Hundreds of new satellites of figure-eight orbit computed with high precision

TL;DR

This paper employs high-precision numerical methods to discover over 700 new satellite orbits of the figure-eight in the three-body problem, including 76 stable ones, with detailed initial conditions provided.

Contribution

It introduces a modified Newton's method combined with high-precision arithmetic to systematically find and verify new figure-eight satellite solutions.

Findings

Over 700 new satellites discovered

76 linearly stable satellites identified

Initial conditions provided with 150 decimal digits

Abstract

Satellites (topological powers) of the famous figure-eight orbit are special periodic solutions of the planar three-body problem. In this paper we use a modified Newton's method based on the Continuous analog of Newton's method and high precision arithmetic for a purposeful numerical search of new satellites of the figure-eight orbit. Over 700 new satellites are found, including 76 new linearly stable ones. 7 of the newly found linearly stable satellites are choreographies. The linear stability is checked by a high precision computing of the eigenvalues of the monodromy matrices. The initial conditions of all found solutions are given with 150 correct decimal digits.