Stability of resonant configurations during the migration of planets and constraints on disk-planet interactions

Abstract

We study the stability of mean-motion resonances (MMR) between two planets during their migration in a protoplanetary disk. We use an analytical model of resonances, and describe the effect of the disk by a migration timescale (T_{m,i}) and an eccentricity damping timescale (T_{e,i}) for each planet (i=1,2 respectively for the inner and outer planet). We show that the resonant configuration is stable if T_{e,1}/T_{e,2} > (e_1/e_2)^2. This general result can be used to put constraints on specific models of disk-planet interactions. For instance, using classical prescriptions for type I migration, we show that when the angular momentum deficit (AMD) of the inner orbit is larger than the outer's orbit AMD, resonant systems must have a locally inverted disk density profile to stay locked in resonance during the migration. This inversion is very untypical of type I migration and our criterion can thus provide an evidence against classical type I migration. That is indeed the case for the Jupiter-mass resonant systems HD 60532b, c (3:1 MMR), GJ 876b, c (2:1 MMR), and HD 45364b, c (3:2 MMR). This result may be an evidence for type II migration (gap opening planets), which is compatible with the large masses of these planets.