The instability mechanism of compact multiplanet systems

Abstract

To improve our understanding of orbital instabilities in compact planetary systems, we compare suites of $N$-body simulations against numerical integrations of simplified dynamical models. We show that, surprisingly, dynamical models that account for small sets of resonant interactions between the planets can accurately recover $N$-body instability times. This points toward a simple physical picture in which a handful of three-body resonances, generated by interactions between nearby two-body mean motion resonances, overlap and drive chaotic diffusion, leading to instability. Motivated by this, 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, rather than 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 $P \gtrsim 1.35$ and that tightly spaced systems ($P \lesssim 1.5$) must possess very low eccentricities ($e \lesssim 0.05$) to be stable for more than $10^9$ orbits.