Dynamical stability for the gravitational evolution of a homogeneous polytrope

The dynamic stability of the spherical gravitational evolution (collapse or expansion) for a homogeneous polytropic gas with any exponent γ, is studied using the lagrangian formalism. We obtain the analytical expression for density perturbations at the first order. In the case γ = 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 γ / = 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 γ < 4/3, and for a collapsing one, for γ > 5/3.