 Gamma-ray compactness problem

Rapidly varying high-energy sources can appear too dense in target photons to transmit their observed gamma rays. For an approximately isotropic stationary region with target-band luminosity $L_2$, radius $R$ and target photon energy ${ϵ}_2$, the target number density is $n\sim L_2/(4\pi R^{2}c{ϵ}_2)$. photon-photon pair production then gives optical depth $\tau \sim n\sigma R$. A luminosity variability size bound can make this very large. relativistic beaming both increases the inferred rest-frame size and reduces the inferred rest-frame radiation density, allowing escape. This radiation-opacity argument is distinct from gravitational stellar compactness and topological compactness.