Gravitational Encounters and the Evolution of Galactic Nuclei. II. Classical and Resonant Relaxation

Direct numerical integrations of the Fokker-Planck equation in energy-angular momentum space are carried out for stars orbiting a supermassive black hole (SBH) at the center of a galaxy. The algorithm, which was described in detail in an earlier paper, includes diffusion coefficients that describe the effects of both random ("classical") and correlated ("resonant") encounters. Steady-state solutions are similar to the Bahcall-Wolf solution but are modified at small radii due to the higher rate of diffusion in angular momentum, which results in a low-density core. The core radius is a few percent of the influence radius of the SBH. The corresponding phase-space density f(E,L) drops nearly to zero at low energies, implying almost no stars on tightly-bound orbits about the SBH. Steady-state rates of stellar disruption are presented, and a simple analytic expression is found that reproduces the numerical feeding rates with good accuracy. The distribution of periapsides of disrupted stars is also computed. Time-dependent solutions are also computed, starting from initial conditions similar to those produced by a binary SBH. In these models, feeding rates evolve on two timescales: rapid evolution during which the region evacuated by the massive binary is refilled by angular-momentum diffusion; and slower evolution as diffusion in energy causes the density profile at large radii to attain the Bahcall-Wolf form.