Interstellar Partition

[Rachelle Shriver]

Weemphasize that the completeness of the partition function, that is, the use of a converged partition function at the typical temperature range of the survey, is very important to decrease the uncertainty on this quantity and thus to derive reliable interstellar molecular densities. In that context, we show how the use of different approximations for the rovibrational partition function together with some interpolation and/or extrapolation procedures may affect the estimate of the interstellar molecular column density. For that purpose, we apply the partition function calculations to astronomical observations performed with the IRAM-30 m telescope towards the NGC 7538-IRS1 source of two N-bearing molecules: isocyanic acid (HNCO, a quasilinear molecule) and methyl cyanide (CH3CN, a symmetric top molecule). The case of methyl formate (HCOOCH3), which is an asymmetric top O-bearing molecule containing an internal rotor is also discussed. Our analysis shows that the use of different partition function approximations leads to relative differences in the resulting column densities in the range 9-43%. Thus, we expect this work to be relevant for surveys of sources with temperatures higher than 300 K and to observations in the infrared. Tables D.1-D.3 are only available at the CDS via anonymous ftp to -strasbg.fr ( ) or via -strasbg.fr/viz-bin/qcat?J/A+A/627/A65

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Molecular astronomy needs accurate spectral analysis of the emission associated with various molecular species in order to identify them and to estimate the physical conditions of the interstellar region they are emitting from (Herbst & van Dishoeck 2009). A comprehensive molecular spectral characterisation is therefore extremely important for astrochemistry because the relative isotopic abundance estimates, the branching ratios, and the rate coefficients along with the activation energy of chemical reactions strongly relies on it (see Shaw 2006). In particular, a precise estimation of the column density of the molecular interstellar species is required in order to investigate the possible chemical reactions taking place in the interstellar medium (ISM). Such estimates are based on molecular spectroscopy that provides transition frequencies, line strengths, and partition functions for a given molecule through accurate laboratory spectral analyses.

Furthermore, the determination of ISM molecular and isotopic abundance ratios provides strong insight into the molecular formation mechanisms that occur in the ISM. To achieve this, it is necessary that the intensity calculation of the molecular species at different temperatures be reliable. It is important to note that the spectroscopic determination of the transition frequencies, the line strengths, and the partition function has to be very accurate due to the high spectral resolution that is now accessible with the present astronomical observatories.

The advent of new infrared and (sub-)millimeter observatories in the last decade (e.g., Atacama Large Millimeter Array (ALMA), Herschel, Stratospheric Observatory For Infrared Astronomy (SOFIA)) has motivated the molecular spectroscopy community to characterize increasingly complex molecules for which spectra were unrecorded until then. The spectroscopic data are gathered through intensive laboratory work, both experimental and theoretical, to predict new and accurate molecular data in the spectral range covered by the observational instruments. In addition, these data permit the exploration of new frequency ranges and enable the prediction of the frequencies via theoretical modeling.

Present databases compile and maintain the spectroscopic data updated in catalogs commonly used by the molecular astronomy community, such as the Cologne Database for Molecular Spectroscopy (CDMS)1 (Endres et al. 2016), the JPL (Jet propulsion Laboratory) database2 (Pickett et al. 1998), the Lovas/NIST catalog3 (Lovas 2004), the Toyama Microwave Atlas for spectroscopists and astronomers4, the SPLATALOGUE database5 (Remijan et al. 2007) and HITRAN6 (Gordon et al. 2017; Gamache et al. 2017). These databases have compiled a huge amount of data provided by spectral analyses performed via intensive laboratory spectral recordings.

So far, spectroscopic studies have made possible the identification of about 200 molecular species7 in star-forming regions and in the ISM. Nevertheless, for a number of molecular species, some reported physical quantities are not always normalized among the different authors. This is the case for example for the partition function values: there are several definitions of the nuclear spin statistical weight and the partition function is not always accounted for. This is also the case of the line strengths: some authors use a definition involving the square molecular dipole moment while others do not.

