“…We use the extragalactic propagation model of Berezinsky, Gazizov, and Grigorieva [27], modified to take account of discrete energy losses of protons as in the paper by Blanton, Blasi and Olinto [28], and assume that protons come from sources distributed uniformly following the expansion of the universe, and lose energy by pion and e + e − production from the cosmic microwave background radiation, as well as from the expansion of the universe. Since the measured composition [29,30] changes from heavy to light within our energy range, we approximate the galactic component of cosmic rays as being the fraction of iron. We take this fraction to be 55% at 10 17 eV, decreasing linearly with log E to 20% at 10 17 eV, then decreasing to zero at 10 20 eV.…”