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TABLE I.—70 Values for Various Sizes of the Semimajor Axis

a 20 2.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.14 2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 2.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 2.75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 T T T I I T I T T I T 0.6 H e * T – H. - - e 0.4. H e ... •T . Fo 0.2 H. . ...’’ * -0.0 1 L 1 l l I —l- —l I —l I 2.2 2.4 2.6 2.8 3.0 3.2

Figure 1.-zo = a sin i as a function of a.

THE PHASE FUNCTION OF THE TROJANS

The average phase function of the PLS asteroids was shown to be of the form F(a) = 1.03T(0) + 0.039|a|- 0.05 (1)

in which T(0) is the opposition effect found for the brighter asteroids, a the phase angle, and F(a) is the brightness variation of the asteroid due to phase; it is zero at phase 4”. In practice, the coefficient of T(0) was determined by only a limited number of asteroids that came close to the opposition point. Of these, the Trojans played an important part; because of their slow motion and their large distance from the Sun, they were always observed at small phase angles. The maximum phase angle for a Trojan in the PLS is 6.5. It follows that the Trojans contribute little to the linear part of the phase function, and heavily to the opposition effect. It is, therefore, surprising that most Trojans yielded negative residuals with respect to expression (1), as may be seen in table II. Here are listed the difference in brightness of the PLS Trojans in September and October 1960, due to phase, the corresponding values based on expression (1), and their difference, O - C.

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Apparently the phase function of the Trojans is flatter than that of the normal asteroids, which indicates a different surface structure. On the other hand, omission of the Trojans in the determination of the phase function will fairly certainly result in a larger value of the coefficient of T(o), so that both the opposition effect and the linear part of the phase function should be steeper for the PLS asteroids than for the numbered ones. This redetermination of F(0) for the PLS has not been made as yet.

DISCUSSION

KOWAL: On August 19 and 20, 1969, plates of the following lagrangian point of Saturn were taken at the Palomar 122 cm Schmidt telescope. The plates are 6.6° in diameter and were centered about 1:5 north of the Ls lagrangian point. No “Trojan" asteroids were found. It is estimated that any object brighter than about 19.5 mag would have been detected.

Because Gehrels (1957) has examined the preceding lagrangian point of Saturn with negative results, it can be stated with confidence that Saturn does not have any “Trojan” asteroids brighter than 19 mag.

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DISCUSSION REFERENCE

Gehrels, T. 1957, Indiana Expedition to South Africa, April-June 1957. Astron. J. 62, 244.

[Editorial note: For further information regarding the PLS, see Kiang's paper, page 187; Kresók's paper, page 197; the “Discussion” following Dohnanyi's paper, page 292; and Lindblad's paper, page 337.]

THE DISTRIBUTION OF ASTEROIDS IN THE DIRECTION PERPENDICULAR TO THE ECLIPTIC PLANE

T. K/AWG
Dunsink Observatory
Ireland

For examining the steady-state distribution of asteroids in the direction perpendicular to the ecliptic plane (the z distribution), we shall assume all orbits to be circular. This assumption is incompatible with the north-south asymmetry found by Nairn (1965); but Kresak (1967) has shown that the asymmetry is caused by a combination of cosmic and human factors and is present only among fainter asteroids, B(a, 0) > 16, where the discovery is grossly incomplete. There is another perhaps even more cogent reason for using only the brighter asteroids: The easily understandable practice of confining asteroid hunting close to the ecliptic plane has meant that among the fainter objects, orbits with high inclinations are underrepresented (Kiang, 1966). Actually, in the range 143 B(a, 0) < 15 where, I estimate, the discovery is 95 percent complete, the sample of inclinations may already be somewhat biased in the same sense. One has to balance this risk, however, with the advantage of a much greater data size; and I shall use all the numbered asteroids with B(a, 0) < 15 as given in the 1962 Ephemeris volume (excluding 13 that are regarded as “lost”).

A very welcome new set of data is provided by the Palomar-Leiden survey (PLS) (van Houten et al., 1970). In this case, important selection effects should and can easily be made. According to the authors (van Houten et al., 1970, p. 360), the area searched extends to a height of 5°9 from the ecliptic. Consider all the orbits with the same radius as for these, the search extends to a heliocentric latitude of b = 5°9(a - 1)/a. Although an orbit with inclination i < b lies entirely within the latitudes +b, an orbit with i > b has only the fraction

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lying in the same range. Hence, as far as the shape of the distribution of i at a given a (in practice, within a small range Aa) is concerned, the correction factor is simply 1/fl. Expression (1) differs a little from expression (3) in the PLS paper (van Houten et al., 1970, p. 361), but appears to be more in line with the assumption of circular orbits.

Among numbered asteroids known at a given time, one always finds a positive correlation between a and is this feature has been reported repeatedly, but the question whether this is due, at least in part, simply to the fact that Earth is inside the ring of asteroids has never been examined. Of course, even if the distribution of i is the same for all a, there will still be a systematic increase of the thickness of the system with increasing distance from the Sun. Here we shall concentrate on the z distribution at different intervals of a.

As may be seen from figure 1, the well-known Kirkwood gaps and other commensurability points divide the main belt (2.0 <a 33.8) quite naturally into nine zones. These will be labeled zones 0 to 8 inclusive. The Hilda group, the Trojans, and the range 1.0 <a & 2.0 will be labeled zones 9, T, and M (for Mars), respectively. Table I lists some statistics of the zones. The next-to-thelast column refers to the numbers of the largest asteroids (B(1,0) < 10) found in the sample. These numbers are very likely to be complete, except the one in zone T. The last column gives the numbers of these objects per unit circle (in AU) of the ecliptic plane. These areal densities are only approximations to the average state of affairs at the corresponding distances from the Sun because of the strong radial asymmetry in the distribution in the ecliptic plane (Kresák, 1967). Because resonance effects obviously dominate the orbits and thus the

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Figure 1.-Frequency distribution of semimajor axes of asteroids in intervals of 0.001 AU. Sample consists of the 1647 numbered asteroids given in the 1962 Ephemeris volume minus the 13 asteroids that are marked as “lost.” The following five fall outside the diagram: 1566 Icarus, 1620 Geographos, 433 Eros, 279 Thule, and 944 Hidalgo with orbital radiia of 1.077, 1.244, 1.458, 4.282, and 5.794 AU, respectively. Commensurability points are marked with arrows, together with the ratios of periods (asteroids Jupiter).

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