Numerical exploration of the probability of capture into the 3:1 mean motion resonance

Abstract

Mean-motion resonances (MMR) 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 resonance in both the radial velocity and the Kepler data is probably due to the difference in the capture behaviors of 2:1 and 3:1 resonances.

To study the capture probability of the 3:1 resonance, numerical three-body integrations with forced migration have been used to examine the dependence of the capture probability on migration rate, planetary masses, and initial orbital eccentricities. First, the numerical results have been confirmed with analytic theory in the adiabatic limit (Borderies & Goldreich 1984) and numerical results of the Hamiltonian model beyond this limit (Mustill & Wyatt 2011) for both the interior and exterior resonances in the circular restricted three-body problem. Then, the numerical exploration of the restricted three-body problem (R3BP) has been extended to cases with 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 large test particle eccentricity. Interestingly, the critical eccentricity of the planet, below which resonance capture is certain, is much larger than the critical eccentricity of test particle which was not expected. Finally, the numerical exploration has been extended to situations with different planetary mass ratio m1/m2. In the cases where both planets are initially on circular orbits, the critical migration rate for certain capture agrees with that of Quillen (2006) in the R3BP. However, it does not change monotonically with m1/m2 and peaks at m1/m2 = 1. For m1/m2 = 1, the resonance capture is certain when the eccentricities of the inner and outer planets are small and decreases as the eccentricities increase. In contrast, the capture probability is low when the eccentricities are small and the capture probability peaks at certain values of the eccentricities in the non-adiabatic limits. The capture probability as a function of planet eccentricities for mass ratios m1/m2 = 0.5 and 2 in the adiabatic limit has also been studied. The capture probability at m1/m2 = 2 shows similar behaviors with m1/m2 = 1 but the capture behaviors at m1/m2 = 0.5 are significantly different from the capture behaviors at m1/m2 = 1.

This research has explored the probability of resonant capture in several new regimes, including the elliptical restricted three-body problem, comparable mass cases in the adiabatic limit and the equal mass case in the non-adiabatic limits. This work enhances our knowledge in the capture behaviors of 3:1 MMR in different limits and is useful in the future studies of the period ratio distribution in extrasolar planet systems.