Vibrational stability of stars in thermal imbalance: A solution in terms of asymptotic expansions

Isentropic oscillations of a star in thermal imbalance are defined as those for which, at every istant, the entropy of each mass element of the configuration in the perturbed motion is equal to that of the same mass element in the unperturbed motion.The solution of the equations describing such isentropic oscillations and written in terms of the infinitesimal displacement r(r ₀,t) is presented in terms of asymptotic expansions up to the first order in the parameter /t _s where is the adiabatic pulsation period for the fundamental mode andt _s , a slow time scale of the order of the Kelvin-Helmholtz time.The solution obtained allows one to define, unambiguously, an isentropic part to the coefficient of vibrational stability of arbitrary stellar models in thermal imbalance, as well as to derive a general formula relating the results of a stability analysis in terms of r and r/r.Application of this general solution to the simple case of homologous motion is also given.