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

摘要

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 istriggered. Chaos occurs where mean motion resonances overlap, as in Wisdom?s pioneering work. But whereasWisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearlycircular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examiningresonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for atest particle in the presence of a planet and then generalize to the case of two massive planets, based on a newapproximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, weexplore the extent to which chaotic two-planet systems eventually result in planetary collisions.