.8231
0.80 ±1.6904; ±0.1038; ± 0.5146;
e = 0.81 -24.6770 -1.2788 -0.6288 -0.0447 0.8262 ± 0.5090;
Evolution of the eigenvalues of the variational equations for large values of e.
stopped these computations when Ae became of the order of 0.001. On the other hand, q.p.o. for e € [0.72,0.84] have been computed using method 8. It can be observed that there is no qualitative difference between the previously computed orbit for e = 0.75 and the one obtained for the same value of e with
Fig. 7.7 Typical shape of the (x,y) projection of the q.p.o. for e = 0.3. The orbit has been computed using method 9.
Results and
Discussion
119
Fig. 7.8 Starting at e = 0.72 two families of q.p.o. have been computed, one using method 9 (left figure) and the other using method 8 (right figure). In the figures, the (x,y) projection of the two orbits correspond to the same value of t — 0.75, and there is no significant difference between them.
L
4
E
M
Fig. 7.9
(x,y) and (x, z) projections of the q.p.o. for e = 0.9.
method 8 (see Figure 7.8). Finally we return to method 9 to finish the family of q.p.o. using r* = 163.251604 which is the value of the time interval corresponding to the orbit for e = 0.84. Figure 7.9 shows the q.p.o. for e = 0.9 on the (x, y)-plane and the (x, z)-plane. Table 7.4 contains the relevant information on the 24 revolutions q.p.o. for the specified values of e. As before, the behavior of the eigenvalues for e e [0,0.5] is regular. There is one pair of real eigenvalues of modulus greater and less than 1 respectively, and four complex eigenvalues for e € [0,0.25]. For 0.25 < e < 0.3 the real eigenvalues collide, become complex and we have three pairs of complex eigenvalues with modulus near 1. The-spectrum remains qualitatively the same in the range 0.3 < e < 0.61. We remark that on this interval, the greatest modulus is roughly 1.5. For e > 0.61, this modulus increases with e. We note that this increase coincides with a strong decrease of p0. For values 0.72 < e < 0.84, that is, the ones computed using method 8, two real and four complex eigenvalues are obtained. On this interval, the greatest modulus varies approximately from 18 to 4. From e = 0.85 on, the spectrum
120
Numerical Determination
to Po zo ±o Vo zo
At Vn
A
Table 7.4
e = 0.30 3.16321202 0.27957673 0.00192582 -0.56778595 0.20333440 -0.00401870 162.98865293 -0.55167942 0.19605646 -1.0346 ±0.3165i -0.4704 ± 0.7542i 0.0857 ±0.9768i
of Suitable Orbits of the Simplified
e = 0.75 3.18754464 0.18775915 0.00620621 -0.38909144 0.10656279 -0.00969274 163.03898525 -0.33339309 0.15087050 7.5382 0.1752 0.2188 ± 0.7591i 0.3466 ± 0.8877i
System
e = 0.90 3.10585883 0.18572728 0.00684658 -0.38230917 0.10574837 -0.00891271 163.25160412 -0.32055969 0.16629437 -12.2592 -0.1472 0.2701 ±0.6178i 0.3922 ± 0.8673i
Initial conditions, final conditions and eigenvalues of the q.p.o. of 24 revolutions.
is qualitatively the same as before, that is, two real and two pairs of complex eigenvalues are obtained. However the greatest modulus oscillates between 5 and 10 approximately. The q.p.o. we present here, have been obtained using some combinations of methods 8 and 9. Of course, the final q.p.o. for e = 1 depends on this combination. Therefore, we can obtain an infinity of q.p.o. around L4 using methods 8 and 9 at different steps of the continuation method on e.
Chapter 8
Relative Motion of Two Nearby Spacecrafts
In this chapter we describe the motion of two nearby spacecrafts which have orbits close to the orbit found in the previous one. Then, the relative distance and orientation of the two spacecrafts is studied. This closes the main objective of the full work.
