Constraints on resonant-trapping for two planets embedded in a protoplanetary disc

Context. A number of extrasolar planet systems contain pairs of Jupiter-like planets in mean motion resonances. As yet there are no known resonant systems which consist of a giant planet and a significantly lower- mass body. Aims. 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. The aim is to examine how the long-term evolution of such a system depends on the mass of the outer planet. Methods. We have performed 2D numerical simulations using a grid-based hydrodynamics code. The planets can interact with each other and with the disc in which they are embedded. We consider outermost planets with masses ranging from 10 M_\oplus to 1 M_{\rm J}. Combining the results of these calculations and analytical estimates, we also examine the case of outermost bodies with masses < 10 M_\oplus. Results. 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 \la3.5 M_\oplus 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_{\rm o} \le 20 M_\oplus 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 \le m_{\rm o} \le 40 M_\oplus or m_{\rm o} = 1 M_{\rm J}, then the 2:1 resonance is established. If 80 \le m_{\rm o} \le 100 M_\oplus, the 3:2 resonance is favoured. Simulations of gas-accreting protoplanets of mass m_{\rm o} \ge 20 M_\oplus, 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 Saturnian value. Conclusions. 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, independent of initial conditions. 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