The distribution and consumption rate of stars around a massive, collapsed object

We consider the steady-state distribution and consumption rate of stars orbiting a massive object at the center of a spherical, N-body stellar system. The distribution of stars is determined by the consumption of low angular momentum stars which pass within a small distance r/sub t/ of the central mass M and by the relaxation processes associated with gravitational stellar encounters. Our method employs an approximate, analytic analysis of the two-dimensional Fokker--Planck equation describing diffusion in energy E and angular momentum J. The basic results are the following: (1) Consumption of low angular momentum stars which have entered the ''loss-cone'' J < or = J/sub min/ approx. = (GMr/sub t/)1/2 dominates the character of the solution inside the critical radius, r/sub crit/>> greater than or equal to r/sub t/, at which the root mean square angular momentum transferred to a star via stellar encounters in one orbital period equals J/sub min/. (2) The total consumption rate of stars by M is roughly the number of stars inside r/sub crit/ divided by the relaxation time at r/sub crit/. (3) A self-consistent solution can be found in which the distribution of stars is almost isotropic for high-J stars and varies only logarithmically with J for low-J stars. (4) The density of core stars has the following form: n(r) approx. =n (r/sub a/)[1+(r/sub a//r)/sup l/], wher the accretion radius r/sub a/ approx. GM/<ν2> >> r/sub t/, <ν2> is the mean-squared velocity dispersion in the core outside r/sub a/, and l decreases slowly from approx. 1.75 at >> r/sub crit/ to approx. 1.60 at r approx. 10r/sub t/. Neglect of loss-cone effects gives a consumption rate too small by roughly the ratio r/sub t//r/sub crit/ and a constant exponent l = 1.75 for >> r/sub t/. These results are applied to massive black holes at the centers of globular clusters and galactic nuclei