Projection of the streamlines of Fig. 5 on the planetocentric x − y plane (left). The blue contours give the boundaries of the disk. Gas approaching the Hill sphere can have a rapid transit phase in the horseshoe region and be released in the inner or the outer disk trapped in the planetary system. Vertical evolution of streamline with respect to time (middle). Evolution of the distance with respect to the planet as a function of time (right). The dashed lines correspond to backward integration: in the left panel we see that gas approaching the Hill sphere comes from U-turn motion passing close to the planet from the inner or outer branch of the horseshoe region.