The definitive orbit of comet 1930 VI basing on 190 observations has been derived. Investigation of the comet orbit evolution for the time span of 200 yr indicated that the 1882 approach (tappr = 1882 Oct. 22.559; δmin = 0.0057 AU) of the comet to Jupiter resulted in considerable element transformations. The perihelion and aphelion distances reduced from 4.1 AU to 1.2 AU and from 12.2 AU to 5.2 AU, respectively, whereas the period of revolution also underwent a decrease from 23.3 yr to 5.8 yr. The next return of the comet to the perihelion is to be expected in 1974 (T= 1974 March, 17.475).

Two apparitions of Comet Westphal in 1852 and 1913 have been linked on the basis of 101 observations, the mean errors being equal to ±2″55. The perturbations from nine major planets and the possible influence of nongravitational forces have been taken into account. The elements and the ephemeride for 1974 have been computed.

The investigation of the 22 apparitions of Comet Encke-Backlund for the time interval 1901–70 was carried out. The integration of equations of motion was performed using the technique devised by Kazimirchak-Polonskaya with automatic step choice, taking into account the perturbations from nine planets (Mercury-Pluto) as well as nongravitational effects.

This paper deals with an iterative version of the general planetary theory. Just as in Airy's Lunar method the series in powers of planetary masses are replaced here by the iterations to achieve improved approximations for the coefficients of planetary inequalities. The right-hand members of the equations of motion are calculated in closed formulas, and no expansion in powers of small corrections to the planetary coordinates is needed. For the N-planet case this method requires the performance of the analytical operations on a computer with power series of 4N polynomial variables, the coefficients being the exponential series of N-1 angular arguments. To obtain numerical series of planetary motion one has to solve the secular system using Birkhoff's normalization or the Taylor series in powers of time. A slight modification of the method in the resonant case makes it valid for the treatment of the main problem of the Galilean satellites of Jupiter.

In a previous paper the authors derived expansions of the derivatives of the disturbing function for the general case including the orbits close to intersection. The present paper deals especially with the case of close commensurability of the mean motions. A new variable v is introduced characterizing the deviation of the mean anomalies from the exact commensurability, and is considered further as an unknown quantity. In the equations of motion the short-period terms are eliminated. The form of expansions of the right-hand sides is chosen basing on the same principles as in the general case. The factors are separated, corresponding to the poles in the case of circular intersecting orbits. For rapidity of calculation the summation in powers of the major semi-axes ratio is made the inner one.

L'ensemble de ce travail fera l'objet d'une publication dans Astronomy and Astrophysics.

In this communication the main equations for the variables: radius vector, longitude, P and Q (variables built from Laplace's perihelium first integral) are given in closed form. These equations are used for deriving the equations of a second-order theory. At this order, the equations for P and Q, are separated and they are integrodifferential linear equations. The equations for the radius vector and for the longitudes, give, after integration, perturbations which are purely trigonometric. The solution shows the features observed in the motion of Jupiter's Galilean satellites. The results are discussed, and extended to include the space variables.

The methods used in deriving conditions for escape, retention and containment for the three-body problem are applied to the n-body case, and similar conditions are obtained. In the n-body problem less stringent conditions are derived, and in the case of retention a further condition is imposed.

The influence of the dynamical figure of the Moon on its rotation with respect to its mass centre (the physical libration) is determined by means of the theorem on the angular moment of a rigid body. In the expansion of the Moon's force function in spherical harmonics all the second and the third order harmonics are taken into consideration. For the determination of the Moon's physical libration components a linear system of differential equations of the second order with constant coefficients is constructed.

The integration displays the essential influence of the new terms in the force function expansion. For evaluation of the disturbed elements of the lunar orbit due to the nonsphericity of the Moon's dynamical figure the Lagrange's equations are solved. The disturbing function is taken in an expansion form in powers of the eccentricity of the lunar orbit and of the inclinations of the Moon's equator and its orbit with respect to the ecliptic. The commensurability of the Moon's mean motion and its angular velocity of rotation produces in the major semi-axis of the lunar orbit secular perturbations of the first order.

A solution of the ideal resonance problem has already been exhibited (Jupp, 1972) explicitly in terms of the ‘mean’ elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the ‘mean’ modulus of the Jacobi elliptic functions is unity; this corresponds to motions on or close to the separatrix of the phase plane of the dynamical system.

A large number of three-body interactions involving one initial binary have been studied by a numerical regularization technique. In each set of experiments some parameters have fixed values, whereas others are selected by uniform sampling of the corresponding distribution functions. Similar statistical results are obtained for different random number sequences at a lower integration accuracy.

This experimental approach permits an approximate determination of the final distributions of eccentricity, velocity and lifetime. These results show a strong dependence on the total angular momentum, total energy and the mass range, whereas other parameters are usually of secondary importance.

