Long-term dynamics of the inner planets in the Solar System

Although the discovery of the chaotic motion of the inner planets in the Solar System (Mercury to Mars) was made more than 30 years ago, the secular chaos of their orbits still demands more analytical analyses. In addition to the high-dimensional structure of the motion, this is probably related to the lack of an adequately simple dynamical model. We consider a new secular dynamics for the inner planets, with the aim of retaining a fundamental set of interactions that explains their chaotic behaviour and at the same time is consistent with the predictions of the most precise orbital solutions currently available. We exploit the regularity in the secular motion of the outer planets (Jupiter to Neptune) to predetermine a quasi-periodic solution for their orbits. This reduces the secular phase space to the degrees of freedom dominated by the inner planets. In addition, the low masses of the inner planets and the absence of strong mean-motion resonances permits us to restrict ourselves to first-order secular averaging. The resulting dynamics can be integrated numerically in a very efficient way through Gauss's method, while computer algebra allows an analytical inspection of planet interactions when the Hamiltonian is truncated at a given total degree in eccentricities and inclinations. The new model matches reference orbital solutions of the Solar System over timescales shorter than or comparable to the Lyapunov time very satisfactorily. It correctly reproduces the maximum Lyapunov exponent of the inner system and the statistics of the high eccentricities of Mercury over the next five billion years. The destabilizing role of the g(1) - g(5) secular resonance also arises. A numerical experiment, consisting of a thousand orbital solutions over one hundred billion years, reveals the essential properties of the stochastic process driving the destabilization of the inner Solar System and clarifies its current metastable state.