Long-term instability of the inner Solar system: numerical experiments

Abstract

ABSTRACT Apart from being chaotic, the inner planets in the Solar system constitute an open system, as they are forced by the regular long-term motion of the outer ones. No integrals of motion can bound a priori the stochastic wanderings in their high-dimensional phase space. Still, the probability of a dynamical instability is remarkably low over the next 5 billion years, a time-scale 1000 times longer than the Lyapunov time. The dynamical half-life of Mercury has indeed been estimated recently at 40 billion years. By means of the computer algebra system trip, we consider a set of dynamical models resulting from truncation of the forced secular dynamics recently proposed for the inner planets at different degrees in eccentricities and inclinations. Through ensembles of 103–105 numerical integrations spanning 5–100 Gyr, we find that the Hamiltonian truncated at degree 4 practically does not allow any instability over 5 Gyr. The destabilization is mainly due to terms of degree 6. This surprising result suggests an analogy to the Fermi–Pasta–Ulam–Tsingou problem, in which tangency to Toda Hamiltonian explains the very long time-scale of thermalization, which Fermi unsuccessfully looked for.