Thermodynamics with an Action Principle Heat and gravitation

Some features of hydro-and thermodynamics, as applied to atmospheres and to stellar structures, are puzzling: 1. It is not understood or universally agreed whether an isolated gas, in an external gravitational field, tends spontaneously to a final state in which the temperature is uniform. 2. Arguments in which a concept of energy plays a role, in the context of hydro-thermodynamical systems and gravitation, are often flawed, and some familiar results concerning the stability of model stellar structures, first advanceded at the end of the 19th century and repeated in the most modern textbooks, are less than completely convincing. 3. 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. 4. The concept of mass in applications of general relativity to stellar structure is unsatisfactory. It is proposed that a formulation of thermodynamics as an action principle is a suitable aproach to adopt for a new investigation of these matters. We formulate thermodynamics of ideal gases in terms of an action principle and study the interaction between an ideal gas and the photon gas, or heat. The action principle provides a hamiltonian functional, not available in traditional approaches where familiar expressions for the energy have no operative meaning. The usual polytropic atmosphere in an external gravitational field is examined under ideal conditions of complete isolation, in order to determine what, if any, incident radiation is required to sustain it. The conclusion is that solar radiation may have little to do with the observed temperature gradient of the earth's atmosphere. This lends support for popular atmospheric models, but a corollary is that an isolated system in a gravitational field may not reach isothermal equilibrium. Independent arguments, involving a centrifuge and the equivalence principle, are adduced to support the latter conclusion. Solar radiation is required to keep the atmosphere warm, but it may have little to do with the existence of the temperature gradient. Inclusion into the dynamics, of the photon gas that coexists with the atomic gas, and a formulation based on an action principle, help to set up a framework for the discussion. A new formulation of the concept of radiative equilibrium is proposed. The choice of boundary conditions is clarified with the help of a properly defined, conserved mass distribution.