On the Dynamical Stability of the Solar System

A long-term numerical integration of the classical Newtonian approximation to the planetary orbital motions of the full solar system (Sun + eight planets), spanning 20 Gyr, was performed. The results showed no severe instability arising over this time interval. Subsequently, utilizing a bifurcation method described by Jacques Laskar, two numerical experiments were performed with the goal of determining dynamically allowed evolutions for the solar system in which the planetary orbits become unstable. The experiments yielded one evolution in which Mercury falls onto the Sun at ~1.261 Gyr from now, and another in which Mercury and Venus collide in ~862 Myr. In the latter solution, as a result of Mercury's unstable behavior, Mars was ejected from the solar system at 822 Myr. We have performed a number of numerical tests that confirm these results and indicate that they are not numerical artifacts. Using synthetic secular perturbation theory, we find that Mercury is destabilized via an entrance into a linear secular resonance with Jupiter in which their corresponding eigenfrequencies experience extended periods of commensurability. The effects of general relativity on the dynamical stability are discussed. An application of the bifurcation method to the outer solar system (Jupiter, Saturn, Uranus, and Neptune) showed no sign of instability during the course of 24 Gyr of integrations, in keeping with an expected Uranian dynamical lifetime of 10^(18) yr