Approximate Analytical Solutions to the Relativistic Isothermal Gas Spheres

TL;DR

This paper develops an improved analytical method combining Euler-Abel transformation and Pade approximation to solve the TOV equation for relativistic isothermal spheres, achieving higher convergence and accuracy, validated by neutron star application.

Contribution

It introduces a novel analytical solution method that enhances convergence and accuracy for the TOV equation in relativistic isothermal spheres, surpassing traditional power series approaches.

Findings

The combined method significantly improves convergence radii.

Results agree well with numerical solutions, with errors around 10^-3.

The approach accelerates solutions by about ten times compared to traditional methods.

Abstract

In this paper, we introduce a novel analytical solution to Tolman-Oppenheimer-Volkoff (TOV) equation, which is ultimately a hydrostatic equilibrium equation derived from the general relativity in the framework of relativistic isothermal spheres. Application of the traditional power series expansions on solving TOV equation results in a limited physical range to the convergent power series solution. To improve the convergence radii of the obtained series solutions, a combination of the two techniques of Euler-Abel transformation and Pade approximation has done. The solutions are given in \exi-\theta and \exi-\mu phase planes taking into account the general relativistic effects \sigma= 0.1, 0.2 and 0.3. An Application to a neutron star has done. A Comparison between the results obtained by the suggested approach in the present paper and the numerical one indicates a good agreement with a maximum relative error of order 10-3, which establishes the validity and accuracy of the method. The procedure we have applied accelerated the power series solution with about ten times than of traditional one.