This work was done in collaboration with Jim Gunn.

We present here some recent (and very preliminary) findings from a study of the early stages of the post-core-collapse evolution of an isolated cluster of identical point “stars”. The method used to follow the behavior of the system is the unified N-body/statistical treatment described in detail by McMillan and Lightman (1984a) and by Lightman and McMillan elsewhere in this volume. Briefly, the method combines the standard “large-N” and “small-N” approaches to the problem in the régimes where they are appropriate by treating the inner regions (r < rN) exactly with a regularized Aarseth N-body code (Aarseth, 1972), while permitting stars at greater and greater radii to retain less and less of their individual identities, ultimately treating the outer portions of the system (r > KrN) in an almost purely statistical fashion.

Results from numerical integrations of random binary-binary encounters have been used to obtain various cross-sections and outcome distributions for the four-body scattering. The initial orbital elements were chosen randomly except the Kepler-energies for which various selected values were used. Rough estimates for mass effects were obtained by simulating encounters of binaries with unequal component masses.

We developed a semi-analytical theory for obtaining the types and energies of the outcome configurations. The theory contains some adjustable parameters, the values of which we deduced by comparing the theory and experiments.

The energy transfer rate by collisions (=outcome is not two binaries) dominates over that due to fly-by's by an order of magnitude, provided that the binaries are hard. The formation of a hierarchical three-body system is fairly common. In a collision of energetically similar very hard binaries the probability is about 20 percent, while it is greater than 50 percent if the binding energies differ by a factor of more than four.

Recent numerical simulations for 1-dimensional systems have shown that the relaxation time due to encounters is far shorter than the generally accepted estimate. To account for this, a new approach to the theory is necessary. The analysis of encounters presented here is characterized by the retention of periodic trajectories in the mean field. The kinetic equation obtained yields a relaxation time scale in qualitative agreement with the simulations. The analysis can be extended to the 3-dimensional case, and preliminary results predict here also a reduction of the relaxation time.

It was von Hoerner (1957) who pointed out that the tidal field of the Galaxy imposed a boundary condition on the size of globular clusters, a fact that had observational consequences when the first deep star counts became available (King 1962). More recently, though, dispute (cf Innanen, et al. 1983) has arisen over the identification of the observed limiting radius from star counts with that predicted by simple dynamical theory (King 1962, 1966; Szebehely 1967). Straightforward application of the classical three body problem seemed to imply that no stable stellar orbits would be found beyond the collinear Lagrangian points. Thus, to first order, the observed limiting radius would be just the radius of the inner Langrangian point L2 from the cluster center. For clusters in eccentric orbits, the tidal radius would be set at perigalacticon, where the tidal force is strongest. Numerical models of stellar orbits about a cluster in a realistic galactic potential by several authors (Jefferys 1976; Keenan and Innanen 1975; Keenan 1981) showed that stable orbits can exist with apocluster distances greater than rL2, due to the presence of non algebraic integrals of the motion that constrained the stars to move in a restricted region about the cluster rather than escaping.

Physical arguments are presented to show that two-body, tidal-capture binaries should form in abundance (Nb ~ 103) during the evolution of globular clusters by the time that core collapse begins. Interactions amongst these binaries and with core single stars will cause ejections from the cluster which pump energy into the system producing a bounce and re-expansion. Detailed numerical Fokker-Planck evolutionary calculations presented here confirm this scenario and indicate that this process is likely to be the dominant energy input for most clusters. During the re-expansion phase rcore ∝ t1/3, rhalf ∝ t2/3 with the core containing several hundred very close binary star systems.

The evolution of a nonisolated globular cluster is presented. The binaries (both, tidally captured and formed in three-body interactions), outflow of mass from stellar envelopes and shocks are considered as sources of energy in the cluster.

The dynamical behavior of a relaxed star cluster containing a massive, central black hole poses a challenging problem for the theorist and intriguing possibilities for the observer. The historical development of the subject is sketched and the salient features of the physical solution and its observational consequences are summarized.

