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

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4. Results

4.1. Model S: the standard model

In this section we describe the result of the standard run with 32768 particles.

Fig. 7 gives as a function of time (in billions of years) the total mass and the total number of stars of the simulation model S. This figure clearly demonstrates the initial epoch of quick mass loss due to stellar evolution followed by a gradual decrease in the total number of stars. Finally the stellar system evaporates.

[FIGURE] Fig. 7. The time evolution of the total mass (solid line) and number of stars (dashed line) of the cluster for the standard, 32768-body run. Both the mass and the number of stars are normalized to their initial value.

4.2. Models IR: constant relaxation time

Table 4 and Fig. 8 give the results of the series of runs with constant relaxation time.


Table 4. Initial conditions and dissolution time in the galactic tidal field for the models with constant relaxation time. The columns give the model name, the number of particles, the tidal and the virial radius (in parsec) and the unit of time in million years. The last two columns gives the life time of the star cluster in units of the time unit [FORMULA] and in addition in billion years. The relaxation time scale at the virial radius of these model is approximately 2.87 Gyr.

[FIGURE] Fig. 8. The mass of the star cluster as a function of time for the runs with constant relaxation time (models IR) of the models with 32k (left solid line), 16k (dashed), 8k (dash-dot), 4k (dotted), 2k (dash-3dot) and 1k (right solid line) runs. The time scaling is cut off at 10 Gyr, at which point model IR1 has still not dissolved completely.

A striking feature of Fig. 8 is the systematic way in which lifetimes are shorter for models with larger total mass. This trend can be explained as follows. For lower mass models, holding the relaxation time fixed implies that the crossing times are longer. Since stars can escape from the system only on a crossing timescale at the tidal boundary, an increase in crossing time tends to increase the dissolution time. In addition, effects of stellar evolution make themselves felt only on a crossing time, which again tends to lengthen the dissolution time for lower mass models.

The latter effect is most notable during the first [FORMULA] years, when the mass of the stellar system decreases dramatically. This initial mass decrease is the result of the presence of massive stars which evolve quickly. After the system has lost about 20% of its initial mass escaping stars become the major channel through which mass is lost until the system dissolves.

By the time half of the mass has been lost, these above arguments, related to the length of the crossing time, become less important, since the time scale for significant change to take place has become longer. The subsequent difference in behavior is linked to the onset of an instability, noted already by FH95, that operates only for high number of stars, where an increase of the ratio [FORMULA] leads to a loss of virial equilibrium, and a consequent rapid dissolution of the cluster.

The strong dependence of the dissolution time on the crossing time implies that we cannot easily extrapolate the result obtained from small-N runs to the evolution of globular clusters with realistic number of particles. From Fig. 8, it is not clear whether or not the lifetime is converging to a particular value. One would hope that, in the limit of [FORMULA], the result would converge to the result of the Fokker-Planck calculation. Even so, we have to face the question whether real globular clusters contain a large enough number of stars to be modeled by Fokker-Planck calculations. So far, we have used up to 32,768 particles, a larger number than any previous N-body simulations intended to model globular clusters. The separation between the relaxation timescale and the crossing timescale is quite large (more than two orders of magnitude), and the lifetime of the system is measured in hundreds of the crossing times. Even so, we still see a fairly strong dependence on the crossing time for the evolution timescale of the total cluster.

In the next subsection, we examine the other way to adjust the timescales, in the series of models with constant crossing times.

4.3. Models IC: constant crossing time

Table 5 and Fig. 9 shows the result of the runs with constant crossing time. Here, the models with [FORMULA] and those with [FORMULA] behave differently, in the sense that the latter models show quick disruption at the end, while the former models do not. Within the models which show the same qualitative behavior, models with longer relaxation time evolve more slowly.


Table 5. Initial conditions and resulting lifetimes of model clusters where the stellar evolution time is scaled to the unit of time of model S ([FORMULA]Myr). The first two columns gives the model name and number of stars. The third and fourth columns give the initial tidal radius and virial radius in parsec followed by the relaxation time (in million years). The last two columns give the dissipation time scale in units of [FORMULA] and in billion years.

[FIGURE] Fig. 9. Mass (normalized to unity) as a function of time (in billion years) for the computations of model IC1 (lower solid line) to IC32 (upper solid line). Line styles are the same as in Fig. 8. The lines for the models IC5 to IC7 are not presented in this figure. Note the different scale along the time axis compared to Fig. 8.

It is natural that systems with shorter relaxation times evolve faster. If the effect of the stellar evolution is not dominant, the main mechanism which drives the evolution of the cluster is two-body relaxation. Thus, the evolution timescale is determined by the relaxation timescale.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998