Perturbations and stability of rotating stars

General properties of the normal modes of linear pulsation of perfect-fluid rotating stars are considered in some detail. With a view toward numerical applications, the discussion is restricted to finite-dimensional versions of the problem. Because the eigenvalue problem is non-selfadjoint, both left- and right-eigenvectors and their Jordan chains are important. These are defined and their mutual orthogonality properties derived. It is then shown that stars which have the time-azimuth reflection symmetry (t,phi) → (-t, -phi) in their unperturbed state have the remarkable property that any right-eigenvector can be changed into the adjoint of the left-eigenvector for the same eigenvalue by the simple operation of complex conjugation and phi-reversal. This property permits a variational principle for the eigenfrequencies of all the normal modes to be formulated. Two versions are given: one is for the quadratic eigenvalue problem; the other is in 'phase space', where the problem is first-order in time and gives a linear eigenvalue equation. (author)