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

2. The model

The simulations presented in this paper were performed by a computer code that consists of two independent parts. One part, the N-body model, integrates the equations of motion of all individual stars while the other part computes the evolution of each single star (see Sect. 3for a description of the numerical implementation).

2.1. Stellar dynamics

The equations of motion of the stars in the stellar system are computed using Newtonian gravity. The numerical integration is performed using a fourth-order individual-time-step Hermite scheme (Makino & Aarseth 1992; see Sect. 3 below).

The radius of the star cluster is limited by the galactic tidal field. We assume for simplicity that the cluster describes a circular orbit around the galactic center. At fixed time intervals, stars that are outside the tidal radius are simply removed. We chose this simple cutoff in order to facilitate direct comparison with the Fokker-Planck results. Shocks due to the passage of the cluster through the galactic plane, twice per orbit, and encounters with giant molecular clouds are neglected.

At regular time intervals, the stellar evolution model updates all stars to the current age of the stellar system and the N-body model is notified of the changes in mass and radius of the stars. Whenever a star loses more than 1% of its mass, the dynamical part of the simulation is reinitialized, to take into account the loss of mass and energy from the system. A treatment of collisions and mergers involving two or more stars is also included.

2.2. Stellar evolution

The evolution of single stars in our model is performed using the stellar evolution model presented by Portegies Zwart & Verbunt 1996, and later dubbed (see Portegies Zwart at al. 1996 and 1997 for a dynamical implementation). These models are based on fitting formulae to stellar evolution tracks of population I stars given by Eggleton et al. (1989) (evolutionary tracks for population II stars are not yet available in this convenient form). These equations give a star's luminosity and temperature as functions of age and initial mass. Other stellar parameters-radius, mass loss and the mass of the core-are then derived from these. Fig. 2 gives the total and main-sequence lifetime of the stars in the model as a function of their mass at zero age.

Fig. 2. Main-sequence lifetime (solid line) and terminal age (dashed line) in billions of years for stars as a function of the zero age mass of the stellar evolution tracks of Eggleton et al. (1989).

We point out here a few important details of the evolutionary model: A star with a mass larger than 8  becomes a neutron star in a supernova, less massive stars become white dwarfs. Neutron stars have a mass of 1.34 ; the mass of a white dwarf is given by the core of the star as it leaves the end of the supergiant branch.

Mass lost by stellar evolution, either in the form of a stellar wind or after a supernova, escapes from the stellar system, as the velocity of the stellar wind or supernova shell exceeds the escape velocity of the star cluster. The escaping mass is presumed to carry away the same specific energy and orbital angular momentum as the mass-losing star has.

2.3. Initial conditions

For the initial model, we followed the specifications of CW90. Each simulation is started at by giving all stars a mass m drawn from a single-component power-law mass function of the form , between lower and upper mass limits and , respectively. From the mean mass , given by the mass function, and the total number of stars in the computation N, the initial mass of the star cluster is computed: .

The initial density profile and the velocity distribution of the stellar system are taken from King models (King 1966). In a King model, the dimensionless depth of the central potential determines the structure of the cluster and thus the ratio of the virial radius to the tidal radius (see e.g. Binney & Tremaine 1987). The velocity distribution is given by a lowered Maxwellian with the same formal velocity dispersion for all stars, independent of mass. The initial positions of the stars are chosen independent of their mass.

The tidal radius is computed assuming that the star cluster initially fills its Roche lobe in the tidal field of the galaxy. In other words, we take the tidal radius of the initial King model as the physical tidal cutoff radius (Takahashi et al. 1997). The mass of the galaxy and distance from the cluster to the center of the galaxy are related by assuming that the star cluster has a circular orbit around the galactic center with a velocity of :

We approximate the tidal radius as the Jacobi radius for the star cluster. The distance from its center of mass to the first Lagrangian point is:

After each diagnostic output, the new tidal boundary of the star cluster is determined, stars outside the tidal boundary are removed from the stellar system. As in CW90, no tidal force is applied to the individual members of the star cluster. The mass of the star cluster declines during its evolution due to mass loss from stellar evolution of the stars and from escaping stars; the tidal radius decreases accordingly.

All our model clusters start in virial equilibrium. The computations are terminated if ten particles remain in the stellar system. The lifetime refers to the age at which the number of stars in the cluster is reduced to this limit.

2.4. System of units

The system of units used in the N-body model are given by , where E is the initial internal energy of the stellar system (Heggie & Mathieu 1986). The transformation from these scaled N-body units to physical units is realized with a set of transformations for mass, length and time.

The total mass of the stellar system determines the mass scaling. Since the star cluster starts filling its Roche lobe, the size scaling is determined by the tidal radius of the initial King model, which is in turn set by the tidal radius of the star cluster in the galactic tidal field. In this paper, we have adopted as the definition of the unit of time which is of order unity in our natural system of units ( of the real crossing time). This corresponds to

Here is the virial radius of the stellar system. Note that is our standard units, whereas the relation between and is determined by the King parameter .

The relaxation time of the cluster is defined as

© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998
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