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Astron. Astrophys. 328, 130-142 (1997)

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2. Star cluster ecology

Stellar evolution plays a role in star cluster evolution similar to the role played by nuclear physics in stellar evolution. In both cases, the microphysical processes play a crucial role in the mechanism of energy generation in the central parts of the system under consideration, a mechanism that tries to balance the energy losses at the outskirts (tidal radius and photosphere, respectively).

In the next two subsections we look separately at the different forms of physics input necessary to follow the evolution of a star cluster. The third subsection then discusses their interconnection. For future reference, initial conditions are discussed in the fourth subsection, while the last subsection provides a brief outline of the series of papers of which this one is the first.

2.1. Stellar dynamics simulations

Great progress has been made in the study of star cluster dynamics, using various approximate methods in which the stars have been treated like a form of fluid, either three-dimensional as in conducting gas sphere models, or six-dimensional as in Fokker-Planck models. In both cases, the main effect of encounters has been taken into account by a form of effective two-body relaxation. We refer to Hut et al. (1992) for a review of these methods.

Unfortunately, both methods have two intrinsic handicaps that make them unsuitable for a detailed quantitative modeling of the evolution of a globular cluster past core collapse. First, they are not set up to deal with the separate evolution of internal and external degrees of freedom of the binaries that play a central role in the energy generation processes in the cluster.

The second problem stems from an introduction of a mass spectrum, as well as a distinction between stars of different radii, such as dwarfs, main-sequence stars, and giants. The root of the problem here is that a gas sphere or Fokker-Planck approach does not follow individual stars, but rather distribution functions. When the number of independent parameters characterizing the distribution functions becomes too large, there will be less than one star left in a typical cell in parameter space - something that clearly invalidates the statistical hypothesis on which these methods are based.

The only solution is to drop the statistical assumption, and to revert to a star-by-star modeling of a globular cluster, through direct N -body calculations. The draw back of such an approach has long been the prohibitive calculational costs involved, and until recently typical production runs only included a few thousand stars. To extend such numbers to include several hundred thousand stars, characteristic of realistic globular clusters, requires an increase of two orders of magnitude in star number, or a factor million in computational cost, from Gigaflops days to Petaflops days (Hut et al. 1988).

Recently, the number of stars modeled in direct N -body calculations has been increased significantly, to [FORMULA], using the GRAPE-4, a form of special-purpose hardware developed by a group of astrophysicists at Tokyo University, running at a speed of 1 Tflops (Makino 1996a). The first scientific results of the GRAPE-4, including the first convincing evidence of gravothermal oscillations in N -body simulations, predicted by Sugimoto & Bettwieser (1983), have been presented by Makino (1996a, b).

The next, and definitive step that will enable any globular cluster to be modeled realistically might take place as early as the year 2000. If funding can be found, there is no technological obstacle standing in the way of a speedup of the current GRAPE-4 machine by a factor of a thousand, during the next five years. Most of this speed-up will come from further miniaturization, allowing a larger number of gates to be mounted on a single chip, and allowing a higher clock speed as well. A Petaflops machine by the year 2000, allowing simulations of core collapse and post-collapse evolution with up to [FORMULA] particles, is thus a realistic goal.

2.2. Stellar evolution population synthesis

The first serious attempts to understand and simulate the evolution of close binaries were made in the mid thirties (Haffner & Heckmann 1937) and late fifties by Crawford (1955), Kopal (1956) and Huang (1956) followed by Morton (1960) and the standard work in binary evolution from Kippenhahn & Weigert (1967). Synthesis of complete populations of single stars became popular in the mid seventies when Tinsley & Gunn (1976) simulated the giant-branch luminosity functions for giant elliptical galaxies. However, it is only recently that detailed studies simulate complete populations of close binaries starting with Dewey & Cordes (1987) who tried to understand the evolutionary sequence of radio pulsars and the presence of an asymmetry in the velocity distribution of single radio pulsars. In later papers, similar evolutionary scenarios for the formation of binary neutron stars were studied in more detail (Tutukov & Yungelson 1993, Lipunov et al. 1995, Portegies Zwart & Spreeuw 1996 and Lipunov et al. 1996), for high mass X-ray binaries and the supernova rate in the galaxy (Tutukov et al. 1992, Lipunov 1994, Dalton & Sarazin 1995, Portegies Zwart & Verbunt 1996) and for lower mass systems with a neutron star (Webbink & Kalogera 1993, Pols & Marinus 1994) or a white dwarf (de Kool 1990, Kolb & Ritter 1992) as the accreting object. The first couragous attempt to combine stellar and binary evolution within the collisional evironment of a globular cluster was performed by Sutantyo (1975), followed more recently by Di Stefano & Rappaport (1992), Sigurdsson & Phinney (1993), Leonard (1994), Davies & Benz (1995) and Davies (1995).

2.3. Ecological networks

Purely stellar-dynamical calculations often rely on rather severe approximations, such as a representation of stars by equal-mass point-masses. And there is a good reason for doing so, since any single deviation from that simple recipe requires other deviations as well. Let us look at one example.

As soon as we introduce a mass spectrum in a star cluster simulation, we will see that the heavier stars start sinking toward the center, on the dynamical friction time scale, shorter than the two-body relaxation time by a factor proportional to the mass ratio of individual heavy stars with respect to that of typical stellar masses. The reason is that relaxation tends toward equipartition of energy, which implies that heavier stars will move more slowly and therefore gather at the bottom of the cluster potential well.

