VARIATIONAL PRINCIPLES FOR STELLAR STRUCTURE Dallas

The four equations of stellar structure are reformulated as two alternate pairs of variational principles. Different thermodynamic representations lead to the same hydromechanical equations, but the thermal equations require, not the entropy, but the temperature as the thermal field variable. Our treatment emphasizes the hydrostatic energy and the entropy production rate of luminosity produced and transported. The conceptual and calculational advantages of the integral formulation over conventional differential formulations of stellar structure are discussed along with the difficulties in describing stellar chemical evolution by variational principles. Submitted to the Astrophysical Journal ∗ e-mail: [email protected]. † e-mail: [email protected]. I. Differential Equations of Stellar Structure 1. Static Stellar Structure as a Non-Equilibrium Steady State In thermostatic equilibrium [1], general statistical properties radically simplify a macroscopic description, but non-equilibrium systems generally do require a microscopic kinetic theory. In special non-equilibrium systems, however, it is possible to leave the microscopic physics implicit in statistical averages, and to proceed with a nominally macroscopic dynamics. Stars in a quasi-static state pass through well-defined spatial structures and temporal stages, so that a macroscopic non-equilibrium thermodynamics can be formulated. In this paper, we apply non-equilibrium thermodynamics to stellar structure, as distinct from stellar evolution. Macroscopic non-equilibrium thermodynamics began with the work of Rayleigh and Onsager; a complete theory was proposed by Prigogine and his followers which applies to an important but restricted class of systems [2]. Later developments have both extended and rivaled this classic work [3,4]. For stars, we may take a generalized Prigogine approach, assuming some type of local statistical equilibrium holds and intensive thermodynamics parameters are defined at least locally in space and time. This assumption validates a macroscopic approach. Non-equilibrium thermodynamics is typically described by conjugate pairs of variables: differences in intensive parameters, the thermodynamic forces, and macroscopic, extensive thermodynamic fluxes. Usually one relates forces to fluxes as cause to effect and assumes a linear or quasi-linear relation between the two. In this paper, we describe the quasi-static stellar structure of a star of steady luminosity as a non-equilibrium steady state (NESS) and present variational principles for both its mechanical and thermal structure. While the usual differential formulation of stellar structure integrates local quantities from point to point, our integral formulation directly calculates global quantities such as total mass, luminosity, radius which are the stellar optical observables. This approach focuses on globally observable quantities and also suggests a procedure for expressing the thermal and mechanical structure of a star analytically in terms of variational parameters with physical significance. 2. Formal Similarity Between Mechanical and Thermal Equations The four first-order differential equations of quasi-static stellar structure occur in two pairs: hydromechanical and thermal [5]. In the Euler representation and assuming spherical symmetry and conductive transport of luminosity, , −dP/dr = Gmρ/r2 , dm/dr = 4πr2ρ , −kdT/dr = l/4πr2 , dl/dr = 4πr2ρ(ε− εν) , (I.1) where the first pair is the mechanical (density-pressure) and the second the thermal (luminosity-temperature) equations. In radiative transport, the thermal conductivity k can be replaced by an equivalent radiative diffusion expression k → 4acT 5/3κρ, where κ is the opacity of matter, ac/4 the Stefan-Boltzmann constant, and the Boltzmann constant kB is set to unity throughout. (Convective forms of luminosity transport are discussed below.) The variables are: r = distance from center; m(r) = cumulative mass from center to r, P (r) = Pm(r)+Pγ(r) = total pressure (matter and radiation), ρ(r) = matter density; l(r) = photon luminosity at radius r, T (r) = common matter-photon temperature, ε(r) = luminosity production per unit mass, εν(r)=neutrino luminosity per unit mass, G =