Proton number-spectra at later times. The solid lines represent the total spectra. The dashed and dotted lines indicate the spectrum upstream and downstream of the forward shock, respectively. The grey vertical lines at 70 GeV, 5 GeV, and 1.5 GeV denote the escape energies at 28 000 years, 44 000 years, and 110 000 years, respectively. The arrow points out the energy bands with a spectral index of 2.2.