Issue 
A&A
Volume 681, January 2024



Article Number  L19  
Number of page(s)  5  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202348947  
Published online  23 January 2024 
Letter to the Editor
HCNH^{+} abundance in cold dense clouds based on the first hyperfine resolved rate coefficients^{⋆}
^{1}
Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes) – UMR 6251, 35000 Rennes, France
email: cheikhtidiane.bop@ucad.edu.sn; francois.lique@univrennes.fr
^{2}
Instituto de Física Fundamental, CSIC, Calle Serrano 123, 28006 Madrid, Spain
email: marcelino.agundez@csic.es
^{3}
Université de Bordeaux – CNRS Laboratoire d’Astrophysique de Bordeaux, 33600 Pessac, France
Received:
13
December
2023
Accepted:
4
January
2024
The protonated form of hydrogen cyanide, HCNH^{+}, holds significant importance in astrochemistry, serving as an intermediate species in ionneutral reactions occurring in the cold molecular clouds. Although it plays a crucial role in the chemistry of HCN and HNC, the excitation rate coefficients of this molecular cation by the dominant interstellar colliders have not been thoroughly investigated, leading to limitations in the radiative transfer models used to derive its abundance. In this work, we present the first hyperfineresolved excitation rate coefficients for HCNH^{+} induced by collisions with both He and H_{2} at low temperatures, addressing a crucial requirement for precise modeling of HCNH^{+} abundance in typical cold dense molecular clouds. Using nonlocal thermodynamic equilibrium (nonLTE) radiative transfer calculations, we reproduced the 1 → 0 and 2 → 1 observational spectra of HCNH^{+} fairly well and derived updated molecular column densities. For the TMC1 molecular cloud, the new HCNH^{+} abundance is twice as large as suggested by previous LTE modeling, whereas the column density of this molecular cation is improved only by 10% in the case of the L483 protostar. The factor of two in the case of TMC1 most likely arises from an error in the early analysis of observational spectra rather than an effect of the LTE assumption, given that the HCNH^{+} lines are predominantly thermalized at densities higher than 2 × 10^{4} cm^{−3}. For multiline studies of clouds of moderate densities, we strongly recommend using the collisional rate coefficients reported in this work.
Key words: molecular data / molecular processes / radiative transfer / scattering / ISM: abundances / ISM: molecules
Hyperfine resolved rate coefficients are available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/vizbin/cat/J/A+A/681/L19
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Protonated hydrogen cyanide, also known as iminomethylium (HCNH^{+}), is the simplest protonated nitrile. It has a dipole moment of ∼0.29 D (Botschwina 1986), large enough to allow its detection in radio astronomy. The first detection of this molecular cation in space took place toward Sgr B2, thanks to the observation of its three lowest rotational lines (Ziurys & Turner 1986). Since then, it has been observed in several cold starforming regions such as the TMC1 dark cloud (Schilke et al. 1991; Ziurys et al. 1992), the DR 21(OH) H II region (Hezareh et al. 2008), the L1544 prestellar core (Quénard et al. 2017), the L483 protostar (Agúndez et al. 2022), and in 16 highmass starforming cores (Fontani et al. 2021). Recently, Gong et al. (2023) reported a comprehensive distribution analysis of HCNH^{+} within the Serpens filament and Serpens South, suggesting that this molecular cation is abundant in cold quiescent regions and deficient toward active starforming regions. These observations present HCNH^{+} as a ubiquitous molecular cation in the cold interstellar medium (ISM) and reinforce the interest in understanding its chemistry.
