The rotation of a dumbbell minor planet

Many small objects in the solar systems look like a long potato. To the lowest order, they can be described as dumbbells (in fact, a couple of asteroids look like a dog-bone). In this contribution, we examine the rotation of a dumbbell under the influence of a point mass. The size of the dumbbell is much smaller than the distance of its centre of mass to the point mass. We studied three cases: the centre of mass undergoes near circular orbits, near elliptic orbits, and straight trajectories (i.e., brief encounters). For near circular orbits, the dumbbell either librates or rotates. For near elliptic orbits, the dumbbell may switch between libration and rotation in an unpredictable way. For straight trajectories, we compute the change of angular velocity in different approaching velocities, approaching angular velocities, phases of rotation and impact parameters.