= 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> $\epsilon_2$, the target <number density> is $n\sim L_2/(4\pi R^2c\epsilon_2)$. <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.
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