The techniques of radar offer a potentially powerful tool for the study of planetoids. It is a new approach, having been applied to extraterrestrial targets only recently in the history of astronomical study. Although the Moon was first, Venus has been observed by radar only since 1961. Since that time the techniques and capability of radar have evolved rapidly and many important new facts about Venus have been gathered. Further, the more distant and difficult targets, Mercury and Mars, have also yielded up secrets to radar probing. Finally, during the close approach of June 1968 Icarus itself was observed by radar from two different observatories (Goldstein, 1969; Pettengill et al., 1969). Review articles on radar studies of the planets are given in Shapiro (1968) and Goldstein (1970).

The word “group” is so general that I would like to suggest that here it is used in its most general way: a group of asteroids is a collection of minor planets that have some feature in common. If we agree on this use of the word “group,” then we can discern the following asteroidal groups:

Families of asteroids were first found by Hirayama (1918, 1923, 1928). More recently Brouwer (1951) and Arnold (1969) have extended greatly the number of families known from the cataloged asteroids. The Palomar-Leiden survey (PLS) (van Houten et al., 1970), which mainly covered very faint, uncataloged objects, found several more families.

Except for the work of Hirayama (1918), which used osculating elements, all of the above studies looked for clusterings of the semimajor axis a, the proper eccentricities é, and proper inclinations í. The calculation of proper elements involves using a theory of secular perturbations to remove the long-period, large-amplitude disturbances of the major planets. The theory used Up to now (Brouwer and van Woerkom, 1950; Brouwer and Clemence, 1961) involved a low-order expansion in the eccentricities and inclinations. There is now a theory available (Williams, 1969) that will accurately handle much higher eccentricities and inclinations than before.

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.

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 14<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”).

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.

Theories on the origin and evolution of asteroids are confronted with three types of experimental tests. The first refers to the dynamic state of the asteroids and consists of orbital and in some cases spin data for bodies as small as about 1 km. (There are reasons to assume that the size spectrum extends to very small objects but nothing is known about them.)

The second type consists of observations of the chemical and structural properties of objects fallen to Earth from space. Here the relationship to asteroids is much more tenuous. Nevertheless, the study of meteorites has provided important insight into the chemical evolution of small bodies in space. As long as one realizes that such data refer only to bodies of special structure, composition, velocity, and other orbital characteristics, they can be useful also for conjectures about asteroids.

Recently Anders (1965) has shown that the smaller asteroids are overabundant relative to their initial numbers because of the collisional fragmentation of larger asteroids. Allowing for this fragmentation, he has derived a reconstructed initial radius distribution. Although the reconstructed distribution only comprises a little more than 100 asteroids and consequently has a large statistical uncertainty as well as the uncertainty introduced by the reconstruction process, Anders finds that when displayed on a log-log plot, the distribution can be represented fairly well by a gaussian curve with a peak near radius R = 30 km. This work has basically been substantiated by Hartmann and Hartmann (1968), although Hartmann (1968) notes that a gaussian distribution underestimates the observed number of more massive asteroids. This suggests that the initial asteroid distribution function was probably broader than a gaussian function, but it still retains a distinct bell-like appearance.

Convincing evidence exists that meteoritic matter was reheated shortly after the initial condensation of the solar nebula for those meteorites thought to be derived from parent bodies. This evidence takes the form of cooling rates carefully determined from diffusion studies of the migration rate of Ni across kamacite-taenite boundaries in iron meteorites (Fish, Goles, and Anders, 1960; Goldstein and Ogilvie, 1965; Goldstein and Short, 1967; Wood, 1964). The notion that the irons condensed directly from the solar nebula requires that these measurements and the existence of large Widmanstätten figures be explained as a condensation event. This seems rather unlikely and, in any event, requires a far more complex explanation than heating and melting in a parent body.

Alfvén and Arrhenius (1970) have considered the development and stability of jetstreams by collisional interactions among grains; and they propose that in a jetstream environment, grains may collect and adhere to form self-gravitating embryos. Gravitational attraction of smaller particles by these embryos may then lead to a net accretion because high relative approach speeds, which could erode or break up embryos, have been minimized during the formation of the jetstream.

It is suggested that aerodynamic forces played a significant role in the selective accumulation of chondrules on asteroids moving with respect to the gas in a primeval solar nebula. Particles smaller than millimeter chondrules would sweep by an asteroid moving in a critical velocity range, whereas larger particles could be accumulated by impact. Theory and calculation cover the case of subsonic velocity and asteroidal diameter up to 50 km for a nebula density up to 10-6 g/cm3, or higher for smaller asteroids.

Data have been accumulating since the beginning of this century that indicate that most, if not all, large asteroids have periodic lightcurves. The variations that are seen have periods of the order of several hours and can be understood as being caused by bodily rotation, accompanied by changes in shape and/or surface properties. Because corresponding color changes are usually absent, the former explanation of a variation in cross section is probably the correct one.

As a result of mutual inelastic collisions, frequent on a geologic time scale, the mass distribution of asteroids undergoes constant change. Using a simplified velocity distribution for asteroids, the redistribution of their masses caused by collisions can be mathematically modeled as a stochastic process and the distribution of asteroidal masses can then be obtained as the solution. This paper is a review of recent progress on this problem.

