STELLAR ENERGY RELAXATION AROUND A MASSIVE BLACK HOLE

Orbital energy relaxation around a massive black hole (MBH) plays a key role in the dynamics of galactic nuclei. Its standard description as diffusion provides a perturbative solution in the weak two-body interaction limit. Our N-body simulations show this fails to describe the short-timescale evolution, which is impacted by extreme events even in the weak limit, and is thus difficult to characterize and measure. We derive a non-perturbative solution for energy relaxation as an anomalous diffusion process, and a robust estimation technique to measure it in N-body simulations, and use these to analyze our numerical results. We empirically validate, for the first time, this theoretical description of energy relaxation around an MBH on all timescales. We constrain the modest contribution from strong encounters, and precisely measure that from the weakest encounters, and thereby calibrate the Coulomb logarithm. This yields a robust analytical estimate for the energy diffusion time, t sub(E). We relate t sub(E) to the time t sub(r) it takes a small density perturbation to return to steady state in a relaxed, single mass stellar cusp, t sub(r) [Asymptotically = to] 10t sub(E) [Asymptotically = to] (5/32)Q super(2)P sub(h)/N sub(h) log Q, where Q = M sub( times )/M sub(*) is the MBH to star mass ratio, and the orbital period P sub(h) and star number N sub(h) are evaluated at the energy scale of the MBH's sphere of influence, E sub(h) = [sigma] super(2) sub([infinity]), where [sigma] sub([infinity]) is the velocity dispersion at infinity. The observed M sub( times )/[sigma] sub([infinity]) correlation then implies that passively evolving stellar cusps around lower-mass MBHs (M sub( times ) [lap] 10 super(7) M sub([middot in circle])) should be dynamically relaxed by the Hubble time. We briefly consider the implications of anomalous diffusion for stars near the Galactic MBH.