Secular Instabilities of Keplerian Stellar Discs

TL;DR

This paper models Keplerian stellar discs around black holes, analyzing their non-axisymmetric secular instabilities, deriving analytical instability conditions, and confirming results with numerical simulations.

Contribution

It introduces idealized waterbag models of stellar discs, derives analytical criteria for their instabilities, and explores their nonlinear evolution and relaxation.

Findings

Waterbags with circular orbits have a single stable mode for each m.

Waterbags excluding circular orbits can have stable or unstable modes.

Numerical simulations confirm linear theory and show nonlinear growth and relaxation.

Abstract

We present idealized models of a razor-thin, axisymmetric, Keplerian stellar disc around a massive black hole, and study non-axisymmetric secular instabilities in the absence of either counter-rotation or loss cones. These discs are prograde mono-energetic waterbags, whose phase space distribution functions are constant for orbits within a range of eccentricities (e) and zero outside this range. The linear normal modes of waterbags are composed of sinusoidal disturbances of the edges of distribution function in phase space. Waterbags which include circular orbits (polarcaps) have one stable linear normal mode for each azimuthal wavenumber m. The m = 1 mode always has positive pattern speed and, for polarcaps consisting of orbits with e < 0.9428, only the m = 1 mode has positive pattern speed. Waterbags excluding circular orbits (bands) have two linear normal modes for each m, which can be stable or unstable. We derive analytical expressions for the instability condition, pattern speeds, growth rates and normal mode structure. Narrow bands are unstable to modes with a wide range in m. Numerical simulations confirm linear theory and follow the non-linear evolution of instabilities. Long-time integration suggests that instabilities of different m grow, interact non-linearly and relax collisionlessly to a coarse-grained equilibrium with a wide range of eccentricities.