in two stages due first to photopair production and second to photopion production, then remains roughly constant, with the thresholds scaling with $\epsilon_{\gamma}^{-1}$ as described in §2 (see also, for example, Stecker 2002, Berezinsky, 2001).

The energy loss lifetime of a particle from synchrotron/Compton emission is simply $$t_{s,c} = \frac{3}{2}\frac{m^4 c^7}{Z^4 e^4 (B^2 + B^2_{eq})}\frac{1}{E},$$ where $m$ is the particle mass and $B_{eq} = \sqrt{6\pi u_{rad}}$ is sometimes called the equivalent isotropic magnetic field for the photon field in terms of Compton losses. For a black body radiation field $B_{eq} = 0.38 T^2~\mu$G.

Inelastic scattering losses are usually dominated by photopion production, which turns on roughly above $E_{\pi} \sim 100~$EeV/T for a black body radiation field. We can approximately model the energy loss lifetime above this threshold as $$t_{\pi} \sim 3\times 10^{16}\frac{1}{T^3}~ {\rm sec}.$$

The simplest constraint, with the escape time set to the free flight time, R/c, is $$\min(t_{\pi},t_{s,c}) > \max(t_{acc},R/c).$$

These limitations can become effective under a variety of circumstances, but especially when the particle accelerator is associated with an intense magnetic field or radiation field (i.e., “hot” in the black body sense) or if the escape time or acceleration time is long. The former condition applies, for 