Perturbations and stability of rotating stars

The normal mode problem for perturbations of a perfect-fluid rotating star is a non-selfadjoint eigenvalue problem. To study the changes in the eigenfrequencies when the operators in the problem change slightly, this paper develops the appropriate lowest-order perturbation theory and discusses two important cases. In the first case the perturbing operator is the inclusion of a 'secular' effect, such as viscosity or radiation reaction. Here the main interest is in unperturbed eigenfrequencies that are dynamically stable, and the perturbed frequencies are analytic functions of the perturbation parameter. Explicit formulae are derived for use in numerical calculations of secular effects in stars. The second case is the study of the onset of dynamical instability itself. Here the perturbing operator is a slight change in the structure of the star. It is found that instability sets in only through eigenvalues which have Jordan chains, and the perturbation theory is non-analytic. It is shown that stable modes do not have Jordan chains, and the marginally stable modes have, in the generic case, chains of length 1. This gives a fairly complete characterization of the manner in which instability arises along a sequence of stars. (author)