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TABLE I.—Division of the Ring of Asteroids Into 12 Zones
B(1,0) < 10
M . . . . . . . . . . . . . . . . . . . . . . . 15 | 1.077 to 1.979 1,879 1.39 0 -
*Sample consists of all numbered asteroids given in the 1962 Ephemeris volume except 13 marked there as “lost,” 279 Thule at a = 4.282 AU,
spatial distribution of the Hilda and Trojan asteroids, and the asteroids in zone M are also rather special, the following discussion on the z distribution will be confined to zones 0 to 8 of the main belt. The frequency distributions of i in zones 0 to 8 observed in the adopted sample of numbered asteroids are listed in table II. Let n(Ai) be the number of objects in the interval Ai in a given zone; then the average number n(Ab) of objects in an interval Ab of the heliocentric latitude, taken without regard to sign, in the same zone is calculated, in practice, according to the formula
is the fraction of a circular orbit of inclination i that is included in the interval Ab. Now let bo.5 and bo.95 denote, respectively, the 50 percent point (the median) and the 95 percent point in the resulting b distribution; then the same percentage points in the corresponding z distribution are given by
In these expressions it is sufficiently accurate to set a equal to the appropriate median value shown in table I. The values 20.5 and 20.95 for each zone, together with their standard errors (s.e.), are listed in table II and are shown in figure 2 (the solid lines). Both show a steady increase with increasing distance from the Sun. Actually, bo.5 and bo.95 also show some increase with increasing a, but, of course, these increases are much less rapid and steady than the ones for 20.5 and 20.95. We now examine the data of fainter asteroids provided by the PLS. The observed frequency distributions are listed in table III. They are corrected for the latitude cutoff as outlined above. The corrected individual frequencies are not shown; only their marginal totals S1 are given. From the corrected i distribution in each zone except zone 8, which has too few objects for a proper determination, the values of 20.5 and 20.95 are derived as before; they are shown as broken lines in figure 2. Two effects are apparent: (1) The thickness of the system of fainter asteroids sampled by PLS also increases with increasing distance from the Sun, and (2) the thickness is noticeably less than that of the system of brighter asteroids at the same distance. It should be remembered that the samples used here are, in one case strictly and in the other approximately, limited by B(a, 0). If we note the actual differences between the solid and broken lines in figure 2 and the differences TABLE II.-Frequency Distribution of Inclinations and 50 and 95 Percent Points of z in Separate Zones
[Data from numbered asteroids with B(a, 0) < 15]
i distributions for zones 0 to 8
Figure 2.—The 50 and 95 percent points in the z distribution in nine intervals of the semimajor axis. Solid lines refer to numbered asteroids with B(a, 0) < 15, and broken lines to those found in the PLS. The “dominoes” along the top edge illustrate the areal densities of the largest asteroids given in the last column of table I.
B(a, 0) - B(1,0) listed in table I, we find that for a system of asteroids limited by constant B(1,0) (i.e., effectively down to certain physical size) there will be a rather reduced rate of increase in thickness with increasing distance. This, of course, assumes that effect (2) is real, which is by no means certain. The correction for the latitude cutoff is based on the value 5:9 for the extension of the search area in PLS. Because vignetting is certainly present on the plates used in PLS, the effective extension may be less than 5°9; thus the i distributions were undercorrected, leading to thickness estimates that are too low. This point should be examined in greater detail. My conclusions are as follows:
(1) The thickness of the system of asteroids of the main belt increases steadily with increasing distance from the Sun. The 50 percent point in the z distribution increases from 0.12 at 2.2 to 0.38 at 3.5 (all values in AU), and the 95 percent point increases from 0.27 at 2.2 to 1.24 at 3.5.
(2) Less certainly, at a given distance from the Sun, the thickness is less for the system of smaller objects than for the system of larger ones.
These statements corroborate and amplify the conclusions on the proper inclinations reached earlier (Kiang, 1966).
i distributions for zones 0 to 8