Stability Boundaries for Resonant Migrating Planet Pairs

Convergent migration allows pairs of planet to become trapped into mean motion resonances. Once in resonance, the planets ’ eccentricities grow to an equilibrium value that depends on the ratio of migration time scale to the eccentricity damping timescale, K = τa/τe, with higher values of equilibrium eccentricity for lower values of K. For low equilibrium eccentricities, eeq ∝ K−1/2. Equilibrium eccentricities also depend on the distance between the planets. Resonances near the planet have lower equilibrium eccentricity. The stability of a planet pair depends on eccentricity so the system can become unstable before it reaches its equilibrium eccentricity. Using a resonant overlap criterion that takes into account the role of first and second order resonances and depends on eccentricity, we find a function Kmin(µp, j) that defines the lowest value for K, as a function of the ratio of total planet mass to stellar mass (µp) and the period ratio of the resonance defined as P1/P2 = j/(j + k), that allows two convergently migrating planets to remain stable in resonance at their equilibrium eccentricities. We scaled the functions Kmin for each resonance of th