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even at moo 19.0. At mo - 18.0 some prevalence of n = 90° over m = 270° is indicated; however, the number of asteroids as bright as this is too small to justify any statistical conclusions. One of the direct consequences of this selection effect is an apparent correlation between m and e, to be distinguished from the virtual correlation produced by the alinement of the lines of apsides by planetary perturbations. Orientation m = A prefers the detection of high-eccentricity orbits; orientation m = A + 180° prefers that of low-eccentricity ones. This effect is shown in figure 2, with characteristic values of eccentricities plotted in a polar diagram of perihelion longitudes. The situation is complicated by the fact that the distribution in eccentricity is at the same time affected by the limitation of the survey in latitude. As already pointed out by van Houten et al. (1970), this limitation tends to eliminate the orbits of high inclinations, which are mostly associated with high eccentricities. On the other hand, the orbits closely alined to Jupiter's line of apsides, which are preferred in PLS by T = \, are also generally above average in eccentricity. Thus the observed distribution of eccentricities in PLS is determined by the interplay of three fundamentally different effects; and without separating them one cannot compare the data with those on the numbered, brighter asteroids. One can only conclude that there is no evidence of any significant difference. At the level of median values, where the selection effects are not extraordinarily prominent, the resemblance of the two samples is very close: eo.50 = 0.144 for the numbered asteroids and eo.50 = 0.147 for the PLS. As we proceed to the high-eccentricity tail of the distribution, the elimination of the asteroids with high-eccentricity, high-inclination orbits by
120 100 60 60
Figure 2.-Medians (p = 0.50) and limiting values of 10 percent occurrence (p = 0.10) of eccentricities of PLS asteroids plotted as a function of perihelion longitude m. The circles indicate comparative values obtained irrespectively of m from 1745 numbered asteroids.
the latitude effect becomes decisive and the proportion drops more rapidly in PLS than among the numbered asteroids. At the beginning of the 10 percent distribution tail, for which the values of eo.10 are plotted in figure 2, the PLS data do not surpass the eccentricity eo.10 = 0.247 of the numbered asteroids even at T = A. The selection effects on the osculating perihelion longitude to obviously are reflected also in the corresponding proper element 3, the difference between 6 and it being normally about 10°, and only rarely exceeding 30°. Thus the longitude limitation affects also the observed structure of the asteroid families in the 6/y plane. As a result of the relationship between the proper elements and the osculating elements, a random distribution in 3 implies a prevalence of certain values of m. The resulting direction of maximum concentration of the osculating perihelia depends on the semimajor axis a In the outer part of the asteroid belt a close alinement to Jupiter's line of apsides takes place. In the inner part, below a = 2.64, the deviation amounts to a few tens of degrees in the retrograde direction, but the maximum concentration is still not far from the PLS area. The elements so, eo corresponding to an orbit of zero proper eccentricity are given in table I for several values of a, which are medians from different asteroid samples. The sets of elements denoted “PLS” and “numbered asteroids” are composed of the median elements of each of these catalogs; the “center of the belt” is an ellipse passing through the median heliocentric distances of the 333 largest asteroids (g < 10.0) in different longitudes (Kresåk, 1967). All asteroid families for which more than 20 members have been identified in PLS are included. The values of induced
oscillations were interpolated from the table of Brouwer and Clemence (1961). The weighted means for the PLS asteroids are no = 354° and eo = 0.038.
It is evident that the degree of alinement to Jupiter's line of apsides is a function of the distribution in semimajor axes. The appreciable differences Aa and Atso between PLS and the numbered asteroids may be due to a relative lack of faint asteroids at a = 3.10 to 3.20, as suggested by van Houten et al. (1970). However, the actual difference is likely to be smaller because the selection of faint asteroids near their perihelia is more efficient for smaller semimajor axes. On the other hand, the ecliptical longitude of the survey plates slightly favors asteroids of greater semimajor axes. Also the elimination of high-inclination objects by the limitation of PLS in latitude (to be discussed in the next section) may affect this difference, especially because of the tendency of asteroids to group into families with discrete values of proper inclination.
An important consequence of the radial asymmetry of the asteroid belt is that the distribution in geocentric distance of the asteroids located, and hence also the gain of the survey, varies with the ecliptical longitude covered. In the case of PLS, including Jupiter's perihelion, the conditions are optimum. The differences between the perihelion opposition magnitude mp, the aphelion opposition magnitude ma, and the mean opposition magnitude mo are expressed by
for statistical purposes we can insert sin” a = 0.5.
we can write the ratio plof the number of asteroids observable at a perihelion opposition to that observable at an aphelion opposition as
The actual relative numbers of asteroids detected in a survey will differ from this, both because of the Law of Areas maintaining the asteroids for a longer time in the remote part of their orbits and because of the variation of the effective field of view with distance. Neglecting the trailing effect, we have
for the relative numbers of asteroids observable in the longitude of perihelion and aphelion, respectively, and
for the relative gain in a survey restricted to a narrow strip along the ecliptic.
The values of mp – mo, m.A - mo, p1, p.2, and p3 for selected types of orbits are listed in table II. The elements used for the computation are the same as in table I.
It must be emphasized that the validity of equation (4) for asteroid families is rather questionable; it appears probable that this distribution law holds only for the asteroidal “sporadic background.” Nevertheless, in relatively narrow intervals of mo involved in the selection effects, it can be adopted as a rough approximation to show, at least, what bias in the observed structure of individual families can be expected.
The data of the third line of table II show the effect of opposition longitude on the total number of asteroids detected in PLS. If the alinement of the lines of apsides of faint asteroids is exactly the same as that of the bright ones, a repetition of the survey under equal conditions, but near the autumnal equinox instead of the vernal equinox, should reveal a total number of asteroids reduced by a factor of 1.26 (i.e., to 79 percent). Inversely, this repetition would yield decisive information on the actual degree of alinement, and would make it possible to determine the actual distribution of eccentricities from the differences in the relation between e and T (fig. 2) against that determined from PLS.
THE EFFECTS OF LIMITATION IN LATITUDE
The principal effects of this type, a strong preference for nodal longitudes Q = N = 0° and Q = N + 180° = 180° and the elimination of orbits of higher inclination at other nodal longitudes, have already been pointed out by the authors of the PLS. The pronounced selection in Q, as well as the gradual diminution of the effect with inclination approaching 0°, is clearly shown in figure 3(a). One important consequence that has not been considered is the transformation of this effect into the system of proper elements.
The poles of the precessional motion of most asteroidal orbits are inclined about 1° from the pole of the ecliptic in the direction north pole - vernal equinox → south pole - autumnal equinox. The weighted mean position of the plane perpendicular to this axis is defined by Q = 88° and io = 0°97 for the PLS asteroids. The elements Q0, io applying to different orbits, as determined from the median values of a using the table of Brouwer and Clemence (1961), are given in the last two columns of table I. The position of the nodes approximately 90° from the area of the PLS makes the data rather sensitive to this deviation. Although the distribution of osculating nodes Q is essentially