3D representation of streamlines with initial conditions on the Hill sphere (see Table 2). The colour bar corresponds to planeto-centric mass flow rate on the Hill surface. The sign is negative for mass inflow and positive for mass outflow. The total integration time is short (about four Jupiter orbits) in order to make this plot clear. The evolution of the same streamlines on a longer integration allows us to follow the dynamics on a projected plane (x/R_H, y/R_H) and to appreciate the evolution of the distance with respect to the planet in Fig. 6.