We describe some aspects of implementing star cluster simulations on HARP. The code NBODY4 employs the Hermite scheme with hierarchical block-steps for direct integration. The algorithms have been optimized for parallel processing with the eight pipeline HARP-2 delivering a peak performance of about 1.7 Gflops for N = 104 particles. Hard binaries are studied by KS regularization which also uses the Hermite scheme, whereas strong interactions between 3–5 particles are treated by chain regularization. Astrophysical processes modelled include mass loss by stellar evolution, two-body tidal interaction, Roche lobe mass transfer, common envelope evolution, magnetic braking and gravitational radiation. Consistent values of stellar radii and evolution type are obtained by fast look-up. A new formulation of collision outcomes yields blue stragglers and other exotic objects. Some recent results for an open cluster model are presented.

The studies of compact binaries containing an accreting white dwarf or neutron star in the dense cores of globular clusters have made considerable progress in the past few years as a result of the high resolution images obtained with HST and ROSAT. It is now clear that cluster cores contain a significant population of these systems which must constrain the similarly large populations of millisecond pulsars as well as dynamical histories of clusters. The population of dim x-ray sources appears to be dominated by cataclysmic variables (CVs) formed by tidal capture and not exchange collisions. Our recent HST/FOS spectra of the first CVs in a cluster core, summarized here in more detail, suggest that cluster cores may contain a significant population of magnetic CVs. The required magnetic WDs may arise in spun-up cores of blue stragglers.

Since the discovery of the first globular cluster pulsar in M28 (Lyne et al. 1987) a total of 33 pulsars have been found to reside within 13 seperate clusters. Many (but not all) of the cluster pulsars have properties similar to the millisecond pulsars in the disk: short period, binarity and low magnetic field strength. The common understanding is that these pulsars are primordial neutron stars (i.e. the remnants of massive stars in clusters) which have been spun up by accretion of matter from a companion. Therefore, in this framework, the cluster pulsars are descendents of Low Mass X-ray Binaries (LMXBs) (Alpar et al. 1982). This hypothesis is by no means accepted by all workers (e.g. Michel 1987, Ray & Kluzniak 1990, Romani 1990, Bailyn & Grindlay 1993). These workers have argued that at least some (if not all) cluster pulsars could be formed by accretion induced collapse of massive white dwarfs. In either case, it is clear from the sensitivity limits of current cluster searches, and the luminosity of field pulsars, that there are currently O(103) extant radio pulsars in the Galactic globular cluster system.

In this review, specifically targeted for astronomers working in the field of globular clusters, not pulsar astronomers, we argue that cluster pulsars have provided us with a new window into the population of long-dead massive stars and the physics of tidal capture. The precision with which pulsars can be timed has created new diagnostics: measurement of the mass distribution in the dense cores, measurement of orbital evolution on short timescales and precise determination of orbital characteristics. It is fair to say that all these diagnostics are unique, and not obtainable by other observations. Despite this, it is our assessment that the typical astronomer who works in the field of globular clusters is apparently unaware of these relevant contributions. Hopefully this review will bridge this gap. A complete copy of the review article may be found at http://astro.caltech.edu/~srk.

Twelve bright (Lx ≳ 1036erg/s) X-ray sources have been detected in globular clusters: they are accreting neutron stars. Five of these are transients. Thirty less luminous sources have been detected in eighteen globular clusters. The luminosity function of these sources does not rise rapidly towards lower luminosities. The sources with Lx ∼ 1033 erg/s are probably transients in their low state; the least luminous sources currently detected may be cataclysmic variables. The old open cluster M 67 harbours chromospherically active binaries, with Lx ∼ 1030−31erg/s. Globular clusters contain fewer of these binaries than estimated by scaling with mass from M 67.

Primordial binaries in globular clusters are important both because their properties are an integral part of the description of the stellar population and because they can strongly influence the dynamical evolution of the cluster (see Hut, this volume).

A key problem in using binaries as a tool for diagnosing cluster evolution is that the tool itself is not very well understood. The theory of binary evolution, despite real successes that can be exploited, has serious problems in many areas relevant to cluster evolution. At least as important but often neglected are connective problems, which arise when theoretical model binaries need to be related to observed classes of object, which often requires poorly understood parts of their physics which can be quite irrelevant to their bulk properties. I shall discuss these issues in general briefly, and then illustrate them with the specific example of X-ray binaries and millisecond pulsars.

There seem to me to be four approaches to the problem of computing the evolution of star clusters. Firstly, one might assume that our knowledge of the evolution of stars can be condensed into a subroutine that can be added to an N-body code. This subroutine would mainly have to give the radius and the time-dependent mass of a star as a function of its initial mass and its age. Secondly, standing this on its head, one might assume that our knowledge of N-body evolution can be condensed into a subroutine that can be added to a stellar evolution code. This subroutine would determine, probably in a Monte-Carlo fashion, whether the star had picked up, or lost, a binary companion, or whether the orbit of its companion was significantly changed; the probabilities would be determined by simple analytic approximations to the time-dependent distribution functions of stars (and binaries) of different masses and ages, and by interaction cross-sections as functions of density and ‘temperature’. Thirdly, if the computing power is available, one might more simply unite an N-body code with a Stellar Evolution (SE) code, and follow both the dynamics and the internal evolution simultaneously. Fourthly, we might hope at some stage to put together simple analytic approximations both from N-body and from SE studies, to develop a unified simple model. I venture to say that it is only the last stage, if it is attainable, that would entitle us to say that we ‘understand’ the evolution of stellar clusters. ‘Understanding’, I think, means that we can extract some essential wisdom from large numerical simulations, and apply it on the back of the proverbial envelope.