8.1
The Selection of Orbits for the Two Spacecrafts
The purpose of the full work is to find: a) Whether or not there are orbits in the Earth-Moon system taking into account the full solar system in its real motion, such that remain (perhaps under a mild station keeping) in a not too big neighborhood of the triangular equilibrium points (of the related RTBP). For these possible orbits it is asked to perform revolutions around the equilibrium point without unwanted loops. The problem as stated before remains somewhat imprecise. However such kind of orbits has not been found till now. If there are such orbits and are (linearly) stable this would give rise to the possibility of powder clouds orbiting around the equilibrium points L4, L5. The existence of such clouds is unknown. It is possible that the orbit, if it exists, is mildly unstable or the stability region is quite small. b) If two particles (spacecrafts) are placed in orbits close to the hypothetical orbit of a) it is requested to know the possibilities for the variation of the mutual distance and the orientation of the line joining them. What we propose here is to use the solutions of the variational problem around the orbit satisfying a). This orbit will be denoted by "reference orbit". Let XQ = (x0,y0,zo,xo,y0,z0) be the initial conditions for the reference orbit for an initial time t0. Let AX0 = (Axo, Ay0, Azo, A±o, Ay0, Azo) be some small amounts. If at t = t0 one of the spacecrafts is placed at X0 + AX0, and the other at X0 - AX0, it 121
122
Relative Motion of Two Nearby
Spacecrafts
is easy to see how the relative position of one spacecraft with respect to the other evolves. Let ^ ( £ 0 , ^ 0 ) be the reference orbit, t € [*o,£o + T] and $t(i0,X0)
=
—
0 4 (i o ,X o ),
the related variational flow. Then, if R(t) denotes the position of the first spacecraft with respect to the second one, we have R(t) = 2$ ( (£ 0 , X 0 )AX 0 , and, in particular, R(t0) = 2AX 0 . To see the possible behaviors we had to consider the different modes of motion in a neighborhood of the reference orbit. This is given by the eigenvalues and eigenvectors of the final variational equations $r(to, -Xo)- This matrix is not a monodromy matrix because we have no periodic orbit. In fact the initial and final points are different in general, but we identify the tangent spaces to R6 at both of them. The six modes allow us to have some idea of the relative motion. This is done in the next section.
8.2
Variations of the Relative Distance and Orientation. Results
We have produced a couple of programs to present the results for different modes. One of them computes the variational matrix at equally spaced times. The other computes the eigenvalues and eigenvectors of $T(£O> -XO), asks for the desired mode and plots the behavior of some selected components. We proceed to describe the possible outputs in detail. The drawing program reads the previously computed variational matrix at some equally spaced points. Then, the eigenvalues are displayed. It is asked which one of the eigenvalues (modes) is to be analyzed. If the eigenvalue is real with modulus greater (smaller) than 1 the mode increases (decreases) on the average. Complex eigenvalues are associated to rotating (on the average) modes, increasing or decreasing according to whether the modulus of the eigenvalue is greater or less than 1. For complex eigenvalues a + i(3 the eigenvectors are complex, u + iv. The real and imaginary components, u and v, span an invariant (under $ T ( * O , - ^ O ) ) plane. In the complex eigenvalue case two components are required to have an initial vector Vjr = m • u + n • v, to be considered as AXo- In the real case we merely take as Vj the associated eigenvector. Then Vi is normalized and we proceed to compute Vi = $t(£o, X0)Vi for t = to,h,..., <M = t0 + T with < i+1 — tj = constant, i = 0,...,M -1. Then three of the components of Vi(t) are plotted against time and each one against the others. The indices of the required components have to be supplied by the program user. Also the norm of Vi(t), or the norm of its projection on the position variables, is plotted against t. As a first example we present the results for the periodic orbit of the intermediate equations of type A. We have taken the initial conditions given in Chapter 5 for orbit A. The pa-
Variations of the Relative Distance and Orientation.
Variables plotted x,y,
Eigenvalue
m
n
1-top
1
1
0
1-bottom
1
0
1
2-top
3
1
0
2-bottom
3
0
1
3-top
5
1
0
x,y, i,\\Vi{t)\\ x,y, i,\\Vi(t)\\ z,z,
3-bottom
5
0
1
z,z,
Figure
±.IIVK*)II x,y,
*,IIVr(*)||
IIVKt)ll l|VK«)|| Table 8.1
Results
123
Respective windows [-1.5,1.5], [-1.0,1.0], [-1.0,1.0], [0.0,2.0] [-1.2,1.2], [-0.8,0.8], [-0.8,0.8], [0.0,1.5] [-1.5,1.5], [-0.75,0.75], [-1.5,1.5], [0.0,1.5] [-1.5,1.5], [-0.6,0.6], [-1.2,1.2], [0.0,1.6] [-1.5,1.5], [-1.5,1.5], [0.0,1.5] [-1.2,1.2], [-1.2,1.2], [0.0,1.2]
Relevant information related to figures 8.1, 8.2, 8.3. Number of revolutions: 12.
