General Relativistic Polytropic Fluid Spheres.
Abstract
This paper discusses solutions of the field equations of general relativity for a compressible fluid sphere in gravitational equilibrium under the assumption that the fluid obeys a polytropic equation of state. With suitable transformations the equations of static equilibrium are equivalent to two coupled first order nonlinear differential equations analogous to the LaneEmden equation in the Newtonian theory of polytropes. Solutions of the equilibrium equations are obtained in terms of the polytropic index n and a parameter whose physical interpretation is the ratio of pressure to energy density at the center of the sphere. The quantity (Tmeasures the deviation from Newtonian gravitational theory. A solution is obtained in closed form for n = 0; this corresponds to the Schwarzschild interior solution for a fluid sphere of uniform density. Solutions for n = 1.0(0.5)3.0 are obtained by numerical integration. The ratio of total mass to invariant radius of a polytropic sphere is found in terms of boundary values of the relativistic LaneEmden functions. Integrals for the gravitational potential energy and rest energy are obtained and evaluated numerically. Properties of the solutions are tabulated for each n and a range of values of . The distributions of density, pressure, mass, and metric tensor components are shown graphically for some typical cases Plots of the massradius relation are given in a form suitable for determination of the internal structure of a polytrope of given mass, radius, and polytropic index. The existence of multiple solutions for some values of mass and radius is a generalrelativistic feature. The maximum ratio of half the gravitational radius to the geometrical radius is 0.214 for n = 1.0 and 0.0631 for n = 3.0. These values are smaller than the limiting ratio 0.340 previously known for the Schwarzschild interior solution (n = 0.0). It appears that models with n = 3.0 and ff> 0 5 are energetically unstable. The gravitational collapse of massive ( 10'M0) starlike objects which may exist near the centers of galaxies is discussed as a possible generalrelativistic mechanism for producing the large amounts of energy ( 10" ergs) associated with strong radio sources.
 Publication:

The Astrophysical Journal
 Pub Date:
 August 1964
 DOI:
 10.1086/147939
 Bibcode:
 1964ApJ...140..434T