General Relativistic Stars: Polytropic Equations of State

In this paper, the gravitational field equations for static sphericallysymmetric perfect fluid models with a polytropic equation of state,$p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\itregular} systems of ordinary differential equations on compact state spaces.The systems are analyzed numerically and qualitatively, using the theory ofdynamical systems. Certain key solutions are shown to form building blockswhich, to a large extent, determine the remaining solution structure. In oneformulation, there exists a monotone function that forces the generalrelativistic solutions towards a part of the boundary of the state space thatcorresponds to the low pressure limit. The solutions on this boundary describeNewtonian models and thus the relationship to the Newtonian solution space isclearly displayed. It is numerically demonstrated that general relativisticmodels have finite radii when the polytropic index $n$ satisfies $0\leq n\lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n