Probability of capture for the 3:1 mean-motion resonance

Abstract

Mean-motion resonances are frequently observed in extrasolar planetary systems. It is generally believed that the resonances result from the convergent migration of planets. The much larger number of systems near the 2:1 resonance compared to 3:1 in both the radial velocity and Kepler data may be due to a difference in the capture behaviors of 2:1 and 3:1 resonances. To study the capture probability of 3:1 resonance, we use numerical three-body integrations with forced migration to examine how the probability depends on migration rate, planetary masses, and initial orbital eccentricities. We first confirm our numerical results with analytic theory in the adiabatic limit (Borderies & Golderich 1986) and numerical results of Hamiltonian model beyond this limit (Mustill & Wyatt 2010) for both the interior and exterior resonances in the circular restricted three-body problem. We then extend our numerical exploration of the restricted three-body problem to non-zero planet eccentricity in the adiabatic limit. The capture probability decreases with increasing planet eccentricity at small test particle eccentricity but does not depend strongly on the planet eccentricity at higher test particle eccentricity. Finally, we extend beyond the restricted problem to different planetary mass ratio m2/m1. In the cases where both planets are initially on circular orbits, we find that the critical migration rate for certain capture agrees with that of Quillen (2006) in the limit that one of the bodies is a test particle but that it does not change monotonically with m2/m1 and peaks at m2/m1 = 1. In the adiabatic regime, the capture probability for comparable masses (m2 m1) shows oscillatory behaviors as a function of eccentricities, which is significantly different from the test particle limits.