Gravitational instability of polytropic spheres and generalized thermodynamics

We complete the existing literature on the structure and stability of polytropic gas spheres reported in the classical monograph of Chandrasekhar (1942). For isolated polytropes with index $1<n<5$, we provide a new, alternative, proof that the onset of 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 $n\ge 3$, the mass-density relation presents some damped oscillations and there exists a limiting mass above which no hydrostatic equilibrium is possible. Like for isothermal gas spheres, the onset of instability occurs precisely at the point of maximum mass. Analytical results are obtained for the particular index $n=5$. We also discuss the relation of our study with extended thermodynamics (Tsallis entropy) recently investigated by Taruya & Sakagami (cond-mat/0107494).