In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

Motions near the 3:1, 4:1 and 5:2 resonances with Jupiter are studied by means of numerical integrations of a semi-analytically averaged Sun-Jupiter-asteroid planar problem. In order to have a model including the very-high-eccentricity regions of the phase space, we adopted a set of local expansions of the disturbing potential, adequate to perform the numerical exploration of regions in the phase space with eccentricities higher than 0.9 (Ferraz-Mello and Klafke, 1991). Individual solutions and qualitative results thus obtained are completely reproduced by numerical integration of the complete equations by filtering off the short-period components of these solutions.

Wisdom's perturbative method is applied to the 5:2 and 7:3 resonances. Some comparisons with Yoshikawa's model are performed: for values of eccentricity up to about 0.3-0.4, agreement exists and it is better for 5:2 resonance. A clear difference between the cases 5:2 and 7:3 is observed: the former one, like in the case 3:1, can show significant variations of eccentricity, even starting from very small values, close to zero, while the latter seems to undergo such variations, but with initial eccentricity not less than a value near 0.1.

Some results concerning the resonances 3:1, 4:1 and 5: 2 are given and compared to results published by other authors.

Results of families of periodic orbits in the elliptic restricted problem are shown for some specific resonances. They are calculated for all mass ratios 0 < μ < 1.0 of the primary bodies and for all values of the eccentricity of the orbit of the primaries e < 1.0. The grid size is of 0.01 for both parameters. The classification of the stability is undertaken according to the usual one and the results are compared with the extensive studies by Contopoulos (1986) in different galactical models.

The subject of this paper is a study of a possible orbital evolution for near-Earth asteroids. The investigation is fulfilled in the frame of the restricted circular three body problem. It is based on the calculations of the Jacobi constant. The osculating elements of some real Apollo-Amor-Aten asteroids are used as the starting parameters. The comparison with the results of other authors is given.

On 1973 November 23, there was a close approach (0.013AU) between (704) Interamnia as the fifth largest minor planet, and (993) Moultona. It was made an attempt to determine the mass of Interamnia from its perturbations on the motion of Moultona, using 82 right ascensions, 72 declinations, and of 1 observation the portion normal to the apparent motion, from altogether 95 observations from 22 oppositions 1923 January to 1991 February with a mean error of 0.″51. For the mass of Interamnia was obtained a result of 0.37 ± 0.17 · 10-10 solar masses. Although its uncertainty, this result gives an independent information about the mass of Interamnia besides the estimation from the diameter and the suspected density, considering the uncertainty of these parameters.

The motion of two small bodies orbiting each other whose barycenter is orbiting around a massive body is studied. The equations of motion are integrated considering the secular part of the disturbing function.

A new value for the Nordtvedt parameter and the mass of Saturn are computed using the ten first Trojan asteroids. From 1262 observations, we find the inverse mass of Saturn 3498.17±0.51 and the Nordtvedt parameter 0.0±0.3.

The recent numerical simulations of Tittemore and Wisdom (1988,1989,1990) and Dermott et al. (1988), Malhotra and Dermott (1990) concerning the tidal evolution through resonances of some pairs of Uranian satellites have revealed interesting dynamical phenomena related to the interactions between close-by resonances. These interactions produce chaotic layers and strong secondary resonances. The slow evolution of the satellite orbits in this dynamical lanscape is responsible for temporary capture into resonance, enhancement of eccentricity or inclination and subsequent escape from resonance. The present contribution aims at developing analytical tools for predicting the location and size of chaotic layers and secondary resonances. The problem of the 1:3 inclination resonance between Miranda and Umbriel is analysed.

We present an iterative method allowing to synthetize a semi-numerical solution for the equations of motion of the resonant Saturn's satellites Titan-Hyperion (limited now to the planar problem). The current theory of Hyperion by Taylor, Sinclair & Message (1987) gives the greatest terms of the long-period part of the solution (depending on two angles: the libration angle τ, and the angular distance of the pericenters ζ). Using it as a first approximation, this solution is substituted numerically in the exact Lagrange equations of motion for Titan and Hyperion, computed for many values of the three angles: τ, ζ and ϕ (the mean synodic longitude). Then, a multivariable Fourier transform allows to reconstruct the equations in these three angles, that is in same form as the initial one with, in addition, the short-period terms. Then, a solution may be obtained and used as a better approximation in an iterative process. Besides a complete determination of the short-period perturbations of Hyperion obtained here completely for the first time, some long-period perturbations of Titan by Hyperion are also found which would be non negligible at the 10 km level.