HCNH^{+} is classified among the most important molecular ions in the ISM since it is the precursor of the widespread HCN and HNC. Ionic compounds play a crucial role in interstellar chemistry, serving as indispensable intermediates in ionneutral reactions that govern gasphase chemistry within cold cores (Agúndez & Wakelam 2013). Extensive research has been conducted to explore the chemistry of HCNH^{+} within dense, cold regions. This molecular cation is mostly formed in low temperature regions (Loison et al. 2014) through the following reactions:
$$\begin{array}{cc}\hfill {\mathrm{HCN}}^{+}/{\mathrm{HCN}}^{+}+{\mathrm{H}}_{2}& \to {\mathrm{HCNH}}^{+}+\mathrm{H},\hfill \end{array}$$(1)
and it undergoes destruction via a dissociative recombination with electrons (Loison et al. 2014; Semaniak et al. 2001):
$$\begin{array}{cc}\hfill {\mathrm{HCNH}}^{+}+{\mathrm{e}}^{}& \to \mathrm{HCN}+\mathrm{H}\hfill \\ \hfill & \to \mathrm{HNC}+\mathrm{H}\hfill \\ \hfill & \to \mathrm{CN}+\mathrm{H}+\mathrm{H}.\hfill \end{array}$$(2)
Based on the reactions in Eqs. (1) and (2) for HCNH^{+} and similar reactions related to HC_{3}NH^{+}, the chemical model of Quénard et al. (2017) was unable to simultaneously reproduce the observed abundance of these protonated molecules toward the cold L1544 prestellar core. The authors pointed out that their model potentially underproduces HCNH^{+}. Fontani et al. (2021) included, in addition to the previous reactions, Eq. (3) in their model as part of the dominant HCNH^{+} formation paths for cold highmass starforming cores:
$$\begin{array}{cc}\hfill {\mathrm{NH}}_{3}+{\mathrm{C}}^{+}& \to {\mathrm{HCNH}}^{+}+\mathrm{H}.\hfill \end{array}$$(3)
They found that changing the initial conditions (the hydrogen column density) or key parameters (the cosmic ray ionization rate) of their chemical model does not lead to a better agreement with the observations. In both attempts, their prediction underestimates the observed HCNH^{+} abundance. In TMC1, Agúndez et al. (2022) found that the [HCNH^{+}]/([HCN]+[HNC]) abundance ratio is underestimated by the chemical model. This finding was seen as a general trend for protonatedtoneutral abundance ratios. More recently, Gong et al. (2023) investigated the impact of the hydrogen number density and the two energy barriers available in the literature for reaction (4), which plays an important role in the HCNH^{+} chemistry:
$$\begin{array}{cc}\hfill \mathrm{HNC}+\mathrm{O}& \to \mathrm{NH}+\mathrm{CO}.\hfill \end{array}$$(4)
Although the authors successfully reproduced the observed abundance of HCNH^{+} in cold highmass starforming cores, it is worth noting the wide range of the reference data they used, [HCNH^{+}]/[H_{2}] = 3 × 10^{−11} − 10^{−9}, which covers all the individual target sources. A selective comparison would suggest a chemical model that yields less HCNH^{+} than observed for five out of the nine studied sources.
The HCN and HNC molecules, which are two of the most important species found in starforming regions, have been extensively studied both from a chemical perspective and in terms of rotational energy transfer. For example, the HCN and HNC abundance profiles derived by Daniel et al. (2013) through nonlocal thermodynamic equilibrium (nonLTE) analysis of observational spectra align well with the chemical predictions of Gérin et al. (2009). Part of this agreement between chemistry and observation can be attributed to the nonLTE modeling of observational spectra, which has been made possible by the availability of accurate HCN and HNC collisional rate coefficients (Ben Abdallah et al. 2012; Sarrasin et al. 2010). In the case of HCNH^{+}, observed abundances have been determined using the LTE approximation, which can lead to discrepancies and consequently contribute to the disagreements with chemical models (Quénard et al. 2017; Fontani et al. 2021; Agúndez et al. 2022; Gong et al. 2023) discussed above. Therefore, it is of high interest to reevaluate the observed HCNH^{+} abundances employing a nonLTE approach, complemented with high accurate collisional rate coefficients.
In the frame of rotational energy transfer, the excitation of HCNH^{+} was first studied by Nkem et al. (2014). They used helium as a projectile and reported rate coefficients for temperatures up to 300 K, considering transitions among the 11 lowlying rotational energy levels. Recently, Bop & Lique (2023) extended the rotational basis considered in the previous work to the 16 first energy levels and also reported the HCNH^{+} scattering data due to collisions with H_{2}, the most abundant species in the ISM. They demonstrated that both orthoH_{2}(j_{2} = 1) and paraH_{2}(j_{2} = 0), where j_{2} represents the rotational quantum number of H_{2}, result in remarkably similar cross sections. This finding supports the idea that only rate coefficients induced by collisions with paraH_{2}(j_{2} = 0) are necessary for modeling the abundance of molecular cations in cold starforming regions.
We revisit the excitation of this molecular cation by paraH_{2} taking into account the influence of the higher rotational energy level of the projectile (j_{2} = 2), as previously done by Hernández Vera et al. (2017) for HCN and HNC. Furthermore, we investigate the hyperfine splitting of the HCNH^{+} collisional rate coefficients resulting from the nonzero nuclear spin of nitrogen, as this effect is clearly resolved in the 1 → 0 observational line spectrum (Ziurys et al. 1992; Quénard et al. 2017).
