The secular equations which were used to exhibit the chaotic behaviour of the solar system (Laskar, 1989) are established here in a Hamiltonian framework. The integration of the former secular equations over 400 Myr showed that the two resonant arguments and given in (Laskar, 1990) present both transitions from libration to circulation. During the circulation of the first argument, temporary libration of the argument is observed, revealing resonance overlap beetween these two resonances, which explains the existence of a large chaotic zone for the motion of the solar system.

The mapping method of Wisdom (1982) has been generalized to encompass all n-body problems with a dominant central mass (Wisdom and Holman, 1991). The new mapping method is presented as well as a number of initial applications. These include billion year integrations of the outer planets, a number of 100 million year integrations of the whole solar system, and a systematic survey of test particle stability in the outer solar system.

We have integrated the motion of the four Jovian planets on Myr timescales in fictitious solar systems in which the orbits differ from those of the real solar system. A change of ≲1% in the major axis of any one of the planets from its real value can lead to chaotic motion with a Lyapunov exponent larger than 10-5 yr−1. A survey of fifty solar systems with initial conditions chosen at random from a reasonable probability distribution shows the majority of them to be chaotic.

We made a numerical study of the General Three-Body Problem in two dimensions, with the intention to obtain some statistical estimates of the outcome of the system after a long time. Two different sets of masses were used. In the first series of experiments we use masses in the ratio of 0.95, 0.04 and 0.01. In the second series, we use masses that are exactly in the Sun-Jupiter-Saturn ratio. To facilitate the discussion, we use the names Sun, Jupiter and Saturn for the three masses, in both cases. In all our experiments, the orbit of Jupiter starts with zero eccentricity and with a unit radius. However, the orbit of Saturn varies in two ways: the initial value of the semi-major axis varies from 1.1 to 3.5 and the eccentricity from 0.0 to 0.75. In total about 4000 cases were run for the two series of masses. All the numerical integrations were done with the method of recurrent power series of order 14, in a heliocentric frame of reference, integrating thus eight simultaneous first-order differential equations. All integrations were performed for a maximum of 12,500 canonical units of time, corresponding to about 2000 revolutions of Jupiter. The cause of termination or type of catastrophe for the system has been determined in all cases. In most cases, this is a close approach of Saturn with Jupiter, followed by ejection of Saturn from the system.

An iterative method for the construction of planetary theories has been developed in order to determine the high order perturbations with respect to the masses. These perturbations are indeed needed to enlarge the validity span of analytical theories up to some million years. The application to the simplified Sun-Jupiter-Saturn problem gives a solution accurate over several ten million years. Throughout the study of the four outer planets we meet with convergence difficulties especially in the determination of fundamental frequencies. One of the results of this study is it shows evidence of long period terms with large amplitude in the mean longitudes: 12 000″ in Saturn longitude, 20 000″ in that of Uranus.

This paper develops a planetary theory in three dimensions with elliptic functions and elliptic integrals. In an earlier treatment, (Williams, Van Flandern, and Wright, 1987) presented a two dimensional planetary theory to the first order of a Picard iteration. The theory did avoid expansions in powers of the ratio of the semi-major axes and it contained only two explicit small divisors, n – n′ and 2n – n′. These advantages are retained in the new theory and in fact no small divisors appear explicitly. Secular terms are removed by adopting an averaging technique rather than continuing the Picard iteration. The Lie series method of (Deprit, 1969) is chosen for the averaging. In order to simplify the Lie operator, the framework for the problem is chosen to be the circular restricted three body problem written in the polar-nodal coordinates of Whittaker. The algorithm is described and a few representative terms are discussed.

The conversion in 1984 from constants, reference frames, time scales, and ephemerides, all essentially based on theories and constants by Newcomb, to the IAU 1976 system of constants, new definitions of time scales, FK5 reference system, and integrated ephemerides has produced the expected accuracy improvements. These changes included a correction to the precession constant of 1.13 arcseconds per century, a correction to the equinox motion of the FK4 of 1.23 arcseconds per century, and a new expression for the Greenwich Mean Sidereal time.

This paper reviews a number of time-dependent phenomena that are relevant to our understanding of the dynamics of planetary rings and that will be investigated using Voyager and Cassini data. A long time baseline may help us decipher the physics of the spokes, understand better the morphology of the F ring and the rigid precession of non-circular ringlets, measure more precisely than has been done so far the satellites' torques and the viscosity of the A ring, and discover small satellites in the Saturnian ring system. Two exciting possibilities are those of determining the recession rates of the small satellites that border the rings, and of observing changes due to viscous diffusion in the irregular structures of the B ring.

