The differential migration of two planets due to planet-disk interaction can result in capture into the 2:1 eccentricity-type mean-motion resonances .
Both the sequence of 2:1 eccentricity resonances that the system is driven through by continued migration and the possibility of a subsequent capture into the 4:2 inclination resonances are sensitive to the migration rate within the range expected for type II migration due to planet-disk interaction .
If the migration rate is fast , the resonant pair can evolve into a family of 2:1 eccentricity resonances different from those found by Lee ( 29 ) .
This new family has outer orbital eccentricity e _ { 2 } \gtrsim 0.4 – 0.5 , asymmetric librations of both eccentricity resonance variables , and orbits that intersect if they are exactly coplanar .
Although this family exists for an inner-to-outer planet mass ratio m _ { 1 } / m _ { 2 } \gtrsim 0.2 , it is possible to evolve into this family by fast migration only for m _ { 1 } / m _ { 2 } \gtrsim 2 . Thommes & Lissauer ( 53 ) have found that a capture into the 4:2 inclination resonances is possible only for m _ { 1 } / m _ { 2 } \lesssim 2 .
We show that this capture is also possible for m _ { 1 } / m _ { 2 } \gtrsim 2 if the migration rate is slightly slower than that adopted by Thommes & Lissauer .
There is significant theoretical uncertainty in both the sign and the magnitude of the net effect of planet-disk interaction on the orbital eccentricity of a planet .
If the eccentricity is damped on a timescale comparable to or shorter than the migration timescale , e _ { 2 } may not be able to reach the values needed to enter either the new 2:1 eccentricity resonances or the 4:2 inclination resonances .
Thus , if future observations of extrasolar planetary systems were to reveal certain combinations of mass ratio and resonant configuration , they would place a constraint on the strength of eccentricity damping during migration , as well as on the rate of the migration itself .