= Beamed gamma-ray optical depth
{title2=$\tau'\propto D^{-(4+2\alpha)}$}
For an isotropically emitting rest-frame blob moving with <Doppler factor> $D$, take the observed target-band <spectral flux density> to satisfy $F_\nu\propto\nu^{-\alpha}$ over the relevant <frequencies>. The sign convention for $\alpha$ matters. At fixed <cosmological redshift>, absorb fixed distance and redshift factors into a reference <optical depth> $\tau_0$.
A light-crossing estimate gives $R'\propto D$ for fixed observed variability time, while a corresponding logarithmic <frequency> bin has observed-to-rest-frame <luminosity> boost $D^4$. At a fixed observed high-energy <photon> <energy>, the <internal pair-production target energy in a relativistic blob> shifts the observed target <energy> by $D^2$. Combining these factors gives
$$
\tau'(D)\sim\tau_0D^{-6}
\frac{F_{\log}(D^2\epsilon_{20})}{F_{\log}(\epsilon_{20})}
=\tau_0D^{-6+2(1-\alpha)}
=\tau_0D^{-(4+2\alpha)},
$$
where $F_{\log}=\nu F_\nu$ and $\epsilon_{20}$ is the reference target <energy> at $D=1$. For example, $\alpha=1$ gives $D^{-6}$, whereas a locally flat $F_\nu$ gives $D^{-4}$. The power-law assumption must cover the target energies swept by changing $D$. Spectral breaks, different emission geometries and external stationary illumination require a different calculation; $D^{-6}$ is not a universal pair-opacity law.
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