The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems

Abstract

Self-gravitating systems have negative specific heats, thus if heat is allowed to flow between two of them, the hotter one loses heat and gets yet hotter while the colder gains heat and gets yet colder. Evolution is thus away from equilibrium. When a single isothermal sphere within a non-conducting box is sufficiently centrally condensed a similar instability arises between the central parts and the outer parts; as a result no equilibrium states exist for an isothermal sphere of energy E (<0) and mass M within a spherical box of radius greater than 0.335 GM2 /(− E ). This is Antonov's discovery that no state of locally maximal entropy exists for stellar systems of given energy and mass contained within a rigid sphere of radius larger than this. The instability is distinct from that found by Ebert which is similar to the Schönberg–Chandrasekhar limit in stars and relates to isothermal spheres at fixed temperature. In fact there are four distinct critical points for instability of isothermal spheres which are related to the turning points of the four total thermodynamic free energies by Poincar^'s theory of linear series of equilibria. This study of the thermodynamics of self-gravitating spheres gives insight on the evolution and the final fate of stellar systems. It also helps in the understanding of some well known phenomena in stellar evolution. It is emphasized that these results prove that the escape of stars from a cluster is not necessary for its evolution but rather that extended systems naturally grow a core–halo structure reminiscent of the internal constitution of a red-giant star.