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

Abstract

We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass $M$ orbiting a black hole of mass $M_{\bullet }\gg M$. Here we describe the collisionless dynamics; Papers II and III are on the (collisional) theory of Resonant Relaxation. The mass ratio, $\epsilon =M/M_{\bullet }\ll 1$, is a natural small parameter implying a separation of time scales between the short Kepler orbital periods and the longer orbital precessional periods. The collisionless Boltzmann equation (CBE) for the stellar distribution function (DF) is averaged over the fast Kepler orbital phase using the method of multiple scales. The orbit-averaged system is described by a secular DF, $F$, in a reduced phase space. $F$ obeys a secular CBE that includes stellar self-gravity, general relativistic corrections up to 1.5 post-Newtonian order, and external sources varying over secular times. Secular dynamics, even with general time dependence, 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 equilibria, $F_0$, through a secular Jeans theorem. A linearized secular CBE determines the response and stability of $F_0$. Spherical, non-rotating equilibria may support 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.

Accepted for publication in MNRAS (final version)

References in corpus (4)

• The Co-Evolution of Galaxies and Supermassive Black Holes: Insights from Surveys of the Contemporary Universe
• Stellar Dynamics around a Massive Black Hole II: Resonant Relaxation
• Three-dimensional Keplerian orbit-superposition models of the nucleus of M31
• Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of Axisymmetric Discs

Cited by in corpus (14)

• The impact of vector resonant relaxation on the evolution of binaries near a massive black hole: implications for gravitational wave sources
• Hidden Planetary Friends: On the Stability Of 2-Planet Systems in the Presence of a Distant, Inclined Companion
• Isotropic-Nematic Phase Transitions in Gravitational Systems
• Relaxation in self-gravitating systems
• Stellar Dynamics around a Massive Black Hole II: Resonant Relaxation
• The Thermodynamics of Rotating Black-Hole Star Clusters
• Isotropic-Nematic Phase Transitions in Gravitational Systems II: Higher Order Multipoles
• The secular evolution of discrete quasi-Keplerian systems. I. Kinetic theory of stellar clusters near black holes
• Dynamical Modeling of Galaxies and Supermassive Black Holes: Axisymmetry in Triaxial Schwarzschild Orbit Superposition Models
• Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of Axisymmetric Discs
• Relativistic Dynamical Stability Criterion of Multi-Planet Systems with a Distant Companion
• Statistical Mechanics of Gravitational Systems with Regular Orbits: Rigid Body Model of Vector Resonant Relaxation
• Secular Instabilities of Keplerian Stellar Discs
• Deformation of the Galactic Centre stellar cusp due to the gravity of a growing gas disc