Stability of spherical stellar systems - I. Analytical results

The so-called 'symplectic method' is used for studying the linear stability of a self-gravitating collisionless stellar system, in which the particles are also subjected to an external potential. The system is steady and spherically symmetric, and its distribution function f₀ thus depends only on the energy E and the squared angular momentum L² of a particle. Assuming that ∂f₀/∂E < 0, it is first shown that stability holds with respect to all the spherical perturbations - a statement which turns out also to be valid for a rotating spherical system. Thus it is proven that the energy of an arbitrary aspherical perturbation associated with a 'preserving generator' δg₁ [i.e., one satisfying ∂f₀/∂L² {δg₁, L²} =0] is always positive if ∂f₀/∂L² ≤0 and the external mass density is a decreasing function of the distance r to the centre. This implies in particular (under the latter condition) the stability of an isotropic system with respect to all the perturbations. Some new remarks on the relation between the symmetry of the system and the form of f₀ are also reported. It is argued, in particular, that a system with a distribution function of the form f₀=f₀(E, L²) is necessarily spherically symmetric.