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 n _ { 3 } : n _ { 2 } : n _ { 1 } =1:2:4 and 1:3:6 are established , as the inner planets , m _ { 1 } and m _ { 2 } , get trapped in a 1:2 resonance and the outer planet m _ { 3 } subsequently is captured in a 1:2 or 1:3 resonance with m _ { 2 } .
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 & Tsiganis 2011 ) ; then mutual inclinations can grow to \sim 35 ^ { \circ } , thus forming a “ 3-D system ” .
On the other hand , we find that trapping of m _ { 2 } in a 2:3 resonance with m _ { 1 } 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 .