Moreover, the internal partition functions can be computed in different ways: a direct sum formula can be used, which involves the exponential of the energy levels, if those energy levels are known; if they are not known one can use various approximations to get the partition function. The main issue is the uncertainty on the partition function and its effect on molecular column densities. When the partition function is computed with the direct sum formula, sometimes it is provided without carrying out an appropriate convergence study in the temperature ranges of the ISM, typically from 9.375 to 300 K. The convergence on the partition function is said to be reached when a complete (full) list of rovibrational energy levels is available at the temperature of a given survey. In that case, due to the integrative nature of partition functions, completeness is more important in general than the accuracy with which those energy levels are estimated (Furtenbacher et al. 2016b). On the contrary, when the partition function is computed using various levels of approximations (because the energy level information is not or not easily available), large uncertainties on the partition function can also occur.

Mangum & Shirley (2015) published a review aimed at describing how to calculate the molecular column density from molecular spectral (rotational or ro-vibrational) transitions. Some years before, Fischer & Gamache (2002) and Fischer et al. (2003) studied, for atmospherical and astrophysical species, the convergence of the internal partition function. However, none of these former studies focused on the implications of using an approximate (or a not fully converged) partition function on the estimates of the interstellar molecular column densities.

The present paper aims at addressing the impact of different levels of approximations of the partition function on estimations of the interstellar molecular physical conditions. Some partition function interpolation and extrapolation procedures commonly used in the literature are also presented along with the analysis of their relevance regarding the temperature range. The interpolation and extrapolation procedures are indeed often used to determine the partition function value at any given ISM temperature. The effect of using an incomplete, that is, not fully converged, partition function is illustrated below via the use of the three following molecules, which represent different molecular geometries: isocyanic acid (HNCO), a quasilinear molecule; methyl cyanide (CH3 CN), a symmetric top molecule; and methyl formate (HCOOCH3), an asymmetric top molecule with a large amplitude internal rotor. Finally, a fourth molecule, hydrogen sulfide (H2 S), serves us to illustrate the effect of anharmonicity on the vibrational contribution of the partition function.

This paper is organized as follows: in Sect. 2 we briefly recall how to estimate the temperature and the interstellar molecular abundance from the molecular spectra under the assumption of local thermodynamic equilibrium (LTE). In Sect. 2 we highlight the need for a complete, that is to say a convergent, partition function to provide more accurate estimates and to decrease the uncertainties of the molecular column densities. In Sect. 3, we outline the various approximations that can be done for the rovibrational partition function calculation and the various interpolation and extrapolation procedures in terms of temperature. Section 4 describes the 30 m astronomical observations of NGC 7538-IRS1. In Sect. 5, we give examples of derived molecular column density estimates from different partition function approximations or interpolation and extrapolation procedures. Finally, the conclusions are set out in Sect. 6.

In this section, we briefly describe how the ISM molecular column densities and/or abundances are derived. For emission lines associated with a given molecule in an astronomical survey, one can derive the total molecular column density N as a function of the integrated intensity W, the partition function and the excitation temperature. The mathematical expression depends on the assumption considered (i.e., optically thin emission, negligible background temperature, LTE, and so on; for further details see Goldsmith & Langer 1999; Mangum & Shirley 2015).

In this section, we present some approximations used for the calculation of the rovibrational partition function of a free molecule with the aim of reaching the correct values (under a convergence study) at the typical temperature range of the ISM.

A molecular partition function8 is defined as a direct sum of exponential terms that involve the energy levels and the temperature of the environment where the molecule is located. Therefore, the direct sum of the partition function can be given as(5)

where Ji and are the rotational angular momentum and the nuclear spin degeneracy of the energy level i, respectively, and Ei is the rovibrational energy usually referred to the ground vibrational state as it is assumed that uniquely the ground electronic state is populated (Fischer & Gamache 2002; Cerezo et al. 2014). This means that an accurate partition function will be determined at any temperature if all energies Ei are accurately known. At the present time, this is unfortunately often not possible because all transitions (and thus all energy levels) have not yet been measured. From a quantum mechanical point of view, one might assume that the molecular parameters derived from the observed transition frequencies can be used to predict the missing energy levels, however their estimates are not always lyingwithin the experimental uncertainty. Therefore, the exponential terms in Eq. (5) are limited to the lowest energies Ei up to a certain upper threshold.

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