Search

In this study we consider the relationship between the survival time and the Lyapunov time for 3-body systems. It is shown that the Sitnikov problem exhibits a two-part power law relationship as demonstrated previously for the general 3-body problem. Using an approximate Poincare map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the 3-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general 3-body problem., Comment: 9 pages, 19 figures, accepted for publication in MNRAS

We present a large multi-epoch high resolution spectroscopic investigation for the search of binary candidates in the Galactic Globular Cluster (GGC) M4. The aim of our work is the identification of the binary candidates, and the determination of the binary fraction and of the binary radial distribution. We present a large multi-epoch high resolution spectroscopic investigation for the search of binary candidates in the Galactic Globular Cluster (GGC) M4. The aim of our work is the identification of the binary candidates, and the determination of the binary fraction and of the binary radial distribution. The average radial velocity of the observed cluster members is $70.29+/-0.07(+/-0.3)(+/-0.1)km/s. The search for variations in radial velocities among the stars with multi-epoch observations yielded 57 binary star candidates. Our radial velocity measurement accuracy allowed us to identify at a 3sigma level binaries with radial velocity variations larger than ~0.3km/s for the target stars with V<15, and larger than ~0.5km/s for the targets with V>15. We identified 4 binary star candidates out of 97 observed targets inside the core radius, and 53 candidates out of 2372 observed stars outside the core radius. Accounting for the incompleteness affecting our survey, the lower limit for the total binary fraction is f=3.0+/-0.3%. The lower limit for the binary fraction in the cluster core is f=5.1+/-2.3%, while outside the core it decreases to f=3.0+/-0.4%. Similarly, we found f=4.5+/-0.4% and f=1.8+/-0.6% for the binary fraction inside and outside the half mass radius., Comment: 13 pages, 12 figures, Accepted for publication in A&A

We examine the longitudinal distribution of the stars escaping from a cluster along tidal tails. Using both theory and simulations, we show that, even in the case of a star cluster in a circular galactic orbit, when the tide is steady, the distribution exhibits maxima at a distance of many tidal radii from the cluster., Comment: 6 pages, 4 figures, accepted for publication in MNRAS

We describe Monte Carlo models for the dynamical evolution of the nearby globular cluster M4. The code includes treatments of two-body relaxation, three- and four-body interactions involving primordial binaries and those formed dynamically, the Galactic tide, and the internal evolution of both single and binary stars. We arrive at a set of initial parameters for the cluster which, after 12Gyr of evolution, gives a model with a satisfactory match to the surface brightness profile, the velocity dispersion profile, and the luminosity function in two fields. We describe in particular the evolution of the core, and find that M4 (which has a classic King profile) is actually a post-collapse cluster, its core radius being sustained by binary burning. We also consider the distribution of its binaries, including those which would be observed as photometric binaries and as radial-velocity binaries. We also consider the populations of white dwarfs, neutron stars, black holes and blue stragglers, though not all channels for blue straggler formation are represented yet in our simulations., Comment: 10 pages, 13 figures. Some numbers and figures may change slightly before journal submission

We outline the steps needed in order to incorporate the evolution of single and binary stars into a particular Monte Carlo code for the dynamical evolution of a star cluster. We calibrate the results against N-body simulations, and present models for the evolution of the old open cluster M67 (which has been studied thoroughly in the literature with N-body techniques). The calibration is done by choosing appropriate free code parameters. We describe in particular the evolution of the binary, white dwarf and blue straggler populations, though not all channels for blue straggler formation are represented yet in our simulations. Calibrated Monte Carlo runs show good agreement with results of N-body simulations not only for global cluster parameters, but also for e.g. binary fraction, luminosity function and surface brightness. Comparison of Monte Carlo simulations with observational data for M67 shows that is possible to get reasonably good agreement between them. Unfortunately, because of the large statistical fluctuations of the numerical data and uncertainties in the observational data the inferred conclusions about the cluster initial conditions are not firm., Comment: 15 pages, 24 figures

Astronomers have constructed models of globular clusters for over 100 years. These models mainly fall into two categories: (i) static models, such as King's model and its variants, and (ii) evolutionary models. Most attention has been given to static models, which are used to estimate mass-to-light ratios and mass segregation, and to combine data from proper motions and radial velocities. Evolutionary models have been developed for a few objects using the gaseous model, the Fokker-Planck model, Monte Carlo models and N-body models. These models have had a significant role in the search for massive black holes in globular clusters, for example. In this presentation the problems associated with these various techniques will be summarised, and then we shall describe new work with Giersz's Monte Carlo code, which has been enhanced recently to include the stellar evolution of single and binary stars. We describe in particular recent attempts to model the nearby globular cluster M4, including predictions on the spatial distribution of binary stars and their semi-major axis distribution, to illustrate the effects of about 12 Gyr of dynamical evolution. We also discuss work on an approximate way of predicting the "initial" conditions for such modelling., Comment: 10 pages,9 figures,invited review, to appear in the proceedings of IAU Symposium 246, 'Dynamical Evolution of Dense Stellar Systems', eds. E. Vesperini, M. Giersz, A. Sills, Capri, Sept. 2007. Sent to the editors 9 October 2007

