Stellar Energy Relaxation around A Massive Black Hole
  [abridged] Energy relaxation around a massive black hole (MBH) is key to
establishing the dynamical state of galactic nuclei, and the nature of close
stellar interactions with the MBH. The standard description of relaxation as
diffusion provides a perturbative 2nd-order solution in the weak two-body
interaction limit. We run N-body simulations and find that this solution fails
to describe the non-Gaussian relaxation on short timescale, which is strongly
influenced by extreme events even in the weak limit, and is thus difficult to
characterize and measure. We derive a non-perturbative solution for relaxation
as an anomalous diffusion process, and develop a robust estimation technique to
measure it in simulations. These enable us to analyze and model our numerical
results, and validate in detail, for the first time, this model of energy
relaxation around an MBH on all timescales. We derive the relation between the
energy diffusion time, t_E, and the time for a small perturbation to return to
steady state, t_r, in a relaxed, single mass cusp around a MBH. We constrain
the contribution of strong encounters, measure that of the weakest encounters,
determine the value of the Coulomb logarithm, and provide a robust analytical
estimate for t_E in a finite nuclear stellar cusp. We find that t_r ~ 10t_E
~(5/32)Q^2P_h/N_h log Q, where Q=M_bh/M_* is the MBH to star mass ratio, the
orbital period P_h and number of stars N_h are evaluated at the energy scale
corresponding to the MBH's sphere of influence, E_h=sigma_inf^2, where
sigma_inf is the velocity dispersion far from the MBH. We conclude, using the
observed cosmic M_bh/sigma correlation, that cusps around lower-mass MBHs
(M_bh<10^7 Mo), which evolved passively over a Hubble time, should be relaxed.
We consider the effects of anomalous energy diffusion on orbital perturbations
of stars observed near the Galactic MBH.
