Pebble accretion rates (red), planetesimal accretion rates (grey), and minimal accretion rates required to sustain a stable gas envelope (black), as a function of the core mass. The curves for the minimal accretion rates are nearly independent of orbital radius between 5−30 AU, but depend strongly on the opacity (Appendix B.3) and on the level of envelope pollution by sublimation of icy pebbles. Labels at the top of the figure indicate H₂O pollution of the atmosphere as a percentage with respect to pure H/He nebular gas, corresponding to the fraction 1 − β from Eq. (11). The critical core mass to collapse the gas envelope can be found at the intersection of an accretion curve with a critical curve. Accretion rate curves that fall in the yellow dashed region are too slow to form the cores of the giant planets before the dissipation of the gas disc (τ_acc = M_c/Ṁ_c>τ_disc = 2 Myr). Red circles mark the mass above which pebble accretion is halted (Eq. (12)) and the gravitational collapse of the gas envelope is triggered.