The equations of motion of the star-members of the cluster averaged on the elliptic orbits are obtained. These equations take into account the tidal forces of the Galaxy. The generalization of the Lagrange-Jacobi equation and Sundman inequality for non-classical scheme of the many-body problems is revised. The dynamical evolution of the moment of inertia is studied. Some theorems which determine the type of the star motion in the cluster are formulated.

It is suggested that inverse cascade that may occur in the turbulent cosmic medium can result in the formation of very large scale structures in the universe, upto the largest scales like the Great Wall. Again clustering of galaxies on all scales is interpreted to be due to these self-organisation processes occurring in a turbulent medium, the largest structures being anisotropic and nearly two dimensional, the smaller structures remaining isotropic. The observed fractal distribution of galaxies is also interpreted on this basis. The direct proportionality between the flow velocity and the linear dimension of the structure may show a way out of the dilemma of missing matter.

The properties of the Trapezium type multiple stars are discussed. The existing data confirm the dynamical instability of the Trapeziums. They show that because of the specific space configurations of components the Trapezium type multiple stars are disintegrated completely or partly (becoming ordinary systems with smaller number of components). The disintegration time is estimated some million years for Trapeziums with negative total energies and much shorter when their total energies are positive. There are some evidences in favour of the existence of Trapeziums possessing positive total energies.

The stability of a galaxy against disruption when it encounters a point mass perturber has been investigated by numerical simulations. The merging process dominates in the case of encounters of binaries of comparable mass and disruption is important in cases where the components have different masses. The disruption of a galaxy occurs when its mean density is less than a critical density - the Roche density. The change in the energy falls steeply near the Roche density in the case of encounters in which the relative orbit of the perturber is circular.

We consider a small stellar system ejected from the interior of a bigger one. The two systems are represented by unequal Plummer models and energy changes in them are derived analytically under the impulse approximation. These are used to obtain the conditions for the Roche instability of the systems. For systems greatly differing in dimension, the density ratio of the two systems is a useful parameter for specifying the Roche limit for the disruption of the smaller system and the mass ratio is a meaningful parameter for specifying the Roche limit for the disruption of the bigger system. Density ratios and mass ratios at Roche limit are given for both systems for various scale-length ratios of the two systems.

The multiplicity factor of stellar encounters in the infinite homogeneous layer with the constant thickness H has been studied. It is seen that simultaneous effect of Chaotically distributed distant field stars leads to the convergence of the collision integral for the great distances in the plane disk and in the layer of the constant finite thickness.

This work is devoted to research of an evolution of dynamical systems with the factors, changed by accident, are investigated by statistical methods. The method of stochastic equations are applied for the study of the probe ensemble of stars, working in a field, where the regular component has spherical symmetry.

It is an assumption of traditional stellar dynamics that Liouville’s equation governs the time evolution of stellar systems. An inevitable consequence of such a premise is that (a) at least some modes of instability of stellar systems may be those of Liouville’s equation; and (b) stellar systems might undergo periodic changes of definite patterns in configuration and velocity spaces. For, Liouville’s equation exhibits eigenmodes of oscillation. While it is not feasible to observe the astronomically long periods of oscillations, the patterns of changes, i.e. the eigenfunctions, may be amenable to observation by analyzing the COD records of brightness and velocity distributions on the visible disks of galaxies and globular clusters. This is our motivation for scrutinizing Liouville’s equation, if not for its own merits. The work is a continuation of a series of papers on the symmetries of Liouville’s equation (Sobouti 1989a, b, Sobouti and Samimi 1989).

A reinvestigation of the linear perturbation theory is presented, which examines the hydrostatic readjustment of an isolated self-gravitating gas sphere to a redistribution of energy. The here presented model describes a stellar system by the common equations of gas in hydrostatic equilibrium but with the effect of the anisotropic velocity distribution on the pressure gradient. We take as equilibrium models the singular isothermal solution with and without anisotropy. The total variation of the Boltzmann entropy resulting from a perturbation of the system caused by a redistribution of heat (i.e. r.m.s. kinetic energy of the stars) is calculated for anisotropic solutions to first order as well as to second order for the isotropic equilibrium. The extremized eigenfunctions which represent the entropy and anisotropy perturbation functions, are determined analytically. They exhibit gravothermal behaviour in the central region where heat is removed. It is also found that the anisotropy readjusts non-thermally in the sense that the system departs from isotropy although the total entropy increases.

A time dependent solution of the collisionless Boltzmann equation (CBE) in two space dimensions first given by Antonov and Nuritdinov is discussed further. The solution is self consistent with a quadratic potential and represents a generalised Freeman bar characterised by 10 parameters. Because of two conserved quantities, they form an 8-dimensional phase space. Geometric and group theoretical aspects of this dynamical system are discussed. Systems of this type can show chaotic oscillations in general. This study raises interesting general questions about time dependent solutions of the CBE.

