Left panel: comparison of the rest-mass density along the φ = 0 direction as obtained with the standard evolution of the energy equation (red solid line) and with the isothermal evolution (blue dashed line), at two different times as shown in the two panels. The data refers to model L.00 with a recoil of 1000 km s^-1. Right panel: mass accretion rate at r = r_min for the isothermal evolution and different values of the kick velocity in the large-size model L.00. This panel should be compared with Fig. 9 and which refers to an evolution of the energy equation.