because the presence of Saturn depletes partially the outer disk, this torque is weakened. Obviously, it is weakened more if the gap at Saturn’s position is deeper, namely if the disk aspect ratio is smaller, as visible in Fig. \[profiles\]. Thus, if the aspect ratio is small enough, the torque received by Jupiter from the inner disk dominates that from the outer disk, and the planet migrates outward, feeling a net positive torque. Indeed, this is what we see happening in Fig. \[MS-Hr\].

The direction of migration of Jupiter determines the subsequent evolution of both planets, once they are locked in resonance. The planets have to move in parallel to preserve the resonant configuration. Therefore there is a competition between the net positive torque received by Jupiter and the net negative torque received by Saturn from the disk. Because these torques are monotonic functions of the planets’ masses, and Jupiter is 3 times heavier than Saturn, in general the positive torque received by Jupiter dominates and the two planets move outward. In the case with $H/r=5$%, however, the net torque felt by Jupiter is close to zero, due to the specific density profile of the disk, and can be effectively 