A Post-Newtonian Study of Differentially Rotating Polytropes

Chandrasekhar's post-Newtonian equations of hydrodynamics are applied to a self-gravitating polytropic fluid in rapid, axisymmetric, differential rotation, and used to construct an iterative method of the "self-consistent field method" type which can be used on the computer to find equilibrium configurations. The equation of hydrostatic equilibrium is integrated to give an algebraic equation which is, in principle, as simple as the analogous Newtonian equation. Expressions are given for the mass and binding energies of configurations, and the domain of validity of the post-Newtonian approximation is discussed. The geometrical characteristics of the rotation are investigated. It is demonstrated that the distributions of angular velocity which occur in equilibrium configurations always have the pro perty that their level surfaces coincide with the level surfaces of angular momentum per unit rest mass, and that these surfaces do not correspond to surfaces whose intrinsic geometries are those of a cylinder. This is different from the situation in Newtonian theory, where the fluid in equilibrium configurations always rotates on cylinders. Subject headings: hydrodynamics - interiors, stellar - relativity - rotation, stellar