We study scenarios where comets are not original members of the Solar System but have been acquired from the surrounding medium through dynamical evolution. This leads to estimates of the present day number density of comets in the interstellar medium which are not in contradiction with observational upper limits. We also consider the dynamical transfer of Oort Cloud comets into short period comets. The process is very sensitive to the inclination of the comet orbit which leads to strong bias in favour of low inclinations in short comet orbits.

The statistical material obtained at the Leningrad observatory amounts to about 3, 104 tripe systems with negative total energy (Anosova and Orlov, 1985; Anosova 1986). From this material the basic qualitative results have been summarized.

The paper deals with the study of the arising and disappearence of collinear (Eulerian) L1, L2, L3, triangular (Lagrangian) L4, L5, coplanar L6, L7, ring L0 and infinitely distant L±∞ solutions in a restricted problem of three variable-mass bodies for different time dependencies of main bodies masses and for some additional conditions imposed on the systems parameters. In this case it is assumed that the motion of variable-mass main bodies is determined by the Gylden-Mestschersky problem. The Bill surfaces in the restricted three-body problem where main bodies masses variate isotropically according to the Mestschersky law are studied. Certain possibilities of applying the results of investigations to nonstationary double stellar systems are discussed.

In extending the results of Henon and Petit (Celes.Mech., 38,67, 1986) an algorithm is suggested to construct the series representing the general encounter-type solution of the spatial eccentric Hill’s problem. The series are arranged in powers of the eccentricity E of Hill’s problem and two integration constants e and k characterizing eccentricity and inclination of the relative motion. A particular non-periodic solution of Henon and Petit corresponding to E = e = k = 0 is taken as an intermediary. The perturbations to this solution are constructed similar to the Lunar theory of Hill and Brown with; the Universal Poissonian Processor. From theoretical point of view Hill’s problem for the encounter case is of particular interest. In distinction from the Lunar problem we do not have here angular arguments with different frequencies. Moreover, the perturbations related with the external eccentricity E (analogous to the perturbations in the motion of the motion of the Moon caused by the eccentricity of the orbit of the Sun) are of resonance character.

The stability of the triangular Lagrangian solutions for the photo-gravitational restricted three-body problem in the three-dimensional case is investigated for the case when the resonances are absent and also when the resonances are present. Stability is proved for most (in the sense of Lebesgue) initial conditions for all μ < μ0 except for the resonance cases.

The paper is concerned with studying the domain of possible motion and a field of the test body velocities in the plane restricted problem of three bodies. The study is based on existence of a quasi-integral of areas (similar to an integral of areas in the problem of two bodies) as well as on the Jacobi integral. The method of constructing the quasi-integrals is a standard one (see, for example, [1],[2].

A systematic investigation has been carried out for periodic solutions for standard-form Hamiltonian systems containing a small parameter/the principal problem of dynamics/. An efficient method of investigation of conditions for periodicity of solutions has been developed. Besides fitting the initial conditions of the action-angle variables, the idea of fitting the values of the parameters of the problem is used. Constructive conditions are obtained for the existence of periodic solutions in both principal and degenerate cases, as well as necessary conditions for their stability; algorithms have been developed for constructing these solutions as series in integer powers of the small parameter. To study particular periodic solutions /by high order resonances/, canonical transformations of the initial equations to a special form are used.

The in-plane motion of a Geosynchronous satellite under the gravitational effects of the sun, the moon and the oblate earth has been studied. The radial deviation (Δr) and the tangential deviation (rcΔθ) have been determined. Here rc represents the synchronous altitude. It has been seen that the sum of the oscillatory terms in Δr for different inclinations is a small finite quantity whereas the sum of oscillatory terms in rcΔθ for different inclinations is quite large due to the presence of the low-frequency terms in the denominator.

After reviewing the original approach leading to the introduction of intermediaries in Satellite Theory, a general procedure to define intermediaries for the Main Problem in this Theory is proposed. This procedure is susceptible to an intuitive interpretation analogous to solving a simple puzzle. The application of this method to the Main Problem allows us not only to recover the well known classical intermediaries but also to obtain several completely new ones, all admitting simple solutions.

Predictability in orbital behaviour of artificial satellites depends on several factors: the accuracy required, the particular dynamical models formulated, the sets of variables chosen to describe them, the numerical or analytical techniques used and, specially, the specific trajectories to be established. In this paper we address the problem of predictability for highly eccentric satellites with (J2 + J22)-perturbation, by using numerical techniques to integrate the equations of motion when expressed in different sets of regular variables.