The Theoretical Frequency Distribution of Photographic Meteors

34.

ASTRONOMY: P. M. MILLMAN

PROC. N. A. S.

throughout the same region as that occupied by the globular clusters; and these clusters appear more clearly than ever to outline our galaxy, as the globular clusters recently discussed by HubbleW and by Shapley and Mohr,6 outline Messier 31 and the Large Magellanic Cloud, Along the galactic plane our direct measures are troubled by absorption, and we can only surmise, from the distribution of the globular clusters and cluster type Cepheids in galactic latitudes 200 to 400, that the dimensions in this direction are much greater than shown here for the directions perpendicular to the plane. The faintest variables in fields 209, 211 and 212 would project on to the galactic plane at a distance of more than 20,000 light years from the sun. The extension of the search to fainter magnitudes in these fields and in fields of somewhat lower latitudes will be an important step in measuring the extent of the Milky Way in its plane. 1 E.g., in H. Mon., 2, 22 (1930). H. B. 874 (1930). Paper read at Ann Arbor meeting of the National Academy of Sciences, November, 1932. 4 H. B. 890 (1932). 6 Ap. J., 76, 44 (1932). 6 H. B. 889 (1932). 2

3

THE THEORETICAL FREQUENCY DISTRIBUTION OF PHOTOGRAPHIC METEORS

By. PETER

M. MILLMAN*

HARvARD COLLEGE OBSERVATORY

Commuiicated December 8, 1932

In considering the frequency distribution of meteors we must first make some simplifying assumptions before the subject can be satisfactorily treated. Let us then assume that: 1. The lutninous part of the paths for all meteors is at an average height of;lb1Jkilometers above the surface of the earth. 2. All meteors are moving with the same constant linear velocity with respect to the earth. 3. If we regard a meteor as a moving point of light, then it radiates a constant quantity of light per second. 4. If the absolute magnitude of a meteor is determined by the light radiated per unit of path when viewed perpendicular to the path at some unit distance, then there is a definite absolute magnitude distribution for meteors entering the atmosphere at the same angle.

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5. All meteors from the same radiant are moving in parallel paths and are distributed with constant density over a cross-section perpendicular to the direction of motion. Furthermore, some law of absolute magnitude distribution must be assumed. Let Nm be the number of meteors brighter than absolute magnitude m. Opik' used the distribution Nm+ i/Nm = 4 for meteors apparently brighter than the third magnitude. In the derivation of the curves which accompany this paper the three distributions Nm+ I/Nm = 3, 4 and 5, have been used. The apparent brightness of any meteor will depend on its distance from the point of observation, the angle its path makes with the line of sight, and the amount of atmospheric absorption present. The curvature of the earth must be taken into account; this was done by first computing frequencies on the assumption that the earth's surface is a plane and then deriving a correction for curvature which was applied to the results. On the assumption of a plane surface the distance of the meteor will vary as sec z, where z is the zenith distance. The brightness will decrease with distance as sec2 z but the angular velocity will decrease as sec z. We may assume that the photographic effect varies inversely as the angular velocity, and hence we have the photographic brightness decreasing as sec z owing to increase of distance from the observer. The angular velocity will also vary as sin i, where i is the angular distance from the radiant. The luminosity of the meteor therefore varies as cosec i. Absorption of the air cuts down the magnitude in proportion to the length of the air path. The photographic absorption coefficient was taken as 0.47 magnitude2 and the length of the air path from Harvard Annals XXIII.3 The effect of distance -was combined with that of absorption and a curve drawn showing the change in apparent magnitude with zenith distance. Another curve was drawn showing the increase in magnitude with angular ii*taxice from the radiant. For a given position of the radiant, the change in apparent magnitude due to the above effects could be computed from these curves for any position in the sky. The corresponding relative numbers of meteors for the three absolute magnitude distributions were then read from curves drawn between log Nm and m To obtain the relative frequency of meteors brighter than any given apparent magnitude another effect must be considered. As the zenith distance increases, a unit solid angle at the point of observation covers a larger area parallel to the earth's surface at any fixed height; its effect will be to increase the frequency of meteors near the horizon. This factor varies as sec3 z and was read from a fourth curve. There remained now to apply the correction due to the curvature of the earth. The amount of this correction was computed from a table given by Opik.4 Frequencies at various points were determined for single

ASTRONOMY: P. M. MILLMAN

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PROC. N. A. S.

radiants at altitudes of 15, 45 and 90 degrees. Lines of equal frequency were drawn by graphical interpolation and the resultant plots are shown in figures 1 and 2. The frequency at the radiant was found by averaging values for distances from half a degree to five degrees from the radiant. Closer than half a degree the cosec law for increase of brightness with approach to the radiant does not hold. Figures 1 and 2 give the most probable relative frequencies over the sky for meteors belonging to a single shower. To find the values for sporadic meteors such distributions must be summed for radiants in all altitudes and azimuths. This was done, mainly by graphical interpolaN-

