= Beamed gamma-ray optical depth
{title2=$\tau'\propto D^{-(4+2\alpha)}$}
For an internally illuminated isotropic rest-frame blob with an unbroken <power-law spectrum>, use $R'\sim Dc\Delta t_{\rm obs}$, a corresponding-bin <luminosity> boost $D^4$, and the <internal pair-production target energy in a relativistic blob>. At fixed observed gamma-ray <energy>,
$$
\tau'(D)\sim\tau_0D^{-6}\frac{F_{\log}(D^2\epsilon_{20})}{F_{\log}(\epsilon_{20})}
=\tau_0D^{-(4+2\alpha)}.
$$
Here $F_{\log}=\nu F_\nu$ and $\epsilon_{20}$ 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.
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