Fourier-Taylor series for the figure eight solution of the three body problem

TL;DR

This paper introduces a Fourier-Taylor series method to analytically approximate the figure eight solution of the three body problem, achieving high accuracy near a specific parameter value, with validation on simpler problems.

Contribution

It develops a novel Fourier-Taylor series approach for approximating complex periodic solutions in celestial mechanics, specifically applied to the figure eight three body problem.

Findings

The Fourier-Taylor series up to 30th order closely approximates the numerical solution.

The method successfully applies to simpler problems, validating its effectiveness.

Unphysical solutions occur except at a specific parameter value where the approximation is accurate.

Abstract

We provide an analytical approximation of a periodic solution of the three body problem in celestial mechanics, the so-called figure eight solution, discovered 1993 by C. Moore. This approximation has the form of a Fourier series whose components are in turn Taylor series w. r. t. some parameter. The method is first illustrated by application to two other problems, (1) the problem of oscillations of a particle in a cubic potential that has a well-known analytic solution in terms of elliptic functions and (2) periodic solutions and corresponding eigenvalues of a generalized Mathieu equation that cannot be solved analytically. When applied to the three body problem it turns out that the Fourier-Taylor series, evaluated up to 30th order, represents un-physical solutions except for a particular value of the series parameter. For this value the series approximates the numerical solution known from the literature up to a relative error of $1.6\times 10^{-3}$.