There is considerable empirical evidence on the behaviour of binary stars in computer simulations of isolated star clusters. Even when there are no binaries initially they form quickly if the mass spectrum of the stars is reasonably realistic. These phenomena are discussed by a combination of analytical and numerical techniques.

There is strong evidence that the occurrence of mutually very nearby stars with identical spectral type and apparent magnitude is in many cases appreciably more frequent than one has reason to expect if there should only be a case of pure random combination. This phenomenon might be associated with the problem of stability of small stellar systems, if we can identify a certain number of the coincidences as some sort of cluster remnants. Besides the fact that there are several physical explanations of the coincidence phenomenon without consideration to dynamical stability conditions, there is also more than one stability aspect which may be regarded as reasonable. In the first instance we have to survey the possibilities that the stability of certain conglomerations of stars - particularly those with two components - may depend on the mass relation and will reach a maximum value for equal masses. As most straightforward we regard the conception that the most massive stars in a cluster (or even a multiple system) form the most stable and hence the most long-lived configuration, so that a cluster remnant most probably will consist of stars of rather equal mass.

To study the dynamical evolution of trapezia, the equations of motion of the member stars in 30 different trapezia were integrated numerically. It was found that two-thirds of the trapezia still remain as such after 106 yr. Thus, the dynamical ages of trapezia containing O-stars are only slightly shorter than their nuclear ages. The properties of the binaries formed during the computations are studied. The numerical results are discussed from the observational point of view.

The motion of triple stellar systems whose components are at rest at the initial epoch is studied. In this study all the components have equal masses, and at the initial epoch the component A is at (-1/2, 0) on the (x, y)-plane, B is at (1/2, 0), and C is anywhere in the region (s) bounded by the positive x-axis, y-axis and a line expressed by (x+1/2)2 + + y2 = 1. By this way all the possible initial configurations are given.

An integral invariant is a generalization of first integrals to differential forms. Although this mathematical technique is more difficult, the integral invariants allow to obtain new properties for systems which have already well-known first integrals. Integral invariant of first order correspond to a ‘local first integral’ near any solution of motion. In this work I obtain an ‘11th local first integral’ for the gravitational n-body problem, or any homogeneous n-body problem as planetary systems. As this local first integral contains a secular term, a discussion of the stability is obtained. The integral invariant is used for the construction of very particular solutions (Levi Civita's or Poincaré's singular solutions). These solutions realize conditional maximum or minimum of the contraction of the system.

It has been suggested that particle positions in expanding N-body gravitational systems may become increasingly correlated. Such incoherent density fluctuations could develop into bound clusters. Similarly superclusters of such clusters may be dynamically created. We present results of explorations of this idea, using computer simulation. Accurate integrations are made of gravitationally interacting systems of a few hundred particles.

The concentric shell model is used to investigate numerically the stability of spherical steady-state stellar systems. Polytropic models with an isotropic velocity distribution are found to be stable almost down to the limiting index n=1/2. ‘Generalized polytropes’, with a distribution function depending on energy and angular momentum, show instability when n is low and the velocity distribution is radially elongated.

Results of some simple ‘thermodynamic’ experiments on self-gravitating n-body systems are reported for a variety of boundary conditions. Systems placed in specularly reflecting enclosures did not show any unusual behavior, even though a variety of conditions was tried in an attempt to start a ‘gravothermal castastrophe’. Similarly, there was no tendency to transfer energy between ‘hot’ and ‘cool’ subclosures within a given cluster. However, systems in ‘isothermal’ enclosures gave up energy to the enclosure at a surprisingly high rate, and sustained the energy-transfer rate as long as the experiment was continued. An explanation of these different behaviors was sought and found in an examination of the premises that underlie certain attempts to construct a thermodynamics for self-gravitating systems. Conventional application of the H-theorem implies violations of the n-body equations of motion and predictions not consistent with observation. Both the ‘gravothermal catastrophe’ and the experiments in an ‘isothermal’ enclosure share this violation of the equations of motion. A new formulation that allows for all the interactions in an n-body system shows that isolated n-body systems need not form binaries or condense into other subaggregates. The virial theorem follows as an ensemble average over the micro-canonical ensemble.

Under the assumption that the inverse square central force law is a good approximation to the gravitational force, at least for large distances, the different possibilities for the evolution of the Universe are sketched. Several of the possibilities lead naturally to a dynamical classification of clusters of galaxies in an expanding universe. In one of the classifications the galaxies must define configurations which are functions of the masses. The virial theorem approach of determining masses of galaxies in a cluster is briefly examined. Some tentative statements concerning a dynamical explanation of the local hypothesis for quasars are advanced. Finally, the role of mathematical probability in predicting the behavior of the Universe is discussed.