This paper is intended for an introduction to the Symposium. In 47 Tuc many milli-second pulsars are found while none has been reported in ω Cen. One might think that in 47 Tuc they have been formed in the collapsed core through tidal capture of a main sequence star by a neutron star. If we use the standard model of gravothermal collapse of globular clusters to integrate the squared stellar density over the core and over its time history, we find, however, the accumulated probability of tidal capture is lower in 47 Tuc than ω Cen. Such contradiction suggests that it will be important to take account of mass segregation as well as stellar evolution in modelling dynamical evolution of star clusters.

Correlations between globular cluster (GC) properties are reviewed, including some new work on cluster tidal radii and densities. Most of the observed correlations can be interpreted within the framework of our current understanding of their dynamical evolution. These correlations provide empirical constraints for models of GC formation and evolution.

We present results from an ongoing program to probe the dense central parts of Galactic globular clusters using multicolor Hubble Space Telescope images (WF/PC-I and WFPC2). Our sample includes the dense clusters M15, 47 Tuc, M30, NGC 6624, M3 and M13. The two main goals of our program are to measure the shape of stellar density profile in clusters (the slope of the density cusp in post core collapse clusters, in particular) and to understand the nature of evolved stellar populations in very dense regions and their variation as a function of radius. The latter includes studies of blue straggler stars and of the central depletion of bright red giants. Our recent WFPC2 study of M15 is described in detail.

HST observations contribute in many ways to a better understanding of the dynamical nature of globular clusters. Unprecedentedly faint photometry gives new determinations of the numbers of low-mass stars. Cluster-to-cluster differences at the faint ends of the mass functions suggest differences in dynamical evolution. Mass segregation is clearly observed, from the envelope inward to the dense cluster center. The distribution of stars in the hitherto unresolved cores gives new data with which to test theories of core collapse, and these core profiles are also sensitive to the number of unseen remnant stars and binaries at the cluster center.

The WFPC2 aboard the Hubble Space Telescope has been used to obtain deep images in three fields at different radial positions in the nearest Globular cluster, M4 (NGC 6121). In this paper, we discuss the white dwarf cooling sequence and show how the dynamical structure of the cluster will affect their cumulative distribution function. We also present the first discussion of our observations of the faint cluster main sequence stars.

Close binaries are widely believed to exist in large numbers in the cores of globular clusters. If present, these binaries are critical sources and sinks of energy that drive the dynamical evolution of their host clusters. I report on HST searches for binaries (based on variability) in the outskirts and cores of several globular clusters; dwarf novae should be particularly easy to find. Dense and loose clusters have been thoroughly searched on timescales ranging from minutes to years. Detailed simulations demonstrate that virtually all binaries with M < 8, amplitudes > 0.1 mag and periods of 2–20 hours should have been found. This includes virtually all known contact binaries. At least 1/3 of all dwarf novae present in several globulars should also have been seen (very easily!) in eruption at M = 4 − 6.

Simple tidal capture theory predicts that dozens of interacting binaries should have been found in our searches; the observed number is typically one or two objects per cluster. Unless tidal capture cataclysmic binaries are rapidly destroyed, ejected, or much fainter than most of their Galactic counterparts, we must conclude that very close binaries in globular cores are rare, and that their total influence on cluster dynamical evolution is less than currently claimed.

This report has two major purposes. First, we summarize here work by our team on the determination of the density of stars near the centers of a large sample of galaxies observed with the Hubble Space Telescope. There appear to be two varieties of elliptical galaxies (and bulges). The stellar densities near the centers of small elliptical galaxies exceed those of globular clusters and the density of the universe at the recombination epoch. The radial dependence of density and implied gravitational force seems inconsistent (at least in the case of the smaller elliptical galaxies) with a long-lived triaxial configuration. It therefore seems likely that the central regions of less luminous elliptical galaxies are axisymmetric. For the more luminous ellipticals, the central densities are far more modest and the presence of a distinct core (defined below) is generally well established. Even in these cases, however, we find few if any galaxies with analytic (Taylor expandable) stellar densities near the center.

In a brief introduction, we first compare, in the framework of globular cluster dynamics, the relative importance of the amounts of information acquired, during the past decades, through proper motion and radial velocity measurements. Next, we review the most recent and important studies based on measurements of radial velocities obtained with single-object and multi-object spectrometers and interpreted with the use of King-Michie and Fokker-Planck models or with non-parametric methods. Then, we present the results obtained from integrated-light spectra in the cores of a few collapsed globular clusters. We conclude with a summary of the most important scientific outputs secured from the radial velocity measurements of stars in globular clusters.

