Acceleration timescale (t_acc, solid black line), upstream diffusion timescale (t_diff, u, short-dashed black line), and dynamical timescale (t_dyn, dotted black line) versus proton energy for T = 10⁴ K, U = 40 km s⁻¹, B = 100 μG, and R = 10⁵ AU. The collisional loss timescale (t_loss, long-dashed line plus colour-coding in the plot legend) depends on the density. The blue, green, and red dots show the value of the proton maximum energy for n = 5 × 10² cm⁻³, 10⁴ cm⁻³, and 2 × 10⁵ cm⁻³, respectively. We note that E_max, p is obtained by the intersection of t_acc with t_diff, u for n = 5 × 10² cm⁻³, while for the other two cases at higher n by the intersection of t_acc with t_loss.