A new analytic approach to infer the cosmic-ray ionization rate in hot molecular cores from HCO⁺, N₂H⁺, and CO observations

¹ Institut de Radioastronomie Millimetrique, 300 rue de la Piscine, 38400 Saint-Martin d’Hères, France
² Research Center for Astronomical Computing, Zhejiang Laboratory, Hangzhou 311100, China
³ INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
⁴ Department of Space, Earth and Environment, Chalmers University of Technology, Gothenburg 412 96, Sweden

^★ Corresponding author; luo@iram.fr; tbisbas@zhejianglab.com

Received: 8 April 2024
Accepted: 11 September 2024

Abstract

Context. The cosmic-ray ionization rate (ζ₂) 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.

Aims. However, measurements of ζ₂ in dense clouds (e.g., n_H ≥ 10⁴ 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).

Methods. We propose a new analytic approach to calculate ζ₂ through HCO⁺, N₂H⁺, and CO measurements. By comparing our method with various astrochemical models and with observations found in the literature, we identify the parameter space for which the analytic approach is applicable.

Results. Our method gives a good approximation, to within 50%, of ζ₂ in dense and warm gas (e.g., n_H ≥ 10⁴ cm⁻³, T = 50, 100 K) for A_V ≥ 4 mag and t ≥ 2 × 10⁴ yr at Solar metallicity. The analytic approach gives better results for higher densities. However, it starts to underestimate ζ₂ at low metallicity (Z = 0.1 Z_⊙) when the value is too high (ζ₂ ≥ 3 × 10⁻¹⁵ s⁻¹). By applying our method to the OMC-2 FIR4 envelope and the L1157-B1 shock region, we find ζ₂ values of (1.0 ± 0.3) × 10⁻¹⁴ s⁻¹ and (2.2 ± 0.4) × 10⁻¹⁶ s⁻¹, consistent with those previously reported.

Conclusions. We calculate ζ₂ toward a total of 82 samples in HMSFRs, finding that the average value of ζ₂ toward all HMC samples (ζ₂ = (7.4±5.0)×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 ζ₂.