unidirectional case, $\theta(N)$ [@roellig2007]. For a plane-parallel shielding medium with $N$ defined perpendicular to its surface, the two shielding functions are related geometrically by the formula $$\label{eq:semi-infinit isotropic shielding functions}
  \theta_\text{iso}(N) =  \int_0^{\frac{\pi}{2}} \theta\left(\frac{N}{\cos \phi}\right)\sin\phi\,d\phi.$$ ]{}

Photodissociation due to cosmic-rays {#sec:CR}
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Cosmic-ray induced emission and photodissociation model
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The ionisation of H$_2$ by cosmic ray collisions leads to the generation of free electrons. These may collisionally excite (or ionise) further H$_2$ whose radiative relaxation generates an ultraviolet flux [@prasad1983; @cecchi-pestellini1992b]. @sternberg1987, @gredel1987, and @gredel1989 estimated the photodissociation properties of several small molecules due to ultraviolet photons arising from this process. Similar calculations are made here for the dissociation of N$_2$ including the detailed effects of line absorption as modelled by the CSE dissociation cross sections.

[ The H$_2$ emission flux between 912 and 1000Å is dominated by line transitions. This emission occurs primarily through the Lyman and Werner bands connecting the ground state with the $B\,{}^1\Sigma_u^+$ and $C\,{}^1\Pi_u$ excited states [@sternberg1987], and with a smaller contribution arising from other excited states [@gredel1989]. Here, we use a model simulating the excitation of 9 electronic states of by cosmic-ray generated electrons and the resulting emission, developed and expounded in detail by 