# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.08947