Fate of identifying noble gas related species in the Crab nebula environment

The first identification of the argonium ion ($\rm{ArH^+}$) towards the Crab Nebula supernova remnant was proclaimed by the Herschel Space observation in the sub-millimeter and far-infrared domain. Very recently the discovery of the hydro-helium cation ($\rm{HeH^+}$) in the planetary nebula (NGC 7027) has been reported by using the Stratospheric Observatory for Infrared Astronomy (SOFIA). Elemental abundance of neon is much more than that of the argon. However, the presence of neonium ions ($\rm{NeH^+}$) is yet to be confirmed in space. Though the hydroxyl radicals ($-\rm{OH}$) are very abundant either in neutral or in the cationic form, hydroxyl cations of such noble gases (i.e., ArOH$^+$, HeOH$^+$, and NeOH$^+$) are yet to be identified in space. Here, we employ a spectral synthesis code to examine the chemical evolution of the hydride and hydroxyl cations of the various isotopes of Ar, Ne, and He in the Crab Nebula filament and calculate their line emissivity and intrinsic line surface brightness under such environment. We successfully explain the observed surface brightness of two transitions of ArH$^+$ ($617$ and $1234$ GHz), one transition of OH$^+$ ($971$ GHz), and one transition of H$_2$ ($2.12$ $\mu$m). We explain the observed surface brightness ratios between various molecular and atomic transitions. We find that our model reproduces the overall observed features when a hydrogen number density of $\sim 10^4-10^5$ cm$^{-3}$ and a cosmic-ray ionization rate per H$_2$ of $\sim 10^{-11}-10^{-9}$ s$^{-1}$ are chosen. The possibility of detecting some hydride and hydroxyl cations in the Crab nebula environment is also highlighted.