HST makes it possible for the first time to study nearly the entire mass range of globular-cluster main sequences, from the turnoff down almost to the theoretical limit for hydrogen ignition. We present main-sequence luminosity functions (LFs) for four clusters that include stars with M < 0.15M⊙ in all cases. We compare these and other LFs that have been obtained with HST for a total of five globulars to date. Two of the three clusters in the sample that have similar metallicities have nearly identical LFs, while the third is relatively deficient in low mass stars. Possible implications of this finding are briefly discussed. Inferred mass functions vary significantly depending on the mass-luminosity relations that are adopted.

High resolution near-infrared imaging and spectroscopy now gives detailed information about the structure, evolution and mass distribution in the nuclear star cluster of the Milky Way. The central parsec is powered by a cluster of luminous and helium rich, blue supergiants/Wolf-Rayet stars. The most likely scenario for the formation of the massive stars is a star formation burst a few million years ago at which time a dense gas cloud may have fallen into the center. The stellar density in the ∼ 0.3 pc radius central core is high enough that collisions with main sequence stars destroy the largest late type giant stars. Radial velocity measurements for about 300 early and late type stars between 0.1 and 5pc radius from the dynamic center now strongly favor the existence of a central dark mass of 2.5 − 3.3 × 106M⊙ (density (109M⊙pc−3, M/L2μm) ∼ 100M⊙/L⊙) within 0.1pc of the dynamic center. This central dark mass cannot be a cluster of neutron stars. It is either a compact cluster of stellar black holes or, most likely, a single massive black hole.

The evolution of spherical single-mass star clusters was followed by numerically solving the orbit-averaged two-dimensional Fokker-Planck equation in energy-angular momentum space. Velocity anisotropy is allowed in the two-dimensional Fokker-Planck model. The development of the anisotropy is discussed in detail.

The revision of the Stodółkiewicz's Monte-Carlo code is presented. It treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The first calculations, for equalmass N-body systems with three-body energy generation accordingly to Spitzer's formulae, show good agreement with the direct N-body calculations for N = 2000 and 10000 particles.

Fluid dynamical techniques to model the dynamical evolution of star clusters, and their successors, gaseous models using an equation of heat conductivity to model relaxation effects, including anisotropy, are presented. The historical merits of such models are reviewed as well as the current status of their credibility, based on quantitative comparisons with other methods, like orbit-averaged Fokker-Planck solutions and direct N-body simulations.

The size of the core is one of the main diagnostics of the evolutionary state of a globular cluster. Much has been learned over the last few years about the behavior of the core radius during and after core collapse, under a variety of different conditions related to the presence or absence of large numbers of binaries. An overview is presented of the basic physical principles that can be used to estimate the core radius. Four different situations are discussed, and expressions are presented for the ratio rc/rh of core radius to half mass radius. The regimes are: deep collapse in the absence of primordial binaries; steady post-collapse evolution after primordial binaries have been burned up; chaotic post-collapse evolution under the same conditions; and post-collapse evolution in the presence of primordial binaries. In addition, modifications to all of these cases are indicated for the more realistic situation where effects of the galactic tidal field are taken into account.

We first review the reasons for carrying out statistical analysis of results from large numbers of N-body simulations, including a summary of previous work. Then we describe some results about the behaviour of N-body systems which have been acquired by this technique. Finally we concentrate on the problems associated with the scaling of N-body data with N, with particular regard to the dissolution of systems in a tidal field when mass is lost by stellar evolution.

We have developed a massively parallel special-purpose computer system for N-body simulations, GRAPE-4 (GRAvity-PipE 4). The GRAPE-4 system is designed for high-accuracy simulations of dense stellar systems. The GRAPE-4 calculates gravitational forces, their derivatives in time and potential energies. It has a hardware for prediction of positions and velocities, which is used for the individual timestep scheme. Using multi-chip module technology, we integrated 1692 chips of 640 megaflops performance. The peak speed of GRAPE-4 is 1.08 teraflops.

We present the first clear evidence that the gravothermal oscillation takes place in N-body systems. We performed direct N-body simulations of systems of point-mass particles with particle numbers from 2,048 to 32,768. In the simulation with 32,768 particles, the central density shows an oscillation with an amplitude of ∼ 103, which is similar to what was observed in more approximate models such as a conducting gas sphere and one-dimensional Fokker-Planck calculations. The amplitude is smaller for a smaller number of particles. The number of particles in the core at the maximum contraction is ∼ 10 for all runs, while the number of particles at the maximum expansion is about 0.01N. For 16,384- and 32,768-body runs, the temperature inversion during the expansion phase is clearly visible.