On The Plane Motions of a Dumb-Bell on a Manifold “Gravity Propeller” In The Generalized Elliptic Sitnikov Problem

The translational-rotational motions of a symmetrical dumb-bell are considered in a generalized elliptical Sitnikov problem. We describe the equations of the dumb-bell motion and their integral manifold “gravitational propeller” which contains the motion such that the dumb-bell barycenter moves along the normal to the motion plane of two primaries, whilst the dumb-bell itself rotates continuously around the normal keeping a constant angle π/2 with normal. We obtained the equation of plane dumb-bell oscillations when its barycenter coincides with the barycenter of primaries. It is shown that this equation coincides with the Beletskii equation if the dumb-bell has an infinitesimal length. Small plane oscillations of dumb-bell are investigated by introducing two small parameters: e (the eccentricity of primaries orbit) and ε (a measure of the deviation of the phase point from the origin). The regions of singular and regular small oscillations and different types of equations for the regular domain are described. We have an increase in the frequency of oscillations to infinity with an increase in the length of the dumb-bell and the tendency of its point masses to pass near the primaries.