Perturbative solution to the Lane-Emden equation: An eigenvalue approach

TL;DR

This paper develops a perturbative eigenvalue method to analytically approximate solutions of the Lane-Emden equation for polytropes, achieving high accuracy across a range of polytropic indices.

Contribution

It introduces a scaled Lane-Emden equation and derives analytical approximants for polytrope properties, improving solution accuracy over existing methods.

Findings

Analytical approximants are accurate within 8.1e-7% for radius and 8.5e-5% for mass for n in [0,1]

Approximate solutions remain within 2% error for n in [1,5)

The method provides a uniformly accurate solution from the origin to the surface of polytropes.

Abstract

Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index $n$. Here we use the known exact solutions of the LEE at $n=0$ and $1$ to solve the equation perturbatively. We first introduce a scaled LEE (SLEE) where polytropes with different polytropic indices all share a common scaled radius. The SLEE is then solved perturbatively as an eigenvalue problem. Analytical approximants of the polytrope function, the radius and the mass of polytropes as a function of $n$ are derived. The approximant of the polytrope function is well-defined and uniformly accurate from the origin down to the surface of a polytrope. The percentage errors of the radius and the mass are bounded by $8.1 \times 10^{-7}$ per cent and $8.5 \times 10^{-5}$ per cent, respectively, for $n\in[0,1]$. Even for $n\in[1,5)$, both percentage errors are still less than $2$ per cent.