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

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5. Discussion

Chernoff & Weinberg obtained a lifetime of 280 Myr for stellar systems with a relaxation time as is chosen in the models S and IR1 to IR16. None of our models disrupts within such a short time span, but the trend of shorter life time for larger mass cluster is clearly visible. However, it seems somewhat unlikely that real globular clusters could have a disruption time of a few hundred Myrs, if we extrapolate our numerical result. The largest number of particles used in our calculations already reaches within a factor of 5 of that of the smaller globular clusters.

As we stated earlier, this rather large discrepancy between the result of our N-body calculation and the result of the Fokker-Planck calculation is surprising. There are several reasons which would cause the evaporation of the Fokker-Planck model to be different from that of N-body system. For example, the Fokker-Planck calculation relies on the assumption of the adiabatic response of the orbits of stars to a change of mass of the stars, assumed to be slow. This assumption is violated for the early stage of evolution, where the stellar evolution time scales are as short as a few Myr.

Another difference is that, in Fokker-Planck calculations, a simple tidal limit value in energy space is used; stars with energy exceeding this value disappear from the cluster. In our study, we remove stars when they reach the tidal radius. Apart from the fact that these two procedures are already different, neither of them are appropriate. In the Fokker-Planck calculation, both the anisotropy and the non-spherical nature of the system, both which might have the effect of significantly enhancing the stellar escape rate, are ignored. Our simple treatment of the tidal boundary allows direct comparison with CW90, but it also has the effect of reducing the escape rate, as no external tidal force is applied to individual stars. This is the main reason why our test models obtained significantly longer life times than those obtained by FH95.

In order to investigate the reason for the discrepancy between different models, both the N-body calculation and Fokker-Planck calculation have to be refined. On the side of the Fokker-Planck calculation, until recently, further refinement has been difficult, since except for Monte-Carlo models no practical implementation for anisotropic models was available. However, recent progress in the two-dimensional Fokker-Planck calculation code (Takahashi 1993, 1995, 1996) has made the study of the effects of anisotropy feasible.

On the N-body side, it is fairly straightforward to try various models for the tidal field, from the simplest one used in our present study to the realistic static model used in FH95 (see Fig. 5). It is even possible to go further to include the dynamic effects of the galactic disk and the bulge. Thus, a more detailed comparison may be possible (see the fascinating "collaborative experiment" reported by Heggie, in preparation).

However, even if the N-body and Fokker-Planck calculations treat the tidal boundary and the anisotropy in the same way, the difference in the dynamical timescale still remains. For the next several years, we will not yet be able to model real globular clusters accurately, since they will continue to fall in between the Fokker-Planck calculations (with an infinitesimal crossing time) and N-body calculations (with too long a crossing time). We thus urgently need some way to interpolate between the two types of results.

The main purpose of the present study is to investigate whether such an extrapolation is feasible. We must conclude that, as yet, there is no obvious way to extrapolate the results from small N values upward. The models with constant relaxation time evolve too slowly, because the crossing time is un-physically long. The models with constant crossing time, on the other hand, evolve too rapidly, because the relaxation timescale is too short. Although we knew these two effects to be qualitatively present, it came to us as a bit of a surprise to see just how important they are, quantitatively.

Consequently, it would be practically impossible to predict the result of a 32k run, from either of the 4k runs, the one from the IC series, as well as the one from the IR series. Even having both in hand would make an extrapolation still dubious. This suggests that for the selected set if initial conditions an extrapolation to real globular clusters, with hundreds of thousands of stars, is still out of the question.

The discrepancy between N-body and Fokker-Planck compuations is partially solved by the introduction of an extra free parameter in the Fokker-Planck models (Takahashi & Portegies Zwart 1998). This parameter provides a timescale on which stars are removed from the stellar system. This treatment requires anisotropic Fokker-Planck models and give excellent agreement with the N-body computations.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998