Stability boundaries for resonant migrating planet pairs

Abstract

Convergent migration allows pairs of coplanar planets to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to equilibrium values that depends inversely on the ratio of migration time-scale to the eccentricity damping time-scale, K = τa/τe. The stability of a planet pair depends on eccentricity so the pair can become unstable before reaching the equilibrium eccentricities. Using a resonant overlap criterion that depends on eccentricity up to second order, we find a function Kmin that defines the lowest value for K, as a function of the ratio of total planet mass to stellar mass that allows two convergently migrating planets to remain stable in resonance. We found that for first-order resonance, Kmin is linear with increasing planet mass and quadratic for second-order resonances. The linear relation continues until the mass approaches a critical mass defined by the 2/7 resonance overlap instability law and Kmin → ∞. Comparing our analytic boundary with an observed sample of two-planet systems, all but one of the systems with measured eccentricities are well inside the stability region estimated by this model. We calculated Kmin for Kepler systems without well-constrained eccentricities and found only weak constraints on K. © 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.