On the migration of a system of protoplanets

The evolution of a system consisting of a protoplanetary disc with two embe dded Jupiter-sized planets is studied numerically. The disc is assumed to b e flat and non-self-gravitating; this is modelled by the planar (two-dimens ional) Navier-Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass M-Jup each, and the radial distances from the star are one and two semimajor axes of Jupiter, respectively. During the evolution a joint wide annular gap is created by the planets. Bo th planets increase their mass owing to accretion of gas from the disc: aft er about 2500 orbital periods of the inner planet it has reached a mass of 2.3 M-Jup, while the outer planet has reached a mass of 3.2 M-Jup. The net gravitational torques exerted by the disc on the planets result in an inwar d migration of the outer planet on time-scales comparable to the viscous ev olution time of the disc. The semimajor axis of the inner planet remains co nstant as there is very little gas left in its vicinity to induce any migra tion. When the distance of close approach eventually becomes smaller than t he mutual Hill radius, the eccentricities increase strongly and the system may become unstable. If disc depletion occurs rapidly enough before the planets come too close t o each other, a stable system similar to our own Solar system may remain. O therwise the orbits may become unstable and produce systems like upsilon An d.