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

Abstract

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 $e_{\mathrm{i}}$ 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, $e_{\mathrm{i}}$ has to pass through a value $e_{\mathrm{i},\mathrm{res}}$ of order 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 $e_{\mathrm{i},\mathrm{res}}\simeq 0.6$, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of $e_{\mathrm{i}}$ is larger than $e_{\mathrm{i},\mathrm{res}}$. We find that, for $q\le 1$, the system cannot enter an inclination-type resonance if the ratio of eccentricity to semimajor axis damping timescales $t_e/t_a$ is smaller than 0.2. This result still holds if only the eccentricity of the outer planet is damped and $q\lesssim 1$. As the disc/planet interaction is characterized by $t_e/t_a\sim 10^{-2}$, 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.

arXiv comment22 pages, 10 figures, accepted for publication in MNRAS

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