The existence and stability of “orbitally uniform” rotations of a vibrating dumbbell on an elliptic orbit

The problem of planar motions of a variable-length dumbbell-shaped body in the central field of the Newton attraction is considered. The rules of varying the dumbbell length, at which the dumbbell with respect to the true anomaly can rotate uniformly around the center of mass, are indicated for the equation of motion of the dumbbell written in a satellite approximation 1–8]. When fulfilling these rules, the necessary stability conditions of mentioned motions are investigated.