There are many problems in attempting to use N-body techniques to model globular star clusters. This talk concentrates on purely dynamical aspects, and the problem of bridging the gap between the largest current simulations (typical N = 32K) and the median globular clusters (N ~ 10^6, though large clusters at birth might easily have had of order 3x10^7 stars). This scaling could perhaps be done purely empirically, but here we concentrate on a theoretical approach. We consider models with a mass spectrum and a steady tidal field.
Consider the evolution of the total mass of a cluster (taken to be the mass of all stars within the tidal radius). A standard application of the theory of relaxation gives the relaxation time as the time scale on which stars escape (e.g. Spitzer 1987, p.52). Therefore the time at which half of the mass has escaped is proportional to t_r, which depends in a known way on N. Unfortunately, simulations with different N do not scale in a way consistent with this prediction (Heggie et al 1998).
It turns out there are at least two complications: (i) the initial conditions used in the simulations quoted above are not self-consistent (they are isolated equilibrium models placed in a tidal field), and have an initial population of "primordial escapers". The time scale on which these escape is different from t_r, and their behaviour complicates the scaling. Unfortunately the use of self-consistent initial conditions (Heggie & Ramamani 1995) does not cure the scaling problem. (ii) as pointed out by T. Fukushige (private communication), relaxation simply gives the stars enough energy to escape, and again the time taken to pass beyond the tidal radius complicates the scaling. To remedy this difficulty, Fukushige and I have been studying the time scale, t_e, on which escape takes place, and recently I have attempted to incorporate this information into a more complete theory of escape.
The essence of the theory is to consider three classes of particle:
(i) those inside the tidal radius without enough energy to escape ("members", mass M_m);
(ii) those inside the tidal radius with enough energy to escape ("Escapers", mass M_e); and
(iii) those outside the tidal radius ("non-members", mass M_n). Again theory can be used to specify the time scale on which stars pass from (i) to (ii) and from (ii) to (iii), provided that the escape time is understood.
A simple mathematical model of this theory works very well, and yields the time evolution of the mass of (i) and (ii) in remarkably good agreement with simulations. In particular M_e begins at 0 (through use of self-consistent initial conditions), rises to a maximum, and then falls as the cluster dissolves. We now concentrate on the maximum value of M_e throughout the evolution, M_max.
As N increases from 1K to 32K, the value of M_max decreases from about 24% to about 10% of the initial mass of the system. (The variation is roughly as N^(-0.23).) How is this to be understood? The mass of these stars is attained by the balance between the time scale on which stars gain enough energy to escape (t_r) and the time scale on which the escape takes place (t_e). It may be expected that their mass scales as the ratio of these time scales, i.e. t_e/t_r. Now it has been customary to assume that t_e scales as the crossing time, t_cr, and so it would follow that M_max varies as (log N)/N.
In fact the work of Fukushige & Heggie shows that t_e does not scale with t_cr. For some stars, t_e is effectively infinite. (This refers to escape from a fixed potential; in fact the potential evolves on a time scale of roughly t_r, and this determines t_e for these stars. These are stars predominantly, but not exclusively, on retrograde orbits.) For such stars, then, t_e/t_r ~ 1. For the remainder, t_e scales as 1/dE^2, where dE is the excessive energy of a star above the minimum needed for escape. Now, because relaxation is a diffusive process, we estimate that the energy gained by a star while inside the cluster, but with enough energy to escape, varies as sqrt(t_e/t_r). It follows that the escape time scale varies as sqrt(t_cr*t_r), i.e. the geometric mean of the crossing and relaxation times. For such stars, then, t_e/t_r ~ sqrt((log N)/N). Though these considerations do not yet give a quantitative explanation of the observed dependence of M_max on N, they suggest that the dependence of M_max on N should lie between N^0 and N^(-0.5).
Whatever the true result, our best guess for M_max for a real globular cluster is of order a few percent. This suggests that a substantial fraction of all stars in a globular cluster have speeds above the escape speed. Such considerations may aid in understanding high-velocity stars that have been observed in at least two clusters (Gunn & Griffin 1979, Meylan et al 1991).
Acknowledgements: To S.J. Aarseth, for the use of his code, to PPARC, for supplying money for a HARP-3, and to the GRAPE team for their kindness, ingenuity and hospitality over many years.
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Meylan G., Dubath P., Mayor M., 1991, ApJ, 383, 587
Spitzer L., Jr., 1987, Dynamical Evolution of Globular Clusters. Princeton University Press