On the consequences of the gravothermal catastrophe

The cores of globular clusters evolve independently of the outer parts and eventually achieve a self-similar evolution with a shrinking almost isothermal portion followed by a power law density profile, $$\rho \propto r ^{-\alpha }$$. It is proved that α is in the range $$2 \lt \alpha \lt 2.5$$ and that both the magnitude of the core energy and the core mass decrease during evolution towards core collapse. In fact $$\mid E_{c}\mid \propto M_{c}^{ \zeta}$$ where $$\zeta = (5 - 2\alpha)/(3 - \alpha)$$. The temporal behaviours of the core density and core velocity dispersion are determined in terms of α. The central density always becomes infinite in finite time but both the core mass and the core energy become zero then. An eigenvalue equation is solved numerically to determine α for a model with energy transport by stellar encounters approximated as heat conduction in a gas. It yields $$\alpha = 2.21, \zeta \simeq 3/4$$ and the detailed density and temperature profiles for the similarity solution of that model.