Diversity and Origin of 2 : 1 Orbital Resonances in Extrasolar Planetary Systems

A diversity of 2 : 1 resonance configurations can be expected in extrasolar planetary systems, and their geometry can provide information about the origin of the resonances. Assembly during planet formation by the differential migration of planets due to planet-disk interaction is one scenario for the origin of mean-motion resonances in extrasolar planetary systems. The stable 2 : 1 resonance configurations that can be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits are (1) antisymmetric configurations with the mean-motion resonance variables θ 1 = λ 1 - 2λ 2 + ϖ 1 and θ 2 = λ 1 - 2λ 2 + ϖ 2 (where λ j and ϖ j are the mean longitudes and the longitudes of periapse, respectively) librating about 0° and 180°, respectively (as in the Io-Europa pair), (2) symmetric configurations with both θ 1 and θ 2 librating about 0° (as in the GJ 876 system), and (3) asymmetric configurations with θ 1 and θ 2 librating about angles far from either 0° or 180°. There are, however, stable 2 : 1 resonance configurations with symmetric (θ 1 ≈ θ 2 ≈ 0°), asymmetric, and antisymmetric (θ 1 ≈ 180° and θ 2 ≈ 0°) librations that cannot be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits. If real systems with these configurations are ever found, their origin would require (1) a change in the planetary mass ratio m 1 / m 2 during migration, (2) a migration scenario involving inclination resonances, or (3) multiple-planet scattering in crowded planetary systems. We find that the asymmetric configurations with large e 2 and the θ 1 ≈ 180° and θ 2 ≈ 0° configurations have intersecting orbits and that the θ 1 ≈ θ 2 ≈ 0° configurations with e 1 > 0.714 have prograde periapse precessions.