Numerical integrations of Kepler-like tightly packed planetary systems show that they are chaotic, with fairly short Lyapunov times, but that they can be stable against planet-planet or planet-star collisions for billions of years. Recent work by Petit et al. showed that this behavior was due to the overlap of three-body mean motion resonances. Caleb Lammers, Sam Hadden and I recently showed that dynamical models that account for small sets of resonances can accurately recover N-body instability times. Thus a simple physical picture emerges, in which a handful of three-body resonances, generated by interactions between nearby two-body mean motion resonances (those on either side of the planet in question), overlap and drive chaotic diffusion, leading to instability. We show that instability times are well described by a power law relating instability time to planet separations, measured in units of fractional semi-major axis difference divided by the planet-to-star mass ratio to the 1/4 power. This is in contrast to the frequently adopted 1/3 power implied by measuring separations in units of mutual Hill radii. For idealized systems, the parameters of this power-law relationship depend only on the ratio of the planets' orbital eccentricities to the orbit-crossing value, and we report an empirical fit to enable quick instability time predictions. This relationship predicts that observed systems comprised of three or more sub-Neptune-mass planets must be spaced with period ratios P≳1.35 and that tightly spaced systems (P≲1.5) must possess very low eccentricities (e≲0.05) to be stable for more than a billion orbits---a prediction for the as yet unknown eccentricities of the Kepler systems.
Host: Wei Zhu
