This paper deals with the long term evolution of the motion of the Moon or any other natural satellite under the combined influence of gravitational forces (lunar theory) and the tidal effects. We study the equations that are left when all the periodic non-resonant terms are eliminated. They describe the evolution of the-mean elements of the Moon. Only the equations involving the variation of the semi-major axis are considered here. Simplified equations, preserving the Hamiltonian form of the lunar theory are first considered and solved. It is shown that librations exist only for those terms which have a coefficient in the lunar theory larger than a quantity A which is function of the magnitude of the tidal effects. The solution of the general case can be derived from a Hamiltonian solution by a method of variation of constants. The crossing of a libration region causes a retardation in the increase of the semi-major axis. These results are confirmed by numerical integration and orders of magnitude of this retardation are given.

The hypothesis that the origin of the resonances among Saturn's satellites is due to orbital evolution is critically reviewed. It is concluded that the hypothesis provides a plausible explanation of the origin of the Mimas-Tethys resonance, but it is unsatisfactory for Bnceladus-Dione, since their resonance is having little effect on the relative evolution rate of these satellites at present.

In this paper the initial results of an investigation of the motion of a rigid body located at the libration point in the planar, restricted three-body problem are given. This problem was analyzed in part by Kane and Marsh (1971), Markeev (1967a,b). However the present investigation is formulated in terms of hamiltonian mechanics. The final results will by used to study nonlinear effects connected with the gravitational influence of the “second” central body.

The paper presents a new theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field. This field is represented through its fourth degree harmonics for the torque due to the Earth (the second degree for the torque due to the Sun). The Moon is assumed to be rigid and its orbital motion is described by the ELP 2000 solution (Chapront and Chapront-Touzé 1981) for the main problem of lunar theory with planetary perturbations and influence of the non-sphericity of the Earth. Comparisons with other theories (Migus 1980 and Eckhardt 1981) are also presented.

Within the framework of linear and regular celestial mechanics, we revise a recent method of Belen'kii (1981). We generalize some of his results, giving a new regularizing function.

We make an application to the zonal earth satellite, considering the hamiltonian function through the harmonic J4. After the angular variable u has been removed, we introduce a new time and we reduce the problem to a linear equation.

A number of applications exist in astronomical research for planetary and lunar ephemerides covering an extended length of time. This paper discusses such a set of ephemerides, DE102/LE51, produced at JPL, covering the time 1411 B.C. to 3001 A.D. The ephemerides are dynamically self-consistent, in that the equations of motion were integrated simultaneously. They also represent the most accurately known positions covering such a time span. They have already been used by a number of different users in a variety of different applications.

It is known that early (i.e. 17th-l8th century) visual observations of Jupiter's Galilean satellites are approximately as accurate as modern visually observed eclipses (Lieske 1982). In the early days the clocks were the primary source of error while in present days problems related to the Earth's and Jupiter's atmospheres have become the primary limiting factor. The early observations were generally made in local apparent time (i.e. related to the real hour angle of the sun). After the development by Huygens in 1656 of the pendulum clock, these early observations, when reduced to a modern UT system, have been shown to be quite accurate.

The methods of the first-order planetary secular perturbations are discussed in this paper. On the basis of Gauss' method, some concrete applicable formulae are derived. And then, orbital evolutions of the nine major planets in solar system during about 2,100,000 years are investigated numerically. The method applied here is in principle suitable for the orbits of arbitrary eccentricities and inclinations.

A new expansion for the gravitational potential of the asteroid belt is derived in this paper on the basis of binomial expansions. Its advantages are: (1)the unified form both for the inner and for the outer regions of the belt; (2)suitability for discussing the motions of the celestial bodies with perihelions within but apohelions beyond the belt; (3)rapidity of convergence.

The perturbations due to the asteroid belt are studied by using our expansion.

In addition to planetary perturbations, the small particles which make up a meteor stream are subject to outward radiation pressure and the Poynting-Robertson effect. New particles can also be generated in a stream through being released from the nucleus of a comet. We summarise the main physical effects, discuss models for meteor stream evolution and give a brief account of the observational data available.

