Parsec-scale cosmic-ray ionisation rate in Orion

Cosmic rays regulate the dynamics and the chemical processes in the densest and coldest regions of the ISM. Still, the determination of the cosmic-ray ionisation rate of H$_2$ (${\zeta^{\rm ion}_{{\rm H}_2}}$) is plagued by uncertainties in the adopted chemical networks and the analysis techniques. This work aims to homogeneously estimate the ${\zeta^{\rm ion}_{{\rm H}_2}}$ at parsec scales towards the Orion Molecular Clouds OMC-2 and OMC-3, probing its variation across a whole star-forming region and a range of column densities never explored before. The most recent ${\zeta^{\rm ion}_{{\rm H}_2}}$ estimates are based on o$-$H$_2$D$^+$, whose abundance we proxy through CO depletion taking advantage of the existing correlation between the two parameters. We therefore employ observations of C$^{18}$O (2$-$1), HCO$^+$ (1$-$0) and DCO$^+$ (3$-$2) towards OMC-2 and OMC-3 to determine the depletion factor, the deuteration fraction and, ultimately, a map of ${\zeta^{\rm ion}_{{\rm H}_2}}$ in these two regions. The depletion factors and deuteration fractions correlate with the total column density of H$_2$, the N$_2$H$^+$ emission and the coldest fields across OMC-2 and OMC-3. The cosmic-ray ionisation rate shows values of ${\zeta^{\rm ion}_{{\rm H}_2}}\sim5\times10^{-18}-10^{-16}$~s$^{-1}$, in agreement with previous o$-$H$_2$D$^+$-based estimates. In addition, it shows an overall decrease for increasing $N(\mathrm{H_2}$), consistently with the predictions from theoretical models. Our approach provides results comparable with theoretical predictions and previous independent studies, confirming the robustness of the analytical framework and the viability of CO depletion as proxy for o$-$H$_2$D$^+$. By exploring the major limitations of the method, we suggest interferometric observations as mandatory to reliably constrain the ${\zeta^{\rm ion}_{{\rm H}_2}}$ also at parsec scales.