The full dynamical problem of a relaxed, self-gravitating, large N-body system containing a massive central black hole has all the necessary ingredients to excite the most dispassionate many-body, computational physicist: it is a time-dependent, multidimensional, nonlinear problem which must be solved over widely disparate length and time scales simultaneously. The problem has been tackled at various levels of approximation over the years. A new 2+1 dimensional Monte Carlo simulation code has been developed in appreciable generality to solve the time-dependent Fokker-Planck equation in E-J space for this problem. The code incorporates such features as (1) a particle “cloning and renormalization” scheme to provide a statistically reliable population of test particles in low density regions of phase space and (2) a time-step “adjustment” algorithm to ensure integration on local relaxation timescales without having to follow typical particles on orbital trajectories. However, critical regions in phase space (e.g. disruption “loss-cone” trajectories) can still be followed on orbital timescales. Numerical results obtained with this Monte Carlo scheme for the dynamical structure and evolution of globular star clusters and dense galactic nuclei containing massive black holes are reviewed.

Recent dynamical integrations of the Einstein field equations for spherical, collisionless (Vlasov) systems in General Relativity suggest a possible origin for the supermassive black holes believed to power quasars and active galactic nuclei. This scenario is discussed briefly.

Direct N-body simulations show that the heat flux from the cusp of stars surrounding an object of mass M ~ 10–100 M⊙ has a profound effect on the evolution of a cluster of N stars of mass m ~ 1 M⊙. A cluster containing an object which is ≳ 5% of its total mass expands, transferring most of the cluster binding energy to a few stars deep in the cusp within several central relaxation times. The results, together with analytic estimates, suggest that core collapse in a globular cluster will be reversed when the core mass is reduced to ~10 M. As is discussed below, this result does not depend on whether or not stars are disrupted by the black hole.

I have considered the rates of formation of binaries in the following processes: two-body tidal encounters, three-body encounters, self-hardening of soft binaries formed in three-body encounters and two-body encounters connected with gravitational wave radiation. I have considered also the cooling and the heating connected with formation of binaries and the subsequent interactions of binaries with single stars. I have found that the rate of formation of heating binaries in two-body tidal encounters differs from that calculated by Ozernoy and Dokuchaev (1982). These differences are connected first of all with the precise calculation of the dissipated energy of two stars passing near-by each other, maximum and minimum binary binding energy and also with the use of the incomplete gamma function instead of gamma function. The processes presently predominat in the globular cluster conditions are the formation of hard binaries in two-body tidal encounters between evolved and unevolved stars and cooling connected with collisions between white dwarfs and main sequence stars and between white dwarfs. In the systems of density greater than 109pc−3 the binaries formed in three-body encounters (and to a smaller extent self-hardening of soft binaries formed in three-body encounters) predominate the globular cluster evolution. In the globular cluster heating by hard binaries exceedes cooling associated with the tidal interactions and collisions of stars.

Many presentations at this meeting have discussed the phenomenon of “core collapse”, and it seems agreed that a stellar system like a globular cluster will, within a few half-mass relaxation times, undergo a runaway increase in central density and achieve a nearly singular density distribution with a logarithmic gradient slightly steeper than −2. The collapse is halted by the formation of binaries when the core has shrunk to contain only a small number of stars, and the system subsequently expands gradually while maintaining a density profile approximating that of a singular isothermal sphere. Light profiles resembling the predicted nearly singular form have been found in a small number of globular clusters (Djorgovski, this meeting), but a puzzle remains in that many more clusters should have undergone core collapse, yet they show quite flat light profiles in their cores.

Tidal stripping and disruption of globular clusters may be responsible for the absence of low density clusters in the inner region of the Galaxy. We have studied these processes by integrating orbits of stars while the cluster is moving through a spherically symmetric galactic potential with constant circular velocity (Vcir =220 km/s). The response of the cluster to the tidal field of the galaxy is calculated in a selfconsistent manner with a collisionless N-body code with N=5000 (van Albada 1982, van Albada and Bontekoe in preparation).

Open clusters provide an opportunity for observational study of several important facets of stellar dynamics which cannot easily be addressed in globular clusters for either fundamental or practical reasons. In particular, open clusters are case studies for:

1. 1) The dynamics of small-N stellar systems. This aspect of open clusters has been excellently reviewed by King (1980), to which the reader is referred. The work of Retterer (1979) and Goodman (this volume) suggests that any differences in the velocity distributions of small-N systems as compared to large-N systems due to the increased importance of close stellar encounters will appear primarily in the high-energy tails of the distributions. This will be reflected in the structure of the cluster halos, where quality data are difficult to obtain. In order to guide observational study, more theoretical work is needed regarding the expected nature and magnitude of any observable deviations from large-N theory.

