Impact of proton–proton losses on the maximum energy. Left: ratio Ė_pp/Ė_DSA as a function of the proton energy for ρ₀. Right: maximum CR energy as limited by proton–proton interactions for a flat density profile with density (ρ₀, black) or a steady-wind profile (Eq. (6); red), for different magnetic fields. E_max is derived by imposing Ė_DSA =Ė_pp at the Bohm limit, and employing the parametrisation from Krakau & Schlickeiser (2015) for the proton–proton losses.