Newtonian restricted three-body gravitational problem with positive and negative point masses

TL;DR

This paper analyzes a Newtonian restricted three-body problem involving both positive and negative point masses, identifying five Lagrange points and their stability properties, including a unique stable point under specific mass conditions.

Contribution

It introduces a novel three-body problem with negative masses and determines the stability and configuration of Lagrange points in this context.

Findings

Five Lagrange points identified, with two non-coplanar due to gravitational repulsion.

All points are linearly unstable except one in a specific mass regime.

The stable point exists when the primary mass exceeds approximately 8.4 times the negative secondary mass.

Abstract

The Newtonian restricted three-body problem involving a positive primary point mass, $m_+$, and a negative secondary point mass, $m_-$, in a circular orbit, and a positive or negative tertiary point mass, $m_3$, with $m_+ > |m_-| \gg |m_3|$, is solved. Five Lagrange points are found for $m_3$, three of which are coplanar with $m_+$ and $m_-$, and two of which are not, a subtle consequence of the gravitational repulsion from $m_-$. All Lagrange points are linearly unstable, except for one point in the regime $m_+ \gtrsim 8.4 |m_-|$, which is linearly stable and collinear with $m_+$ and $m_-$.