  In the cosmos, any two bodies share a gravitational attraction. When in
proximity to one another in empty space, their motions can be modeled by
Newtonian gravity. Newton found their orbits when the two bodies are infinitely
small, the so-called two-body problem. The general situation in which the
bodies have varying shapes and sizes, called the full two-body problem, remains
open. We find relative equilibria (RE) and their stability for an approximation
of the full two-body problem, where each body is restricted to a plane and
consists of two point masses connected by a massless rod, a dumbbell. In
particular, we find symmetric RE in which the bodies are arranged colinearly,
perpendicularly, or trapezoidally. When the masses of the dumbbells are
pairwise equal, we find asymmetric RE bifurcating from the symmetric RE. And
while we find that only the colinear RE have nonlinear/energetic stability (for
sufficiently large radii), we also find that the perpendicular and trapezoid
configurations have radial intervals of linear stability. We also provide a
geometric restriction on the location of RE for a dumbbell body and any number
of planar rigid bodies in planar orbit (an extension of the Conley
Perpendicular Bisector Theorem).
