Dynamical stability for the gravitational evolution of a homogeneous polytrope

Abstract

URL: http://www-spht.cea.fr/articles/s00/008 Stabilité dynamique de l'évolution gravitationnelle d'un polytrope homogèneThe dynamic stability of the spherical gravitational evolution (collapse or expansion) for a homogeneous polytropic gas with any exponent $\gamma ,$ is studied using the lagrangian formalism. We obtain the analytical expression for density perturbations at the first order. In the case $\gamma =~4/3,$ the Jeans'criterion is easily generalized to a self-similar expanding background. The collapsing case is found to be always unstable. The stability of density modes obtained for $\gamma \not = 4/3$ does not introduce any conditions on the wavelength perturbation, but only a criterion on the polytropic index. As a result, stability is obtained for an expanding gas provided $\gamma 5/3.