In this paper we discuss the influence of the deformation of a planet caused by a first satellite on the orbit of a second satellite of the same planet, where both satellites are supposed to be involved in an orbit-orbit resonance. Numerical results are given for seven orbit-orbit resonances in the Solar System.

We explain the high values of the acceleration (≈ 2° cy−2) found in the longitude of Mimas by Kozai and Dourneau when they fit to observations their current theory of the Mimas' motion. In fact, we have found that very long-period terms are missing in these theories; their expansion in powers of t well agrees with the observed acceleration. Effects of tidal dissipation are far smaller and could be determined only after accounting of these long-period terms.

It is considered the problem of two natural satellites around a spheric planet. Only gravitational forces act on the system. The satellites move in a central field disturbed by the mutual atraction between them. The disturbing function is developed in power series of the small parameter (ratio of mass of satellites and planet), of the eccentricities and inclinations (assumed small) of the satellites' orbits. The mean motions are supposed to be commesurable in the ratio 2:1. It is, also, assumed that the critical angles are of inclination type. The critical angles of eccentricity type circulates and may be eliminate as fast variables like the short-periodic terms. The hamiltonian is truncated up to the second order in the inclinations.

The goal of our work is to determine the influence of Enceladus in the motion of Helene. We constructed a model considering an oblated central body (Saturn) and three satellites (Enceladus, Dione and Helene) under the action of central forces. The development of the potential was made assuming small eccentricities, null inclinations and two resonances present in this system (2:1 between Enceladus and Dione and 1:1 between Dione and Helene). The mean Hamiltonian preserves terms derived of the oblateness of the central body and also terms of both resonant arguments. The auxiliary Hori's system generated by this Hamiltonian is completely integrable. In the solution for Helene we included the perturbative effects of Dione and Enceladus which extend the usual teatment.

For the first order resonance, the problem of the motion of two small masses around a primary body can be of three different types: eccentricity, inclination or eccentricity-inclination. The eccentricity type resonance problem has been the subject of several works since Poincaré(1902). The inclination type resonance problem was studied by Greenberg(1973) who used a particular reference system to obtain an integrable auxiliary system. Sessin and Ferraz-Mello(1984) studied the eccentricity type resonance problem considering the eccentricities of the orbits of the two small masses. Sessin(1991) study the inclination type resonance problem for an arbitrary reference system. In this paper we will study a dynamical system that includes both types of resonance. This study is based in the models developed by Sessin and Ferraz-Mello(1984) and Sessin(1991). The resulting system of differential equation is non-integrable; thus, the families of trivial periodic solutions are studied.

In this paper we consider the motion of a satellite whose mean motion is commensurable with the frequency of the rotational motion of the planet in the ratio p:p+q.

The generalized canonical version of Hori's method is discussed using some properties of generalized canonical system and, then, applied in solving the main problem of satellite theory.

Variations in the influx rate of incoming Oort cloud comets, leading to the occurrence of comet showers, are reviewed with special emphasis on the dynamical processes that produce them. It is found that comet showers as intense as 10-100 times the background comet flux may occur at average intervals of several 107 years, being very close stellar passages at distances ≈ 104AU and, perhaps, penetrating encounters with intermediate-size molecular clouds the trigger mechanisms. Penetrating encounters with giant molecular clouds and stellar passages at distances of a few 103AU may trigger comet showers about 102 − 103 times more intense than the background comet flux at average intervals of a few 108 years. The impact cratering record on the Earth does not show any clear evidence of past comet showers. Some orbital properties of new and dynamically young comets suggest that the influx rate of Oort cloud comets is currently near its background level.

The gravitational processes affecting the dynamics of comets are reviewed. At great distances from the Sun the motion of comets is primarily affected by the vertical component of the galactic field, as well as by encounters with stars and giant molecular clouds. When comets move in the region of the planets, encounters with these can strongly affect their motion. A good fraction of all periodic comets spend some time in temporary libration about mean motion resonances with Jupiter; some comets can be captured by this planet as temporary satellites. Finally, there is a small number of objects with orbital characteristics quite different from those of all other short-period comets.