Trapping in three-planet resonances during gas-driven migration

We study the establishment of three-planet resonances-similar to the Laplace resonance in the Galilean satellites-and their effects on the mutual inclinations of the orbital planes of the planets, assuming that the latter undergo migration in a gaseous disc. In particular, we examine the resonance relations that occur, by varying the physical and initial orbital parameters of the planets (mass, initial semi-major axis and eccentricity) as well as the parameters of the migration forces (migration rate and eccentricity damping rate), which are modeled here through a simplified analytic prescription. We find that, in general, for planetary masses below 1.5 M_J, multiple-planet resonances of the form n₃ : n₂ : n₁ = 1 : 2 : 4 and 1:3:6 are established, as the inner planets, m₁ and m₂, get trapped in a 1:2 resonance and the outer planet m₃ subsequently is captured in a 1:2 or 1:3 resonance with m₂. For mild eccentricity damping, the resonance pumps the eccentricities of all planets on a relatively short time-scale, to the point where they enter an inclination-type resonance (as in Libert and Tsiganis 2011); then mutual inclinations can grow to ~35°, thus forming a "3-D system". On the other hand, we find that trapping of m₂ in a 2:3 resonance with m₁ occurs very rarely, for the range of masses used here, so only two cases of capture in a respective three-planet resonance were found. Our results suggest that trapping in a three-planet resonance can be common in exoplanetary systems, provided that the planets are not very massive. Inclination pumping could then occur relatively fast, provided that eccentricity damping is not very efficient so that at least one of the inner planets acquires an orbital eccentricity higher than e = 0.3.