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

¹ Observatoire de l’Université de Genève, 51 chemin des Maillettes, 1290 Sauverny, Switzerland
e-mail: jean-baptiste.delisle@unige.ch
² ASD, IMCCE-CNRS UMR 8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France
³ CIDMA, Departamento de Física, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal

Received: 8 April 2015
Accepted: 12 June 2015

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 for the inner and outer planets, respectively). We show that the resonant configuration is stable if T_e,1/T_e,2> (e₁/e₂)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 greater 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 atypical 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 evidence of type-II migration (gap-opening planets), which is compatible with the high masses of these planets.