On the leier influence on a dumbbell motion in the central Newtonian force field

We consider the system moving in the Newtonian central force field and consisting of a dumbbell satellite and a particle. The particle coasts along on the cable with ends placed in the dumbbell endpoints. We call such cable a «leier». (Dutch maritime term «leier» means the rope with both fixed ends). We suppose the system mass center describes circular orbit, the particle mass is small in comparison with the dumbbell mass and the cable length is small in comparison with orbit radius. Assuming that the cable don't leave the orbit plane we study the dumbbell rotations forced by the particle. We note that the particle sufficiently influence the dumbbell motion only in vicinity of the dumbbell rotation separatrix. We claim that there exist a set of the dumbbell unstable asymptotic motions tending to librations about the orbit tangent. Initial conditions for these motions compose a surface in the system phase space. We deduce an equation approximating this surface. We consider this equation as a criterion for the direction of the dumbbell rotation from the vicinity of unstable equilibria. We deduce formulae approximating the dumbbell near-separatrix motion if the cable is rather long and the dumbbell is composed of equal masses. Using numerical procedures, we analyse the dumbbell motion in two-dimensional transections of four-dimensional space of initial conditions if the cable is rather short.