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.
