The ionising effect of low energy cosmic rays from a class II object on its protoplanetary disk

We investigate the ionising effect of low energy cosmic rays (CRs) from a young star on its protoplanetary disk (PPD). We consider specifically the effect of $\sim3\,$GeV protons injected at the inner edge of the PPD. An increase in the ionisation fraction as a result of these CRs could allow the magnetorotational instability to operate in otherwise magnetically dead regions of the disk. For the typical values assumed we find an ionisation rate of $\zeta_\mathrm{CR} \sim 10^{-17}\mathrm{s^{-1}}$ at $1\,$au. The transport equation is solved by treating the propagation of the CRs as diffusive. We find for increasing diffusion coefficients the CRs penetrate further in the PPD, while varying the mass density profile of the disk is found to have little effect. We investigate the effect of an energy spectrum of CRs. The influence of a disk wind is examined by including an advective term. For advective wind speeds between $1-100\mathrm{km\,s^{-1}}$ diffusion dominates at all radii considered here (out to 10$\,$au) for reasonable diffusion coefficients. Overall, we find that low energy CRs can significantly ionise the midplane of PPDs out to $\sim\,1\,$au. By increasing the luminosity or energy of the CRs, within plausible limits, their radial influence could increase to $\sim2\,$au at the midplane but it remains challenging to significantly ionise the midplane further out.