2. 2) The effects of a large stellar-mass spectrum. While globular clusters probably also have a spectrum of masses spanning an order of magnitude or more, the extension of the mass spectrum to higher masses in open clusters permits the dynamical consequences of a mass spectrum to be more easily studied. In addition, many studies have pointed out the significance of stellar mass lośs in massive stars for the evolution of clusters (e.g., Terlevich 1983). However, as yet the theory regarding the effects of such mass loss has not provided unique observational predictions which might act as critical tests, so this important issue will not be discussed further here.

3. 3) A spectrum of cluster ages. Open clusters span the complete spectrum of age from birth to dissolution. Except for the classic work of Wielen (1971 and this volume) in which he analyses the observed distribution of cluster ages, this feature of open clusters has been little exploited. Yet by mapping appropriate diagnostics with cluster age open clusters have the potential to provide empirical measures of the fundamental dynamical time-scales, e.g. the relaxation time, equipartition time and mixing time. Also, study of the very youngest clusters provides appropriate initial conditions for evolutionary models as well as insight into star- and cluster-formation processes.

4. 4) The effect of the external environment. Open cluster halos are molded by the galactic tidal field. In addition, open clusters are subject to Impulsive encounters with clouds in the interstellar medium. The effects of these environmental influences can be examined in numerous ways, including detailed analyses of the structure of cluster halos and observations of the variation with radius of anisotropy in the cluster velocity distributions.

The dynamical evolution of open star clusters has been studied successfully by numerical N-body simulations. We compare in detail the theoretically predicted lifetimes with those derived from the observed age distribution of open clusters. Rare but efficient encounters between clusters and giant molecular clouds are probably responsible for the sudden disruption of many open clusters. Massive black holes from the galactic corona would, on the average, affect only old open clusters.

Using numerical N-body calculations we have simulated the dynamical evolution of young clusters as they emerge from molecular clouds. Starting with initially virialized systems of stars and gas we follow the evolution of these systems from the time immediately after the stars have formed in a cloud until a time long after all the residual star-forming gas has been dispersed. In the models stellar systems were composed of 50, and in some cases 100, stars and these stars were represented as point masses. The stellar mass function followed a power law with an index of −2.5 and ranged over two decades in mass (Scalo 1978). Gas in the models was represented as an extra term in the gravitational potential function governing stellar motions, and was set to follow a density distribution corresponding to a spherically symmetric Plummer potential function (Plummer 1911). Starting with these initial conditions, stellar motions were then integrated and evolution of each stellar system was followed as gas was dispersed from the vicinity of the stars as a function of time.

We explore the bright central diffuse objects of the closest visible giant HII regions 30 Dor (LMC) and NGC 3603 (Galaxy), with the aid of CCD imagery in 1′ seeing. Both central objects are interpreted as dense cluster cores composed of normal stars with masses ranging up to ≈ 100 M⊙. Each object appears to contain at least one close massive binary. Being significantly denser, the core of NGC 3603 seems to be dynamically more evolved than the core of 30 Dor.

N-body simulations of dynamical evolution of open clusters have been computed with the purpose of comparing them with observations. Special effort has been put into reproducing conditions present in galactic clusters. Most of the models contain 1000 bodies with masses following a power-law mass function of slope a = −2.75 and mean mass 0.5M⊙. Neutron stars or white dwarfs (depending on the initial stellar mass) are generated by instantaneous changes in individual masses, when stars reach the end of their main sequence life. Close approaches between particles are treated by a two-body regularization technique that allows to follow binary evolution in detail. Two types of tidal perturbation are considered: a smooth linearized galactic tidal field is simulated assuming that the clusters move in a circular orbit at 10kpc from the galactic centre; transient shocks are simulated by encounters with extended interstellar clouds of different mass-spectrum, densities and space concentration. It is found that the combined action of evolutionary mass loss and binaries (when the cluster has a realistic mass function) is enough to arrest the core collapse. Tidal heating shapes the halo of the cluster. There is good agreement with the observed density and velocity distribution of open clusters and with reported changes in their mass function.

The first results of an extensive proper motion study of the Pleiades cluster are presented. A total of 166 exposures covering a 3 by 3 degrees area are now incorporated. The accuracies of the centennial proper motions range from in the central region to in the outermost region.

Six proposed Space Telescope programs involving globular clusters are described. The projects appropriate for galactic clusters are: the detection of white dwarfs, the study of the faint end of the Population II luminosity function, the measurement of mass segregation, and the search for a cusp in the density distribution caused by core collapse or by a massive black hole. The two programs that involve extragalactic globular clusters are: the determination of the luminosity function of clusters around different galaxies and the measurement of tidal radii of clusters surrounding elliptical galaxies.