Structure of Polytropic Stars in General Relativity

TL;DR

This paper models the internal structure of polytropic stars within General Relativity, deriving solutions and stability conditions, and demonstrating the unique determination of matter density from thermodynamic parameters.

Contribution

It provides analytic and numerical solutions for relativistic polytropic stars and establishes the unique link between thermodynamic parameters and matter density.

Findings

Derived a single equation for mass distribution in relativistic stars.

Presented analytic and numerical solutions for star structure.

Proved the matter density is uniquely determined by thermodynamic parameters.

Abstract

The inner structure of a star or a primordial interstellar cloud is a major topic in classical and relativistic physics. The impact that General Relativistic principles have on this structure has been the subject of many research papers. In this paper we consider within the context of General Relativity a prototype model for this problem by assuming that a star consists of polytropic gas. To justify this assumption we observe that stars undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due to the emission of this energy it is worthwhile to consider an idealized model in which the gas is polytropic. To find interior solutions to the Einstein equations of General Relativity in this setting we derive a single equation for the cumulative mass distribution of the star and use Tolman-Oppenheimer-Volkoff equation to derive formulas for the isentropic index and coefficient. Using these formulas we present analytic and numerical solutions for the polytropic structure of self-gravitating stars and examine their stability. We prove also that when the thermodynamics of a star as represented by the isentropic index and coefficient is known, the corresponding matter density within the star is uniquely determined.