On the Co-orbital Motion in the Three-Body Problem: Existence of Quasi-periodic Horseshoe-Shaped Orbits

Janus and Epimetheus are two moons of Saturn with very peculiar motions. As they orbit around Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies approach each other and their mutual gravitational influence lead to a swapping of the orbits: the outer moon becomes the inner one and vice-versa. This behavior generates horseshoe-shaped trajectories depicted in an appropriate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results on the “horseshoe motion” have been obtained even in the restricted three-body problem. Adapting the idea of Arnol’d (Russ Math Surv 18:85–191, 1963) to a resonant case (the co-orbital motion is associated with trajectories in 1:1 mean motion resonance), we provide a rigorous proof of existence of 2-dimensional elliptic invariant tori on which the trajectories are similar to those followed by Janus and Epimetheus. For this purpose, we apply KAM theory to the planar three-body problem.