The structure of this paper is as follows: Sect. 2 provides a concise overview of the scattering calculations. Section 3 is dedicated to the astrophysical modeling, while Sect. 4 presents the concluding remarks.
2. Collisional rate coefficients
Accurate potential energy surfaces (PESs; 4D PES for HCNH^{+}H_{2} and 2D PES for HCNH^{+}He), which are computed using the explicitly correlated coupled cluster method with single, double, and triple noniterative excitation [CCSD(T)F12] (Knowles et al. 1993, 2000) in conjunction with the augmentedcorrelation consistentpolarized valence triple zeta Gaussian best set (augccPVTZ) (Dunning 1989), are available in the literature (Bop & Lique 2023). These authors computed cross sections for the 16 lowlying rotational energy levels of HCNH^{+} due to collisions with both He and paraH_{2} (hereafter denoted as H_{2}) using the “exact” closecoupling quantum mechanical approach (Arthurs & Dalgarno 1960), implemented in the MOLSCAT scattering code (Hutson & Green 1994).
The rotational energy levels were calculated using the spectroscopic constant of H_{2} [B_{0} = 59.322 cm^{−1} and D_{0} = 0.047 cm^{−1}] and HCNH^{+} [B_{0} = 1.2360 cm^{−1} and D_{0} = 1.6075 × 10^{−6} cm^{−1}] (Huber 2013; Amano et al. 2006). The calculations were performed from a total energy of 2.5 cm^{−1} to 800 cm^{−1}, using a fine step size. To better treat the couplings between open and closed channels for convergence reasons, it was necessary to take into account the 31 lowlying rotational energy levels of HCNH^{+} (j_{1} = 0 − 30) in the calculations. The hybrid log derivativeairy propagator was used to solve the coupled equations (Alexander & Manolopoulos 1987). The integration limits were adjusted automatically for each total angular momentum (J), and the switching point from the log derivative to the airy integrator was set to 16 a_{0}. The integration step was maintained below 0.2 a_{0} by adjusting the STEP parameter depending on the collision energy. Here, we address two important aspects of HCNH^{+}–H_{2} collisions that have not been considered before: (i) the effect of the H_{2} rotational basis, specifically the inclusion of j_{2} = 2, and (ii) the HCNH^{+} hyperfine structure arising from the nonzero nuclear spin of nitrogen.
2.1. The impact of the H_{2} rotational basis
The excitation of HCNH^{+} by collisions with H_{2} has been investigated using only j_{2} = 0 (Bop & Lique 2023). The authors roughly estimated the mean deviation of the cross sections due to the inclusion of j_{2} = 2 in the calculations to ∼20%. Considering the huge anisotropy of the HCNH^{+}−H_{2} potential energy surface and its deep global minimum of ∼1426.6 cm^{−1}, revisiting the collisional excitation of HCNH^{+} by H_{2} with the most accurate level of precision becomes a necessary endeavor.
In Fig. 1, we evaluate the influence of the H_{2} rotational basis in the HCNH^{+} collisional cross sections for selected total energies. The incorporation of j_{2} = 0 − 2 into the H_{2} rotational basis results in deviations of up to a factor of three compared to the restriction to j_{2} = 0, while using an extended basis (j_{2} = 0 − 4) instead of j_{2} = 0 − 2 leads to moderate improvements, that is, deviations less than a factor of 1.5. For a total energy of 250 cm^{−1}, the root mean square errors obtained when using j_{2} = 0 and j_{2} = 0 − 4 in comparison with the use of j_{2} = 0 − 2 are 65% and 9%, respectively. Therefore, restricting the H_{2} rotational basis to the ground level is insufficient to derive accurate collisional rate coefficients and the inclusion of j_{2} = 0 − 4 slightly improves the results obtained using j_{2} = 0 − 2, but the computational cost increases by a factor of approximately ten. Therefore, all calculations are performed including j_{2} = 0 − 2 in the H_{2} rotational basis.
Fig. 1. Comparison of the HCNH^{+} cross sections for selected total energies. We note that σ_{J = 0 − 5} is the sum of partial cross sections over total angular momenta up to J = 5. The stars depict the impact of including j_{2} = 0 − 2 in comparison to the constraint of the H_{2} rotational basis to j_{2} = 0, while the empty circles estimate the influence of a more exhaustive H_{2} rotational manifold (j_{2} = 0 − 4) with respect to j_{2} = 0 − 2. The dashed diagonal lines delimit an agreement region of a factor of 1.5. 