200

N-rn

+5 ~

RADIAT AT

120

80 40

9T 086

70

6r 4030 20 60 15

10

FIGURE 1

The photographic frequency of meteors with radiant at the zenith plotted against altitude h.

tion, assuming radiants distributed evenly over the whole sky. In the summation the radiants near the zenith must be given higher weight since here the meteors are coming down perpendicular to the atmospheric surface and there will be a greater number per unit area per second. For this reason meteors from the different radiants should be weighted as cos Z, where Z is the zenith distance of the radiant. Dr. Opik kindly placed some of his unpublished work at the disposal of the writer. According to this, which was based on both observation and theory, meteors of the same mass and velocity have luminous paths which vary in length approximately as sec Z, and hence their luminosity per unit length varies as cos Z. This variation in the length of path means

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37

a change of magnitude of 2.5 log cos Z, and when this is translated into frequencies for the three absolute magnitude distributions used we have factors (cos Z)120, (cos Z)150, (Cos Z) 1.75. Combining these expressions h

RADIANT :s AT h-4 5

h

h

RADIANT

RADIANT

in' 45e

2fl,

FIGURE 2

Curves showing the relative chance of photographing a meteor of any given shower in different positions in the sky. Each plot covers half of the visible hemisphere.

with the factor mentioned above we see that meteor frequencies from different radiants should be weighted as (cos Z)2.20, (COS Z)2.60, (COS Z)2.75, for Nm+i/Nm = 3, 4 and 5, respectively.

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PRoe. N. A. S.

The results of this summation are given in table 1 and plotted in figure 3(a). The relative frequencies are reduced to a scale equalling 100 at v the zenith. TABLE 1 FREQUENCY OF PHOTOGRAPHIC METEORS

Nm+ i/Nm = 3

ZBNITH DISTANCE

4

100 100 100 98 96;1 80

0 15 30 45 60 75

100 100 90 76 54 28

5 100 98 86 56 28 8

The above values hold for chart photographs of meteors, where the position angle of the meteor trail has no effect on the quality of the photograph. In the case of meteor spectrum photographs, no spectrum will l

l

(d) l

12

l

l

l_-

L

120

C-~~~~~~~-q

40

e0

;0

d0

S0

40

30

20I

to

h 0

FIGURE 3 Frequency of photographic meteors for radiants distributed evenly over the sky plotted against altitude h.

be recorded if the meteor happens to cross the photographic field perpendicular to the refracting edge of the prism. Since there is a preferential downward motion of meteors near the horizon, while at the zenith the motions are at random, the greater zenith distances, on this

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account, will have a higher weight when we are attempting to photograph meteor spectra. The amount of this effect was evaluated by dividing the sky into sections and computing the proportion of meteors coming from radiants in each section for points 15 and 45 degrees above the horizon. The effective dispersion of a spectrum photograph will vary as cos 0, where 0 is the angle the meteor trail makes with the refracting edge of the prism. The weight of a photograph was taken to vary as the effective dispersion, and it was found that for meteor spectrum photography the frequency for points at altitudes of 15 and 45 degrees should be multiplied by 1.36 and 1.12, respectively, for all three absolute magnitude distributions. The resultant values are listed in table 2 and plotted in figure 3(b). TABLE 2 Z3NITH DISTANCE

Nm+ i/Nm C 3

FREBQUNCY OF PHOTOGRAPHIC METEORS 4

0

100

100

15 30 45 60 75

102 106 110 116 110

102 96 86 66 36

5

100 100 90 62 34 10

It will be seen that the apparent frequency distribution of photographic meteors depends greatly upon the absolute magnitude distribution used in the calculations. The assumption of a uniform distribution of radiants over the sky is of course only true on the average. There is a natural clustering of radiants near the apex of the earth's way. At 6:00 P. M., with the antapex high in the sky, the frequency at small zenith distances will fall slightly. After midnight, when the apex is near the eastern horizon, the frequencies at great zenith distances, especially in the east, will be increased slightly. At 6:00 A. M., with the apex high in the sky, frequencies at small zenith distances will be increased. * AGASSIZ FELLOW. 1 Opik, E., Bull. Harvard Col. Observ. 879, 5 (1930). 2 King, E. S., Annals Astron. Observ. Harvard Col., 59, 114 (1912). 3 Pickering, E. C., Ibid., 23, 95 (1890). 4

Opik, E., Pubi. Tartu Observ., 25, 7 (1922).