Stellar Dynamics around a Massive Black Hole I: Secular Collisionless Theory

S. Sridhar, Jihad R. Touma

Published 2015-09-08Version 1

We present a theory in 3 parts, of the long-term (or secular) evolution of stellar systems orbiting within the sphere of influence of massive black holes in galactic nuclei. Here we describe the secular collisionless dynamics of a (Keplerian) stellar system of mass $M$ orbiting a black hole of mass $M_\bullet \gg M$. The stellar distribution function (DF) $f$ obeys the collisionless Boltzmann equation (CBE) in 6-dim phase space. The small mass ratio, $\varepsilon = M/M_\bullet \ll 1$, implies a separation of time scales in the motions of stars: the fast Kepler orbital periods and the secular time scale which is longer by a factor $\varepsilon^{-1}$. We orbit-average the CBE over the fast Keplerian orbital phase using the Method of Multiple Scales. Then $f$ is expressed as the sum of a secular DF $F$ in a 5-dim (Gaussian Ring) space, and small fluctuations that remain of $O(\varepsilon)$ over secular times. $F$ obeys a secular CBE that includes stellar self-gravity, general relativistic corrections up to 1.5 post-Newtonian order, and external sources. Secular dynamics conserves the semi-major axis of every star. This additional integral of motion promotes extra regularity of the stellar orbits, and enables the construction of secular equilibrium DFs ($F_0$) through a Secular Jeans theorem. Secular equilibria allow for varied spatial geometries including figure rotation. A linearized secular CBE determines the linear response and stability of $F_0$. Spherical, non-rotating equilibria may support small-amplitude, long-lived, warp-like distortions. We also prove that an axisymmetric, zero-thickness, flat disc is secularly stable to all in-plane perturbations, when its DF $F_0$ is a monotonic function of the angular momentum at fixed energy.