2.2. Hyperfine resolved rate coefficients
Fully exploiting the information embedded within hyperfine resolved observational spectra requires an explicit description of the hyperfine splitting in the collisional rate coefficients. In this work, we only consider the coupling between the HCNH^{+} rotation and the nitrogen nuclear spin (I = 1). This coupling results in a slight splitting of each HCNH^{+} rotational level into three hyperfine components, with the exception of the ground energy level, which remains unsplit. The hyperfine components are identified by a quantum number F defined as I − j_{1}≤F ≤ I + j_{1}. We employed the nearly exact recoupling method (Alexander & Dagdigian 1985) in the scattering matrix, which produced the results presented in the previous section. In this manner, we computed hyperfine resolved rate coefficients for the HCNH^{+} 25 lowlying energy levels, (j_{1}, F) ≤ (8, 9), at low temperatures (T = 5 − 30 K).
Figure 2 displays the HCNH^{+} hyperfine resolved rate coefficients obtained using both He and H_{2} as collision partners at 10 K, the typical temperature of cold starforming regions. The magnitude of the H_{2}induced rates vary between ∼10^{−10} and ∼10^{−12} cm^{−3} s^{−1}, whereas the Herates drop drastically down to ∼10^{−14} cm^{−3} s^{−1}. The hyperfine resolution does not alter the existing disparity between the He and H_{2}rate coefficients, as discussed by Bop & Lique (2023), for the rotational transitions. The new insight into this plot is the unveiled propensity rule, Δj_{1} = ΔF, which applies to both projectiles. The data presented in this section are available in electronic supplementary material via the CDS and they will be accessible through databases such as Basecol, LAMDA, and EMAA.
Fig. 2. HCNH^{+} hyperfine resolved rate coefficients for the (8, 9) → (${j}_{1}^{\prime}$, F′) transitions at 10 K. The blue (red) line stands for collisional data obtained using H_{2} (He) as a projectile. 
3. Modeling astronomical lines of HCNH^{+}
The collisional rate coefficients calculated here can be applied to model the lines of HCNH^{+} in those astronomical sources where lines are narrow, so that the hyperfine structure can be resolved. In cold dense clouds, line widths are typically below 1 km s^{−1} (e.g., Agúndez et al. 2023), and thus if observed with a good enough spectral resolution, the hyperfine structure of the lowj_{1} lines of HCNH^{+} can be resolved.
Protonated HCN has been observed in different types of molecular clouds (e.g., Schilke et al. 1991). Here we focus on two cold dense clouds, TMC1 and L483, where lowj_{1} lines of HCNH^{+} have been recently observed with a high spectral resolution. In the case of TMC1, the j_{1} = 1 → 0 and j_{1} = 2 → 1 lines have been observed with the 30 m telescope of the Institut de RadioAstronomie Millimétrique (IRAM) with a spectral resolution of 49 kHz. The observations of the j_{1} = 1 → 0 line at 74.1 GHz are part of a 3 mm line survey (Marcelino et al. 2007; Cernicharo et al. 2012), while those of the j_{1} = 2 → 1 line at 148.2 GHz are part of the Astrochemical Surveys At IRAM (ASAI) program (Lefloch et al. 2018). In the case of L483, the j_{1} = 1 → 0 line at 74.1 GHz was observed with the IRAM 30 m telescope with a spectral resolution of 49 kHz during a 3 mm line survey of this cloud (Agúndez et al. 2019, 2021). The observed lines are shown in Fig. 3.
Fig. 3. Lines of HCNH^{+} in TMC1 (j_{1} = 1 → 0 and j_{1} = 2 → 1 in the left panels) and L483 (j_{1} = 1 → 0 in the right panel). Black histograms correspond to the observed line profiles, while the red lines correspond to the synthetic line profile calculated with the LVG model (see text). 
