angular distribution factor: $$\label{eq:better}
\frac{{\textrm{d}}\Phi(\theta)}{{\textrm{d}}\Omega}
=\frac{\cos\theta\;\textrm{Br}_\gamma}{2\pi\tau}\int_D {\textrm{d}}\cos\alpha\;
\frac{n(l\sin\theta\csc\alpha) l \sin\theta}
{\gamma^3\sin^3\alpha(1-\beta \cos\alpha)^2}\,.$$ Here $l \approx 1\,\textrm{A.U.}$ is the Sun-detector distance and $n(r)$ is the number density of $V$ at distance $r$ from the center of the Sun. Given our assumptions, $$n(r)=\frac{C_\odot \cdot \textrm{Br}_V}{4\pi v r^2}\exp\{-r/(v\tau\gamma)\}\,,$$ where $v\simeq c(1-1/(2\gamma^2))$ is the velocity of $V$.

A few words about the range of the integral for ${\textrm{d}}\cos\alpha$, which we have denoted as $D$ above, are in order. If the Sun were transparent, $D$ would simply be the domain $[-1,\, +1]$. In the current case, however, the Sun is effectively opaque since photons traveling through the interior are instantaneously absorbed or degraded in energy through their interactions with solar material. Thus one should simply discard those photons from consideration, which leads to a lower bound for the integral at $\cos\alpha=\sqrt{1-(l\sin\theta)^2/R^2_\odot}$ whenever $l\sin\theta<R_\odot$. Consequently, $$D=
\begin{cases}
\left[\sqrt{1-(l\sin\theta)^2/R^2_\odot}\,,\;+1\right],
    & \quad l\sin\theta<R_\odot\;; \\
\left[-1\,,\; +1\right],& \quad l\sin\theta\geq R_\odot\;.
\end{cases}$$ The lower integration limits are not important for $\gamma\gg 1$, as almost all the photons detected come from within a small angle $\sim 1/\gamma$ and the flux is completely negligible when $\theta\approx \theta_\odot$. We illustrate ${\textrm{d}}\Phi/{\textrm{d}}\Omega$ for a representative value of $\gamma=10^3$ in Fig. \[fig:spec\_smeared\]. Given a realistic 