Some mathematical models of the formation of meteor streams are developed. Some of the testable predictions of these models are compared with observations.

Steady state number of the short-period(SP) comets captured from nearly parabolic(NP) orbits with 60°≤ i(inclination) ≤ 120° and 0.5≤q (perihelion distance) ≤1.5 AU is calculated. Due to smallness of the q and slowness of the capture process, almost all these SP comets become completely extinct. Combining annual flux of the observed NP comets of high inclination with the capture probability from NP to SP orbits and the ejection rate in SP orbits obtained by Monte Carlo simulations, we find that the steady state number of the extinct SP comets in high-inclination orbits is at least 1-2 hundred. If this number can be considered to correspond to the observed one of Apollo-type objects with i ≥ 60°, it is concluded that only less than a few percent of extinct comets leaves sizable non-volatile cores or shells. The number of asteroid-like bodies deduced from the extinct comets with small perihelion distance and high inclination has less ambiguity than that of low inclination, because contribution from asteroid belt is negligible and the source comets are visible(free from observational selection) through the whole course of orbital evolution.

The orbital evolution of comet P/Lexell has been characterized by a sequence of three close planetary encounters within 12 years; the first of these encounters was with Jupiter, 1767, and lead to a reduction of the perihelion distance from 2.9 AU to 0.67 AU. Then the comet encountered the Earth at a distance of 0.015 AU, in 1770, and was discovered. The final, very close encounter with Jupiter, in 1779, removed the comet from observability, sending it into an orbit of perihelion distance in excess of 5 AU (Kazimirchak-Polonskaya, 1967).

The possible effects of non-gravitationnal forces on the motion of the comet P/Boethin are investigated for various values of the orbital period. A time interval of 2000 years backward and forward is treated. The authors find in all cases that the comet librates temporarily around the 1/1 resonance with Jupiter as a remote jovian satellite during at least two centuries.

The numbered asteroids are classified into families by a new method proposed by the author(Kozai, 1979a) and their characteristics are studied by several aspects. It seems to the author that there are two kinds of the families, one being very compact in the phase space and containing many faint as well as a few bright asteroids and one being rather loose and bounded by secular and mean motion commensurable regions.

Using Hill's modified stability criterium, regions of orbital elements are established for conditions of stability. The model of the three-dimensional restricted problem of three bodies is used with the Sun and Jupiter as the primaries. Four different cases are studied: direct and retrograde, outside and inside asteroidal orbits. The directions of the asteroidal orbits refer to the synodical reference frame and the positions refer to Jupiter's orbit. The orbital parameters of the asteroids are the semi-major axis (a), the eccentricity (e), and the inclination from Jupiter's orbital plane (i). The effects of the other orbital elements are not investigated in this paper. The argument of the perihelion and the longitude of the ascending node are fixed at Ω = ω = 90° and the time of perihelion passage is T = 0 for all orbits.

The utilization of two different stability criteria, namely, Hill's modified stability criterium and the method of surface of section, has been employed for asteroid orbits. The idea is to compute different criteria of stability for the same asteroids in order to compare the results and see the practical interest of the computations for researches about evolutionary trends of individual asteroids, groups and families of asteroids.

The stability of the asteroid orbits has been studied by the method of surface of section. Families of simple symmetric periodic orbits of the asteroid and their stability have been computed and this served as a guide for the selection of the energy levels for the surface of section. In this way all possible cases for the structure of phase space have been obtained. It was found that the region in phase space around the resonant orbits at the resonances 1/3, 3/5, 5/7,.... is unstable, but small stability regions of doubly symmetric periodic orbits near the above resonances are also present. At the resonances 1/2, 2/3, 3/4, .... it was found that there exist two separate regions in phase space at about the same resonance 1/2, 2/3, 3/4,...., respectively, one being stable and the other unstable. At certain energy levels only the stable region appears. The above results are consistent with the observed distribution of the asteroids.