An efficient approach for searching three-body periodic orbits passing through Eulerian configuration

TL;DR

This paper introduces an efficient symmetry-based numerical method to discover a large number of three-body periodic orbits passing through Eulerian configurations, including many new solutions, with high precision details provided.

Contribution

The paper presents a novel symmetry property-based approach for systematically finding three-body periodic orbits, significantly expanding the known solutions with high accuracy.

Findings

Produced 6,333 distinct solutions including 12,431 initial conditions

Discovered 35 free-fall solutions among the new orbits

Identified all linearly stable solutions as previously known ones

Abstract

A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on two previously established symmetry types on the shape sphere, each solution is presented by one or two distinct initial conditions (one or two points in the search domain). A numerical search based on Newton's method on a relatively coarse search grid for solutions with relatively small scale-invariant periods $T^{*}<70$ is conducted. The linear systems at each Newton's iteration are computed by high order high precision Taylor series method. The search produced 12,431 initial conditions (i.c.s) corresponding to 6,333 distinct solutions. In addition to passing through the Eulerian configuration, 35 of the solutions are also free-fall ones. Although most of the found solutions are new, all linearly stable solutions among them (only 7) are old ones. Particular attention is paid to the details of the high precision computations and the analysis of accuracy. All i.c.s are given with 100 correct digits.