The most detailed discussion of this problem considers the influence of the following collisional processes on the asteroidal mass distribution: (1) loss of asteroids by catastrophic breakup, (2) creation of new objects from the fragments of a catastrophically disrupted one, (3) erosive reduction in the masses of individual asteroids, and (4) erosive creation of new objects (i.e., production of secondary ejecta during erosive cratering by projectiles not large enough to catastrophically disrupt the target object). The main result is that after a sufficiently long period of time the population of asteroids may reach a quasi-steady-state distribution, regardless of the initial distribution. This final distribution is a product of a slowly decreasing function of time by a power law of index 11/6 for masses smaller than the largest asteroids. For the largest asteroids, an additional factor is included that expresses the influence on the distribution of the absence of masses larger than those observed. The observed distribution of bright asteroids from the McDonald asteroidal survey and that of faint ones from the Palomar-Leiden asteroidal survey are each individually consistent with the theoretical distribution, although they differ from each other by a numerical factor.

This paper is concerned with some aspects of determining the evolution of the size distribution of a finite number of mutually colliding and fragmenting particles such as the asteroids or interplanetary dust. If n(m, t) is the number of particles per unit volume per mass interval at time t, then n = dn/dt is the rate at which that number changes with time. This rate can be calculated if the laws are known according to which the colliding bodies erode one another and fragment and if the influence of collisions on the motion of the particles is known. To reduce the complexity of the problem, one assumes that the speed of approach between the bodies is always the same vcoll and that they, as well as the debris, occupy a fixed volume (“particles in a box”). Only collisions between two bodies are considered, and the way in which erosion and fragmentation occurs at a given value of vcoll depends only on their masses. The particles are assumed to be spherical.

It has been generally assumed in the past that the fragmentation of asteroidal bodies and the production of meteorites are solely the result of collision events. (See Dohnanyi, 1969; Hartmann and Hartmann, 1968; Wetherill, 1967.)

A possible mechanism of noncollisional fragmentation will be proposed below, its proper framework of applicability will be defined, and evidence suggesting and supporting its existence will be adduced. Briefly, it is shown that the presence of even trace amounts of hydrogen in meteoritic metal phases (Edwards, 1955) may have caused the parent bodies of iron meteorites to undergo, spontaneously, delayed brittle fracture under the action of prolonged slow stresses, the imprint of which has been recorded in the phase structure of meteorites (Baldanza and Pialli, 1969). This phenomenon, termed “hydrogen embrittlement,” has been amply documented in the literature on the metallurgy of ferrous metals (Bernstein, 1970;Tetelman, 1969).

Unlike planets and satellites, smaller bodies in the solar system, like asteroids, comets, and meteoroids, are affected by nongravitational forces due to collisions, viscosity, and in some cases electromagnetic forces. The way such forces change the orbits of the bodies seems not to have been analyzed until recently. For example, it is generally believed that collisions between asteroids will make their orbits spread over an increasing volume of space and that collisions inside meteor streams will make their cross sections increase. At least under certain conditions the reverse is true, as shown by the papers of Baxter and Trulsen at this symposium. Furthermore, besides the usual picture of meteoroids being emitted by comets, we should also discuss the reverse process; viz, comet formation by bunching in a meteor stream.

An inelastic collision integral is used in a Boltzmann-type equation for a distribution of particles in Kepler orbits. A Fokker-Planck equation is found that leads to radial density clustering.

Jetstreams probably played an important role at an intermediate stage of the formation of the solar system (Alfvén and Arrhenius, 1970). A Jetstream is defined here as a collection of grains moving in neighboring elliptical orbits around a central gravitating body and with the dynamics modified by the action of complicated collision processes among the grains themselves.

Three main types of collisions will take place in such a stream. Hyper-velocity impacts will lead to fragmentation of the grains involved. At lower impact velocities, the particles will retain their identities after the collision even if they might be deformed to some degree depending on impact velocity and internal structure of the grains. With still lower impact velocity, accretion can take place, the grains sticking together after collision to form larger grains. A common feature of these collision processes are that they will be partially inelastic. A certain fraction of the kinetic energy of the colliding particles will be spent on changing their internal structure. The internal kinetic energy in an isolated Jetstream thus will tend to decrease with time. The mass spectrum of the grains also will vary during the lifetime of a stream, the probability for accretive processes increasing with time.

A study of asteroid orbits is made to determine if there exist groupings of similar orbits in the asteroid population. A computer program based on Southworth's D criterion for similarity in meteor orbits is used. The program successfully sorted out the asteroid families listed by Hirayama, Brouwer, and van Houten et al. A number of new families were detected, several of which appear to be more significant than the minor Brouwer families. Asteroidal streams (jetstreams) are studied and a list of such streams is presented.

The Jetstream concept was introduced by Alfvén in 1969. Since then, the subject has been studied from various aspects by Danielsson (1969), Arnold (1969), Alfvén and Arrhenius (1970), Lindblad and Southworth, and Trulsen. In an attempt to define a Jetstream, we may say that it is a group of objects moving in space with almost identical orbits. The largest objects in the Jetstream may have any size, but the group must include a vast number of very small objects and their density must be large enough for the objects to interact. This means that collisions between the particles give rise to viscosity in the stream. Other interactions (e.g., by electromagnetic forces) are not excluded a priori.