Modelling the ArH$^+$ emission from the Crab Nebula

We have performed combined photoionization and photodissociation region (PDR) modelling of a Crab Nebula filament subjected to the synchrotron radiation from the central pulsar wind nebula, and to a high flux of charged particles; a greatly enhanced cosmic ray ionization rate over the standard interstellar value, $\zeta_0$, is required to account for the lack of detected [C I] emission in published Herschel SPIRE FTS observations of the Crab Nebula. The observed line surface brightness ratios of the OH$^+$ and ArH$^+$ transitions seen in the SPIRE FTS frequency range can only be explained with both a high cosmic ray ionization rate and a reduced ArH$^+$ dissociative recombination rate compared to that used by previous authors, although consistent with experimental upper limits. We find that the ArH$^+$/OH$^+$ line strengths and the observed H$_2$ vibration-rotation emission can be reproduced by model filaments with $n_{\rm{H}} = 2 \times 10^4$ cm$^{-3}$, $\zeta = 10^7 \zeta_0$ and visual extinctions within the range found for dusty globules in the Crab Nebula, although far-infrared emission from [O I] and [C II] is higher than the observational constraints. Models with $n_{\rm{H}} = 1900$ cm$^{-3}$ underpredict the H$_2$ surface brightness, but agree with the ArH$^+$ and OH$^+$ surface brightnesses and predict [O I] and [C II] line ratios consistent with observations. These models predict HeH$^+$ rotational emission above detection thresholds, but consideration of the formation timescale suggests that the abundance of this molecule in the Crab Nebula should be lower than the equilibrium values obtained in our analysis.