Rayleigh’s stability criterion for circular orbits in Newtonian gravity and general relativity

The study of circular orbits under the action of central forces or in spherically symmetric spacetimes is a well-treated subject in introductory courses of clasical mechanics and general relativity, respectively. The stability of these orbits is usualy presented by means of the analysis of maxima and minima of the effective potential for different values of angular momentum. We present, in a didactic way, Rayleigh’s stability criterion, which allows an analysis equivalent to the one regarding the effective potential behavior, but more practical, taking into account the behavior of only one function: the angular momentum of circular orbits as a function of radius. We argue that this criterion, which is usually not presented in the disciplines mentioned above, may be naturally included in the respective syllabuses, contributing to the students’ learning process.

Keywords:
Newtonian gravity; General relativity; Circular Orbits