The Equations Governing Linear Perturbations in a Quasi-Static Star

Abstract

A spherically symmetric quasi-static star is considered, and an inertial frame of reference is adopted whose origin coincides with the star’s mass centre. With respect to this frame of reference, an orthogonal system of spherical coordinates is introduced. The quasi-static star is supposed to be in hydrostatic equilibrium. The variation of the gravitational potential inside the star is determined both by the resolution of Poisson’s second-order differential equation and from the general integral solution of Poisson’s equation. Next, the linearised equations that govern small perturbations in a quasi-static star are presented in an Eulerian form. They consist of the linearised equation of motion, the equation expressing the mass conservation of the perturbed elements, an equation relating the perturbations of pressure, mass density, and entropy, and finally Poisson’s perturbed differential equation.