We describe a fully automated gravitational scattering package capable of determining cross sections and reaction rates for binary-single-star scattering, and present some applications to systems of astrophysical interest.

Evolution with time of isolated hierarchical triple stars, stable from a dynamical point of view, is studied using analytical and numerical approaches. The results are applied to the study of the evolutionary cycle of hierarchical systems formed by dynamical capture in open clusters (up to 10,000 stars).

The number density of stars in the solar neighbourhood is sufficiently low that encounters between two stars will be extremely rare. However, in the cores of globular clusters, and glactic nuclei, number densities are sufficiently high (∼ 105 stars/pc3 in some systems) that encounter timescales can be comparable, or even less than, the age of the universe. In other words, a large fraction of the stars in these systems will have suffered from at least one close encounter or collision in their lifetime.

The hydrodynamics of collisions and mergers of main-sequence stars is discussed in the light of recent 3-D calculations using the smoothed particle hydrodynamics (SPH) method. Theoretical models for the formation of blue stragglers are reviewed in the context of recent comparisons between the observed properties of blue stragglers in dense globular clusters and the predictions of those models.

The probability for collisional interaction of three body binaries is calculated as a function of the physical radius and mass of the stellar objects and the depth of the cluster potential well. For typical cluster parameters, there is a significant chance of physical collision for objects as small as white dwarfs. One consequence of the collisions is to lower the amount of heat produced from hardening a binary, thereby diminishing the efficiency of the three-body heating mechanism.

We review the tidal capture process and in particular the chaotic orbital evolution which follows capture. We discuss the formation of low-mass X-ray binaries in globular clusters via tidal capture and speculate on the possibility that some field low-mass X-ray binaries were formed this way in open clusters which have since dispersed, or in existing old open clusters which are not accessible to observation because of obscuration by dust or because they are indistinguishable from the rich background of galactic stars.

We propose that proto-globular cluster clouds form in a collapsing protogalactic cloud as a consequence of thermal instability. The clouds are photoionized and heated by nearby massive stars. Most are not self-gravitating, but are confined by the residual hot gas in the protogalactic cloud. Their masses evolve as they undergo cohesive collisions with each other and erosion due to interaction with the residual halo gas. Collisions may also trigger thermal instability and fragmentation within protocluster clouds. The resulting cloudlets are pressure confined, and fall toward the center of the protocluster cloud due to inverse buoyancy. Their mass distribution is also regulated by coagulation and erosion. While most cloudlets have substellar masses, the largest become self-gravitating, and collapse to form protostellar cores without further fragmentation. The initial stellar mass function is established as these cores capture additional residual cloudlets. Energy dissipation from the mergers ensures that the cluster will remain bound in the limit of low star formation efficiency. Dissipation also promotes the formation and retention of the most massive stars in the cluster center.

We have examined the dynamical evolution of stellar system containing massive remnant stellar component. If individual mass of remnant stars is much heavier than that of normal stars which comprise most of the mass in the cluster, remnant stars quickly form a subsystem within the core of cluster of ordinary stars. The subsystem evolves on its own relaxation time scale which is very short. However, the post collapse expansion driven by the three-body binary heating becomes very slow because the expansion energy of the compact subcluster can be easily absorbed by surrounding cluster. The gravitational radiation can lead to the merger of binaries when binaries become very hard. A central seed black hole might form if repeated merger becomes very efficient. Otherwise, relatively stable two-component phase of central compact cluster of remnant stars surrounded by larger cluster of low mass stars would last for a long time.

We first review reasons why dark matter is an interesting issue in connection with star clusters. Next we consider to what extent the presence of dark matter is consistent with their dynamics and structure. We review various model-dependent and model-independent methods which have been applied to two well studied clusters, NGC 6397 and 47 Tuc. We suggest that about half of the mass in each object is still unobserved, possibly in the form of a mixture of low-mass stars and white dwarfs.

Some observable relationships between globular cluster parameters appear as a result of long time dynamical evolution of the cluster system. These relationships are inapplicable to the studies of the globular clusters origin.

My first reaction, at the end of this conference, is that it has not produced nearly as much “Confrontation of Theory and Observations” as one might have wished. We have all made our separate presentations, but no one has yet made the connections that are so much to be desired. Perhaps it is just that we need time to absorb what we have heard, and react to it.

The center of the stellar association in 30 Doradus, R136 (HD 38268), is an unusually high concentration of massive and bright O, B and Wolf-Rayet stars and represents the closest example of an intense starburst region.

We present preliminary results from an ongoing study of the central 5 arcmin2 of NGC 6624 based on short F336W, F439W, and F555W (UBV) Hubble Space Telescope WFPC2 exposures (Yanny et al. 1996). NGC 6624 is a dense, metal rich, post core collapse globular cluster at a distance of 8.1 kpc. Nearly 5000 stars with V ≳ 21 (1.5 mag below the turnoff) are detected within the 34′′ × 34′′ area of the PC1 CCD which imaged the cluster center. Individual stars brighter than V = 20 are easily identifiable in the central image section shown in Fig. 1. Image simulations and similar data on the denser cluster M15 indicate that the effects of incompleteness and photometric error are negligible on the V < 19.5 sample of stars in NGC 6624. The stellar surface density profile derived from such a sample is well approximated by a power law of index α = −0.85 (Fig. 2). The density profile shows no hint of flattening towards smaller radii in the radial range over which it is reliably measured (r ≳ 0′′.3). A B vs. U – V color-magnitude diagram of the r < 15′′ region (Fig. 3) shows a well defined blue straggler sequence. Our preliminary findings are consistent with an earlier study of this cluster by Sosin and King (1995) using pre-repair FOC images.