We outline the steps needed in to calibrate the Monte Carlo code in order to perform large scale simulations of real globular clusters. We calibrate the results against $N$-body simulations for $N = 2500$, 10000 and for the old open cluster M67. The calibration is done by choosing appropriate free code parameters., Comment: 5 pages, 8 figures, to appear in the proceedings of the 246th IAU symposium on 'Dynamical Evolution of Dense Stellar Systems', eds. E. Vesperini, M. Giersz, A. Sills, Capri, Sept. 2007

Black holes and neutron stars present extreme forms of matter that cannot be created as such in a laboratory on Earth. Instead, we have to observe and analyze the experiments that are ongoing in the Universe. The most telling observations of black holes and neutron stars come from dense stellar systems, where stars are crowded close enough to each other to undergo frequent interactions. It is the interplay between black holes, neutron stars and other objects in a dense environment that allows us to use observations to draw firm conclusions about the properties of these extreme forms of matter, through comparisons with simulations. The art of modeling dense stellar systems through computer simulations forms the main topic of this review., Comment: 23 pages, to appear in Prog. Theor. Phys

Though about 80 pulsar binaries have been detected in globular clusters so far, no pulsar has been found in a triple system in which all three objects are of comparable mass. Here we present predictions for the abundance of such triple systems, and for the most likely characteristics of these systems. Our predictions are based on an extensive set of more than 500 direct simulations of star clusters with primordial binaries, and a number of additional runs containing primordial triples. Our simulations employ a number N_{tot} of equal mass stars from N_{tot}=512 to N_{tot}=19661 and a primordial binary fraction from 0-50%. In addition, we validate our results against simulations with N=19661 that include a mass spectrum with a turn-off mass at 0.8 M_{sun}, appropriate to describe the old stellar populations of galactic globular clusters. Based on our simulations, we expect that typical triple abundances in the core of a dense cluster are two orders of magnitude lower than the binary abundances, which in itself already suggests that we don't have to wait too long for the first comparable-mass triple with a pulsar to be detected., Comment: 11 pages, minor changes to match MNRAS accepted version

We study the motion of stars in a star cluster which revolves in a circular orbit about its parent galaxy. The star cluster is modelled as an ellipsoid of uniform spatial density. We exhibit two 2-parameter families of self-consistent equilibrium models in which the velocity at each point is confined to a line in velocity space. We exhibit the link between this problem and that of a uniform rotating ellipsoidal galaxy. With minimal adaptation, Freeman's bar models yield a third family., Comment: 4 pages, 5 figures, accepted by MNRAS

At least 10-15% of nearby sun-like stars have known Jupiter-mass planets. In contrast, very few planets are found in mature open and globular clusters such as the Hyades and 47 Tuc. We explore here the possibility that this dichotomy is due to the post-formation disruption of planetary systems associated with the stellar encounters in long-lived clusters. One supporting piece of evidence for this scenario is the discovery of freely floating low-mass objects in star forming regions. We use two independent numerical approaches, a hybrid Monte Carlo and a direct $N$-body method, to simulate the impact of the encounters. We show that the results of numerical simulations are in reasonable agreement with analytical determinations in the adiabatic and impulsive limits. They indicate that distant stellar encounters generally do not significantly modify the compact and nearly circular orbits. However, moderately close stellar encounters, which are likely to occur in dense clusters, can excite planets' orbital eccentricity and induce dynamical instability in systems which are closely packed with multiple planets. We compute effective cross sections for the dissolution of planetary systems and show that, for all initial eccentricities, dissolution occurs on time scales which are longer than the dispersion of small stellar associations, but shorter than the age of typical open and globular clusters. Although it is much more difficult to disrupt short-period planets, close encounters can excite modest eccentricity among them, such that subsequent tidal dissipation leads to orbital decay, tidal inflation, and even disruption of the close-in planets., Comment: 57 pages, 14 figures, 4 tables, major revision by authors, accepted for publication at the Astrophysical Journal (ApJ)

