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Houten, C. J. van, Houten-Groeneveld, I. van, Herget, P., and Gehrels, T. 1970, Palomar-Leiden Survey of the Faint Minor Planets. Astron. Astrophys. Suppl. 2, 339–448.
Kiang, T. 1966, Bias-Free Statistics of Orbital Elements of Asteroids. Icarus 5,437–449.
Kresåk, L. 1967, The Asymmetry of the Asteroid Belt. Bull. Astron. Inst. Czech. 18, 27–36.
Nairn, F. 1965, Spatial Distribution of the Known Asteroids. Res. Inst. Rept. T-9, Astron. Sci. Center, Ill. Inst. Tech.
VAN HOUTEN (submitted after meeting): Formula (3) in the PLS is based on circular orbits. Kiang indeed found an error: (a cos i-1) should be (a – 1); fortunately, for most asteroids cos i = 1.
Kiang's expression (1) should include the correction for the length of the arc traversed by the asteroid during the observation period, as explained in the PLS. The correct
in which n is the mean daily motion of the asteroid and At the period over which the observations extend.
This additional correction term is only important for large inclinations, and therefore it is not certain how this influences the data derived by Kiang. For that reason a comparison is made in table D-I with my own results given in an earlier paper' in which Zo is given. This can be transformed into Z by multiplication with a factor of 0.64. If the distribution of z is gaussian,
My values of zos were obtained by interpolation in table I of my paper. It follows that no systematic difference exists between my values and those of Kiang for the first four zones; whereas for zones 5 through 8, Kiang's values appear about 6 percent too low. This difference is hardly meaningful. Accordingly, Kiang's conclusion that the PLS asteroids are more concentrated toward the ecliptic than the numbered asteroids must be accepted as being correct. His tentative
TABLE D-I.—Comparison of zo.5 in the Eight Kiang Zones
Kiang zone van Houten, Kiang Difference
conclusion that vignetting effects may be the cause of this difference should be rejected on account of the small field effects of the Palomar 122 cm Schmidt plates. More likely the explanation should be sought in the remark made by Kiang (1966): “Large values of i are especially associated with values of the node around 90°....” Because in the PLS the nodal values of the high-inclination asteroids cluster around 0° and 180°, it can be expected that the PLS material is deficient in minor planets with large inclination, which is in agreement with Kiang's conclusion.
DISCUSSION REFERENCE Kiang, T. 1966, Bias-Free Statistics of Orbital Elements of Asteroids. Icarus 5,437-449.
ORBITAL SELECTION EFFECTS IN THE PALOMAR-LEIDEN ASTEROID SURVEY
The selection effects appearing in the list of minor planet orbits based on the Palomar-Leiden survey are discussed. In addition to purely geometrical effects produced by the limitation of the survey in time and position, the arrangement of orbits by perturbations plays an important role. Some apparent differences from the orbits of brighter asteroids can be easily explained. As an example of the operation of the selection effects, it is shown that the asteroidal jetstream believed to exist within the Nysa family is spurious.
The results of the Palomar-Leiden survey (PLS) (van Houten et al., 1970) will undoubtedly be, for years to come, the basic reference on the orbits of the faintest asteroids detectable by the present techniques. The PLS results provide an excellent counterpart to the list of numbered minor planets in the Ephemeris nearly as extensive and extended in the mass scale about three orders of magnitude lower. Although the accuracy of the orbits is insufficient for a recovery, it is quite satisfactory for statistical purposes. The only drawback is the inevitable limitation of the survey in time and position, which introduces selection effects rather different from those applying to the catalog of numbered asteroids. A correct appraisal of these effects is a prerequisite of any comparison of the two samples.
The important selection effect coming from the relation between absolute magnitude, mean distance, and mean opposition magnitude is common to both samples and will not be considered here. Selection effects special for PLS can be divided into two groups: those produced by the limitation of the survey in longitude (or time) and those produced by the limitation in latitude (or declination). Each of these consists of two components: one independent of the particular longitude interval covered by the survey and the other dependent on it.
In general, the former component produces primary effects, some of which have already been cited by the authors of the survey. Nevertheless, in some respects the latter component is also very significant, especially in the particular position of the survey areas chosen for PLS. The plates were taken, as in the previous McDonald survey, near the vernal equinox where a small number of background stars makes the searches more efficient than in opposition areas of lower galactic latitude. By coincidence, this is at the same time the region of maximum clustering of asteroid perihelia due to the perturbational alinement of their lines of apsides to that of Jupiter. Moreover, the survey area is situated about midway between Jupiter's nodes on the ecliptic, approximately in the same longitude as the poles of the precessional motion of the orbital planes produced by secular perturbations. Plates centered on the ecliptic deviate here about 1° north from the great circle of the central plane of the asteroid belt. This deviation is not negligible compared with the 6° half-width of the strip covered by the survey.
In the following analysis of osculating elements, only first- and second-class orbits (1119 in number, Q = 1 and Q = 2 in table 7 of PLS) will be used. Where proper elements are introduced, the data are restricted to the first-class orbits only (967 asteroids of table 9 in PLS).
THE EFFECTS OF LIMITATION IN LONGITUDE
Because the detectability of an asteroid depends on its apparent brightness at the time of exposure, an immediate consequence of a time-limited survey is a preference for those asteroids that happen to be near their perihelia. This preference for mean anomalies near M = 0°, clearly borne out by the PLS catalog, should be reflected in the orbital elements as a maximum occurrence of the perihelion longitude m = \, or about 0° to 30°, and a minimum at m = A + 180°, or about 180° to 210°. The strength of this effect obviously depends on eccentricity e, it vanishes as e approaches zero. The observed distribution of the perihelion longitudes of the PLS asteroids (Q = 1 and Q = 2) is shown in figure 1. The asymmetry is pronounced indeed, with about four times as many asteroids recorded near their perihelia as near their aphelia. As expected, the asymmetry decreases with decreasing eccentricity to a rather uniform distribution at e < 0.10. The only unexpected feature is the double maximum, with two lobes displaced about 30° to 40° on either side of the expected position. The reason for this duplicity is not quite clear. It may be noted that the errors in co and m due to measuring errors would also tend to disperse it to both sides of t = A and m = A + 180°. The importance of this effect should increase with decreasing e, in accordance with the edged outline of the distribution at e < 0.15. However, the angle of displacement appears too large for this interpretation as far as first- and second-class orbits are concerned. The selection effect of a time-limited survey on perihelion longitudes can be eliminated if the actual plate limit (in apparent magnitude) is replaced by an artificial limit of mean opposition magnitude, up to which the search is essentially complete. Unfortunately, this considerably reduces the number of orbits available. From figure 1(b) we see that asteroids with m0 × 19.0 are those that contribute substantially to the asymmetry. Some traces of the effect remain