Gravitational instability of polytropic spheres and generalized thermodynamics

We extend the existing literature on the structure and stability of polytropic gas spheres reported in the classical monograph of Chandrasekhar (1932). For isolated polytropes with index 1 < n<5, we provide a new, alternative, proof that the onset of gravitational instability occurs for n = 3 and we express the perturbation profiles of density and velocity at the point of marginal stability in terms of the Milne variables. Then, we consider the case of polytropes confined within a box of radius R (an extension of the Antonov problem for isothermal gas spheres). For n greater than or equal to 3, the mass-density relation presents damped oscillations and there exists a limiting mass above which no hydrostatic equilibrium is possible. As for isothermal gas spheres, the onset of instability occurs precisely at the point of maximum mass in the series of equilibrium. Analytical results are obtained for the particular index n = 5. We also discuss the relation of our study with generalized thermodynamics (Tsallis entropy) recently investigated by Taruya & Sakagami (2002).