Stellar dynamics around a massive black hole – II. Resonant relaxation

• Published in: Monthly notices of the Royal Astronomical Society Vol. 458; no. 4; pp. 4143 - 4161
• Main Authors: Sridhar, S., Touma, Jihad R.
• Format: Journal Article
• Language: English
• Published: London Oxford University Press 01.06.2016
• Subjects: Astronomy; Black holes; Black holes (astronomy); Correlation; Dynamical systems; Dynamics; Gravity; Mathematical analysis; Normal distribution; Stars & galaxies; Statistical mechanics; Stellar system evolution; Stellar systems; Wakes; Galaxy: centre; galaxies: nuclei; galaxies: kinematics and dynamics
• ISSN: 0035-8711; 1365-2966
• DOI: 10.1093/mnras/stw543

We present a first-principles theory of resonant relaxation (RR) of a low-mass stellar system orbiting a more massive black hole (MBH). We first extend the kinetic theory of Gilbert to include the Keplerian field of a black hole of mass M •. Specializing to a Keplerian stellar system of mass M ≪ M •, we use the orbit-averaging method of Sridhar & Touma to derive a kinetic equation for RR. This describes the collisional evolution of a system of N ≫ 1 Gaussian rings in a reduced 5-dim space, under the combined actions of self-gravity, 1 post-Newtonian (PN) and 1.5 PN relativistic effects of the MBH and an arbitrary external potential. In general geometries, RR is driven by both apsidal and nodal resonances, so the distinction between scalar RR and vector RR disappears. The system passes through a sequence of quasi-steady secular collisionless equilibria, driven by irreversible two-ring correlations that accrue through gravitational interactions, both direct and collective. This correlation function is related to a ‘wake function’, which is the linear response of the system to the perturbation of a chosen ring. The wake function is easier to appreciate, and satisfies a simpler equation, than the correlation function. We discuss general implications for the interplay of secular dynamics and non-equilibrium statistical mechanics in the evolution of Keplerian stellar systems towards secular thermodynamic equilibria, and set the stage for applications to the RR of axisymmetric discs in Paper III.