-1cm
\label{CRchem}
Each time a CR proton ionises a hydrogen atom, an electron with average energy \(\langle E \rangle = 35{\,{\rm eV}}\) is produced \citep{SpitzerTomasko1968}. Including the ionization energy of \(13.6{\,{\rm eV}}\), the CR proton loses approximately \(50{\,{\rm eV}}\) per scattering. This necessarily places a limit on how many scatterings a CR proton can undergo before losing all its energy to ionisation, as well as limiting the distance it may travel. This distance may be described by a penetration depth \[D_p(n, \epsilon) \approx \frac{\beta c \epsilon} {-({\rm d}\epsilon / {\rm d}t)_{\rm ion}}\] where \citep{Schlickeiser2002} \[- \left( \frac{{\rm d}\epsilon} {{\rm d}t} \right)_{\rm ion}(n, \epsilon) = 1.82\times10^{-7}\,{\rm \small eV\,s}^{-1} n_{{\mathrm{H}}} f(\epsilon),\] \[f(\epsilon) = (1 + 0.0185 \,{\rm ln}\beta )\, \frac{2 \beta^2}{\beta_0^3 + 2 \beta^3},\] and \[\beta = \sqrt{1 - \left( \frac{\epsilon}{m_{\rm \tiny H}c^2}+1 \right)^{-2}}.\] Here, \(({\rm d}\epsilon / {\rm d}t)_{\rm ion}\) is the rate at which a CR proton loses energy to ionisation. \(\beta_0\) is the cutoff below which the interaction between CRs and the gas decreases sharply; we use \(\beta_0=0.01\), appropriate for CRs travelling through a neutral IGM \citep{StacyBromm2007}. As \(D_p(n, \epsilon)\) is the mean free path of CRs of energy \(\epsilon\) travelling through a gas with number density \(n\), we may define an effective cross-section \(\sigma_{CR}(n,\epsilon)\) for the interaction \[\sigma_{CR}(n,\epsilon) = \frac{1}{n D_p(n, \epsilon)}.\]
As the CR penetration depth is \(\gg\) than the box size everywhere except approaching the centre of the star-forming minihalo, we may estimate the column density \(N\)—and thus, the CR attenuation along a given line of sight—using the same technique described in \citet{Hummeletal2015}. The gas column density approaching the centre of the accretion disk varies by roughly an order of magnitude between the polar (\(N_{\rm \small pole}\)) and equatorial ( \(N_{\rm \small equator}\)) directions, with the column density along these lines of sight well fit by \[{\rm log}_{10}(N_{\rm \small pole}) = 0.5323\, {\rm log_{10}}(n) + 19.64\] and \[{\rm log}_{10}(N_{\rm \small equator}) = 0.6262\, {\rm log_{10}}(n) + 19.57,\] respectively. We assume every line of sight within 45 degrees of the pole experiences column density \(N_{\rm \small pole}\) while every other line of sight experiences \(N_{\rm \small equator}\) \citep{Hosokawaetal2011}, allowing us to calculate an effective optical depth such that \[e^{-\tau_{\rm \small CR}} = \frac{2 \Omega_{\rm \small pole}}{4\pi} e^{-\sigma_{\rm \small CR} N_{\rm \small pole}} + \frac{4\pi - 2 \Omega_{\rm \small pole}}{4\pi} e^{-\sigma_{\rm \small CR} N_{\rm \small eq}},\] where \[\Omega_{\rm \small pole} = \int_0^{2\pi}{\rm d}\phi \int_0^{\pi/4}{\rm sin}\theta \,{\rm d}\theta = 1.84\,{\rm sr}.\]