Orbits of the three-body problem with large potential

TL;DR

This paper constructs solutions to the planar three-body problem with large potential energy, featuring a tight binary and a distant third body, avoiding triple collision with specific energy and angular momentum conditions.

Contribution

It demonstrates the existence of solutions maintaining high potential energy and a specific orbital structure, expanding understanding of possible three-body configurations.

Findings

Solutions with arbitrarily large potential energy exist.

Configurations include a tight binary and a distant third body.

Solutions avoid triple collision under given conditions.

Abstract

Consider the planar three-body problem with masses positive $m_1,m_2,m_3$ position vector $q(t) = (q_1(t),q_2(t),q_3(t))\in\mathbb{R}^6$. Let $$U(q) = \frac{m_1m_2}{r_{12}}+\frac{m_1m_3}{r_{13}}+\frac{m_2m_3}{r_{23}}$$ where $r_{ij}=|q_i-q_j|$. Assume that the angular momentum is nonzero so that triple collision is impossible and fix any negative energy.. Then given any constant $K>0$ there are solutions with $U(q(t))\ge K$ for all $t\in\mathbb{R}$. These solutions will have a single close approach to triple collision. The configuration will always be a tight binary with $m_1, m_2$ close and the distance from the binary to $m_3$ diverging as $t\rightarrow\pm\infty$.