I believe I have solved, at least in principle, the long-standing problem of the Kirkwood gaps, and have incidentally initiated a new approach to questions of orbital stability. I shall begin with the concept I call hyperperiod. A given periodic dynamical system S with period P may or may not have a latent long period - the hyperperiod P∗. If P∗ exists, then any small displacement or variation, actual or virtual, once-for-all or recurrent, will induce a displacement y which will be periodic with period P∗ and will be of bounded amplitudes. We can then say that S is stable. If P∗ dose not exist, then y will eventually become indefinitely large - and we say that S is unstable.

In 1962 the author published a paper on secular perturbations of asteroids with high eccentricity and inclination (Kozai, 1962) and found that for such asteroids the secular perturbations are more complicated than those estimated by the classical linear theory. Before that the author(Kozai, 1954) tried to extend the classical theory to include terms of squares of disturbing masses, eccentricities and inclinations in expressions of secular motions of angular variables and found that the sum of the longitudes of the perihelion and the ascending node is not as stable quantity as the classical theory predicted, and, therefore, it cannot be used to estimate the ages of asteroid families. Then Williams(1969) and Yuasa(1973) extended the classical theory further.

The attitude toward the systems of interplanetary objects is largely different from that accustomed in stellar astronomy. The stellar systems are viewed from the outside, with their shape, population, and internal structure apparent at first glance. The motions are difficult to recognize, if ever, and so the individual orbits are degraded to a background mechanism maintaining the observed organization and rotation of the system. In contrast to this, we dispose of copious information on the dynamics of individual interplanetary objects, but the bodies to which the orbital data refer are sampled under the influence of strong selection effects. Lacking a consistent survey of their systems, we tend to think of them in terms of the biased statistics in the phase-space of orbital elements, rather than in terms of the real three-dimensional distributions. No wonder that we often come across serious misrepresentations: the asteroid belt depicted as a plane ring divided by the Kirkwood gaps like the ring of Saturn; the Trojan clouds as small spherical systems resembling globular star clusters; or the meteor streams as elongated rings of uniform width and population all around. And we seem to receive these gross misrepresentations with surprisingly little annoyance.

“Slow-moving Object Kowal” was recognized by the first author on 1977 November 1 on plates obtained with the 122-cm Schmidt telescope at Palomar two weeks earlier (Kowal 1977a). The object, asteroidal in appearance and of photographic magnitude about 18, was found to have moved less than 3′ between similar 75-minute exposures made on October 18 and 19. For something only 8° from opposition this retrograde motion was significantly slower than the 8′-15′ daily motion of normal minor planets at opposition and immediately suggested that the object was located at almost the distance of Uranus. There was a slight possibility that the object was closer and moving almost directly toward or away from the earth, but the fact that the two trails were identical in length to ± 6 percent suggested that this was unlikely. That the object was well beyond the main belt of minor planets was confirmed when Gehrels (1977) succeeded in identifying it close to the extrapolated position near the corner of an exposure obtained with the same telescope on October 11.

In order to provide the necessary background for this report, this section summarizes the previously published “Theory of the Trojan Asteroids, Part I”. Treating the system as the case of 1:1 resonance in the restricted problem of three bodies, the author constructs a formal long-periodic solution of 0(m), where m is the mass-parameter of the system, assumed to be sufficiently small.

This paper is a brief review of results presented elsewhere (Van Flandern 1977, 1978). The conclusion of these results is that at least the comets, and probably also the asteroids and meteorites, originated in the breakup of a major planet in the present location of the asteroid belt, at an epoch of just 5×106 years ago. Although there are many “well-known facts” about the solar system which seem to contradict this conclusion, these contradictory “facts”, upon closer examination, are often not so convincing as we have been inclined to assume; in each instance so far suggested there is a plausible alternative interpretation of the data which is supportive of the breakup hypothesis. A compelling contradictory argument has not yet surfaced. In view of this, and in consideration of the strength of the arguments favoring the hypothesis, it will be necessary to judge the conclusion on the merits of the case, without the intervention of the apriori decision that it cannot be true.

The different hypotheses proposed so far on the origin of comets have been reviewed in detail last year (Delsemme 1977). In particular, I mentioned why the orbital statistics seem to be consistent with only one hypothesis, namely that comets were accreted at unknown but moderate distances (10 to 1000 A.U.) within the protosolar nebula, and ejected later into a sphere whose radius is some 50,000 A.U., usually called the Oort's cloud. Safronov (1977) and Cameron (1977) have different scenarios to do just that. Even with improvements like that of Dermott and Gold (1978) the choice between these scenarios will remain impossible until we have a convincing model describing the gravitational collapse of an interstellar cloud into a planetary system.

Based on the number of ‘new’ comets seen on near-parabolic orbits, one can predict the number of comets that should be found on definitely elliptical orbits on their subsequent returns. We show here that about three out of four of these returning comets are not observed.

