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 which follow 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 are getting closer and their mutual gravitational influence leads to a swapping of the orbits. The outer moon becoming the inner one and vice-versa, this behavior generates horseshoe-shaped trajectories depicted in an adequate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far no 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 (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. To this aim, we apply KAM theory to the planar three-body problem.