Testing analytical methods to derive the cosmic-ray ionisation rate in cold regions via synthetic observations

¹ Centre for Astrochemical Studies, Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, 85749 Garching bei München, Germany
e-mail: eredaelli@mpe.mpg.de
² Departamento de Astronomía, Facultad Ciencias Físicas y Matemáticas, Universidad de Concepción, Av. Esteban Iturra s/n Barrio Universitario, Casilla 160, Concepción, Chile
³ INAF, Istituto di Radioastronomia – Italian node of the ALMA Regional Centre (It-ARC), Via Gobetti 101, 40129 Bologna, Italy
⁴ Dipartimento di Chimica, Università degli Studi di Roma “La Sapienza”, P.le Aldo Moro 5, 00185 Roma, Italy
⁵ Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy
⁶ INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
⁷ INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy

Received: 14 March 2023
Accepted: 15 February 2024

Abstract

Context. Cosmic rays (CRs) heavily impact the chemistry and physics of cold and dense star-forming regions. However, the characterisation of their ionisation rate continues to pose a challenge from the observational point of view.

Aims. In the past, a few analytical formulas have been proposed to infer the cosmic-ray ionisation rate, ζ₂, from molecular line observations. These have been derived from the chemical kinetics of the involved species, but they have not yet been validated using synthetic data processed with a standard observative pipeline. In this work, we aim to bridge this gap.

Methods. We performed a radiative transfer on a set of three-dimensional magneto-hydrodynamical simulations of prestellar cores, exploring different initial ζ₂, evolutionary stages, types of radiative transfer (for instance assuming local-thermodynamic-equilibrium conditions), and telescope responses. We then computed the column densities of the involved tracers to determine ζ₂, employing a recently proposed method based on the detection of H₂D⁺. We compared this approach with a previous method, based on more common tracers. Both approaches are commonly used.

Results. Our results confirm that the equation based on the detection of H₂D⁺ accurately retrieves the actual ζ₂ within a factor of two to three in the physical conditions explored in our tests. Since we have also explored a non-local thermodynamic equilibrium (non-LTE) radiative transfer, this work indirectly offers insights into the excitation temperatures of common transitions at moderate volume densities (n ≈ 10⁵ cm⁻³). We also performed a few tests using a previous methodology that is independent of H₂D⁺, which overestimates the actual ζ₂ by at least two orders of magnitude. We considered a new derivation of this method, however, we found that it still leads to high over-estimations.

Conclusions. The method based on H₂D⁺ is further validated in this work and demonstrates a reliable method for estimating ζ₂ in cold and dense gas. On the contrary, the former analytical equation, as already pointed out by its authors, has no global domain of application. Thus, we find that it ought to be employed with caution.