The "standard Model" for Massive Stars in General Relativity

Abstract

Using general relativity, this paper describes models of very massive hot stars composed of ideal gas and radiation, in which the ratio of gas pressure to total pressure is assumed to be constant throughout a particular configuration. Under this assumption, a relation between pressure and energy density is obtained which is similar in form to an adiabatic equation of state with n = 3, but which includes the internal energy density of the gas as well as the radiation. The solutions of the general-relativistic equations of hydrostatic equilibrium (for spherical symmetry) appropriate to this equation of state depend on two parameters: the ratio a of central pressure to central rest-energy density, and the ratio p of gas pressure to total pressure Expressions are given which relate the variables describing the internal structure of the configurations to the solutions of the differential equations of equilibrim The mass and radius are expressed in terms of a and p but depend on quantities that have to be found by solving the equilibrium equations. The central values of the temperature, rest-mass density, and pressure are expressed in terms of a and p directly. Integrals for the rest energy, proper kinetic energy, and energy of radiation are obtained, and the integral for the rest energy appears in an expression for the binding energy of tJie system. Stability against radially symmetric perturbations is determined using Chandrasekhar s variational principle. For p 1, corresponding to the most massive objects, the equations of equilibrium have the same form as those for an adiabatic sphere with n = 3, and the solutions depend on the parameter a only. The results from numerical integration of the equilibrium equations are presented as contours of constant mass and radius in the log,, a-log,, p-plane. The region of stability in this plane is obtained by numerical calculation from the variational principle. For p I, instability occurs for small values of a, and a post-Newtonian approximation is adequate to describe the stable configurations (The equivalent Newtonian models are all stable if the ratio of specific heats of the gas is greater than ,4.) When p is not small compared to unity, the equations must be integrated to give a two-parameter family of solutions. The binding energy can be calculated correctly only when p is not set equal to zero. For large masses and radii the instability sets in along a curve in the log,, a-log,, p-plane where the binding energy as a function of a, with fixed p, has a relative maximum. For a marginally stable object of mass 10' Alo, the radius is 0.5 light-year. A post-Newtonian approximation giving the internal structure of the configurations for small a and p is presented in the Appendix. The boundary between stable and unstable solutions, as calculated in this approximation, is given by the equation logi, p,, = 1.1985 + log,, a.