arXiv subjects

Douglas C. Heggie

Publications and source records attributed to Douglas C. Heggie.

At least 55 records · Page 3Linked to original sources

arXiv2006

McScatter: a Simple Three-Body Scattering Package with Stellar Evolution

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.

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arXiv2005

The Classical Gravitational N-Body Problem

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.

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arXiv2003

Parameters of core-collapse

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, $ξ^{-1}$, of the central relaxation time, and produces a density profile in which $ρ\propto r^{-α}$, where $α$ is a constant. The parameters $α$ and $ξ$ 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.

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arXiv2002

A Simple Dynamical Model for Omega Cen

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.

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arXiv2002

Long-term evolution of isolated N-body sytems

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.

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arXiv2001

The Gravitational Million-Body Problem

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.

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arXiv2001

Two collaborative experiments in star cluster evolution

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'').

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arXiv2000

Mass Loss from Globular Clusters

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.

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arXiv1996

The Effect of Encounters on the Eccentricity of Binaries in Clusters

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 $δe$ induced by the encounter declines in general as a power-law, $δe\propto (a/\rp)^{5/2}$, 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.

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arXiv1996

Exchange Cross Sections for Hard Binaries

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.

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arXiv1995

Statistics of Small-N Simulations

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.

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arXiv1995

Dark Matter in Globular Clusters

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.

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arXiv1995

Statistics of N-Body Simulations. III. Unequal Masses

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.

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arXiv1995

THE ORBITAL ECCENTRICITIES OF BINARY MILLISECOND PULSARS IN GLOBULAR CLUSTERS

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.

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arXiv1994

Pre-Collapse Evolution of Galactic Globular Clusters

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.

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arXiv1994

Statistics of N-Body Simulations. II. Equal Masses after Core Collapse

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.

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arXiv1993

Gravothermal Oscillations

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.

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arXiv1993

Gravothermal Expansion in an $N$-Body System

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.

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