A new analytic approach to infer the cosmic-ray ionization rate in hot molecular cores from HCO$^+$, N$_2$H$^+$, and CO observations

The cosmic-ray ionization rate ($\zeta_2$) is one of the key parameters in star formation, since it regulates the chemical and dynamical evolution of molecular clouds by ionizing molecules and determining the coupling between the magnetic field and gas. However, measurements of $\zeta_2$ in dense clouds (e.g., $n_{\rm H} \geq 10^4$ cm$^{-3}$) are difficult and sensitive to the model assumptions. The aim is to find a convenient analytic approach that can be used in high-mass star-forming regions (HMSFRs), especially for warm gas environments such as hot molecular cores (HMCs). We propose a new analytic approach to calculate $\zeta_2$ through HCO$^+$, N$_2$H$^+$, and CO measurements. Our method gives a good approximation, to within $50$\%, of $\zeta_2$ in dense and warm gas (e.g., $n_{\rm H} \geq 10^4$ cm$^{-3}$, $T = 50, 100$ K) for $A_{\rm V} \geq 4$ mag and $t \geq 2\times10^4$ yr at Solar metallicity. The analytic approach gives better results for higher densities. However, it starts to underestimate the CRIR at low metallicity ($Z = 0.1Z_\odot$) and high CRIR ($\zeta_2 \geq 3\times10^{-15}$ s$^{-1}$). By applying our method to the OMC-2 FIR4 envelope and the L1157-B1 shock region, we find $\zeta_2$ values of $(1.0\pm0.3)\times10^{-14}$ s$^{-1}$ and $(2.2\pm0.4)\times10^{-16}$ s$^{-1}$, consistent with those previously reported. We calculate $\zeta_2$ toward a total of 82 samples in HMSFRs, finding that the average value of $\zeta_2$ toward all HMC samples ($\zeta_2$ = (7.4$\pm$5.0)$\times$10$^{-16}$ s$^{-1}$) is more than an order of magnitude higher than the theoretical prediction of cosmic-ray attenuation models, favoring the scenario that locally accelerated cosmic rays in embedded protostars should be responsible for the observed high $\zeta_2$.