The stability of the equilibrium of supermassive and superdense stars

The second variation of the mass-energy (by constant baryon number) is analysed to inquire into the stability of the equilibrium for a spherical mass under the action of its own gravitation. The equations of general relativity are used. For zero density at the surface, the sphere is stable if (and only if) a solution of a second order equation (Jacobi equation) has only one zero in the interior (at center) of the sphere. The results obtained in this way coincide of course with those obtained by means of the Chandrasekhar-Misner-Zapolsky variational principle, but the solution of the Jacobi equation is more simple and straightforward. In particular, the stability of the several modes are analysed without making any choice of a trial function for the Lagrangian displacement.The method is applied to the GrattonR-polytropic model of supermassive stars.Work done in the Laboratorio Astrofisico, Frascati-Roma, Italy.