Evolution of eccentricity and orbital inclination of migrating planets in 2:1 mean motion resonance

We determine, analytically and numerically, the conditions needed for a system of two migrating planets trapped in a 2:1 mean motion resonance to enter an inclination-type resonance. We provide an expression for the asymptotic equilibrium value that the eccentricity ei of the inner planet reaches under the combined effects of migration and eccentricity damping. We also show that, for a ratio q of inner to outer masses below unity, ei has to pass through a value ei, res of the order of 0.3 for the system to enter an inclination-type resonance. Numerically, we confirm that such a resonance may also be excited at another, larger, value ei, res ≃ 0.6, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of ei is larger than ei, res. We find that, for q ≤ 1, the system cannot enter an inclination-type resonance if the ratio of eccentricity to semimajor axis damping time-scales te/ta is smaller than 0.2. This result still holds if only the eccentricity of the outer planet is damped and q ≲ 1. As the disc/planet interaction is characterized by te/ta ∼ 10− 2, we conclude that excitation of inclination through the type of resonance described here is very unlikely to happen in a system of two planets migrating in a disc.