A New Analytical Solution to the Relativistic Polytropic Fluid Spheres

TL;DR

This paper develops an accelerated power series method with convergence improvements for solving the relativistic TOV equation describing polytropic fluid spheres, achieving accurate solutions across a broad parameter range.

Contribution

It introduces a novel combination of Euler-Abel transformation and Padé approximation to enhance series convergence for the TOV equation solutions.

Findings

Series converges for 0<=n<=1.5 with all sigma values.

The combined method achieves convergence for 0<=n<=3.

Maximum relative error compared to numerical solutions is about 0.001.

Abstract

This paper introduces an accelerated power series solution for Tolman-Oppenheimer-Volkoff (TOV) equation, which represents the relativistic polytropic fluid spheres. We constructed a recurrence relation for the series coefficients in the power series expansion of the solution of TOV equation. For the range of the polytropic index 0<=n<=0.5, the series converges for all values of the relativistic parameters (sigma), but it diverges for larger polytropic index. To accelerate the convergence radii of the series, we first used Pad\'e approximation. It is found that the series is converged for the range 0<=n<=1.5 for all values of sigma. For n>1.5, the series diverges except for some values of sigma. To improve the convergence radii of the series, we used a combination of two techniques Euler-Abel transformation and Pad\'e approximation. The new transformed series converges everywhere for the range of the polytropic index 0<=n<=3. Comparison between the results obtained by the proposed accelerating scheme presented here and the numerical one, revealed good agreement with maximum relative error is of order 0.001.