also an important limit to acceleration models, as we shall see. Collisions between protons and photons can generate leptonic pairs if total energy in the center of momentum frame exceeds the proton mass by more than the pair mass. The proton energy threshold, $E_{\pm}$ for this photopair production when the incident photon energy is $\epsilon_{\gamma}$ is approximately $E_{\pm} \approx 10^3/E_{\gamma}(eV)$ GeV. For the CMB $\epsilon_{\gamma}\sim 10^{-3}$eV, so the photopair production threshold energy is about 0.01 EeV.

The losses are much more serious, however, if the collision energy is enough to excite the $\Delta^+$ resonance near 1.2 GeV in the center of momentum frame, since that decays to produce pions, typically carrying away of order 10% of the total energy. This threshold is roughly $E_{\pi}\sim 5\times 10^7/\epsilon_{\gamma}(eV)$ GeV. For CMB photons the effective threshold is just below 100 EeV. The cross section rises sharply, so that protons near the threshold lose most of their energy in the CMB in less than a few $\times 10^8$ years. Thus their effective propagation length is less than about 100 Mpc. The significance of this was pointed out in the 1960s independently by Greisen (1966) and by Zatsepin & Kuz’min (1966). If UHECRs are of 