The dynamical evolution of a cluster in which N, the number of stars, is large, results primarily from encounters between pairs of stars; these tend to modify the system so that it approaches a more probable configuration. In statistical equilibrium, the distribution of stars among states of different energy, Ei, per unit mass would be determined to a first approximation by the probability Pi, where

here gi is the statistical weight of each state, proportional to the amount of phase space available, and B is a constant. Evidently dynamical evolution tends a) towards stellar states which are more tightly bound by their mutual gravitational attraction and have a correspondingly lower Ei, and b) towards the free state, where the volume of available phase space is very large. The only relatively stable state for an isolated cluster is one in which a central, non-evolving massive configuration (a compact object, or tight binary, or a hierarchical multiple system) has all the initial binding energy of the cluster and all other stars have escaped.

This review describes work on the evolution of a stellar system during the phase which starts at the end of core collapse. It begins with an account of the models of Hénon, Goodman, and Inagaki and Lynden-Bell, as well as evaporative models, and modifications to these models which are needed in the core. Next, these models are related to more detailed numerical calculations of gaseous models, Fokker-Planck models, N-body calculations, etc., and some problems for further work in these directions are outlined. The review concludes with a discussion of the relation between theoretical models and observations of the surface density profiles and statistics of actual globular clusters.

The past decade has seen the development of powerful numerical methods for studying star cluster evolution by direct integration of the Fokker-Planck equation. Cohn's basic algorithm for spherical systems of identical point masses and its application to the study of core collapse is reviewed. Merritt's extension of this method to treat systems containing a mass spectrum, and Goodman's extensions to include strong scattering and rotation are discussed. Results from direct Fokker-Planck computations of core collapse in single mass and multi-mass isotropic clusters and single mass anisotropic clusters are presented to illustrate the method. Calculations of pre- and post-collapse evolution with a central black hole and with heating by hard binaries are reported. Prospects for future development of the direct Fokker-Planck method will be discussed, with emphasis on the central goal of developing physically realistic models for interpreting Hubble Space Telescope observations of globular cluster structure.

The mathematical treatment of encounters is discussed briefly, and it is pointed out that in a spherical system with an isotropic distribution function and equal-mass stars, the relevant equations can be analytically orbit-averaged. Even in small systems, close encounters have little effect on core collapse. Near a black hole, however, close encounters may cause an accumulation of stars on large, very eccentric orbits.

Presence of stars with disparate masses causes great differences in the dynamical evolution of star clusters from the evolution of single component clusters. One remarkable effect is acceleration of the evolution. Another effect is destabilization or stabilization. In two-component clusters equipartition at the cluster centre is nearly achieved if Spitzer's (1969) condition is satisfied. In multi-component clusters equipartition at the cluster centre is achieved if either the range of stellar mass is very narrow or the mass spectrum is very steep. Global equipartition is never achieved. Post-collapse evolution of multi-component clusters is discussed briefly and some remained problems are presented.

We can understand physics of self-gravitating system in terms of gaseous models in so far that their global natures and effects of self-gravity are concerned. Here summarized what are known in idealized gaseous models. They include gravothermal collapse/expansion in linear and non-linear regimes, and post-collapse evolution with gravothermal oscillation. Also discussed are their relations with discrete system and with treatment in statistical mechanics.

Stellardynamical systems are modelled by gaseous spheres. The properties of an isothermal equilibrium configuration with a central singularity are discussed in the context of post-collapse evolution. For the numerical calculations heating effects are taken into account. Evolution towards the equilibrium configuration is monotonically only if much energy is delivered from hardening processes of binary stars. Since the equilibrium configuration is gravo-thermally unstable a minute emission of energy is sufficient to turn gravothermal contraction into gravothermal expansion. In such case equilibrium is approached by oscillations of the core.

Binaries provide an energy source in dense stellar systems. Exothermic gravitational interactions in star clusters can play a role similar to that of nuclear reactions in single stars. These gravitational interactions can be modeled in a laboratory setting, in the form of numerical binary-single star and binary-binary scattering experiments. Gravitational cross sections obtained this way can be applied to model star cluster evolution, just as nuclear cross sections are used as input data in stellar evolution calculations. References are given to detailed descriptions of gravitational cross sections, and a useful new example of an application is given: the rate at which hard binaries form in a homogeneous stellar background, as the solution of an integral equation describing the combined effects of creation, destruction, hardening and softening of binaries.

The main principles for direct integration of large point-mass systems are outlined. Most particles are advanced by the Ahmad-Cohen neighbour scheme, using fourth-order force polynomials and individual time-steps. Close encounters and persistent binaries are handled by two-body regularization, whereas extreme triple and quadruple configurations are treated as unperturbed systems by special regularization techniques. As an illustration of these methods, we discuss some recent results of an isolated system containing 1000 particles of unequal mass, with special emphasis on the post-collapse phase.

We summarize the methods of a new “hybrid” computer code for stellar dynamics. All particles in the inner spatial region are followed exactly via a direct N-body code and all particles in the outer spatial region are treated statistically via a distribution function and Fokker-Planck type methods. An intermediate region, with features of both, allows exchange of particles and energy between the outer and inner regions. We apply our code to the period just before core collapse and just after and summarize the results.

