angular resolution, it is clear that all events associated with the decays of highly boosted mediators will be concentrated in the angular bin covering the solar center.

The energy distribution of $\Phi$ follows from a similar integral. In general, we can write the full differential distribution in the form, $$\begin{split}
\label{eq:spectrum_gen}
\frac{{\textrm{d}}^2 \Phi(\theta, p)}{p^2{\textrm{d}}p \,{\textrm{d}}\Omega}
=&\frac{l\sin\theta \cos\theta \textrm{Br}_\gamma}
{\gamma\tau}\int_D {\textrm{d}}\cos\alpha \cdot
\frac{f(p,\alpha)\, n(l\sin\theta\csc\alpha)}{\sin^3\alpha}\\
=&\frac{l\sin\theta \cos\theta\; \textrm{Br}_\gamma}{\tau}
\int_D \frac{{\textrm{d}}\cos\alpha}{\sin^3\alpha} \cdot
(1-\beta \cos\alpha) f'[\gamma p(1-\beta \cos\alpha)]
n(l\sin\theta\csc\alpha)\,,
\end{split}$$ where $D$ is the same domain explained above. Specializing to the case of two-body decays of $V$ to photons, we have $$\label{eq:spectrum}
\frac{{\textrm{d}}^2 \Phi(\theta, p)}{p^2{\textrm{d}}p\, {\textrm{d}}\Omega}
=\frac{(\gamma^2-1)\sin\theta\cos\theta l\; \textrm{Br}_\gamma}
{2\pi\gamma\tau(2\gamma p_0 p-p_0^2-p^2)^{3/2}}\cdot\frac{p}{p_0}\cdot
n\left(\frac{\sqrt{\gamma^2-1}\sin\theta l p}
        {\sqrt{2\gamma p_0 p-p_0^2-p^2}}\right)\,,$$ where $p_0\equiv m_V/2=m_\chi/(2 \gamma)$ is the photon energy measured in the rest frame of $V$. When $l\sin\theta\geq R_\odot$, this expression is valid as long as $$p\in \left[\;
p_0\sqrt{\frac{1-\beta}{1+\beta}}\;,\;
p_0\sqrt{\frac{1+\beta}{1-\beta}}
\;\right]\;.$$ and is understood to vanish for $p$ outside this range. For $l\sin\theta < R_\odot$, the lower limit is again modified to $p_0 \gamma^{-1}(1-\beta\sqrt{1-l^2\sin^2\theta/R^2_\odot}\,)^{-1}$ due to absorption.

The resulting photon spectrum varies dramatically within a tiny range of $\theta$ that for the Lorentz boosts considered here is typically well below the angular sensitivity of any gamma ray 