= Solution
For a blob in a <relativistic jet>, take two emission events separated by time $\Delta t_e$ in the galaxy frame. The blob moves transversely by $u\Delta t_e\sin\theta$, while its second <photon> has $u\Delta t_e\cos\theta$ less distance to travel to the distant observer. Therefore the arrival interval is $\Delta t_{\rm obs}=\Delta t_e(1-\beta\cos\theta)$ and
$$
\boxed{\beta_{\rm app}\equiv\frac{u_t}{c}=\frac{\beta\sin\theta}{1-\beta\cos\theta}}.
$$
Differentiating with respect to viewing angle gives $d\beta_{\rm app}/d\theta=\beta(\cos\theta-\beta)/(1-\beta\cos\theta)^2$. The maximum occurs at $\cos\theta=\beta$ and equals $\beta\gamma$. Hence <apparent superluminal motion> is possible precisely when $\beta\gamma>1$, or \b[$\beta>1/\sqrt2$]. This is an arrival-time effect; the material and every local signal still travel slower than <light>.
\Image[/past-exam-of-the-relativistic-astrophysics-and-cosmology-course-of-the-university-of-cambridge/2013/paper-1-jet-arrival.png]
{title=Original arrival-time geometry and apparent transverse speeds for subluminal jet velocities}
For $\gamma\gg1$, $1-\beta\simeq1/(2\gamma^2)$ and $1-\beta\cos\theta\simeq(\gamma^{-2}+\theta^2)/2$. Thus the <Doppler factor> is
$$
D\simeq\frac{2\gamma}{1+\gamma^2\theta^2}\longrightarrow\boxed{2\gamma}\quad(\theta\ll\gamma^{-1}).
$$
The <Lorentz-invariant specific intensity> gives $I_\nu/\nu^3=I'_{\nu'}/\nu'^3$. Since $\nu=D\nu'$, a <power-law spectrum> with $I'_{\nu'}\propto\nu'^{-\alpha}$ gives $I_\nu(\nu)=D^3I'_{\nu'}(\nu/D)\propto D^{3+\alpha}$. This is <Doppler boosting of a moving blob>: the <spectral flux density> of an unresolved, intrinsically isotropic blob has the same boost. Comparing identical approaching and receding blobs, with no differential absorption,
$$
\frac{S_1}{S_2}=\left(\frac{D_1}{D_2}\right)^{3+\alpha}
=\boxed{\left(\frac{1+\beta\cos\theta}{1-\beta\cos\theta}\right)^4}\quad(\alpha=1).
$$
A spatially steady continuous jet can have a different boost exponent because the emitting material contributing at one arrival time differs; it must not silently replace the specified blob model.
Using the stipulated peak of the <photon-photon pair production> <cross section>, not the exact head-on threshold, gives
$$
\epsilon_{20}=\frac{2(m_ec^2)^2}{\epsilon_1}
=\frac{2(5.11\times10^5)^2}{5\times10^{11}}\,\mathrm{eV}
=\boxed{1.04\,\mathrm{eV}}.
$$
The relevant <photons> are near <infrared>. Interpret the quoted $F_2$ as target-band <energy> flux, or <energy> flux per logarithmic <frequency> interval, consistent with its units $\mathrm{W\,m^{-2}}$; it is not a <spectral flux density> per hertz. For isotropic emission $L_2=4\pi d^2F_2$. In a stationary source of radius $R$, the order-of-magnitude <photon> <energy density> is $L_2/(4\pi R^2c)$, so
$$
n\sim\frac{F_2d^2}{R^2c\epsilon_{20}},\qquad
\tau\sim\frac{F_2d^2\sigma_T}{5Rc\epsilon_{20}}.
$$
The <luminosity variability size bound> gives $R\lesssim c\Delta t$. Taking the largest allowed radius gives the smallest stationary-source <optical depth> in this estimate:
$$
\boxed{\tau_0\sim\frac{F_2d^2\sigma_T}{5\epsilon_{20}c^2\Delta t}}.
$$
This assumes the <gamma rays> and target <photons> occupy the same region, with an approximately isotropic target field and an order-unity geometric escape factor. A smaller region makes the <gamma-ray compactness problem> worse.
Here $d=3\times10^{25}\,\mathrm m$, $R\sim3\times10^{11}\,\mathrm m$, and $\epsilon_{20}=1.67\times10^{-19}\,\mathrm J$. Consequently
$$
\boxed{\tau_0\simeq3.2\times10^7\gg1}.
$$
Such a stationary co-spatial source would absorb almost all of its high-energy <photons>.
Let primes denote the blob <rest frame> and continue to ignore <cosmological redshift>. For fixed observed gamma-ray <energy>, $\epsilon'_1=\epsilon_1/D$. The co-moving target <energy> at the assumed cross-section peak is therefore $\epsilon'_2=D\epsilon_{20}$. The <internal pair-production target energy in a relativistic blob> acquires a second <Doppler shift> when this same target population is observed along our <line of sight>:
$$
\boxed{\epsilon_{2,\rm obs}=D\epsilon'_2=D^2\epsilon_{20}}.
$$
It would be incorrect to transform only the <gamma ray> while treating the internal target <radiation> as a stationary external field. The target <photons> responsible for collisions span directions in the blob frame; the observed <energy> here labels the corresponding part of the internally emitted <radiation spectrum>.
<relativistic beaming> both increases the allowed rest-frame source radius, $R'\lesssim Dc\Delta t$, and lowers the inferred rest-frame <luminosity>, $L'_{\log}\sim L_{{\rm iso},\log}/D^4$. The <beamed gamma-ray optical depth> then follows from $\epsilon'_2=D\epsilon_{20}$:
$$
\tau'(D)\sim\tau_0D^{-6}
\frac{F_{\log}(D^2\epsilon_{20})}{F_{\log}(\epsilon_{20})}.
$$
For an unbroken <power-law spectrum> $F_\nu\propto\nu^{-\alpha}$, $F_{\log}=\nu F_\nu\propto\nu^{1-\alpha}$, so $\tau'\propto D^{-(4+2\alpha)}$. In particular, $\alpha=1$ gives the stated $D^{-6}$. Spectral breaks, an external target field or non-coincident emission regions change this inference. Under the requested scaling, transparency requires
$$
D\gtrsim\tau_0^{1/6}\simeq17.8,\qquad
\boxed{\gamma_{\min}\simeq8.9}.
$$
This minimum uses the most favourable nearly axial view, $D\simeq2\gamma$. The exact axial relation gives $\gamma_{\min}=(D_{\min}+D_{\min}^{-1})/2\simeq8.94$; a less favourable viewing angle requires a larger <Lorentz factor>. The numerical bound is conditional on the stipulated opacity scaling and internal-source geometry.
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