viscosity at $r=1$ corresponds to $\alpha=3.5\times 10^{-3}$ in a Shakura and Sunyaev (1973) prescription.]{} This simulation allows the planets to sculpt the disk, opening gaps around their orbits, and it sets a new surface density profile of the disk. When we do simulations with different disk parameters, we start from this profile, and let the planets evolve for additional 400 Jovian orbits still without feeling the disk’s perturbations, so that the disk profile adapts to the new situation. Only at this point we release the planets, letting them evolve under the effects of the disk and of their mutual perturbations. This procedure allows us to avoid possible spurious initial migrations, that might occur if the initial gas distribution is inconsistent with the presence of the planets.

Dependence on the disk aspect ratio {#H/r}
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In a first series of runs, we have fixed the value of the viscosity ($\nu=10^{-5.5}$, in our units), and we have studied the evolution of Jupiter and Saturn as a function of the disk aspect ratio $H/r$. MS01 found that the aspect ratio simply changes the outward migration speed, by a quantity proportional to $(H/r)^{-3}$. We find that the value of the aspect ratio can have 