Can cosmic rays explain the high ionisation rates in the Galactic Centre?

The $\mathrm{H}_2$ ionisation rate in the Central Molecular Zone, located in the Galactic Centre region, is estimated to be $\zeta\sim2\times10^{-14}~\mathrm{s}^{-1}$, based on observations of H$_3^+$ lines. This value is 2-3 orders of magnitude larger than that measured anywhere else in the Galaxy. A high cosmic-ray density has been invoked to explain the unusually high ionisation rate. However, this excess is not seen in the $\gamma$-ray emission from this region, which is produced by high-energy cosmic rays. Therefore, an excess is expected only in the low-energy cosmic-ray spectrum. Here, we derive constraints on this hypothetical low-energy component in the cosmic-ray spectra and we question its plausibility. To do so, we solve numerically the cosmic-ray transport equation in the Central Molecular Zone, considering spatial diffusion, advection in the Galactic wind, reacceleration in the ambient turbulence, and energy losses due to interactions with matter and radiation in the interstellar medium. We derive stationary solutions under the assumption that cosmic rays are continuously injected by a source located in the Galactic Centre. The high-energy component in the cosmic-ray spectrum is then fitted to available $\gamma$-ray and radio data, and a steep low-energy component is added to the cosmic-ray spectrum to explain the large ionisation rates. We find that injection spectra of $p^{-7}$ for protons below $p_{enh,p}c\simeq780~\mathrm{MeV}$ and $p^{-5.2}$ for electrons below $p_{enh,e}c=1.5~\mathrm{GeV}$ are needed to reach the observed ionisation rates. This corresponds to an extremely large cosmic-ray power of the order $\sim10^{40-41}~\mathrm{erg}\,\mathrm{s}^{-1}$ injected at the Galactic Centre. We conclude that cosmic rays alone can not explain the high ionisation rates in the Galactic Centre region.