Gravothermal catastrophe: The dynamical stability of a fluid model^⋆

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
e-mail: mattia.sormani@physics.ox.ac.uk
² Università degli Studi di Milano, Dipartimento di Fisica, via Celoria 16, 20133 Milano, Italy
e-mail: giuseppe.bertin@unimi.it

Received: 30 October 2012
Accepted: 21 January 2013

Abstract

A re-investigation of the gravothermal catastrophe is presented. By means of a linear perturbation analysis, we study the dynamical stability of a spherical self-gravitating isothermal fluid of finite volume and find that the conditions for the onset of the gravothermal catastrophe, under different external conditions, coincide with those obtained from thermodynamical arguments. This suggests that the gravothermal catastrophe may reduce to Jeans instability, rediscovered in an inhomogeneous framework. We find normal modes and frequencies for the fluid system and show that instability develops on the dynamical time scale. We then discuss several related issues. In particular, (1) for perturbations at constant total energy and constant volume, we introduce a simple heuristic term in the energy budget to mimic the role of binaries. (2) We outline the analysis of the two-component case and show how linear perturbation analysis can also be carried out in this more complex context in a relatively straightforward way. (3) We compare the behavior of the fluid model with that of the collisionless sphere. In the collisionless case the instability seems to disappear, which is at variance with the linear Jeans stability analysis in the homogeneous case. We argue that a key ingredient for understanding the difference lies in the role of the detailed angular momentum in a collisionless system. A spherical stellar system is expected to undergo the gravothermal catastrophe only in the presence of some collisionality, which suggests that the instability is dissipative and not dynamical. Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of self-gravitating inhomogeneous gaseous systems.