(Simplified) We present a theoretical framework which establishes how the core radius of a star cluster varies with the mass of an assumed central black hole. Our result is that the ratio of core to half-mass radius varies as the 3/4 power of the ratio of the black hole to cluster mass, when the system is well relaxed. The theory compares favourably with a number of simulations of this problem, which extend to black hole masses of order 10% of the cluster mass. Though strictly limited as yet to clusters with stars of equal mass, our conclusion strengthens the view that clusters with large core radii are the most promising candidates in which to find a massive black hole., Comment: 4 pages, 2 figures, to appear in PASJ Letters, PASJ Vol. 59, No. 3. (to be issued on June 25, 2007). Minor revisions

We describe a simple computer package which illustrates a method of combining stellar dynamics with stellar evolution. Though the method is intended for elaborate applications (especially the dynamical evolution of rich star clusters) it is illustrated here in the context of three-body scattering, i.e. interactions between a binary star and a field of single stars. We describe the interface between the dynamics and the two independent packages which describe the internal evolution of single stars and binaries. We also give an example application, and introduce a stand alone utility for the visual presentation of simulation results., Comment: 16 pages, Accepted for publication in New Astronomy. Source codes available at: http://manybody.org/manybody/McScatter.html and http://www.manybody.org/manybody/roche.html

In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behavior of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, with values of N ranging from 256 to 16384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters. In this range, we find that the steady-state `deuterium main sequence' is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff (1994), although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10%, the binary heating in the core during the post collapse phase almost saturates (becoming nearly independent of f), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker-Planck simulations by Gao et al. (1991), this is of little dynamical consequence. At variance with the study of Gao et al. (1991), we see no sign of gravothermal oscillations before 150 halfmass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations., Comment: 14 pages, 22 figures, MNRAS accepted

[abridged] We extend our analysis of the dynamical evolution of simple star cluster models, in order to provide comparison standards that will aid in interpreting the results of more complex realistic simulations. We augment our previous primordial-binary simulations by introducing a tidal field, and starting with King models of different central concentrations. We present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, and N values from 512 to 16384. We also attempt to extrapolate some of our results to the larger number of particles that are necessary to model globular clusters. We characterize the steady-state `deuterium main sequence' phase in which primordial binaries are depleted in the core in the process of `gravitationally burning'. In this phase we find that the ratio of the core to half-mass radius, r_c/r_h, is similar to that measured for isolated systems. In addition to the generation of energy due to hardening and depletion of the primordial binary population, the overall evolution of the star clusters is driven by a competing process: the tidal disruption of the system. We find that the depletion of primordial binaries before tidal dissolution of the system is possible only if the initial number is below 0.05 N, in the case of a King model with W_0=7 and N=4096 (which is one of our longest living models). We compare our findings, obtained by means of direct N-body simulations but scaled, where possible, to larger N, with similar studies carried out by means of Monte Carlo methods., Comment: 15 pages, 18 figures, matches MNRAS accepted version, some sections reorganized but no major changes

We review some modern applications of the theory of few-body encounters between binaries and single stars. Then we focus on a new treatment of adiabatic encounters, in a regime which is of importance in encounters between a star and a planetary system in a star cluster., Comment: 8 pages, 2 figures. To appear in Few-Body Problem, Annales Universitatis Turkuensis, C. Flynn, ed. Small correction to Fig.1

Let a number, N, of particles interact classically through Newton's Laws of Motion and Newton's inverse square Law of Gravitation. The resulting equations of motion provide an approximate mathematical model with numerous applications in astrophysics, including the motion of the moon and other bodies in the Solar System (planets, asteroids, comets and meteor particles); stars in stellar systems ranging from binary and other multiple stars to star clusters and galaxies; and the motion of dark matter particles in cosmology. For N=1 and N=2 the equations can be solved analytically. The case N=3 provides one of the richest of all unsolved dynamical problems -- the general three-body problem. For problems dominated by one massive body, as in many planetary problems, approximate methods based on perturbation expansions have been developed. In stellar dynamics, astrophysicists have developed numerous numerical and theoretical approaches to the problem for larger values of N, including treatments based on the Boltzmann equation and the Fokker-Planck equation; such N-body systems can also be modelled as self-gravitating gases, and thermodynamic insights underpin much of our qualitative understanding., Comment: 29 pages, 4 figures, to appear in Encyclopaedia of Mathematical Physics, Elsevier, 2006; proof corrections and one new reference inserted

