Beamed gamma-ray optical depth (${\tau }^{\prime }\propto D^{-(4+2\alpha )}$)

For an internally illuminated isotropic rest-frame blob with an unbroken power-law spectrum, use $R^{\prime }\sim Dc\Delta t_{\mathrm{obs}}$, a corresponding-bin luminosity boost $D^4$, and the internal pair-production target energy in a relativistic blob. At fixed observed gamma-ray energy,Here $F_{\log }=\nu F_{\nu }$ and ${ϵ}_{20}$ is the target energy at $D=1$. The frequently used $D^{-6}$ is the $\alpha =1$ case, not a universal law for every spectral energy distribution. Geometry, spectral breaks and external illumination affect the result.