rameter e used in Chapter 7 is set equal to 0. Hence we consider the I.E. The time interval has been taken equal to 12 revolutions of the orbit. All the eigenvalues are complex. We ask for the modes associated to the eigenvalues numbers 1, 3 and 5. For each of them the values (m,n) = (1,0) and (0,1) are used. The results are displayed in Figures 8.1 to 8.3 according to the data shown in Table 8.1. According to the eigenvalues of orbit A the three pairs of eigenvalues are associated to "frequencies" equal to 0.398, 1.035 and 1.094, respectively, if the frequency of the periodic orbit is taken equal to 1. Hence in a time interval equal to 12 revolutions of the periodic orbit the modes should turn 4.78, 12.42 and 13.12 "revolutions" , respectively. This is in good agreement with what is displayed in Figures 8.1, 8.2 and 8.3. At the light of the results shown, the more appealing modes are associated to the second couple of eigenvalues. The size of ||Vr(i)|| varies by a factor slightly greater than 2, and the orientation is changing smoothly (not exactly monotonically) with the mentioned frequency 1.035. Now we describe the results for the orbit of the simplified equations, i.e. e = 1, for 12 "revolutions". We recall that the eigenvalues are as follows: Two of them are real, negative and of modulus greater than 1. The other four are disposed in two pairs of complex conjugate eigenvalues, one of them of modulus less than 1 and the other close to 1. Figures 8.4 to 8.6 show several plots according to Table 8.2. Here ||V/(f)||* means the norm of the first components x, y, z. For simplicity the * has been suppressed on the vertical axis label of the figures. Figures 8.4 (top) and 8.5 (top) are, in some sense, equivalent to 8.1 (x on the top) and 8.1 (x on the bottom). The mode associated to eigenvalue 3 (Figures 8.5 bottom and 8.6 top) is again the more regular. It is the analogous of mode 3 for
124
Relative Motion of Two Nearby
Eigenvalue
m
n
4-top
1
1
0
4-bottom
1
1
0
5-top
2
1
0
Variables plotted x,y, i,\\Vi{t)\\* z,y, *,\\Vi(t)\\ x,y,
5-bottom
3
1
0
z,\mw x,y,
6-top
3
0
1
6-bottom
5
1
0
Figure
x,y, *,\\Vi(t)W x,y,
*,\\vi(t)\r Table 8.2
Spacecrafts
Respective windows [-8.0,8.0], [-8.0,8.0], [-4.0,4.0], [0.0,10.0] [-0.3,0.3], [-4.0,4.0], [-0.3,0.3], [0.0,10.0] [-6.0,6.0], [-3.0,3.0], [-0.15,0.15], [0.0,6.0] [-4.0,4.0], [-2.0,2.0], [-0.1,0.1], [0.0,4.0] [-3.0,3.0], [-1.5,1.5], [-0.1,0.1], [0.0,3.0] [-0.1,0.1], [-0.05,0.05], [-1.0,1.0], [0.0,1.0]
Relevant information related to figures 8.4, 8.5, 8.6. Number of revolutions: 12.
e = 0 displayed in Figure 8.2. Finally the mode associated to the eigenvalue 5 is related to the almost vertical oscillation that for e = 0 is exactly vertical. For the more regular behavior, the mode associated to the third eigenvalue, the relative variation of the modulus of the spatial components, ||V/(£)||*, 1S greater than for e = 0. Concretely, as shown in Figures 8.5 (bottom), 8.6 (top) the quotient HV>(t)llm«/IIVr(*)llmin a l o n S t h e f u l 1 t i m e i n t e r v a l i s roughly 20. Finally we present the results for 24 "revolutions" and e = 0.9. This corresponds to the orbit described in Chapter 7. Figures 8.7 to 8.10 display the plots with parameters given in Table 8.3. In a similar way to what is found in the previous cases, Figure 8.7 (top and bottom) for eigenvalues 1 and 2, respectively, are reminiscent of the first mode in the periodic case. Now the associated eigenvalues are negative, one of modulus greater than 1 and the other less than 1. For the eigenvalue number 5, as in the case of 12 revolutions for e = 0 and e = 1, what is obtained is, essentially, a vertical oscillation (Figure 8.10). The remaining Figures 8.8 to 8.9 correspond to a couple of complex conjugate eigenvalues with modulus slightly less than 1. Different initial vectors are taken in this invariant plane under D$T- For them the ratio ||V/(*)||max t o ll^Wllmin ranges from 70 to 250. The nicer picture corresponding to m = 1, n = 1 gives a projection on the (x, j/)-plane which consists, essentially, of revolutions around the origin. The vector (x(t),y(t),z(t)) (first 3 components of Vi(t)) is almost contained in the (a;,j/)-plane. Table 8.4 below gives, for the different figures, the components of the initial vector, Vi.
