On the Dumb-Bell Equilibria in the Generalized Sitnikov Problem

This paper discusses and analyzes the dumb–bell equilibria in a generalized Sitnikov problem. This has been done by assuming that the dumb–bell is oriented along the normal to the plane of motion of two primaries. Assuming the orbits of primaries to be circles, we apply bifurcation theory to investigate the set of equilibria for both symmetrical and asymmetrical dumb–bells.
We also investigate the linear stability of the trivial equilibrium of a symmetrical dumb–bell in the elliptic Sitnikov problem. In the case of the dumb–bell length $l\geqslant 0.983819$, an instability of the trivial equilibria for eccentricity $e \in (0,\,1)$ is proved.