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

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 $m_{-}$.

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