<
5'
cr
3
00
cr re"
0
r—>
3
3
1
3
1
3
3
3
3
O O O
a> CT>
O ~J ~J
r—»
1
o o
o o
o o
o o o o
-4 CO
F—»
O
o
i—i
Cn
o o o o
oo 4^
1
1—1
h^ eo OS OS to O CO
r—
l
o o o o
o
to
to
o
o
o
en
CO o o o CO to h-' h-' CO 1—1 h-' to
o
a>
o en
o o
-J o to -J
o o
r—>
o o
o o
o o
o o
( ^ -
os eo oo ^J CO CO 00 o CC.
1 1 o o o o o o o o CO -4 o o o o
4*.
IX)
t-)
«s
o o to OO en eo eo o H-
1 1 1 1 1 1 1 o o o o o o o o o o o o o to to o to o o en to to o o I—' t—' OS r—> OS
to r — » h-» 00 00 CO 4^ to
o
o o o o o o o o o o o 4^ Cn 05 O
o o o O i-i o o -a o GO h-' o
o
cn ax CO en o eo en cn as oo OS ^i co eo
1
to
1 1 1
o to o eo en o
o o o o o o o o o o o o o o o o to ~a t—! 4^ o o o o o eo o oo 1—' to 4^ t—' -~j 4^ OS -J
--I to H-' CO o
t—1
1 1 1 1 1 1 1 o o o o o o o O o o o o o o o 00 eo to CT> o 4^ to eo 4^ o as CO h-» h-»
1 1
1—I
O
1
o o t£. Cn
1
o O o O o o to o O o o oo to to r—> to
1 o OJ o O
eo o
1 1 1 1 1 1 1 1 1 o o o o o o o o o o o o o o o> o to o o (—1 to 4^ eo en Oi Os o o o H-i Os
H-» Cn (O eo eo o 4^ 4^ t—' to CO tO CO On CO 4^ 00
o O
1 1 1 1 O O o o o o O o O o o o o O o o o o 00 0O O 00 os -a en o O ~J eo 00 0O eo o *>. oo oo OS h-» to eo Ol !""> ^1 o O eo h-' oo eo1 o Cn h-' --J en ( » t—' to 00 I- O o oo oo eo co o o CO oo OS -~-i H
1
3
*1
00 00 00 00 00 00 00 00 00 00 00 00 00 OO 00 00 oo 00 I—> CO CO 00 00 ~a --J CI 05 en en 4^ eo eo to to h- L h-' TO " rV c+- rr r+ rr c+ rr r+- rr crh rr r+ rr c+- rr c+- p U o U o X) o X}u o o o X) O u CD o 1CJ3 o X)
-
"
en
'—' top
|_! tO • en to -
p o 1o
^
'z—' *z—'
to en
~1-" "O
I-1 O to en
11
^—^ ^ *
e-f.
3"
is! H = «S
o
1—'
en
o
r—'
126
Relative Motion of Two Nearby
Spacecrafts
A
1/ r y
S
AN.. \ VJ ^ ^
A
iy
A
AN \ 'tJ ^ ^
V
A\
A \jJ
W
r-ri
l A A \J^
\/
Fig. 8.1 Modes associated to the eigenvalue number 1 for (m,n) (bottom).
\
= (1,0) (top) and ( m , n ) = (0,1)
Variations of the Relative Distance and Orientation.
Results
127
-JS.
V
\J y
VIMVWVW'
J
A
v
v
W/\AAA/ \y
-A
vv/*
UUUX/U
-A
v
n
A
A-t
\7\7
yrnnry Lfi
!'
r\
VT
U-VJ WWV\i nrvv\ MA/V
,- Y
Fig. 8.2 Modes associated to the eigenvalue number 3 for (m, n) = (1, 0) (top) and (m, ra) - (0, X) (bottom).
