Gravothermal Catastrophe: the dynamical stability of a fluid model

TL;DR

This paper investigates the dynamical stability of a self-gravitating isothermal fluid sphere, linking thermodynamic and dynamical criteria for gravothermal catastrophe, and compares fluid and collisionless models to understand instability mechanisms.

Contribution

It demonstrates that the conditions for gravothermal catastrophe align with Jeans instability criteria and extends the analysis to multi-component systems and collisionless cases.

Findings

Instability develops on the dynamical time scale.

Conditions for catastrophe coincide with thermodynamic predictions.

Collisionless systems may not exhibit the same instability as collisional fluids.

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 be carried out also 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 to understand the difference (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) lies in the role of the detailed angular momentum in a collisionless system. 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.