General Relativistic Stars : Polytropic Equations of State

TL;DR

This paper reformulates the equations governing static relativistic stars with polytropic equations of state into dynamical systems, analyzing their solutions' structure, and determining conditions for finite or infinite radii based on the polytropic index.

Contribution

It introduces a novel dynamical systems approach to analyze relativistic star models with polytropic equations of state, revealing solution structures and radius conditions.

Findings

Relativistic models have finite radii for n ≤ 3.339.

Models with n ≥ 5 have infinite radii.

Intermediate n values show mixed finite and infinite radii solutions.

Abstract

In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic solutions towards a part of the boundary of the state space that corresponds to the low pressure limit. The solutions on this boundary describe Newtonian models and thus the relationship to the Newtonian solution space is clearly displayed. It is numerically demonstrated that general relativistic models have finite radii when the polytropic index $n$ satisfies $0\leq n \lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n<5$, there exists a 1-parameter set of models with finite radii and a finite number, depending on $n$, with infinite radii.