Evolution of planetary systems in resonance

We study the time evolution of two protoplanets still embedded in a protoplanetary disk. The results of two different numerical approaches are presented and compared. In the first approach, the motion of the disk material is computed with viscous hydrodynamical simulations, and the planetary motion is determined by N-body calculations including exactly the gravitational forces exerted by the disk material. In the second approach, only the N-body integration is performed but with additional dissipative forces included such as to mimic the effect of the disk torques acting on the disk. This type of modeling is much faster than the full hydrodynamical simulations, and gives comparative results provided that parameters are adjusted properly. 
Resonant capture of the planets is seen in both approaches, where the order of the resonance depends on the properties of the disk and the planets. Resonant capture leads to a rise in the eccentricity and to an alignment of the spatial orientation of orbits. The numerical results are compared with the observed planetary systems in mean motion resonance (GJ 876, HD 82943, and 55 Cnc). We find that the forcing together of two planets by their parent disk produces resonant configurations similar to those observed, but that eccentricity damping greater than that obtained in our hydrodynamic simulations is required to match the GJ 876 observations.