More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits

TL;DR

This paper numerically discovers over 600 new families of Newtonian periodic three-body orbits with equal masses and zero angular momentum, significantly expanding the known solutions and suggesting a quasi Kepler's third law.

Contribution

The authors numerically identify more than 600 new families of periodic three-body orbits, vastly increasing the known solutions in this classical problem.

Findings

Discovered 695 families of periodic orbits, including known and new ones.

Proposed a quasi Kepler's third law for these orbits.

Provided visualizations and online resources for the orbits.

Abstract

The famous three-body problem can be traced back to Isaac Newton in 1680s. In the 300 years since this "three-body problem" was first recognized, only three families of periodic solutions had been found, until 2013 when \v{S}uvakov and Dmitra\v{s}inovi\'c [Phys. Rev. Lett. 110, 114301 (2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known Figure-eight family found by Moore in 1993, the 11 families found by \v{S}uvakov and Dmitra\v{s}inovi\'c in 2013, and more than 600 new families that have been never reported, to the best of our knowledge. With the definition of the average period $\bar{T} = T/L_f$, where $L_f$ is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law $ \bar{T}^* \approx 2.433 \pm 0.075$ for the considered case, where $\bar{T}^*= \bar{T} |E|^{3/2}$ is the scale-invariant average period and $E$ is its total kinetic and potential energy, respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere" can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm