Constraints on resonant-trapping for two planets embedded in a   protoplanetary disc
  We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. We consider outermost planets with masses ranging from 10 earth masses to 1 M_J. We also examine the case of outermost bodies with masses < 10 earth masses (M_E). Differential migration of the planets due to disc torques leads to different evolution outcomes depending on the mass of the outer protoplanet. For planets with mass < 3.5 M_E the type II migration rate of the giant exceeds the type I migration rate of the outer body, resulting in divergent migration. Outer bodies with masses in the range 3.5 < m_o < 20 M_E become trapped at the edge of the gap formed by the giant planet, because of corotation torques. Higher mass planets are captured into resonance with the inner planet. If 30 < m_o < 40 M_E or m_o=1 M_J, then the 2:1 resonance is established. If 80 < m_o < 100 M_E, the 3:2 resonance is favoured. Simulations of gas-accreting protoplanets of mass m_o > 20 M_E, trapped initially at the edge of the gap, or in the 2:1 resonance, also result in eventual capture in the 3:2 resonance as the planet mass grows to become close to the mass of Saturn. Our results suggest that there is a theoretical lower limit to the mass of an outer planet that can be captured into resonance with an inner Jovian planet, which is relevant to observations of extrasolar multiplanet systems. Furthermore, capture of a Saturn-like planet into the 3:2 resonance with a Jupiter-like planet is a very robust outcome of simulations. This result is relevant to recent scenarios of early Solar System evolution which require Saturn to have existed interior to the 2:1 resonance with Jupiter prior to the onset of the Late Heavy Bombardment. 
Arnaud Pierens, Richard P. Nelson