Bias versus variance when fitting multi-species molecular lines with a non-LTE radiative transfer model

Application to the estimation of the gas temperature and volume density

¹ Université de Toulon, Aix Marseille Univ, CNRS, IM2NP, Toulon, France
e-mail: antoine.roueff@univ-tln.fr
² IRAM, 300 rue de la Piscine, 38406 Saint-Martin-d’Hères, France
³ LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, 75014 Paris, France
⁴ Instituto de Física Fundamental (CSIC). Calle Serrano 121, 28006, Madrid, Spain
⁵ National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
⁶ Laboratoire d’Astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, Allée Geoffroy Saint-Hilaire, 33615 Pessac, France
⁷ Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, Grenoble 38000, France
⁸ Chalmers University of Technology, Department of Space, Earth and Environment, 412 93 Gothenburg, Sweden
⁹ Univ. Lille, CNRS, Centrale Lille, UMR 9189 - CRIStAL, 59651 Villeneuve d’Ascq, France
¹⁰ LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, 92190 Meudon, France
¹¹ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université Paul Sabatier, Toulouse cedex 4, France
¹² Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile
¹³ Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Sorbonne Paris Cité, Paris, France
¹⁴ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
¹⁵ Department of Earth, Environment, and Physics, Worcester State University, Worcester, MA 01602, USA
¹⁶ Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA, 02138, USA
¹⁷ School of Physics and Astronomy, Cardiff University, Queen’s buildings, Cardiff CF24 3AA, UK

Received: 2 January 2024
Accepted: 19 February 2024

Abstract

Context. Robust radiative transfer techniques are requisite for efficiently extracting the physical and chemical information from molecular rotational lines.

Aims. We study several hypotheses that enable robust estimations of the column densities and physical conditions when fitting one or two transitions per molecular species. We study the extent to which simplifying assumptions aimed at reducing the complexity of the problem introduce estimation biases and how to detect them.

Methods. We focus on the CO and HCO⁺ isotopologues and analyze maps of a 50 square arcminutes field. We used the RADEX escape probability model to solve the statistical equilibrium equations and compute the emerging line profiles, assuming that all species coexist. Depending on the considered set of species, we also fixed the abundance ratio between some species and explored different values. We proposed a maximum likelihood estimator to infer the physical conditions and considered the effect of both the thermal noise and calibration uncertainty. We analyzed any potential biases induced by model misspecifications by comparing the results on the actual data for several sets of species and confirmed with Monte Carlo simulations. The variance of the estimations and the efficiency of the estimator were studied based on the Cramér-Rao lower bound.

Results. Column densities can be estimated with 30% accuracy, while the best estimations of the volume density are found to be within a factor of two. Under the chosen model framework, the peak ¹²CO (1 – 0) is useful for constraining the kinetic temperature. The thermal pressure is better and more robustly estimated than the volume density and kinetic temperature separately. Analyzing CO and HCO⁺ isotopologues and fitting the full line profile are recommended practices with respect to detecting possible biases.

Conclusions. Combining a non-local thermodynamic equilibrium model with a rigorous analysis of the accuracy allows us to obtain an efficient estimator and identify where the model is misspecified. We note that other combinations of molecular lines could be studied in the future.