Analysis of the Fractional Relativistic Polytropic Gas Sphere

TL;DR

This paper formulates fractional relativistic TOV equations for polytropic gas spheres, introduces analytical solutions, and investigates how fractional and relativistic parameters influence stellar models, including white dwarf mass limits.

Contribution

It presents a novel fractional TOV framework with analytical solutions and explores their effects on stellar configurations, extending classical models with fractional calculus techniques.

Findings

Fractional and relativistic parameters significantly affect stellar volume and mass.

Maximum white dwarf mass approaches Chandrasekhar limit under certain parameters.

Enhanced convergence methods improve solutions of fractional TOV equations.

Abstract

Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman-Oppenheimer-Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Rieman Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions to solve the fractional TOV equations yields a limited physical range to the convergent power series solution. Therefore, the two techniques of Euler-Abel transformation and Pade approximation have been combined to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters (\sigma), we calculated twenty fractional gas models for the polytropic indexes n=0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n=0.5 are that the sphere's volume and mass decrease with increasing \sigma and the fractional parameter (\alpha). For n=1, the volume decreases when \sigma=0.1 and then increases when \sigma=0.2 and 0.3. The volume of the sphere reduces as both \sigma and \alpha increase for n=1.5 and n=2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n=3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters (\alpha=1, \sigma=0.001). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit=1.63348 M at \alpha=0.95 and \sigma=0.001.