Stability Analysis of Relativistic Polytropes

TL;DR

This paper investigates the stability of relativistic polytropic spheres in astrophysics, identifying critical parameters for stability and comparing analytical and numerical solutions to validate findings.

Contribution

It provides new critical relativistic parameters for stability across various polytropic indices, confirming previous results and enhancing understanding of relativistic star stability.

Findings

Stable polytropes occur below critical relativistic parameters.

Unstable solutions arise when parameters exceed critical values.

Analytical and numerical solutions agree within a small error margin.

Abstract

A main question in astrophysics and cosmology has been the severe stability of the astrophysical objects, whether a particular equilibrium configuration is stable. In this article, we study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state , considering structures with the polytropic indices and illustrates the results for the relativistic parameters . We determined the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n=1 (0.5)-2.5, stable relativistic polytropes occur for less than the critical values 0.42, 0.20, 0.10, and 0.04 respectively, while unstable relativistic polytropes are obtained when the relativistic parameter is greater than the same values. When n=3.0 and, energetically unstable solutions have occurred. The results of critical values obtained in this paper for different polytropic indices are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order.