The dynamical evolution of a spherical star system is attractive to the theorist for two reasons. In the first place the physical problem, with some idealization, appears to have an appealing, if somewhat deceptive, simplicity; a large number of point masses, attracting each other with inverse-square gravitational forces, are assumed to move in a quasi-steady system with spherical symmetry. In the second place, the observed globular clusters appear to conform rather closely to this ideal of a spherical star system; to analyze the changes of these clusters with time, resulting from the dynamical interaction of stars with each other, is surely an important aspect of the theory of galactic evolution.

It is pointed out that in advanced phases of evolution the cores of clusters ought to evolve homologously and reasons are sought as to why this evolution should be at constant core energy. It is pointed out that continued evolution after infinite core densities have been achieved is an important area for future research.

Spatial and time correlations of the force acting on a star are derived for finite gravitational systems. It is shown that their behavior is qualitatively different than that for infinite mediums.

The dissolution time for a binary system is considered. We explain why Chandrasekhar's dissolution time differs from that given by Ambartsumian and Oort in that it does not depend on the velocities of the field stars. We show that the difference lies in the definition of what constitutes ‘relative change in velocity’ of the two stars in the binary. Indeed, using the general approach of Chandrasekhar and von Neumann (appropriately modified) we derive a velocity dependent dissolution time.

Dispersion relations have been obtained and analyzed for a non-uniform, non-rotating spherical gravitational system. A restriction to short wavelengths makes it possible to consider a linearized form of the collisionless Boltzmann equation, differing from that for homogeneous systems by the appearance of a term expressing the effect of the mean self-gravitational field upon the motion. The mean field affects the radially directed wave perturbations, with a breaking of symmetry.

Inwardly and outwardly directed modes have quite different propagation characteristics, inward modes being preferentially propagated. Locally, the stability of the system is enhanced due to the effect of the mean field.

Many self-gravitating stellar systems are satellites of larger galaxies and must therefore be subjected to the tidal field of the parent system. Examples are the globular clusters and dwarf elliptical galaxies, which are satellites of our Galaxy. Most previous studies of tidal effects have been highly simplified, e.g. clusters in circular planar galactic orbits (Bok, 1934), or have assumed that the tidal field acts to limit the size of a star cluster without any effects on its internal structure or stability (Spitzer and Shapiro, 1972; Spitzer and Thuan, 1972).

Numerical results on escape and binary evolution in N-body systems are summarized and some results of a new calculation with initial binaries are described.

Numerical experiments on the gravitational N-body problem are reviewed for the case of non-isolated systems. The effect of an external field on expanding cubical systems is discussed.

Tensor generalizations of the virial theorem were checked in a 100-body integration. The virial theorem was remarkably well satisfied, and the calculation confirmed the generalized Lagrange-Jacobi identities. The potential energy tensor, the kinetic energy tensor, and the virial tensor showed surprisingly long correlation times of about of a crossing time.

Using information on the rates at which binaries suffer encounters in a stellar system (Heggie, 1974a), we here study the effects of such processes on the evolution of the system itself. First considering systems with no binaries initially, we show that low-energy pairs attain a quasi-equilibrium distribution comparatively quickly. Their effect on the evolution of the cluster is negligible compared with that of two-body relaxation. In small systems energetic pairs may form sufficiently quickly to exercise a substantial effect on its development and on the escape rate, but in large systems their appearance is delayed until the evolution of the core is well advanced. In that case they appear to be responsible for arresting the collapse of the core at some stage.

Binaries of low energy, even if present initially in large numbers, are likely to have at most only a temporary effect on the evolution of the system. High-energy pairs are not easily destroyed, and so, if present initially, their effect is persistent. It competes with two-body relaxation especially when the fraction of such pairs and the total number-density are high, as in the core, where, in addition, binaries tend to congregate by mass segregation. When encounters with binaries become important, being mostly ‘super-elastic’ they enhance escape and lead to ejection of mass from the core into the halo, thus accelerating the rate at which mass is lost by tidal forces. It is difficult to decide observationally whether globular clusters possess sufficiently large numbers of binaries for these effects to be important.

