*Astron. Astrophys. 337, 363-371 (1998)*
## 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.
© European Southern Observatory (ESO) 1998
Online publication: August 17, 1998
helpdesk.link@springer.de |