On Free Fall in the Three Body Problem

TL;DR

This paper explores the complex dynamics of three particles under Newtonian gravity during free fall, revealing various solution types including periodic, collision, and breakup trajectories, and illustrating the system's singularities and sensitivities.

Contribution

It provides a detailed analysis of singularities, periodic solutions, and breakup scenarios in the three-body free fall problem, highlighting new insights into system behaviors and potential escape criteria.

Findings

Periodic solutions can occur without collisions.

Singularities can mimic elastic bounces.

Three main types of final trajectories identified.

Abstract

The free fall of three particles under Newtonian attraction allows to illustrate some of the complexities of the general three body problem. The total collapse or singularity that occurs when starting from one of the five central configurations (two triangular and three collinear) generates periodic solutions and the singularity mimics an elastic bounce. Periodic solutions without collisions where found by Standish : three particles fall from an initial triangle to each other and without colliding, come later to rest on another triangle where the motion reverses. Singularities where the motion ends, are illustrated by equal particles starting from an isosceles triangle. The lack of continuity in neighbouring solutions is illustrated by particles starting from a nearly equatorial triangle. Although the total energy is negative, an elliptic-hyperbolic break up of the system where all three particles go to infinity. is possible. Two particles are tightly bound in elliptic motion, their CoM recedes to infinity while in an hyperbolic motion with the third particle. The famous historic case of the Pythagorean triangle shows that such a break up may happen after a long time and several close passages. The break-up in a elliptic hyperbolic system occurs in a very short time period around a very close passage. Progress in the understanding the interactions between the particles when they are very close,can lead to sharper escape criteria. This review suggests that for the free fall, there are only three types of final trajectories : a) periodic with or without collisions, b) ending in a ternary collision and c) a break-up in an elliptic-hyperbolic system.