We point out a strong time-evolution of the mass-to-light conversion factor \eta commonly used to estimate masses of dense star clusters from observed cluster radii and stellar velocity dispersions. We use a gas-dynamical model coupled with the Cambridge stellar evolution tracks to compute line-of-sight velocity dispersions and half-light radii weighted by the luminosity. Stars at birth are assumed to follow the Salpeter mass function in the range [0.15--17 M_\sun]. We find that $\eta$, and hence the estimated cluster mass, increases by factors as large as 3 over time-scales of 20 million years. Increasing the upper mass limit to $50 M_\sun$ leads to a sharp rise of similar amplitude but in as little as 10 million years. Fitting truncated isothermal (Michie-King) models to the projected light profile leads to over-estimates of the concentration par ameter c of $\delta c\approx 0.3$ compared to the same functional fit applied to the proj ected mass density., Comment: Draft version of an ApJ letter

Star clusters and dwarf galaxies gradually dissolve as they move in the potential of their host galaxy. Once their density falls below a certain critical density (which is comparable with the background density of the galaxy) it is often assumed that their evolution is completed. In fact the remnant of such a system forms a distribution of stars which are unbound to each other and which move on similar orbits in their host potential. With this motivation we study the evolution of an idealised unbound system and follow its expansion and dissolution in the tidal field of a model galaxy. Initially the stars are uniformly distributed (with a density below the critical density) within an ellipsoidal volume. The system itself travels on a circular orbit within a galaxy modelled as an isothermal sphere. The initial velocities of the stars are chosen by assuming that they move on (three-dimensional) epicycles with guiding centre at the centre of the ellipsoid, though the usual epicyclic theory is altered to account for the self-gravity of the system. This is believed to be the first exact equilibrium model of a stellar system in a tidal field. Our main task is to study the stability of such configurations and the time-scale of their dissolution, as a function of the initial density and size of the ellipsoid. If the time of dissolution is measured by an increase of the half-mass radius of 50 %, we find that systems of low density (about 1 % of the background density) and small radius (50 pc on an orbit of radius 10 kpc) can survive for about 20 galactic orbits. For small systems we show that the lifetime is approximately proportional to the inverse square root of the density., Comment: 14 pages, 7 figures, accepted by A&A

This is a summary paper of MODEST-2, a workshop held at the Astronomical Institute ``Anton Pannekoek'' in Amsterdam, 16-17 December 2002. MODEST is a loose collaboration of people interested in MOdelling DEnse STellar systems, particularly those interested in modelling these systems using all the available physics (stellar dynamics, stellar evolution, hydrodynamics and the interplay between the three) by defining interfaces between different codes. In this paper, we summarize 1) the main advances in this endeavour since MODEST-1; 2) the main science goals which can be and should be addressed by these types of simulations; and 3) the most pressing theoretical and modelling advances that we identified., Comment: Accepted by New Astronomy

This paper considers the phenomenon of deep core collapse in collisional stellar systems, with stars of equal mass. The collapse takes place on some multiple, $\xi ^{-1}$, of the central relaxation time, and produces a density profile in which $\rho \propto r^{-\alpha}$, where $\alpha$ is a constant. The parameters $\alpha$ and $\xi$ have usually been determined from simplified models, such as gas and Fokker-Planck models, often with the simplification of isotropy. Here we determine the parameters directly from N-body simulations carried out using the newly completed GRAPE-6., Comment: 5 pages, 3 figures, accepted by MNRAS. Reference added to Table 1

We construct a simple dynamical model of the massive globular cluster Omega Cen. The model includes simple treatments of dynamical evolution in a galactic tide, and evolution of single stars. Binary stars and rotation are neglected. The model approximately fits observational data on the surface brightness profile, the profile of radial velocity dispersion, and the main sequence mass function at two radii., Comment: 6 pages, 1 figure, MNRAS, accepted; uses mn2e style

We report results of N-body simulations of isolated star clusters, performed up to the point where the clusters are nearly completely dissolved. Our main focus is on the post-collapse evolution of these clusters. We find that after core collapse, isolated clusters evolve along nearly a single sequence of models whose properties are independent of the initial density profile and particle number. Due to the slower expansion of high-N clusters, relaxation times become almost independent of the particle number after several core collapse times, at least for the particle range of our study. As a result, the dissolution times of isolated clusters exhibit a surprisingly weak dependence on N. We find that most stars escape due to encounters between single stars inside the half-mass radius of the cluster. Encounters with binaries take place mostly in the cluster core and account for roughly 15% of all escapers. Encounters between single stars at intermediate radii are also responsible for the build up of a radial anisotropic velocity distribution in the halo. For clusters undergoing core oscillations, escape due to binary stars is efficient only when the cluster center is in a contracted phase. Our simulations show that it takes about 10^5 N-body time units until the global anisotropy reaches its maximum value. The anisotropy increases with particle number and it seems conceivable that isolated star clusters become vulnerable to radial orbit instabilities for large enough N. However, no indication for the onset of such instabilities was seen in our runs., Comment: 14 pages, 20 figures, MNRAS in press, V2: Order of authors changed in author-field

We review what has been learned recently using N-body simulations about the evolution of globular clusters. While simulations of star clusters have become more realistic, and now include the evolution of single and binary stars, the prospect of reaching large enough N is still a distant one. Nevertheless more restricted kinds of simulations have recently brought valuable progress for certain problems of current observational interest, including the origin and structure of tidal tails of globular clusters. In addition, such simulations have forced us to rethink some basic aspects of stellar dynamics, including, in particular, the process of escape. Finally we turn to faster, approximate methods for studying star cluster dynamics, where the role of N-body simulations is one of calibration., Comment: 12 pages, 5 figures, to appear in Proc. IAU Symp. 208, Astrophysical Supercomputing using Particle Simulations, Tokyo (Japan), July 10-13, 2001; eds. J. Makino and P. Hut, ASP, forthcoming. Two figures are at low resolution in this version

One of the fundamental aspects of statistical behaviour in many-body systems is exponential divergence of neighbouring orbits, which is often discussed in terms of Liapounov exponents. Here we study this topic for the classical gravitational N-body problem. The application we have in mind is to old stellar systems such as globular star clusters, where N~10^6, and so we concentrate on spherical, centrally concentrated systems with total energy E<0. Hitherto no connection has been made between the time scale for divergence (denoted here by t_e) and the two-body relaxation time scale (t_r), even though both may be calculated by consideration of two-body encounters. In this paper we give a simplified model showing that divergence in phase space is initially roughly exponential, on a timescale proportional to the crossing time (defined as a mean time for a star to cross from one side of the system to another). In this phase t_e is much less than t_r, if N is not too small (i.e. N is much more than about 30). After several e-folding times, the model shows that the divergence slows down. Thereafter the divergence (measured by the energies of the bodies) varies with time as t^{1/2}, on a timescale nearly proportional to the familiar two-body relaxation timescale, i.e. t_e ~ t_r in this phase. These conclusions are illustrated by numerical results., Comment: 19 pages, 3 figures, accepted by J. Stat. Phys; new figures, numerical results and references; improved text; fig.1 at low resolution in this version

We summarise the specification and some previously unpublished results of the collaborative experiment (``KyotoI'') that was carried out at the time of the Kyoto General Assembly of the IAU. As the subject has advanced considerably since then, and codes have become more elaborate, we describe an agreed proposal for a new experiment (``KyotoII'')., Comment: 10 pages, 6 figures, to appear in "Astrophysical Supercomputing Using Particles", eds J. Makino and P. Hut, Proc. IAU Symposium 208, Tokyo, July 10-13, 2001

This didactic paper is motivated by the problem of understanding how stars escape from globular star clusters. One formulation of this problem is known, in dynamical astronomy, as Hill's problem. Originally intended as a model for the motion of the moon around the earth with perturbations by the sun, with simple modifications it also serves as a model for the motion of a star in a star cluster with perturbations by the galaxy. The paper includes introductory sections on the derivation of the equations of motion of Hill's problem, their elementary properties, and extensions to deal with non-point masses and non-circular orbits. We then show how the rate of escape may be calculated numerically and estimated theoretically, and discuss how this simple picture is modified if the stars in a cluster are also undergoing two-body relaxation. Finally we introduce some established ideas for obtaining the distribution of escape times., Comment: 22 pages, 11 figures, to appear in Steves B.A., Maciejewski A., eds, "The Restless Universe: Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galactic Systems", Institute of Physics Publishing, Bristol, 2001

We consider both observational and theoretical issues related to dynamical mass loss from the old globular star clusters of the Galaxy. On the observational side the evidence includes tidal tails and extratidal extensions, kinematic effects, effects on the mass function, and influences on the statistical properties of surviving objects. Even for isolated clusters, the theoretical issues are not fully understood. The effects of a steady tide (i.e. for a cluster in a circular orbit) include the imposition of a tidal boundary, and lowering of the escape energy. Less familiar, however, are the effects of induced mass loss. Even the definition of an ``escaper'' is not straightforward. When mass loss is driven by relaxation, as in N-body models, the rate of loss of mass does not scale in a simple way with the relaxation time. Reasons for this include the very long time scales on which stars escape even with energies above the escape threshold. For the realistic case of unsteady tides it is still unclear under what circumstances mass loss is dominated by relaxation or tidal heating., Comment: 12 pages, 6 figures; to appear in Dynamics of Star Clusters and the Milky Way, eds S. Deiters, B. Fuchs, A. Just, R. Spurzem, R. Wielen; ASP Conf. Ser

