A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.