Evolution in time of the semi-major axis (black), the perihelion (red), and the aphelion (orange) of the planets that remain within the HZ of each simulation in both the first (left panel) and the second set (right panel) of simulations. The blue and light-blue shaded areas represent the conservative and the optimistic HZ, and the dashed gray line represents the position of the snow line. A color figure is available in the electronic version.