Equilibrium Configurations of Tethered Three-Body Systems and Their Stability

The paper considers planar motion of a three-body tethered system whose center of mass moves in a circular orbit around the Earth. The equilibrium configurations of the system and their stability are examined and contrasted with those of the classical circular restricted three-body problem. It is observed that there are four classes of equilibrium configurations of tethered three-body systems: in two of them the bodies are collinear, while in the other two, the bodies lie in a triangular configuration. Analysis of the stability of these equilibrium configurations shows that the triangular configurations are unstable, while one of the collinear configurations (the one along the local vertical) is stable. An integral of motion is obtained, which turns out to be the Hamiltonian of the system per unit mass. The zero-velocity curves based on this Hamiltonian are presented.