into $U_\rho(\Delta)=-(M_p m)/\sqrt{\Delta^2+\rho^2}$, where $\Delta$ is the distance between the planet of mass $M_p$ and a fluid element of mass $m$, and $\rho$ is called the [*smoothing length*]{}. The choice of an appropriate value for the smoothing length is the subject of a vast debate. Essentially, two recipes are used: either $\rho$ is chosen proportionally to the Hill radius of the planet, or proportionally to the local thickness of the disk.

In the simulations presented above, our choice of $\rho$ was equal to 60% of the planet’s Hill radius $R_H=a_p(M_p/3)^{1/3}$, where $a_p$ is the semi major axis of the planet and $M_p$ is normalized relative to the mass of the star. We have decided to redo the simulation with $H/r=3$%, adopting $\rho=0.7 H$, where $H$ is the thickness of the disk at the distance of the planet (namely 0.03 $a_p$). In this case, the new value of $\rho$ for Saturn and Jupiter is, respectively, 73% and 50% of those previously adopted. The new simulation is compared with the previous one in Fig. \[comp-tch\]. Because the simulations do not start exactly in the same way, for a more meaningful comparison we have plotted the evolution of the planets only from the 