of density $n_\text{H}=10^3$cm$^{-3}$ is 30000AU, assuming the UV-to-visual extinction factor of 3.53 discussed in Sec. \[sec:shielding functions\]. For the larger densities found near the midplane of a protoplanetary disc [@jonkheid2006] this distance is accordingly reduced, e.g., $n_\text{H}=10^8$cm$^{-3}$ and $\sim1$AU. In principle, the various cosmic-ray induced photodissociation rates calculated below will be overestimated for objects which are not significantly larger than this mean-free-path, due to the escape of some UV photons. However, the significance of cosmic-ray induced photon fluxes is generally limited to regions fully-shielded from external UV radiation, and such regions also necessarily shield the escape of internally-generated photons. ]{}

The dust grains in a protoplanetary disc are likely larger than those in interstellar space, which have radii $<1\,\mu\mathrm{m}$ [@mathis1977]. @li2003a deduced properties of the dust grains in one protoplanetary system, HD4796A, by reference to observed infrared and sub-millimetre wavelength emission. Their modelling required a lower limit of 1$\mu$m on the grain size distribution. The N$_2$-dissociating cosmic-ray induced UV flux considered here has wavelengths shorter than 1000Å and will not be as strongly absorbed by grains larger than this. However, a population of polycyclic aromatic hydrocarbons (PAHs) has been inferred in a number of discs [e.g., @li2003b; @geers2006; @geers2007; @maaskant2013] 