On broadened definitions of instability for stars in thermal imbalance

Abstract

The classical theory of stability of dynamical systems is employed to demonstrate that traditional definitions of pulsational instability cannot be directly applied to stars in thermal imbalance. In particular, it is shown that, for the case of thermal imbalance, pulsational displacements and pulsational velocities have separate and distincte-folding times. This being true, a broadened set of definitions becomes necessary, and we formulate such a set, again with reference to the classical theory. In accordance with the new definitions, it is argued that the development of observable pulsations requires as a necessary condition infinitesimal instability of both absolute displacement and velocity. If either one is unstable without the other, this constitutes a class of (probably) non-pulsational instability, not previously treated in the astrophysical literature. Finally, it is shown that the stability of stars in thermal imbalance may be evaluated according to the present definitions by employing either of two existing theories — the energy approach due to Demaret 91974; 1975; 1976) or the small perturbation technique of Coxet al. (1973).