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5.2 Fokker-Planck

The computational limitations of N-body simulations can be sidestepped by describing the system in terms of distribution functions f (x, v,t) m with the number of stars of mass m at time t in the range 3 (x,x + d x) and 3 (v,v + d v) given by 3 3 dN = fmd xd v. This description requires that either the phase-space element 3 3 d xd v be small enough to be infinitesimal yet large enough to be statistically meaningful, or that fm be interpreted as the probability distribution for finding a star of mass m at a location in phase space. The evolution of the cluster is then described by the evolution of fm. The gravitational interaction is provided by a smoothed gravitational potential f, which is determined by
sum [ integral ] \~/ 2f = 4p mi fmi(x,v,t)d3v . (27) i
The effect of gravitational interactions is modeled by a collision term G[f] (see [19117] for specific descriptions of G). The dynamics of the globular cluster are then governed by the Fokker-Planck equation:
@f @f @t-+ v . \~/ f - \~/ f .@v- = G[f ]. (28)

In the Fokker-Planck approach, the mass spectrum of stars is binned, with a separate f m for each bin. Increasing the resolution of the mass spectrum requires increasing the number of distribution functions and thus increasing the complexity of the problem. The inclusion of additional physical variables such as binaries adds further complexity. Methods for numerically solving the Fokker-Planck equation use either an orbit-averaged form of Equation (28View Equation[25], or a Monte Carlo approach [50555689Jump To The Next Citation Point].

The two time scales involved in the evolution of fm are tcross (which governs changes in position) and trelax (which governs changes in energy). The orbit-averaged form of Equation (28View Equation) derives from the realization that changes in position are essentially periodic with orbital period T ~ t « t cross relax. Thus, one can average over the rapid changes in position and retain the slow changes in the phase space coordinates that occur over relaxation times. When one does this, the Fokker-Planck equation is reduced to an equation involving three phase-space variables and time. The most appropriate variables are the energy and angular momentum. The orbit-averaged solutions of the Fokker-Planck equation cannot easily handle the effect of binaries and the binary interactions that occur during the evolution of a globular cluster [52]. These effects are usually inserted by hand using statistical methods. The advantages of the orbit-averaged approach are that one can generalize it to handle anisotropy in velocity, thus allowing study of the effects of the galactic gravitational field and tidal stripping. One can also include the rotation of the cluster [90].

Monte Carlo simulations do not actually solve the Fokker-Planck equation, but rather rely on a representative sample of n test stars which are followed numerically with random velocity perturbations applied as are appropriate to the diffusion coefficients in G. The advantage of the Monte Carlo approach is that one can handle binary interactions and stellar evolution directly (and then extrapolate the results of the n test stars to the N actual stars in the cluster). However, like Fokker-Planck, the Monte Carlo method currently requires the cluster to be spherically symmetric, so gravitational effects of the galaxy must be treated in an ad hoc way.

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