the source. In [@pfrommer2004] a framework is outlined for incorporating the fireball model for very high energy cosmic ray proton interactions with the ICM as well as pion production threshold effects. The resulting differential source function $q_\gamma$ (with units of inverse energy and volume) is $$\begin{aligned}
\label{eq:crspec}
q_\gamma(r,E_\gamma) \textrm{d}E_\gamma\textrm{d}V \simeq& \sigma_{pp}\  c\  n_N(r)\ \xi^{2-\alpha_p}\ \frac{\tilde{n}_{CR_p}(r)}{\textrm{GeV}}\ \frac{4}{3\alpha_p}\left(\frac{m_{\pi^0}c^2}{\textrm{GeV}}\right)^{-\alpha_p} \nonumber \\ &\times \left[\left(\frac{2E_\gamma}{m_{\pi^0}c^2}\right)^{\delta_\gamma}+\left(\frac{2E_\gamma}{m_{\pi^0}c^2}\right)^{-\delta_\gamma}\right]^{-\alpha_\gamma/\delta_\gamma}\ \  \textrm{d}E_\gamma\textrm{d}V,\end{aligned}$$ where \_[pp]{} = 32(0.96+e\^[4.4-2.4\_p]{}) models the effective inelastic p-p cross section and $n_N$ is the target nucleon density in the ICM. The quantity $\tilde{n}_{CR_p}$ with the dimensions of the cosmic ray proton number density, has a normalization chosen so that the kinetic cosmic ray proton energy is proportional to the thermal energy density of the ICM. Also, $\xi = 2$ is the pion multiplicity, and $\delta_\gamma = 0.14\alpha_p^{-1.6}+0.44$ is a shape parameter for the $\pi^0$-threshold physics [@pfrommer2004]. By integrating over all solid angles and dividing by $n_N$ and $\tilde{n_{CR_p}}$ (to be independent of a model’s spatial dependence) the final differential gamma-ray source function (i.e. the gamma-ray flux per unit energy and unit time, per unit target and impinging cosmic-ray flux) is obtained.

The spatial distribution of cosmic ray sources is modeled in terms of the ratio of the 