Oort (1950) first suggested that the source of the long-period comets is a large spherical cloud of comets surrounding the solar system and extending roughly halfway to the nearest stars. The observational evidence for this is the distribution of original inverse semi-major axes of the long-period comets which shows a large spike of comets at very small positive values of 1/ao, less than 10−4 AU−1. Attempts to model the evolution of these comets by Oort in his original paper, by Kendall (1961), Shteins (1961), and Whipple (1962) were successful in recreating the general shape of the 1/ao distribution. However in each case the authors were unable to match the observed ratio of new comets from the Oort cloud versus older comets evolving under the influence of planetary perturbations.

Although the problem of the origin of comets has not yet been settled, there are only two possibilities; one is to regard that comets formed within the solar system when or soon after planets formed, and the second possibility is that comets have been captured from interstellar space by planetary perturbations or by perturbations of the galactic nucleus. To decide upon the possibility, one has to take into account statistical features of cometary orbits.

Subtle effects influencing the results of orbit determination have to be taken into consideration when recomputing the cometary orbits for the Catalogue. It is very important to apply the numerical integration method, taking into account the specific conditions of the comet's motion, which almost eliminates a cumulation of numerical errors of integration. In this point of view the recurrent power series integration is adapted for cometary orbital computations. The optimal value of step-size of integration according to the required accuracy is calculated at every integration step. The method gives excellent results for “normal” comets as well as in the cases of close approaches of a comet to the Sun and to the planets.

In 1975 a remarkably large number of short-period comets were discovered - only the most recent number from 1977 is comparable. While the average discovery rate has been 0.8-1.0 new short-period comets per year (Kresák 1974), in 1975 there were six discoveries. In five of the cases IAU Circulars soon afterwards contained indications that close encounters with Jupiter had recently taken place (Marsden 1975, Kastel' 1975). For two of the comets, P/Kohoutek and P/Smirnova-Chernykh, also pre-encounter orbital elements were outlined, suggesting that substantial reductions of the perihelion distances had occurred.

Original nearly parabolic orbits of comets are known to be evolved toward short-periodic elliptic orbits as statistical results of hundreds of encounters with Jupiter. There seems to be two methods to handle the process, namely, the method by exact numerical integrations for each orbit (Everhart, 1972) and random walk approach by using probability distributions of perturbations after single encounters (Lyttleton and Hammersley, 1963; Shteins, 1972). Since both methods need a great number of input parabolic comets to have only a few tens of short-periodic ones, the second method may save time compared with the first one, which is in turn more accurate. The purpose of this paper is to clarify the characteristics of single-encounter effects, in order to develope the second method more elaborately and extensively.

P/Halley In a previous paper (Kiang 1973), I derived a differential equation which governs the behaviour of the residual O - C in the time of perihelion passage of a comet. The work was stimulated by the discovery (Brady 1972) that a long periodicity of about 600 years seemed to be present in the residual for P/Halley. The derivation was based on the following assumptions:

1. 1. The system consists of only the Sun, Jupiter and P/Halley, and Jupiter is assumed to move in a circular orbit with a period P′ = 11.8614 yr.

2. 2. P/Halley is assumed to move in a fixed Keplerian orbit with a period P = (13/2) P′ exactly, so that the whole system is strictly recurrent with a period Po = 13 P′ = 154.198 yr.

It is generally believed that the structure of dust tails of comets can be described in terms of the mechanical theory. The basic parameters of this theory are the repulsive force and initial emission velocity of particles. However, the effect of initial velocity does not seem to have been considered seriously in the discussion of the overall structure of dust tails. Finson and Probstein (1968) have recently proposed an elaborated method to analyse the tail brightness profiles by introducing the distribution function of particles. The initial velocity effect was, however, taken into account only in an approximate way. They assumed that a group of particles are emitted isotropically with a single speed, and afterwards forms a spherical shell expanding uniformly in time. The isotropic emission is suggested by the fluid-dynamical considerations on the gas-dust interaction in the inner head region, but the subsequent spherical expansion implies merely a simplification for the calculation to obtain the line-of-sight integrals analytically. Due to this simplification, the validity of the Finson-Probstein method is seriously limited, particularly when a tail after perihelion passage is concerned.

The n-body computer program by Schubart and Stumpff (1966) has been slightly modified to study the gravitational interaction between two fragments of a split comet nucleus in the sun's gravitational field. All calculations refer to the orbit of Comet West (1976 VI), the velocity of separation of the fragments is assumed to be equal in magnitude to the velocity of escape from the parent nucleus, and the numerical integration of the relative motion of one fragment (called the companion) with respect to the other (principal fragment) is carried over the period of 200 days from separation.

The effect of binary encounters is examined by restricting considerations to a sphere of finite radius R, instead of an infinite one assumed hitherto. The sphere is centered at an assigned test-particle with velocity ∇o. The corresponding modified assumption is that particles lying outside the sphere are scattered randomly with an average number density n and their velocity distribution is Maxwellian with an r.m.s. of in the space velocity. As for the particles' masses the same M is presumed.

Distributions of the original reciprocal semimajor axes l/a and the perihelion distances q of eighty “new” comets (Marsden et al, 1978) are discussed from the view point of a hypothesis of the interstellar origin (Hasegawa, 1976).