Gravitational instability of finite isothermal spheres

We investigate the stability of bounded self-gravitating systems in the canonical ensemble by using a thermodynamical approach. Our study extends the earlier work of Padmanabhan [Astrophys. J. Supp. 71, 651 (1989)] in the microcanonical ensemble. By studying the second variations of the free energy, we find that instability sets in precisely at the point of minimum temperature in agreement with the theorem of Katz [ Mon. Not. R. astr. Soc. 183, 765 (1978)]. The perturbation that induces instability at this point is calculated explicitly; it has not a ``core-halo'' structure contrary to what happens in the microcanonical ensemble. We also study Jeans type gravitational instability of isothermal gaseous spheres described by Navier-Stokes equations. We show analytically the equivalence between dynamical stability and thermodynamical stability and the fact that the stability of isothermal gas spheres does not depend on the viscosity. This confirms the findings of Semelin et al. [astro-ph/9908073] who used numerical methods. We also give a simpler derivation of the geometric hierarchy of scales inducing instability discovered by these authors. The density profiles that trigger these instabilities are calculated analytically; they present more and more oscillations that also follow a geometric progression. This suggests that the system will fragmentate in a series of `clumps' and that these `clumps' will themselves fragmentate in substructures. The fact that both the domain sizes leading to instability and the `clumps' sizes within a box follow a geometric progression with the same ratio suggests a fractal-like behaviour. This gives further support to the interpretation of de Vega et al. [Nature, 383, 56 (1996)].