Energy Loss of High-Energy Cosmic Rays in Pair-Producing Collisions with Ambient Photons

General and essentially exact formulas for the energy-loss process $A+\ensuremath{\gamma}\ensuremath{\rightarrow}A+{e}^{+}+{e}^{\ensuremath{-}}$ are given for high-energy cosmic-ray nuclei traversing an isotropic radiation field of arbitrary energy spectrum. These formulas are obtained by integrating the appropriate differential cross section in the rest frame of the cosmic-ray particle. Results are then given for the case of a Planckian photon distribution at arbitrary temperature. The effect of this energy loss on a universal flux of cosmic rays is then investigated along with the possibility that the observed "knee" at high energies is due to pair production. Finally, the energy-loss formula is applied to quasistellar objects, and it is shown that if these objects are at cosmological distances, they are opaque to particles above a certain energy which depends upon the size of the object.