128
Relative Motion of Two Nearby
Spacecrafts
&WVWWVWW^wVYYV\M/VYVviV i
•
.4,
,t
to/wwwww .Ml
.
„
,
,
L _ _ _
,
,
it
Fig. 8.3 Modes associated to the eigenvalue number 5 for (m, n) = (1,0) (top) and (m, n) = (0,1) (bottom).
Variations of the Relative Distance and Orientation.
Fig. 8.4
Results
Modes associated to the eigenvalue number 1 for (m,n)
= (1,0).
129
130
Fig. 8.5
Relative Motion of Two Nearby
Spacecrafts
Modes associated to the eigenvalue number 2 (top) and 3 (bottom) for (m,n)
= (1, 0).
Variations of the Relative Distance and Orientation.
r
Results
131
x
A,- • w / - / V \ A / \ / ^ A A ^ / V .
Fig. 8.6 Modes associated to the eigenvalue number 3 for (m,n) number 5 for (m,n) = (1,0) (bottom).
= (0,1) (top) and the eigenvalue
132
Relative Motion of Two Nearby
A.
WVA ^
. I ^ I / ^ .
/\
Kw V W V
^r^Alr\sK\K^ w*w VUU
(A-
V
'• '
iVA,
j V
Spacecrafts
/^
-^=^7-
ZV,
sJJUu
r\
[y
-^^
v
T^
rZ
^ t
Fig. 8.7 Modes associated to the eigenvalue number 1 for ( m , n ) = (1,0) (top) and the eigenvalue number 2 for (m,n) = (1,0) (bottom).
Variations of the Relative Distance and Orientation.
Results
133
W A /wywV ^ v ^ w /W WVV^ (AAAAA -tf-V v U i, J
A VN/WW'
^Wvi^V^ f ^
•/A./Mvn
...,y
UM/W^
KJWWVWW
r\ ^l,^iA,^/vy ^ww^r^T^T^i A
A/\.
/s
t ^%M ^"V^'V^W L J\
^w^/wywi/1%
(UA
rIMi ...y
\>V Fig. 8.8 Modes associated to the eigenvalue number 3 for (m, n) = (1,0) (top) and the eigenvalue number 3 for ( m , n ) = (0,1) (bottom).
134
Relative Motion of Two Nearby
WV
A N
A
Spacecrafts
AA-M^A.^ V v
^ v ' T V j/y'WV v vv^v AAMAfio^
pp^^w
r\i
i^/WV^
u
^Wt
\J
AAAAA-.A^AA^/^.A''^* tW^vVvvvVvi/V^v^^"^
VMftMAAA#^VwVW'
77
[A
IAWVWVWWAM Kp„*rH/\rAf\J\sMf\,t
U/v/vvu'V^ Fig. 8.9 Modes associated to the eigenvalue number 3 for (m, n) = (1,1) (top) and the eigenvalue number 3 for ( m , n ) = ( 1 , - 1 ) (bottom).
Comments
Fig. 8.10
8.3
on the Applicability
135
of the Results
Modes associated to the eigenvalue number 5 for (m,n)
= (1,0).
Comments on the Applicability of the Results
The results given show that there are orbits around L\ (or L5) which perform revolutions around the equilibrium point in a more or less large strip. In what respects to the variations of distance and orientation between two massless particles with orbits close to the reference orbit and symmetrical with respect to it, we have found that the behavior is quite irregular. However there is one mode associated to a couple of complex conjugate eigenvalues for which the orientation changes with some regularity. With the exclusion of some small loops, the relative position vector revolves around the origin. The "period" of revolution is close to a synodical month. The ratio of maximal to minimal distance is of the order 100 to 1. The velocity of variation of the orientation and of the distance can be relatively large. For instance, if the maximum distance is 10 km and the minimum one 100 m, to fix the ideas, the rate of change can be bigger than 1 km/day and the rate of change of the orientation in close approaches can be bigger than 1 rad/day.
