Sun-Jupiter-Saturn System May Exist: A Verified Computation of Quasiperiodic Solutions for the Planar Three-Body Problem

Abstract

In this paper, we present evidence of the stability of a model of our Solar System when taking into account the two biggest planets, a planar (Newtonian) Sun-Jupiter-Saturn system with realistic data: masses of the Sun and the planets, their semiaxes, eccentricities and (apsidal) precessions of the planets close to the real ones. (We emphasize that our system is not in the perturbative regime but for fixed parameters.) The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM-iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem.