Adiabatic Fluid Spheres in General Relativity.
Abstract
This paper describes solutions of the spherically symmetric hydrostatic equations of general relativity for a compressible fluid obeying a pressureenergy density relation of the form P = Kpgl+lIfl, t = PgG2 + flP. This equation is satisfied by an ideal gas undergoing an adiabatic process. Relations of this form are also obtained for degenerate gases in which the constituent particles are nonrelativistic or extremely relativistic (in the sense of special relativity). Equations of equilibrium analogous to the LaneEm den equation in the Newtonian theory of polytropes are derived. Numerical solutions of these equations have been obtained in terms of the ratio a of pressure to rest energy density at the center of the configuration. The mass and radius are found in terms of a and the adiabatic constant K. The metric components, pressure, density, and speed of sound are expressed in terms of the solutions to the hydrostatic equations. Integrals for the gravitational potential energy and binding energy are derived and evaluated numerically. The values of a at which instability against radial perturbations sets in are found using Chandrasekhar's variational principle. For n < 3 the mass as a function of a rises to a first maximum and then oscillates with diminishing amplitude about an asymptotic value. The objects become unstable at the first maximum in the mass For n = 3, the configurations are unstable for all values of a. The binding energy also reaches a maximum at the point of instability. As a increases, the binding energy eventually becomes negative, but there are unstable configurations with positive binding energy. For n = 3, the binding energy is always negative. A graphical method is given for determining the parameter a, and hence the internal structure, of a configuration of specified mass and radius. No more than one characteristic value of a yielding a stable configuration can be found, and for some combinations of mass and radius this boundaryvalue problem is shown to have no solution. (In the Newtonian theory of polytropes, there is always an equilibrium configuration with given mass, radius, and polytropic index.) Applications are made to degenerate stars, including white dwarfs in which the electron gas is extremely relativistic (n = 3), and neutron stars in which the neutrons are nonrelativistic (n = ). The essential features of the neutron stars described by Oppenheimer and Volkoff are obtained with n = , using the appropriate value for K. Objects composed of baryons interacting with each other through a vector meson field, as described by Zel'dovich, are considered under the case n = 1. Using reasonable estimates for the baryon charge and the mass of the vector meson, the maximum mass attained by a Zel'dovich baryon star is 3 Mo. Finally, an appendix gives a postNewtonian approximation that adequately describes configurations for which a <= 102.
 Publication:

The Astrophysical Journal
 Pub Date:
 November 1965
 DOI:
 10.1086/148435
 Bibcode:
 1965ApJ...142.1541T