Arnold Diffusion in the Full Three-Body Problem

Abstract

The full three-body problem, on the motion of three celestial bodies under their mutual gravitational attraction, is one of the oldest unsolved problems in classical mechanics. The main difficulty comes from the presence of unstable and chaotic motions, which make long-term prediction impossible. In this paper, we show that the full three-body problem exhibits a strong form of instability known as Arnold diffusion. We consider the planar full three-body problem, formulated as a perturbation of both the Kepler problem and the planar circular restricted three-body problem. We show that the system exhibits Arnold diffusion, in the sense that there is a transfer of energy -- of an amount independent of the perturbation parameter -- between the Kepler problem and the restricted three-body problem. Our argument is based on the topological method of correctly aligned windows, which is implemented into a computer assisted proof. We demonstrate that the approach can be applied to physically relevant masses of the bodies, choosing a Neptune-Triton-asteroid system as an example. In this case, we obtain explicit estimates for the range of the perturbation parameter and for the diffusion time.