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

TL;DR

This paper analytically and numerically investigates the conditions under which migrating planets in 2:1 resonance can develop significant inclination, finding that typical disc conditions make such inclination excitation unlikely.

Contribution

It provides new analytical expressions for eccentricity equilibrium and identifies key damping ratio thresholds for inclination resonance onset in migrating planetary systems.

Findings

Inclination resonance requires inner planet eccentricity > 0.3 or 0.6.

Resonance unlikely if eccentricity damping is much faster than migration.

Typical disc conditions prevent inclination excitation in migrating planets.

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 $e_{\rm i}$ 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, $e_{\rm i}$ has to pass through a value $e_{\rm i,res}$ 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 $e_{\rm i, res} \simeq 0.6$, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of $e_{\rm i}$ is larger than $e_{\rm i,res}$. 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.