If stars would live forever, there would be a large overconcentration of heavy stars in the core of a star cluster. However, in reality there is an important counter-effect: heavy stars burn up much faster than lighter ones. They may or may not leave degenerate remnants, that may or may not be heavier than the average stellar mass in the cluster (a quantity that decreases in time). Clearly, it would be grossly unrealistic to introduce a mass spectrum without removing most of the mass of the heaviest stars on the time scale of their evolution off the main sequence and across the giant branches.

Another reason for introducing finite life times for stars comes from abandoning the very restrictive point mass model. As soon as we do that, giving our stars a finite radius will give rise to stellar collisions. The heavier stars produced in the collision of two turn-off stars, for example, will burn up on a time scale an order of magnitude smaller than the age of the cluster. Again, we have to take this into account to be consistent, especially since the merger products themselves are prime candidates for further merging collisions.

The need to let many stars shed most of their mass, together with the fact that most of the energy in a globular cluster is locked up in binaries, poses a formidable consistency problem. Since binaries play a central role in cluster dynamics, consistency requires that we follow their complex stellar evolution, which involves mass overflow (which can be stable or unstable, and can take place on dynamical or thermal or nuclear time scales) and the possibility of a phase of common-envelope evolution. On top of all that, we will have to find simple recipes for the hydrodynamic effects occurring in three-body and four-body reactions, and in occasional [FORMULA] reactions, which are bound to occur in dense cluster centers.

To sum up: there does not seem to be a half-way stopping point, at which we can expect to carry out consistent cluster evolution simulations. Either we study the interesting but unrealistic mathematical-physics problem of an equal-mass point particle model, or we opt for a realistic model with some set of stellar-evolution recipes. The main question here is: what is the simplest set that is still consistent?

2.4. Initial conditions

In most stellar dynamics simulations of star clusters, the Plummer model is used as a standard model to specify the initial conditions for the distribution of the point particles. While not very realistic, this choice has had the advantage of making comparisons between different runs, as well as between different approaches, relatively straightforward. Of course, when attempts are made to model particular star clusters, other models have often replaced the Plummer model as a starting point. King models, for example, are already more realistic in that they provide a form of spatial cut-off that can be interpreted as a tidal radius.

For similar reasons, we will use a standard model for our simulations that combine stellar dynamics and stellar evolution. In most cases, the use of our standard model will be mostly for illustrative reasons, to provide a gauge for comparison between our various results, as well as between our results and that of others. For historical reasons, we choose our standard model to be based on a Plummer model for the macroscopic initial star distribution, and a Salpeter model for the initial mass function.

An additional advantage of these simple choices is that they limit the number of free parameters. The Plummer model, for example, contains only one free parameter, N, the number of stars in the system (apart from a choice of mass and length scales, that are irrelevant in the point particle case). In contrast to the Plummer model, our standard model can be expected to form a multi-parameter family. As soon as we abandon the point-mass approach, we have to deal with microscopic as well as macroscopic mass and length scales.

Of these various scales, the macroscopic quantities can be chosen independently, while the microscopic ones can be fixed, statistically, by specifying a mass distribution together with appropriate cut-off masses at the high and low end. In general, an arbitrary functional form for the mass distribution function can lead to an arbitrarily large number of parameters. Interestingly, our standard model definition allows us to limit the total number of free parameters to three.

Starting from the macroscopic side, we can take the total mass M and the half-mass radius [FORMULA] of the Plummer model as our first two free parameters. With a Salpeter choice of powerlaw distribution function, the third free parameter can be chosen in the form of the lower mass cut-off [FORMULA]. The higher-mass cut-off [FORMULA] could be specified independently, but this is not strictly necessary: since the Salpeter distribution function converges at the high-mass end, we can simply parcel out the total mass M over different stellar masses, between star masses of [FORMULA] and [FORMULA], and we will naturally be left with a single most massive star. This procedure is not unrealistic: nature probably limits the number of high-mass stars in medium-size galactic clusters in a similar way.

In fact, we can go even further, and make the following somewhat arbitrary but natural choices: [FORMULA]  pc, [FORMULA]. This leaves only the total mass M to be specified, or equivalently, the total number of stars N. For future convenience, we will refer to this `most standard' model as our reference model. For systems with a few thousand stars, we are dealing with a typical open cluster, with velocity dispersions of order 1 km/s, while for a few hundred thousand stars, we have a reasonable approximation to a globular cluster, for which typical stellar velocities are an order of magnitude higher.

In addition to this standard model, the various papers in this series will also contain the results of more realistic models. However, we will typically provide at least one run from a standard model, in order to provide comparison material for the more detailed models.

2.5. Stepping stones

In the current series of papers, our goal is to provide a series of ecological simulations, based on a flexible stellar dynamics code coupled to a comprehensive set of stellar evolution modules. These modules in turn are based on recipes that govern the behavior of both single star and binary star evolution, as well as interactions between larger numbers of stars.

In order to present results that can be reproduced and critically assessed by other groups, we clearly document the recipes used, as well as their coupling to the dynamics. With this aim, we give a detailed description of our approach in the first few papers in this series, which will form stepping stones towards a full-fledged ecological star cluster evolution code.

The present paper starts off with rather extreme approximations for the stellar dynamics, as well as the stellar evolution parts of our simulations. With respect to the former, we start with a laboratory-type situation, in which we consider a homogeneous distribution of stars, kept constant in time. With respect to the latter, we consider a population of single stars only. Paper II relaxes the second assumption, by introducing a population of primordial binaries, and allowing the formation of new binaries as well (see Portegies Zwart et al. 1997). Later papers will subsequently relax the former assumption, with the ultimate goal of using a self-consistent N -body code.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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