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Astron. Astrophys. 337, 363-371 (1998)

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1. Introduction

Theoretical models for the evolution of star clusters are generally too idealized for comparison with observations. However, detailed model calculations with direct N-body methods are not feasible for real globular clusters, even with fast special-purpose computers such as GRAPE-4 (Makino et al. 1997) or advanced parallel computers (Spurzem & Aarseth 1996).

If we could scale the results of N-body simulations with relatively small numbers of particles (such as [FORMULA]) to real globular clusters, then it would become feasible to perform computations with relatively small numbers of particles and still derive useful qualitative conclusions about larger, more massive systems. However, to determine the proper scaling is difficult because the ratio between two fundamental time scales, the relaxation times and the dynamical time, is proportional to N. In typical globular clusters, this ratio exceeds [FORMULA] and the two time scales are well separated. In N-body simulations, the ratio is generally much smaller.

The inclusion of realistic effects such as mass loss due to stellar evolution and the effect of galactic tidal fields (with the galaxy approximated as a point mass, but also with the inclusion of disc shocking) further complicate the scaling problem (see, e.g., Heggie 1996). A proper treatment of stellar evolution is particularly problematic, since its characteristic timescale changes as stars evolve.

Chernoff & Weinberg (1990, CW90) performed an extensive study of the survival of star clusters using Fokker-Planck calculations which included 2-body relaxation and some rudimentary form of mass loss from the evolving stellar population. In their simulations the number of particles is not specified. Their models are defined by the initial half-mass relaxation time and by the initial mass function of the cluster. Since their models do not specify the number of stars per cluster, each of their model calculations corresponds to a one-dimensional series of models, when plotted in a plane of observational values, such as total mass versus distance to the galactic center (Fig. 1). All points of the solid line in that figure correspond to a single calculations by CW90, since they have an identical relaxation time. As we will see later, it is useful to consider other series of models, for which the crossing time is held constant while varying the mass. An example of such a series is indicated by the dashed line in Fig. 1. The shapes of these lines are derived under the assumption of a flat rotation curve for the parent galaxy.

[FIGURE] Fig. 1. Cluster mass versus the distance to the galactic center. The solid line indicates the model parameters for which the relaxation time is constant (iso relaxation time); the dashed line indicates the initial conditions for which the crossing time of the star cluster is constant (iso crossing time)

The main conclusion of CW90 was that the majority of the simulated star clusters dissolve in the tidal field of the galaxy within a few hundred million years. Fukushige & Heggie (1995, FH95) studied the evolution of globular clusters using direct N-body simulation, using the same stellar evolution model as used by CW90. They used a maximum of 16k particles and a scaling in which the dynamical timescale of the simulated cluster was the same as that of a typical globular cluster, corresponding to one of vertical lines in Fig. 1.

FH95 found lifetimes much longer than those in CW90's Fokker-Planck calculations, for the majority of the models used in CW90. However, the reason for the discrepancy is rather unclear, because the calculations of FH95 and those of CW90 differ in several important respects. The relaxation times differ because FH95 held the cluster crossing time fixed in scaling from the model to the real system. However, the crossing times themselves are also different, since the crossing time is by definition zero in a Fokker-Planck calculation. Finally, the implementation of the galactic tidal field is also quite different. CW90 used a simple boundary condition in energy space (spherically symmetric in physical space), in which stars were removed once they acquired positive energy, but the underlying equations of motion included no tidal term. FH95 adopted a much more physically correct treatment, including tidal acceleration terms in the stellar equations of motion and a proper treatment of centrifugal and coriolis forces in the cluster's rotating frame of reference (see FH95).

In order to study the behavior of star clusters with limited numbers of stars, and to compare with the results of the Fokker-Planck simulations of CW90, we selected one of their models and perform a series of collisional N-body simulations in which the evolution of the individual stars is taken into account. According to CW90 the results should not depend on the number of stars in the simulation as long as the relaxation time is taken to be the same for all models. It is, among others, this statement which we intend to study. We find that for this set of initial conditions Fokker-Planck models do not provide a qualitatively correct picture of the evolution of star clusters. The effects of the finite dynamical time scale are large, even for models whose lifetime is several hundred times longer than the dynamical time.

The main purpose of this paper is to study the survival probabilities of star clusters containing up to a few tens of thousands of single stars, in order to gain a deeper understanding of the influence of the galactic tidal field and the fundamental scaling of small N clusters to larger systems. Only single stars are followed; primordial binaries are not included. The computation of gravitational forces is performed using the GRAPE-4 (GRAvity PipE, see (Ebisuzaki et al., 1993), a special-purpose computer for the integration of large collisional N-body systems). Hardware limitations (speed as well as storage) restrict our studies to [FORMULA]k particles.

The dynamical model, stellar evolution, initial conditions and scaling are discussed in Sect. 2. Sect. 3 reviews the software environment and the GRAPE-4 hardware, and discusses the numerical methods used. The results are presented in Sect. 4 and discussed in Sect. 5; Sect. 6 sums up.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998