In this paper a cluster is modelled as a smooth potential (due to the cluster stars) plus the steady tidal field of the Galaxy. In this model there is a minimum energy below which stars cannot escape. Above this energy, however, the time scale on which stars escape varies with the orbital parameters of the star (mainly its energy) in a way which we attempt to quantify, with both theoretical arguments and computer simulations. Within the limitations of the model we show that the time scale is long enough to complicate the interpretation of full N-body simulations of clusters, and that stars above the escape energy may remain bound to the cluster for about a Hubble time., Comment: 15 pages, 11 figures, plain TeX, submitted to MNRAS

We consider the use of N-body simulations for studying the evolution of rich star clusters (i.e. globular clusters). The dynamical processes included in this study are restricted to gravitational (point-mass) interactions, the steady tidal field of a galaxy, and instantaneous mass loss resulting from stellar evolution. With evolution driven by these mechanisms, it is known that clusters fall roughly into two broad classes: those that dissipate promptly in the tidal field, as a result of mass loss, and those that survive long enough for their evolution to become dominated by two-body relaxation. The time scales of the processes we consider scale in different ways with the number of stars in the simulation, and the main aim of the paper is to suggest how the scaling of a simulation should be done so that the results are representative of the evolution of a `real' cluster. We investigate three different ways of scaling time. One of these is appropriate to the first type of cluster, i.e. those that dissipate rapidly, and similarly a second scaling is appropriate only to the second (relaxation-dominated) type. We also develop a hybrid scaling which is a satisfactory compromise for both types of cluster. Finally we present evidence that the widely used Fokker-Planck method produces models which are in good agreement with N-body models of those clusters which are relaxation dominated, at least for N-body models with several thousand particles, but that the Fokker-Planck models evolve too fast for clusters which dissipate promptly., Comment: 24 pages, 3 figures, MNRAS, in press

We review recent progress in computational methods for studying the dynamical evolution of large star clusters. After describing some highlights of a new comparison of different models, including N-body models, we mention in detail recent developments in three codes which allow models with anisotropic distributions of velocity to be studied: (i) a Fokker-Planck code using finite differences, (ii) a Monte Carlo code and (iii) a gaseous model code., Comment: 6 pages, postscript, etc., also available at http://www.maths.ed.ac.uk/people/douglas/preprints.html To appear in Andersen J., ed, Highlights of Astronomy Vol. 11. Kluwer, Dordrecht. Pagination differs from that in published version

The simulation of rich star clusters presents challenging problems of several kinds, including the design of suitable hardware and software, and numerous theoretical problems in stellar dynamics and stellar physics. Great progress has been made possible in recent years through the widespread use of GRAPE hardware. Simulations are, however, still too small to be applied to real star clusters without scaling. How this is done is partly an issue of stellar dynamics, and it has thrown into focus a number of fundamental theoretical problems in this field., Comment: 10 pages, postscript, also available at http://www.maths.ed.ac.uk/people/douglas/preprints.html To appear in T. Ebisuzaki, ed, Proc. Int. Symp. on Supercomputing - New Horizon of Computational Science, Tokyo, Japan, 1-3 September, 1997. Kluwer, Dordrecht

In this paper we show the results of a large set of N-body simulations modelling the evolution of globular clusters driven by relaxation,stellar evolution,disk shocking and including the effects of the tidal field of the Galaxy. We investigate the evolution of multi-mass models with a power-law initial mass function (IMF) starting with different initial masses, concentrations, slopes of the IMF and located at different galactocentric distances. We show to what extent the effects of the various evolutionary processes alter the shape of the IMF and to what extent these changes depend on the position of the cluster in the Galaxy. Both the changes in the global mass function and in the local one (measured at different distances from the cluster center) are investigated showing whether and where the local mass function keeps memory of the IMF and where it provides a good indication of the current global mass function. The evolution of the population of white dwarfs is also followed in detail and we supply an estimate of the fraction of the current value of the total mass expected to be in white dwarfs depending on the main initial conditions for the cluster (mass and position in the Galaxy).Simple analytical expression by which it is possible to calculate the main quantities of interest (total mass, fraction of white dwarfs, slope of the mass function) at any time t for a larger number of different initial conditions than those investigated numerically have been derived., Comment: 20 pages LaTeX (MNRAS style), 23 figures, MNRAS (in press)

