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The steady-state distribution and consumption rate of stars orbiting a massive object at the center of a spherical N-body stellar system are reexamined, concentrating on the manner in which stars gradually diffuse inward toward the center by repeated small-angle gravitational encounters with each other. The analysis incorporates the two-dimensional Fokker-Planck equation in energy (E) and angular-momentum (J) space, for which the approximate velocity-diffusion coefficients computed for the case of a Maxwellian field-star distribution are employed. The distribution of the stars as a function of J is determined for small J by considering the rate at which stars of fixed E enter the loss cone; a Fokker-Planck equation in E alone and containing an energy-dependent 'sink' term is obtained by integrating the two-dimensional equation over J and substituting an expression for the stellar consumption rate; and the approximate steady-state solution to this equation in the 'self-similar' regime is found along with the total rate at which the central object consumes stars. The results are applied to a massive black hole at the center of a globular cluster (M15) and an elliptical galaxy.
Lightman et al. (Sat,) studied this question.