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

Jean Teyssandier (1) , Caroline Terquem (1)

1 IAP - Institut d'Astrophysique de Paris

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_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_i has to pass through a value e_{i, 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 e_{i, 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 e_i is larger than e_{i, 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 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 ≲ 1. As the disc/planet interaction is characterized by t_e/t_a ∼ 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.

Keywords: celestial mechanics; planets and satellites: dynamical evolution and stability; planetary systems; Astrophysics - Earth and Planetary Astrophysics

Domains: Sciences of the Universe [physics]; Astrophysics [astro-ph]

Jean Teyssandier, Caroline Terquem. Evolution of eccentricity and orbital inclination of migrating planets in 2:1 mean motion resonance. Monthly Notices of the Royal Astronomical Society, 2014, 443, pp.568-583. ⟨10.1093/mnras/stu1137⟩. ⟨insu-03645320⟩