Galactic globular clusters, which are ancient building blocks of our Galaxy, represent a very interesting family of stellar systems in which some fundamental dynamical processes have taken place on time scales shorter than the age of the universe. In contrast with galaxies, these clusters represent unique laboratories for learning about two-body relaxation, mass segregation from equipartition of energy, stellar collisions, stellar mergers, and core collapse. In the present review, we summarize the tremendous developments, as much theoretical as observational, that have taken place during the last two decades, and which have led to a quantum jump in our understanding of these beautiful dynamical systems., Comment: A review of 171 pages (TeX, no special macros) in press in The Astronomy and Astrophysics Review. Only 7 figures available in postscript form

We present the first comprehensive fitting formula for exchange reactions of arbitrary mass ratios. In a comparison with numerical results, this expression is shown to be accurate in the hard binary limit to within 25\% for most mass ratios. The result will be useful in forming quantitative estimates for the branching ratios of various exchange reactions in astrophysical applications. For example, it can be used to construct quantitative formation scenarios for unusual objects in globular clusters, such as binaries containing a pulsar., Comment: 25 pages, TeX + 2 ps-figures, to appear in ApJ

We present results on the exchange cross section for the interaction between a hard binary and a field of single stars, for arbitrary masses. The results are based partly on extensive numerical scattering experiments, and partly on analytic estimates of the mass-dependence of the cross section. They can be used to estimate the rate of exchange in an arbitrary mixture of masses, provided that the binary is hard., Comment: LaTeX, 2 pages (LaTeX, crckapb.sty), to appear in the proceedings of IAU Symposium 174, Dynamical Evolution of Star Clusters, eds. P. Hut and J. Makino (Dordrecht: Kluwer)

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., Comment: 10 pages uuencoded compressed postscript; to appear in P. Hut and J. Makino, eds, Dynamical Evolution of Star Clusters - Confrontation of Theory and Observations, Proc. IAU Symposium 174

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., Comment: 10 pages in uuencoded compressed postscript; to appear in P. Hut and J. Makino, eds, Dynamical Evolution of Star Clusters - Confrontation of Theory and Observations, Proc. IAU Symposium 174

We describe results from large numbers of $N$-body simulations containing from $250$ to $1000$ stars each. The distribution of stellar masses is a power law, and the systems are isolated. While the collapse of the core exhibits the expected segregation of different masses, we find that the post-collapse evolution is, at a first approximation, homologous. This is quite surprising because there is no reason for supposing that mass segregation should not continue to have a substantial effect on the evolution of the cluster. In fact the spatial distribution of the mean stellar mass is nearly static throughout the post-collapse regime, except for the overall expansion of the systems, and this helps to explain why the post-collapse evolution is nearly self-similar. Self-similarity is also exhibited by the distribution of anisotropy and the profile of departures from equipartition, which show little change during the post-collapse phase. The departures from energy equipartition and isotropy are small in the core and increase with radius. During post-collapse evolution massive stars (mainly) are removed from the system by binary activity. This effect dominates the preferential escape of low-mass stars due to standard two-body relaxation processes., Comment: 20 pages, plain tex, figures available from d.c.heggie@ed.ac.uk; submitted to MNRAS

We derive analytical expressions for the change in the orbital eccentricity of a binary following a distant encounter with a third star on a hyperbolic or parabolic orbit. To establish the accuracy of these expressions, we present detailed comparisons with the results of direct numerical integrations of the equations of motion for the three bodies. We treat with particular care the difficult case of a binary with zero initial eccentricity. In this case, we show that the eccentricity $\delta e$ induced by the encounter declines in general as a power-law, $\delta e\propto (a/\rp)^{5/2}$, where $a$ is the binary semi-major axis and $\rp$ is the periastron distance of the encounter. This power-law arises from the octupole-level secular perturbation of the binary. In contrast, non-secular quadrupole-level perturbations induce an eccentricity change that declines exponentially with $\rp$. These non-secular effects can become dominant at sufficiently small $\rp$, for a sufficiently high relative velocity, or for a sufficiently massive perturber. We also derive cross sections for eccentricity change and compare our results with those of previous studies based on numerical scattering experiments. Our results have important implications for a number of astrophysical problems including, in particular, the evolution of binary millisecond pulsars in globular clusters., Comment: final version with minor revisions, uses MNRAS TeX macros, 23 pages, to appear in MNRAS, also available from http://ensor.mit.edu/~rasio/papers/

