Resonant driven overstabilities in stellar systems

Abstract

In this paper we show that the linearized equations describing a neutral oscillation inside a spherical stellar system allow a conserved quantity (the wave energy) in the absence of resonant stars. Resonant stars act as sources or sinks of wave energy when present. We further show that the normal mode equation can be interpreted with this simple physical picture. We study spherical and disc systems in which the stars are all on nearly circular orbits, and derive local WKB dispersion relations, as well as a full global mode analysis. We prove that spherical shell models are unstable to a purely growing instability, and examine its counter-part in disc systems. We show that this mode is caused by the destabilization of neutral Jean's oscillations due to counter-rotating stars. We also examine more realistic systems, and prove that spherical systems in which |$2\Omega-k$| reaches a maximum value, have a spectrum of oscillations which are necessarily destablized by resonant stars. We prove the existence of a spectrum of discrete normal modes in rotating discs, and discuss mechanisms for destabilizing them. Numerical simulations of spherical systems are also presented which support the theoretical conclusions. We find these normal oscillation modes for the axisymmetric case, with overstability growing on a time-scale much longer than the oscillation period. The numerical scheme conserved orbital phase space sufficiently accurately that we were able to find the existence of beats in the quadrupole response during the linear stage, which suggested that more than one eigenmode dominated the response. The system became more centrally condensed as the amplitude of the axisymmetric response grew, in agreement with the theoretical expectation of a negative energy mode losing energy as it grows.