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Figure 2.-Asteroid sin is plotted against semimajor axes. Open circles are for objects that can encounter Mars during the age of the solar system. The two high-inclination regions are isolated from the main belt by secular resonances.

the one that starts at 2.03 AU runs between the Phocaea region and the main belt. A third strong resonance caps the belt off at about 30° inclination. The gap at 2.5 AU is the Kirkwood gap at the 3:1 commensurability with Jupiter. The Hungaria and Phocaea regions are not large families but segments of the belt isolated by resonances." The former region contains one normal-sized family and the latter contains two. Numerous families can be recognized in the data. Arnold's (1969) work is strongly confirmed. Many of the families are found to contain a relatively large object, and this large object is always at the edge of the family in a, e', sin i' space. These families with large objects would seem to be debris from a major cratering event on the large asteroid. The families without large objects presumably result from total disruption of the parent asteroid. The distinction between families with and without a large object is somewhat artificial. The two classes really merge. Of course, total disruption is the limiting case of

"See, however, p. 363 of PLS.


cratering. The above opinions obviously rely on the collision theory of the origin of asteroid families.


This work was one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract no. NAS 7-100, sponsored by the National Aeronautics and Space Administration.


Arnold, J. R. 1969, Asteroid Families and Jet Streams. Astron. J. 74, 1235-1242. Brouwer, D. 1951, Secular Variations of the Orbital Elements of Minor Planets. Astron. J. 56, 9–32. Brouwer, D., and Clemence, G. M. 1961, Secular Perturbations. Methods of Celestial Mechanics, ch. 16, pp. 507-529. Academic Press, Inc. New York. Brouwer, D., and van Woerkom, A. J. J. 1950, The Secular Variations of the Orbital Elements of the Principal Planets. Astron. Papers Amer. Ephemeris, vol. 13, pt. 2, pp. 85-107. Hirayama, K. 1918, Groups of Asteroids Probably of Common Origin. Astron. J. 31, 185-188. Hirayama, K. 1923, Families of Asteroids. Jap. J. Astron. Geophys. 1, 55-105. Hirayama, K. 1928, Families of Asteroids. (Second Paper.) Jap. J. Astron. Geophys. 5, 137-162. Houten, C. J. van, Houten-Groeneveld, I. van, Herget, P., and Gehrels, T. 1970, Palomar-Leiden Survey of Faint Minor Planets. Astron. Astrophys. Suppl. Ser. 2(5), 339-448. Öpik, E. J. 1963, The Stray Bodies in the Solar System. Pt. 1. Survival of Cometary Nuclei and the Asteroids. Advances in Astronomy and Astrophysics, vol. 2, pp. 219-262. Academic Press, Inc. New York. Williams, J. G. 1969, Secular Perturbations in the Solar System, pp. 1-270. Ph. D. Dissertation, UCLA.


MARSDEN: Does the 9:2 resonance with Jupiter (or 2:3 resonance with Mars) really have a decisive influence on the motions of the Hungaria asteroids? WILLIAMS: These are only approximate commensurabilities and probably unimportant as far as the existence of these objects is concerned. Their mean motions spread over a large range, 1270 to 1410 arcsec/day, whereas the commensurabilities in question lie at 1346 and 1258 arcsec/day. The Jovian commensurability is of seventh order and should be very small, whereas Mars has such a small mass that it is hard for it to have a significant influence. Actually the Hungaria asteroids lie in a small island of stability between two of the secular resonances and the Mars crossing boundary. I think they demonstrate that the belt was once much larger but that Mars has swept out, by collisions and close approaches, the regions that are now empty. BRATENAHL: The histogram of N versus a is remarkable in showing how sharply Mars defines the inner boundary of the asteroid belt. Do you have any estimate of the lifetime of Eros, and is that limited by an impact on Mars or on which planet? WILLIAMS: The Mars crossers typically have lifetimes of 108 to 109 yr and may impact any of the terrestrial planets. If Eros could evolve into an Earth crosser through secular perturbations, then its lifetime might be an order of magnitude smaller. ANONYMOUS: Can an explanation be given of the mechanism by which observational selection can give rise to an apparent jetstream?

WILLIAMS: I will give an example of a selection effect for the Flora family due to secular perturbations. These perturbations cause a bias in the eccentricities that has an approximately sinusoidal dependence on the longitude of perihelion and an amplitude of 0.05. This causes the average perihelion distance of 1.9 AU to have peak variations of +0.1 AU. The objects with perihelia of 1.8 AU will be 0.7 mag brighter than those objects with perihelia of 2.0 AU, the two extremes being 180° apart in the sky. Because an asteroid is discovered in the vicinity of its perihelion, more small objects will be seen in the direction of the closer perihelia. Using the factor of 2.5 per mag for the differential number density from the PLS gives a factor of 2.59.7 = 1.9 in the ratio of the number of objects discovered at the two extremes. To be cataloged, an object must be seen at a minimum of three different oppositions. Because the discoveries at the different oppositions are usually independent of one another, the peak factor among cataloged objects will be 1.93 = 6.8. Averaging the sinusoidal bias over a 180° range of longitude of perihelia, in the direction of the minimum perihelia, results in about four times as many cataloged objects as in the opposite half of the sky. Such a concentration would be considered to be evidence of a jetstream. There are also seasonal selection effects due to weather and altitude of the ecliptic.

The above is only an order of magnitude calculation, but it illustrates the severity of the selection effects among the cataloged asteroids fainter than mean opposition photographic magnitude 15.0.


Sterrewacht te Leiden

The Palomar-Leiden survey (PLS) was set up as an extension to fainter magnitudes of the McDonald survey. The latter is, therefore, the more important survey as far as asteroid statistics are concerned. The main result of the PLS is that no clearcut differences exist between the fainter asteroids found in this survey and the numbered asteroids in the distribution functions of eccentricity, inclination, and semimajor axis and that the statistical relations found in the McDonald survey have a continuous extension in the PLS material. I would, therefore, propose not to summarize the results of the PLS, which would appear to be a tedious job, but to give here some new results that should properly have been included in the publication, but, for reasons of pressures to publish as soon as possible, were not.


In the PLS a derivation is given of the density distribution perpendicular to the plane of the ecliptic, under the assumption that this distribution is independent of the distance to the Sun. Whether this was really the case was not checked. This is done here: the average value of zo = a sin i has been determined as a function of a, and this value is assumed to be proportional to the average distance of the asteroids to the plane of the ecliptic at a distance to the Sun equal to a. The results are given in table I and depicted in figure 1. It is seen that the assumption of constancy of Z0 is wide off the mark; Zd varies approximately linearly with a, with the exception of a bulge near a = 2.6, which is caused by the new Io family. The value of zo at a = 3.3 is about four times as large as at a = 2.2.

This result shows that the average value of the orbital inclination is a function of the distance to the Sun, the nearby asteroids having, on the average, considerably smaller inclinations than those in the outer parts of the asteroid belt. It was checked that this is also the case for the numbered asteroids, so this result should not be new.

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