Analytical Study of the Co-orbital Motion in the Circular Restricted Three-body Problem

Abstract

In the restricted three-body problem (RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the co-orbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits (Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.