 Beamed gamma-ray optical depth

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 $R^{\prime }\propto D$ 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 giveswhere $F_{\log }=\nu F_{\nu }$ and ${ϵ}_{20}$ 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.