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Chapter 9
Summary
9.1
Objectives of the Work
First of all we state the problem and the objectives of this study. It is well-known that the restricted three-body problem has triangular equilibrium points. Those points are linearly stable in the planar case for values of the mass parameter, fi, of the RTBP below the Routh's critical value /xi, with the exception of the values /X2 and /i 3 . It is also known that in the spatial case it is nonlinearly stable, not for all the initial conditions in a neighborhood of the equilibrium points £4, L5 but for a set of measure relatively big. This follows from the celebrated KolmogorovArnold-Moser theorem (see, for instance [3]). In fact there are neighborhoods of computable size for which one obtains "practical stability" [7] in the sense that the massless particle remains close to the equilibrium point for a big time interval (some millions of years, say). The question is which part of this stability subsists when the idealized RTBP is substituted by the system Earth-Moon with its real motion and under the very strong influence of the Sun and the milder perturbations due to planets, solar radiation pressure, no spherical shape of the Earth and Moon, etc. What is known about the problem ? Despite a somewhat large bibliography (see Chapter 1 and the references) few things are known (at least as far as the authors have been able to find). What has been done concerning the problem follows two approaches: a) Numerical simulation of a more or less accurate model of the real solar system. Usually the starting point has been taken at one of the equilibrium points L4, L5. The results are slightly confusing. Depending on the initial epoch chosen, the orbit escapes in a few months or behaves according to the pattern that we proceed to describe. First the particle spirals from the equilibrium point outwards until it reaches a size of the order of magnitude of the Earth-Moon distance. Then the particle spirals inwards going close again to the equilibrium point. The behavior repeats itself several times 137
138
Summary
or, eventually, escapes after some of these big oscillations, when a closed encounter with one of the primaries is produced. b) Study of periodic or quasi-periodic orbits of some much simpler problem. This can be the bicircular model (see Chapter 2) or a coherent system close to the bicircular one and still periodic or a Hamiltonian system retaining a few leading terms of the equations. In this case, the methods of perturbation theory, mainly those based on Lie series lead to auxiliary Hamiltonians much more simple that can be studied analytically. The other cases can be studied in turn numerically or semianalytically. The results are again confusing: small changes in the approach produce big changes in the size of the periodic orbits or they even disappear. This is a consequence of the lack of convergence of the methods used. The concrete questions for this study have been: a) Is there some orbit of the real solar system which looks like the periodic orbits of the previous b) ? That is, are there orbits performing revolutions around L4 covering, eventually, a thick strip ? Furthermore we would be pleased if those orbits be quasi-periodic. However there is no guarantee that such orbits exist or they be quasi-periodic. b) If the orbit of a) exists and two particles (spacecrafts) are put close to it, how does the mutual distance and orientation change with time ?
9.2
Contribution to the Solution of the Problem
We describe shortly our main results towards the solution of the problem. a) All the relevant bibliography has been examined. The results have been stated in what is known on the problem in Chapter 1. They do not mean a big step towards the solution of the problem. b) We have considered five systems of differential equations. We list them by increasing order of complexity: i) The restricted three-body problem, planar and circular. It is autonomous. ii) The bicircular problem, where the Moon is taken in a circular orbit around the Earth and the Sun in a circular orbit around the EarthMoon barycenter. The massive bodies are not dynamically coherent. We study the motion of a particle under the action Earth+Moon+Sun. It is a periodic system. hi) The system that we call "intermediate equations", I.E. This is the system closer to the real one that is still periodic with the same period of the bicircular one, that is, the synodical period of the Moon.
Contribution
to the Solution of the Problem
139
iv) The system that we call "simplified equations". It is a system rather close to the real one, written in some useful normalized coordinates. It is no longer periodic but the terms which appear in it are quasiperiodic. The basic frequencies are the mean motions of the following angles: The Moon mean longitude, the longitude of the lunar perigee, the longitude of the mean ascending node, and the Sun elongation. It contains a few hundreds of trigonometric terms. v) Finally, the real equations of motion under the gravitational attraction of the full solar system. c) For the bicircular problem (either around L 4 or L5) periodic orbits have been found. Five periodic orbits have been located around each of the equilibrium points. One of them is quite small, two are medium size with the synodic period of the Moon. The other two have triple period and are somewhat larger. There is numerical evidence that there are not more periodic orbits of the given periods in a neighborhood of £4,5 of moderate size. Furthermore these orbits have been related to orbits of the RTBP, the well-known short period and long period families around £4,5 and a family obtained by triplication of the short period family. d) For the intermediate equations we have also found periodic orbits of the same type of the ones found in c). Despite the vector fields are close, it is not possible to pass directly from the periodic orbits of the bicircular problem to those of the I.E. A continuation method is required. All the orbits, except the small one, increase in size in a significant way. A reason for this is described, as well as the important consequences. The main one is the lack of convergence of any procedure using series expansions around £4,5. This explains previous confusing results appearing in the literature. Two approaches are proposed: either a semianalytic procedure (i.e., to work numerically with the coefficients of functions of prefixed form) or purely numerically, trying to continue the periodic orbits of the I.E. to orbits of the simplified ones. e) A semianalytic procedure has been introduced that can give an approximate solution of quasi-periodic type of the simplified equations, S.E. The algorithm is reminiscent to Picard's method for the solution of ordinary differential equations. We introduce modifications to avoid secular terms and we compute the function to integrate, when a previous iterate is known, by Fourier analysis using a FFT routine. The main difficulties of this approach are the big requirements of memory and computing time, as well as the convergence of the procedure that requires the use of a continuation parameter. It has been only partially successful. f) A parallel shooting method has been introduced to obtain numerically the desired orbits. As starting point, the intermediate periodic orbit has been taken (concretely for L4 and the orbit called A). With minor modifica-
140
Summary
tions, in the method one can start at an approximate quasi-periodic orbit, obtained as proposed in e), if it is available. Denote generically by w one of the variables x,y,z. If wsa is the semianalytic solution, then the standing equations, using a continuation parameter e are written as simplified equations "| or > + (1 real equations J
e)wsa.
