On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions

This paper is devoted to the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of the paper is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.     

Boltzmann equation approach to the N-body problem with masses varying according to the Eddington-Jeans law

We study special aspects of the N-body problem with masses varying according to the Eddington-Jeans law with powers n=2 and 3. Our main result is that a particular set of variables can be found that allows one to write the pertinent Boltzmann and Poisson equations in a fashion similar to that corresponding to the usual fixed mass situation.     

Qualitative features of the evolution of some polar satellite orbits

Two special cases of the problem of the secular perturbations in the orbital elements of a satellite with a negligible mass produced by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun are considered. These cases are among the integrable ones in the general nonintegrable evolution problem. The first case is realized when the plane of the satellite orbit and the rotation axis of the planet lie in its orbital plane. The second case is realized when the plane of the satellite orbit is orthogonal to the line of intersection between the equatorial and orbital planes of the planet. The corresponding particular solutions correspond to those polar satellite orbits for which the main qualitative features of the evolution of the eccentricity and pericenter argument are described here. Families of integral curves have been constructed in the phase plane of these elements for the satellite systems of Jupiter, Saturn, and Uranus.     

The use of radar observations of Near-Earth Asteroids in the determination of the dynamical equinox

Near-Earth Asteroids (NEAs) are Solar system special class objects attracting the attention of astronomical community especially during several of the last decades. To some extent the NEAs have an advantage over the minor planets of the main belt: due to close and regular approaches to the Earth the radar observations of NEAs can be obtained for the greater number of objects than for those of the main belt of the minor planets. In this paper the use of all available radar observations together with optical ones resulting in the combined data analysis solution is discussed for different problems such as the asteroid orbits and catalog orientation parameters determination. In particular the problem of the motion of the dynamical equinox in the Hipparcos reference system is considered. About 13000 radar and optical observations of 24 NEAs and main belt minor planets have been used to obtain the precise asteroid orbits, FK5 catalogue orientation parameters and the motion of the dynamical equinox from 1750 till 2000 in the Hipparcos system.     

Simple methods for orbit determination and identification of asteroids. I

The many discoveries of main belt asteroids with insufficient observations for reliable orbits cause special difficulties for identifications and data storage; mainly because of the mixing of objects due to their motion in α and δ. This problem may be reduced by substituting positions by slowly and smoothly varying quantities. A rational system of preliminary designations will be proposed, and surveys and compact ephemerides are discussed. All computations are very simple and performed on a small, programmable pocket computer.     

Application of two special orbits in the orbit determination of lunar satellites

Using inter-satellite range data,the combined autonomous orbit determination problem of a lunar satellite and a probe on some special orbits is studied in this paper.The problem is firstly studied in the circular restricted three-body problem,and then generalized to the real force model of the Earth-Moon system.Two kinds of special orbits are discussed:collinear libration point orbits and distant retrograde orbits.Studies show that the orbit determination accuracy in both cases can reach that of the observations.Some important properties of the system are carefully studied.These findings should be useful in the future engineering implementation of this conceptual study.     

Equilibrium solutions of the restricted problem of 2 + 2 axisymmetric rigid bodies

The restricted problem of 2 + 2 homogenous axisymmetric ellipsoids such that their equatorial planes coincide with the orbital plane of the centers of mass is considered. The equilibrium solutions of this problem are shown to exist. Six of these solutions are located about the collinear points of the restricted problem of three axisymmetric ellipsoids. A special case of this problem is studied and sixteen solutions are found in the neighborhood of the triangular Lagrangian points.     

On the limiting polarization of radio-waves

We discuss the limitations of various expressions governing the determination of the limiting polarization of radio-waves and slightly extend earlier work, which was limited to the case where the wave-vector is parallel to the density gradient, to the special case where the magnitic field is parallel to the density gradient, but the wave-vector can be at an angle to both. We consider the problem especially in connection with the problem of pulsar emission.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Particular solutions of the singly averaged Hill problem

We consider the case of averaging the perturbing function of the Hill problem over the fastest variable, the mean anomaly of the satellite. In integrable special cases, we found solutions to the evolutionary system of equations in elements.     

Finiteness of spatial central configurations in the five-body problem

We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.     

CORIOLIS力和离心力摄动对Robe问题平动点位置和线性稳定性的影响

研究了Coriolis力和离心力摄动对Robe限制性三体问题主要平动点位置及线性稳定性的影响，给出了Robe限制性三体问题主要平动点的摄动位置和Coriolis力和离心力摄动对主要平动点位置和线性稳定的影响量级．改进了Shrivastava的结果．     

On the solution of the problem of three fixed centres

For a special choice of the initial conditions a solution of the plane restricted problem of four bodies i.e. the problem of the motion of a passively gravitating material pointP attracted according to Newton's law by three fixed point massesP ₁,P ₂ andP ₃ has been obtained.     

On the continuation of periodic orbits from the restricted to the general three-body problem

Some properties of the characteristic surface of a family of symmetric periodic orbits of the general three-body problem, corresponding to a fixed value of the ratio of the masses of two of the bodies, are studied in view of recent theoretical and numerical results. Periodic orbits of the planar circular restricted problem with period equal to an integer multiple of 2 are of special interest for the structure of a characteristic surface.     

Periodic solutions about the collinear lagrangian solution in the general problem of three bodies

The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame.The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses.Finally, the fundamental family of periodic solutions is extended by numerical integration all the wap up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated.     

A model for binary-binary close encounters and collisions from a dynamical point of view

The goal of this paper is to provide a model for binary-binary interactions in star clusters, which is based on simultaneous binary collision of a special case of the one-dimensional 4-body problem where four masses move symmetrically about the center of mass. From the theoretical point of view, the singularity due to binary collisions between point masses can be handled by means of regularization theory. Our main tool is a change of coordinates due to McGehee by which we blow-up the singular set associated to total collision and replace it with an invariant manifold which includes binary and simultaneous binary collisions, and then gain a complete picture of the local behavior of the solutions near to total collision via the homothetic orbit.     

Equations of motion of the restricted problem of three bodies with variable mass

In this paper we have deduced the differential equations of motion of the restricted problem of three bodies with decreasing mass, under the assumption that the mass of the satellite varies with respect to time. We have applied Jeans law and the space time transformation contrast to the transformation of Meshcherskii. The space time transformation is applicable only in the special casen=1,k=0,q=1/2. The equations of motion of our problem differ from the equations of motion of the restricted three body problem with constant mass only by small perturbing forces.     

Dynamics of a particle under the gravitational potential of a massive annulus: properties and equilibrium description

This paper studies the main features of the dynamics around a massive annular disk. The first part addresses the difficulties finding an appropriated expression of the gravitational potential of a massive disk, which will be used to define the differential equations of motion of our dynamical system. The second part describes the main features of the dynamics with special attention to equilibrium of the system.     

Periodic orbits in the Planar General Three-Body Problem

The article contains a numerical study of periodic solutions of the Planar General Three-Body Problem. Several new periodic solutions have been discovered and are described. In particular, there is a continuous family with variable masses, extending all the way from the elliptic restricted problem to the general problem with three equal masses. All our examples have special symmetry properties which are described in detail. Finally we also suggest some important applications to the natural satellites of the solar system.     

Stability of resonant planetary orbits in binary stars

This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets, moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets). The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general, instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement with the existing qualitative predictions