Tidal locking and the gravitational fold catastrophe

The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally-locked circular orbits, is regulated by a single dimensionless control parameter which depends on the properties of the two bodies and on the total angular momentum of the system. The method described in this work results in compact expressions for the radius of the circular orbit and the tidally-locked spin/orbital frequency. The limiting case in which one of the two orbiting objects is point-like is studied in detail. An analysis of the effective potential, which in this limit depends on only two parameters, allows one to clearly visualize the properties of the system. The notorious case of the Mars' moon Phobos is presented as an example of a satellite that is past the no return point and, therefore, will not reach a stable or unstable tidally-locked orbit.