For e = 0 we have the semianalytic solution (or, as done in practice, the periodic intermediate orbit) and for e = 1 the real solution. Two families of methods have been presented to solve the parallel shooting continuation problem. The difficulties encountered and how and why to over come them, has been discussed. g) Due to limitations in CPU time we have presented the solution for 12 revolutions and an approximate solution for 24 revolutions. Those CPU limitations can be dramatically reduced if the ephemerides are available as a table (JPL tapes for instance) and there is memory enough to store the vector field of the starting orbit at a suitable number of epochs. Furthermore, the extension to longer periods is only a matter of computing time, the program being implemented in general. h) The orbit (orbits in fact) found in g) has been taken as reference orbit. By means of the variational equations the behavior of nearby particles, as well as the mutual distance and relative orientation, has been studied. There are modes unsuitable due to big changes in the distance or to excessive loops in the revolution of one particle around the other, or to the almost vertical character of the oscillations. However, in any case a node has been found that is not so bad. Anyway the changes in distance and in the orientation angle can be fast (as much as 1 km/day and 1 rad/day for two particles running at a mutual distance ranging in [0.1,10] km. It is really remarkable that the orbits found are only mildly unstable. Several modes are stable and the instabilities (if any) produce an increase by a factor of 10 in a couple of years, at most.
9.3
Conclusions
As a final conclusion of the work, there is evidence that orbits moving in a somewhat big annulus around L4 and L5 exist, that these orbits have small components out of the plane of the Earth-Moon system, and that they are at most mildly unstable. The mutual distance of two points starting close to these orbits changes by an important factor (at most 1 to 100), and the orientation changes in a regular way, unless some small loops are present in the projection of the relative motion on the (x,y)-pl&ne or this projection comes too close to the origin.
141
Outlook
In any case we believe that it can be a useful place to locate one or two spacecrafts for scientific purposes because of the nice properties concerning stability. The station keeping necessary to maintain the orbit in its right place can be reduced to an unimportant amount. 9.4
Outlook
In this work we have introduced a methodology that in short can be termed as "increasing order of complexity in the models and continuation method to reach one after the other". The standing question is: what remains to do for the practical applications ? The answer, and this should be the objective of further work to complete the solution, is to use the final orbit (orbits) given in this work and to refine them using the suitable model for the solar system. An eventual continuation can help to end this purpose. There is also a recommendation concerning the final equations to impose when the orbit is determined for some number of revolutions. It would be desirable that the initial and final velocities (in the (x, t/)-plane) are rather close, especially if 228 revolutions (some 19 years) are requested. Finally, concerning the problem of relative motion of two nearby particles, when the reference orbit is available, if the couple A3, A4 = A3 is taken with parameters (m,n) = (1,0) and (0,1) we obtain solutions Xj(t), yj(t), Zj(t), j = 3 , 4 . A normalized linear combination can be taken equal to a(x3,y3,z3)T
+ P(x4,y4,z4)T,
a2 +/3 2 = 1,
and values of a, /? be determined such that ll^llmax
\\v\LJ be minimum, where ||V||* = (x2 + y2 + z2)1/2. the rate of change of the orientation.
A similar approach can minimize
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