Low-mass binary millisecond pulsars (LMBPs) are born with very small orbital eccentricities, typically of order $e_i\sim10^{-6}$--$10^{-3}$. In globular clusters, however, higher eccentricities $e_f\gg e_i$ can be induced by dynamical interactions with passing stars. Here we show that the cross section for this process is much larger than previously estimated. This is because, even for initially circular binaries, the induced eccentricity $e_f$ for an encounter with pericenter separation $r_{\rm p}$ beyond a few times the binary semi-major axis $a$ declines only as a power-law, $e_f\propto(r_{\rm p}/a)^ {-5/2}$, and {\it not\/} as an exponential. We find that {\it all\/} currently known \lmbps in clusters were probably affected by interactions, with their current eccentricities typically an order of magnitude or more greater than at birth., Comment: uuencoded compressed postscript with figures, 8 pages

This paper is concerned with collisionless aspects of the early evolution of model star clusters. The effects of mass loss through stellar evolution and of a steady tidal field are modelled using $N$-body simulations. Our results (which depend on the assumed initial structure and the mass spectrum) agree qualitatively with those of Chernoff \& Weinberg (1990), who used a Fokker-Planck model with a spherically symmetric tidal cutoff. For those systems which are disrupted, the lifetime to disruption generally exceeds that found by Chernoff \& Weinberg, sometimes by as much as an order of magnitude. Because we do not model collisional effects correctly we cannot establish the fate of the survivors. In terms of theoretical interpretation, we find that tidal disruption must be understood as a loss of {\sl equilibrium}, and not a loss of {\sl stability}, as is sometimes stated., Comment: 23pp. Plain TeX, and 13 illustrations available on request from d.c.heggie@ed.ac.uk, submitted to MNRAS

This paper presents and analyses statistical results from a large number of $N$-body simulations of isolated systems with equal masses, in which $250\le N\le 2000$. It concentrates on the phase starting around the end of core collapse, when binaries start to play a crucial role. Interactions of hard binaries are consistent with the Spitzer (1987) cross section. The evolution of the half-mass radius after core collapse nearly follows classical theory, The evolution of the inner parts of the system around the time of core bounce is consistent with these simplified models provided that the continuous production of energy, as usually assumed, is replaced by a model of {\sl stochastic} energy production. Similarly, post-collapse evolution of the core requires a modest recalibration of the coefficient of energy generation, especially for small $N$. The distributions of velocity dispersion and anisotropy become remarkably homologous soon after core bounce. The bound mass of the systems very nearly follows a power law with time. A small number of escapers, presumed to be those associated with binary activity, dominate the energy which is carried off. The ``internal" energy of escaping binaries is consistent with theoretical expectations., Comment: 32pp and 21 figures

Gravothermal oscillations occur in several models for the post-collapse evolution of rich star clusters. This paper reviews the main literature, and presents some new N-body results which appear to exhibit this phenomenon., Comment: E-mail: d.c.heggie@ed.ac.uk; 11 pages

This paper describes the numerical evolution of an $N$-body system with a slight ``temperature inversion''; i.e. the maximum velocity dispersion occurs not at the centre but further out. Fluid models predict that the core of such a system expands on a time-scale of thousands of central relaxation times, and here this behaviour is qualitatively confirmed for an $N$-body system of over 3000 bodies. With certain qualifications, this demonstrates the existence in N-body systems of one of the fundamental mechanisms which, in fluid models, drive the gravothermal oscillations discovered by Bettwieser & Sugimoto., Comment: 25pp and 12 figures (available from d.c.heggie@ed.ac.uk), tex

This review concentrates almost entirely on globular star clusters. It emphasises the increasing realisation that few of the traditional problems of star cluster astronomy can be studied in isolation: the influence of the Galaxy affects dynamical evolution deep in the core, and the spectrum of stellar masses; in turn the evolution of the core determines the highest stellar densities, and the rate of encounters. In this way external tidal effects indirectly influence the formation and evolution of blue stragglers, binary pulsars, X-ray sources, etc. More controversially, the stellar density appears to influence the relative distribution of normal stars. In the opposite sense, the evolution of individual stars governs much of the early dynamics of a globular cluster, and the existence of large numbers of primordial binary stars has changed important details of our picture of the dynamical evolution. New computational tools which will become available in the next few years will help dynamical theorists to address these questions., Comment: 10 pages, 3 figures, TeX

We study the dynamical evolution of idealised stellar systems by averaging results from many $N$-body simulations, each having modest numbers of stars. For isolated systems with stars of uniform mass, we discuss aspects of evolution up to the point of core collapse: relaxation and its $N$-dependence, the evolution of the density profile, the development of the velocity dispersion and anisotropy, and the rate of stellar escape., Comment: 24 pages

This paper generalises King's models for tidally truncated star clusters by including approximately the non-spherical symmetry of the tidal field and the resulting non-spherical distortion of the cluster., Comment: 22 pages