Aarseth has shown by means of n-body calculations that, in star systems with a range of particle masses, the most massive stars quickly form a binary which soon takes up a large fraction of the total binding energy of the cluster. Similar effects appear in other kinds of physical systems as well; mesic atoms behave in much the same way. The phase volumes of two otherwise equivalent stellar systems, each dominated by a tightly bound binary, favor exchange to incorporate the more massive star in the binary by a factor equal to the cube of the ratio of masses.

In addition to its traditional roles of checking theories and applying them, observation also plays an important role in guiding theory and indicating the form that it should take. This paper reviews types of observation, types of stellar system (omitting the highly flattened ones), and the relations between them. Observational limitations are indicated, as well as places where observations of particular sorts are needed, either for guidance of theories or for their application.

Stellar dynamical theories for collisional systems are tested by numerical N-body simulations of isolated star clusters with N up to 500. For the dynamical evolution of the density distribution in star clusters, good agreement is found between Monte Carlo results and N-body models. There remains a discrepancy between theory and simulations in the rate of stellar escape from clusters and/or in the mechanism by which escapers are produced. Simulations of non-isolated star clusters are compared with observations of open clusters. The observed age distribution of open clusters can be explained by the dynamical dissolution of clusters. Stars of low mass do not escape relatively more frequently than stars of average mass.

This paper consists of two independent parts.

1. (1) The Monte Carlo method for computing the evolution of spherical stellar systems has been modified so that the computation can be continued after the time of formation of the central singularity. Results are presented for systems with equal and unequal star masses. The initial core-halo formation is followed by a general expansion of the cluster, while the central singularity absorbs a growing fraction of the total negative energy.

2. (2) Theoretical expressions of the ‘diffusion coefficients’, which describe the effect of encounters in a stellar system, contain a factor In(γN) where N is the number of stars and γ is a constant usually taken to be of the order of 0.4. A reconsideration of the ‘non-dominant terms’ leads to a substantially lower value, of the order of 0.15 for equal masses and 0.075 for unequal masses with a typical distribution. This correction improves the agreement between N-body and Monte Carlo simulations of spherical systems.

Dynamical masses have been determined for 10 globular clusters. Comparison of the dynamically determined M/L values with those calculated from extrapolations of the observed upper main sequence/giant branch luminosity functions for M3 and M5 indicate that low mass stars (∼0.2 M⊙) comprise a large fraction of the total number.

H. A. Pels and J. H. Oort have discussed the results of a search for fainter Hyades that had been made by G. Pels. The search extended over a region of about 15° radius around the Hyades centre. It was made from proper motions determined from a comparison of Astrographic Catalogue positions with those on plates taken with the 34-cm Leiden refractor. The data extend to about 13m. In part of the field comparisons were also made with plates taken for the Astrographic Charts, in order to extend the luminosity distribution to somewhat fainter magnitudes.

If the galaxies composing the Coma Cluster were initially ‘Ostriker-Peebles’ galaxies (i.e., isothermal spheres extending to 250–500 kpc), the hidden mass is likely to be tidally torn-off halos. Owing to dynamical friction, the remnant cores tend to spiral in to the center of the cluster. This happens as long as the hidden mass, whatever its origin, consists of objects much less massive than galaxies.

The conditions under which double galaxies may be formed by tidal capture are considered. Estimates for the increase in the internal energy of colliding galaxies due to tidal effects are used to determine the magnitudes Vcap and Vdis of the maximum relative velocities at infinite separation required for tidal capture and tidal disruption respectively. A double galaxy will be formed by tidal capture without tidal disruption of a component if Vcap > Vi and Vcap > Vdis where Vi is the initial relative speed of the two galaxies at infinite separation. If the two galaxies are of the same dimension, formation of double galaxies by tidal capture is possible in a close collision either if the two galaxies do not differ much in mass and density distribution or if the more massive galaxy is less centrally concentrated than the other. If we assume, as statistics suggest, that the mass of a galaxy is proportional to the square of its radius, it follows that the probability of the formation of double galaxies by tidal capture increases with the increase in mass of the galaxies and tidal disruption does not occur in a single collision for any distance of closest approach of the two galaxies.

We present results on four different mediums characterised by their ‘density conservation’ in a two dimensional space (phase space for unidimensional plasma and self gravitating systems, configuration space for two dimensional Navier Stokes fluid and guiding center rod plasma).