-1cm

Accounting for this attenuation of the CR background and following the treatment of \citet{StacyBromm2007} and \citet{InayoshiOmukai2011}, the cosmic ray heating rate \(\Gamma_{\rm \small CR}\) and ionisation rate \(k_{\rm \small CR}\) are given by \[\Gamma_{\rm \small CR} = \frac{ E_{\rm \small heat}}{50\,{\rm \small eV}} \int_{\epsilon_{\rm min}}^{\epsilon_{\rm max}} \left| \left( \frac{{\rm d}\epsilon} {{\rm d}t} \right)_{\rm ion} \right| \frac{dn_{\rm \tiny CR}}{d\epsilon} e^{-\tau_{\rm \small CR}} d\epsilon,\] and \[k_{\rm \small CR} = \frac{\Gamma_{\rm \small CR}}{ n_{\rm \small H} E_{\rm \small heat}},\] where \(\epsilon_{\rm min} = 10^6{\,{\rm eV}}\), \(\epsilon_{\rm max}= 10^{15}{\,{\rm eV}}\), \(n_{\rm \small H}\) is the number density of hydrogen, and \(E_{\rm \small heat}\) is the energy deposited as heat per interaction \citep{Schlickeiser2002}. While CRs lose about \(50{\,{\rm eV}}\) per interaction, only about \(6{\,{\rm eV}}\) of that goes towards heating in a neutral medium \citep{SpitzerScott1969,ShullvanSteenberg1985}; we set \(E_{\rm \small heat}\) accordingly.

Here we assume the incident CR background is composed solely of protons, and all interactions occur with hydrogen only. While this neglects the slight difference in average CR energy loss per interaction for hydrogen (\(36{\,{\rm eV}}\); \citealt{BakkerSegre1951}) as compared to helium (\(40{\,{\rm eV}}\); \citealt{WeissBernstein1956}), the resulting error in the employed heating and ionisation rates is sufficiently small for our purposes (see \citealt{JascheCiardiEnsslin2007} for a more rigorous treatment).

The resulting heating and ionisation rates for both the \({{u}_{\rm \small CR}}=u_0\) and \(10^5\,u_0\) cases are shown in Figure \ref{fig:khrates}, along with the expected rates in the absence of attenuation. Also shown are the highest and lowest X-ray heating and ionisation rates considered in \citet{Hummeletal2015}. Compared to X-rays, CRs produce significantly less heating per ionisation event and penetrate to much higher densities before being attenuated; the X-ray heating and ionisation rates fall off much faster as the gas becomes optically thick.