Satellite

Quantum mechanics

Chaotic

Hamiltonian system

Astrophysics

Phase space

Computer science

Orbital elements

Orbital mechanics

Classical mechanics

Terrestrial planet

Outer planets

Thermodynamics

Physics

Nonlinear system

Planetary system

Astronomy

Lyapunov exponent

Artificial intelligence

Planet

Solar System

Celestial mechanics

Astronomy and Astrophysics

Although the discovery of the chaotic motion of the inner planets in the solar system dates back to more than thirty years ago, the secular chaos of their orbits still dares more analytical analyses. Apart from the high-dimensional structure of the motion, this is probably related to the lack of an adequately simple dynamical model. Here, we consider a new secular dynamics for the inner planets, with the aim of retaining a fundamental set of interactions responsible for their chaotic behaviour, while being consistent with the predictions of the most precise orbital solutions currently available. We exploit the regularity in the secular motion of the outer planets, 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. On top of that, the smallness of the inner planet masses and the absence of strong mean-motion resonances permits 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 for analytical inspection of planet interactions, once the Hamiltonian is truncated at a given total degree in eccentricities and inclinations. The new model matches very satisfactorily reference orbital solutions of the solar system over timescales shorter than or comparable to the Lyapunov time. 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.

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