We report on an investigation of the core collapse of an initially bound system of 1024 point masses based on a new numerical method which incorporates a recursive center of mass reduction of the N-body system and includes regularization of the two body encounters. A tree which organizes the point masses into a logarithmic hierarchy is the primary feature of the numerical algorithm. The spatial structure of the tree allows for rapid force calculation while the associated temporal structure allows for variable time steps. Simulations are calculated for several core collapse times in an effort to follow the end point evolution of the system. The point-like nature of the interactions are maintained throughout the simulations.

Pease and Shapley (1917) first remarked on the apparent flattening of several Galactic globular clusters, a view that has been confirmed by many subsequent studies. Tidal stresses, internal rotation, and velocity anisotropies can cause deviations from sphericity in stellar systems. In general, we might expect globular clusters to have some angular momentum at the time of formation and, if they collapsed from flattened initial conditions, to have anisotropic pressure support. Since the velocity distributions within the clusters can be altered by a variety of internal and external processes, their shapes are expected to evolve. In this article, we review the methods for measuring ellipticities and the results that have emerged from such studies. Our main purpose, however, is to discuss the processes that determine the shapes of globular clusters and the ways in which they change with time.

Equlibrium spherical stellar systems exhibiting instabilities on a dynamical timescale were first studied by Henon (1973), using a spherically symmetric N-body code. We have re-examined Henon's models using an improved code which includes non-radial forces to quadrupole order. In addition to the radial instability reported by Henon, two new non-radial instabilities are also observed. In one, found in models with highly circular orbits, the mass distribution exhibits quadrupole-mode oscillations. In the other, seen in models with highly radial orbits, the system spontaneously breaks spherical symmetry and settles into a tri-axial ellipsoid. These instabilities, which are driven by fluctuations of the mean field, offer some analogies to the well-known dynamical instabilities of a cold disk of stars. While our models are rather artificial, they indicate that dynamical instabilities may be more common in spherical systems than had been thought.

Spherical stellar systems show during their secular evolution the development of velocity anisotropy in their halo (cf. e.g. Hénon, 1971). The present study examines the general reasons for generation of anisotropy in stellar systems by means of a gaseous star cluster model including anisotropy. Moment equations of the Boltzmann equation are considered for spherical symmetry in coordinate space but not in velocity space closed in third order by a heat flux equation. The coefficient of heat conductivity is tailored to describe the flux of energy due to the cumulative effect of distant gravitative encounters and generalized to include effects of anisotropy and external gravitation by a massive central object (Bettwieser et al., 1984).

The long-term evolution of N-body systems with high N (>104) cannot yet be studied by direct integration of the equations of motion. Instead, the study of systems with smaller N can give insight into the higher N limit in two ways. First, we can measure the dependence on N of all quantities of interest for a range of N-values, and extrapolate the results to higher N. Second, even a very high-N system will eventually undergo core collapse, at which point the central dynamics will be dominated by a subset of stars with small N (10–100).

We determine if the orbits of the galactic globular clusters can be circular, using a spherical logarithmic potential. If they are not, we calculate their perigalactic distances and we deduce a pseudo-eccentricity. Using a numerical and geometrical method, we obtain the minimum of all the possible pseudo-eccentricities for each cluster. The corresponding perigalactic distances are compared with the values given by Innanen et al. (1983) with a different method.

In the late 70s, the functional integral formulation equivalent to the Fokker-Planck equation was worked out (Graham). We apply this functional integral formulation to gravitationally interacting systems, whose dynamics may be analyzed by separating the forces operating on a particle into a mean field force and fluctuations due to random collisions at intermediate range (scattering at small angles). In this poster the formalism is presented for short periods (in the stochastical meaning) to systems with isotropic distribution background in velocity space (different spatial densities are possible). Later the functional integral for the local change for the distribution function is evaluated in the steepest descent approximation. In the end we point out the applicability of the method to slowly evolving globular star clusters near thermal equilibrium. In conditions of slow evolution we can express the evolution of the orbits in terms of local deviation from equilibrium.

We present stationary analytical models for the structure of spherical stellar systems with several mass components. Evolution is simulated by increasing the central velocity of escape. We compare our models to the results of numerical simulations. An observable kinematical effect of mass segregation is pointed out.

The presence of an initial binary population in a stellar cluster can have a dominant effect on the dynamical evolution of the whole system. We discuss simulations, carried out with a direct N-body method for several models with N = 300 equal mass objects, with 20% of all the stars as initial binaries having different binding energy. These calculations show that binaries with values of the binding energy, h, in the range 5–10 times the mean kinetic energy, x, are the most strongly interacting ones with the field stars and between themselves. The main result of these interactions is an enhanced expansion of the core and also of the whole cluster. The binary heating prevents the gravitational collapse for a time longer than the collapse time for systems without binaries. Energetic binaries with h greater than 25x concentrate at the center, due to mass segregation but these binary-binary interactions produce disruption and escape of the components, with smaller heating effect. Binary-binary interactions occur predominantly in the inner regions and are mainly responsible for the core heating and for the escaping stars. Single star-binary encounters, instead, occur more frequently in the outer regions, and only contribute to the strong expansion of the halo. In all the simulations there was no formation of new persistent hard binaries.

Recently, Israel and Kandrup (1984; Kandrup 1984 a,b,c,d) have formulated a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object here is to indicate how that formalism may be used to construct a theory of ‘collisional’ stellar dynamics, valid for a collection of point mass stars in the limit that incoherent radiative effects may be neglected.