I outline some of the main processes that shape planetary rings. Then I focus on two outstanding issues, the role of self-gravity in the precession of narrow rings and the dynamics of Neptune's arcs. By airing these well-defined but unsolved problems, I hope to encourage others to join me in the quest for their solutions.

Since fifteen years, space and ground-based observations have completely renewed our image of planetary rings. Simultaneously, a wealth of theoretical and numerical models have flourished. Collisions between ring's particles and gravitational perturbations of nearby satellites should explain most of the ring's structures. However, important questions are still unanswered. We do not understand why the rings are so dissimilar. We do not know the ring's origin and their stability over billions of years. Most of the ring's complex structures, the existence of arcs, and color and optical depth variations are not yet explained. Among all the ring mysteries, the uniform precession of narrow ringlets and the azimuthal brightness asymmetries should receive a high priority.

The influence of the perturbing satellite on the planetary ring at isolated Lindblad resonances is studied with numerical computer simulations, combining the Aarseth's force polynomial method for orbit integrations with the calculation of particle-particle impacts. Observed angular momentum exchange between the satellite and the dissipative, non-selfgravitating ring agrees with the Goldreich-Tremaine formula for gravitating ring within 20%, verifying that the exerted torque is not sensitive to the details of the dominant physical processes. The theoretically predicted angular momentum luminosity reversal was also observed.

The dynamical evolution of dense planetary rings, such as Saturn's rings, is mainly governed by the mutual impacts between macroscopic icy particles. The local equilibrium state is determined by the energy loss in partially inelastic impacts and the viscous gain of energy from the systematic velocity field. Due to frequent impacts the time-scale for the establishment of local energy equilibrium is very short, as compared to the time-scale for radial evolution, which is determined by viscous spreading, and in some cases also by the angular momentum exchange with external satellites. Therefore, local and radial behaviour can, to a large extent be studied separately. This fact is utilized by the local simulation method (Wisdom and Tremaine, 1988; Salo, 1991), following the orbital evolution in a small co-moving region inside the rings with periodic boundary conditions. Compared to previous simulation methods (Salo, 1987) this enables much higher surface density. With the presently attainable number of particles (up to several thousands), realistic modeling of dense regions is possible, taking simultaneously into account the particle size distribution, rotation of particles, as well as vertical self-gravity. By combining several local simulations with different surface densities, it is possible to deduce the expected radial behaviour as well.

The dynamical evolution of the planetary rings is simulated by means of a numerical model in which particles interact through mutual attraction and inelastic collisions. We use a mixed simulation: a deterministic integration of the N - body problem for large distances (“particle-mesh” method with an expansion of density and potential in spherical harmonics) and a Monte Carlo treatment for the close encounters. The implementation is done in the Connection Machine in order to be able to make a detailed simulation using a greater number of particles (of the order of 105). The deterministic calculation of the action of a shepherding satellite on the particles will allow us to study the effect of resonances on the formation and the evolution of the sharp edges of the rings.

In this paper the author presents evidences showing that for most of the asteroids the motions are stable in the sense that they never approach major planets very closely and explains about mechanisms to avoid very close approaches by investigating the variations due to the secular perturbations of the eccentricities as functions of the arguments of perihelion, particularly, for asteroids with high eccentricities and inclinations. It is believed that some kinds of dynamical evolution processes have made the asteroid motions stable. The author shows also that there were some kinds of collisions among asteroids in the past which produced families and present distribution of asteroids as there are very faint asteroids only near Kirkwood gaps.

Generalizing the secular perturbation theory of Milani and Knežević (1990), we have determined in the a — e — I proper elements space the locations of the secular resonances between the precession rates of the longitudes of perihelion and node of a small body and the corresponding eigenfrequencies of the secular perturbations of the four outer planets. We discuss some implications of the results for the dynamical evolution of small solar system bodies. In particular, our findings include: (i) the fact that the g = g6 resonance in the inner asteroid belt lies closer than previously assumed to the Flora region, providing a plausible dynamical route to inject asteroid fragments into planet-crossing orbits; (ii) the possible presence of some low-inclination “stable islands” between the orbits of the outer planets; (iii) the fact that none of the secular resonances considered in this work exists for semimajor axes > 50 AU, so that these resonances do not provide a mechanism for transporting inwards possible Kuiper–belt comets.

We propose here the first results of the semi analytical method of Henrard(1990) applied to the calculation of proper elements for the asteroids.

Let us consider the planar case of the circular restricted three-body problem. The mass of central body is unit, the radius of the circular orbit of perturbing point P′ (of mass μ = 1/1047.39 according to Jupiter's mass) is also unit and the other unit is such that the gravitational constant is equal to 1. The Keplerian elements of the perturbed mass point P (asteroid), with semi-major axis α < 1, are given by usual notations; the elements of P‘ are indicated with a prime (’).