LINEAR-STABILITY OF SPHERICAL COLLISIONLESS STELLAR-SYSTEMS

The linear stability analysis is presented here in a self-contained form, and several general issues, related to the symmetry properties of the relevant equations and to the boundary conditions are formulated. A numerical code for the study of the linear stability of collisionless spherical stellar systems with no radial truncation is constructed and applied to survey a family of astrophysically interesting anisotropic equilibrium models. The code is flexible, in that it can accept any reasonable initial distribution function f=f(E, J(2)) and is not restricted to a specific mass model. We have focused on the l=2 modes and on the so-called radial orbit instability. Marginal stability has been identified, corresponding to a value of 2K(r)/K-T = 1.58 for the ratio of the total radial to tangential kinetic energy, somewhat on the low side with respect to a generally accepted stability criterion. As the ratio 2K(r)/K-T increases, the number of unstable modes and the value of their growth rates are found to increase considerably, so that for negative-temperature models we have found up to six modes, with a growth rate much higher than the inverse half-mass crossing time and matching the timescales available in the innermost regions of the galaxy. For these negative-temperature models the density perturbations are very concentrated and indicate that the system would evolve rapidly through sizable poloidal motions which are bound to redistribute both the orbits and the mass in the central parts of the galaxy. The results shown are briefly compared with those derived from N-body simulations.