Heat and gravitation. II. Stability

Abstract

Some features of hydro- and thermodynamics, as applied to atmospheres and to stellar structures, are puzzling. 1. The suggestion, first made by Laplace, that our atmosphere has an adiabatic temperature distribution, is confirmed for the lower layers, but the reason why it should be so is difficult to understand. 2. The standard treatment of relativistic thermodynamics does not allow for a systematic treatment of mixtures, such as the mixture of a perfect gas with radiation. 3. The concept of mass in applications of general relativity to stellar structures is less than completely satisfactory. 4. Arguments in which a concept of energy plays a role, in the context of hydro-thermodynamical systems and gravitation, are not always convincing. It was proposed that a formulation of thermodynamics as an action principle may be a suitable approach to adopt for a new investigation of these matters. In this second article of a series we propose to base criteria of stability on the hamiltonian functional that is provided by the variational principle, to replace the reliance that has often been placed on ad hoc definitions of the "energy". We introduce a new virial principle that is formulated entirely within the Eulerian description of hydrodynamics, which allows a simpler derivation of a well known stability criterion for polytropic stellar configurations. Boundary conditions are based entirely on mass conservation. The new approach is tested on isothermal and polytropic atmospheres and then used to initiate a new study of stars. Traditional results for polytropic, spherical configurations are confirmed, but our study gives new insight and new results for the case that radiation is taken into account. © 2010 The Author(s).