Small perturbations in flat galaxies

Abstract

The aim of this series of papers is to develop straightforward methods of computing the response of flat galaxies to small perturbations. This Paper I considers steady state problems; Paper II considers time varying perturbations and the effects of resonances; and Paper III applies the methods developed in Papers I and II to a numerical study of the stability of flat galaxies. The general approach is to study the dynamics of each individual orbit. The orbits are described by their apocentric and pericentric radii,r a andr p , and the distribution function of an equilibrium model is a function ofr a andr p . The mass density and potential corresponding to a distribution function is found by means of an expansion in Hankel-Laguerre functions; the coefficients of the expansion being found by taking moments of the mass density of the individual orbits. This leads to a simple method of constructing equilibrium models. The response to a small perturbation is found by seeking the response of each orbit. When the perturbations are axisymmetric and slowly varying, the response can be easily found using adiabatic invariants. The potential is expanded in a series of Hankel-Laguerre functions, and the response operator becomes a discrete matrix. The condition that the model is stable against adiabatic radial perturbations is that the largest eigenvalue of the response matrix should be less than one. An analytic approximation to the response matrix is derived, and applied to estimate the eccentricity needed for stability against local perturbations.