To model the lines of HCNH^{+}, we carried out excitation and radiative transfer calculations under the large velocity gradient (LVG) formalism (Goldreich & Kwan 1974). The code used is similar to MADEX (Cernicharo et al. 2012). We implemented the rate coefficients calculated here for inelastic collisions of HCNH^{+} with H_{2} and He, where the hyperfine structure of HCNH^{+} is taken into account. The adopted abundance of He relative to H_{2} is 0.17, based on the cosmic abundance of helium, which implies that collisional excitation is dominated by H_{2}, with He playing a minor role. We adopted the physical conditions of TMC1 and L483 from the study of Agúndez et al. (2023). For TMC1 we adopted a gas kinetic temperature of 9 K and a volume density of H_{2} of 1.0 × 10^{4} cm^{−3}, while for L483 the adopted gas temperature is 12 K and the H_{2} volume density 5.6 × 10^{4} cm^{−3}. The adopted line width, 0.46 km s^{−1} for TMC1 and 0.39 km s^{−1} for L483, was taken directly from the arithmetic mean of the values measured on the spectrum of the j_{1} = 1 → 0 line. We then varied the column density of HCNH^{+} until matching the velocityintegrated intensity of the observed lines. The calculated line profiles are compared to the observed ones in Fig. 3. The column densities derived for HCNH^{+} in TMC1 and L483 are 4.2 × 10^{13} cm^{−2} and 2.4 × 10^{13} cm^{−2}, respectively. Previous determinations of the column density based on LTE are 1.9 × 10^{13} cm^{−2} in TMC1 (Schilke et al. 1991) and 2.7 × 10^{13} cm^{−2} in L483 (Agúndez et al. 2019). The values determined here differ with respect to the previous values by a factor of two for TMC1 and by 10% for L483.
To understand the different improvements resulting from nonLTE modeling in comparison to the previous LTEbased abundances of HCNH^{+} obtained for TMC1 and L483, we investigated the deviation of the brightness temperature (T_{B}) with respect to LTE as a function of the gas density. Figure 4 shows that the use of He as a collision partner tends to delay the thermalization of the lines, whereas employing H_{2} as a projectile, which is the major gas component in cold dense clouds, suggests that LTE assumption is valid for densities larger than 2 × 10^{4} cm^{−3}. Furthermore, the j_{1} = 2 → 1 line from Schilke et al. (1991) is reported to have an intensity (in main beam temperature) of 0.48 K, which aligns well with our T_{B} of 0.50 K obtained by correcting the 0.40 K antenna temperature with the beam efficiency of the IRAM 30 m telescope. We thus reinterpreted the observations of HCNH^{+} by employing the LTE assumption. As depicted in Table 1, using this approximation to determine the HCNH^{+} column density introduces an error of approximately 5% in both TMC1 and L483. The factor of two observed in the case of TMC1, when comparing the LVGbased column density with the data reported by Schilke et al. (1991), is likely to result from the assumptions incorporated in their analysis rather than an effect of the adopted LTE approximation.
Fig. 4. Density dependence of the HCNH^{+} brightness temperature ratio for the j_{1} = 1 → 0, 2 → 1, and 3 → 2 lines. Solid and dashed lines were obtained using the collisional rate coefficients of H_{2} and He, respectively. The dashdotted line refers to thermalization. The line width was set to 1.0 km s^{−1}. Thanks to the optically thin regime, these ratios are valid for column densities lower than 5 × 10^{13} cm^{−2}. 
Column density of HCNH^{+} (in 10^{13} cm^{−2}) under the LTE assumption and the LVG formalism for TMC1 and L483.
As discussed in the Introduction, the observed HCNH^{+} column densities are underestimated by predictions from chemical models. In the case of TMC1, for example, Agúndez et al. (2022) found that their chemical model underestimates the [HCNH^{+}]/([HCN]+[HNC]) abundance ratio by a factor of ten. Since we revealed that the observed HCNH^{+} column density in this region is actually twice as large, the protonatedtoneutral abundance ratio turns out to be a factor of 20 higher than predicted by the chemical model. Although this matter remains unresolved, we dispel any doubts that could implicate the observations, clearly identifying the chemical models as the sole factor. We suggest for future modelings using the new experimental rate constants of reactions in Eq. (1) measured for temperatures down to 17 K (Dohnal et al. 2023). Employing the latter results in the chemical models for cold dense clouds is more reasonable than the early reaction rates which were measured at room temperature (Wakelam et al. 2012).
4. Conclusion
We computed the first hyperfine resolved rate coefficients of HCNH^{+} induced by collisions with He and H_{2}. We used the most accurate recoupling method based on nuclear spinindependent scattering matrices calculated by the mean of the closecoupling quantum mechanical approach. When employing H_{2} as the collision partner, we considered the coupling with the first excited rotational energy level of paraH_{2}, thereby improving the previously available nuclear spinfree rotational rate coefficients.
Based on the new rate coefficients, we modeled HCNH^{+} emission lines observed toward TMC1 and L483 using nonLTE radiative transfer calculations under the LVG formalism. With column densities of 4.2 × 10^{13} cm^{−2} and 2.4 × 10^{13} cm^{−2} for TMC1 and L483, respectively, the synthetic spectra reproduced the observed ones quite well. The updated HCNH^{+} abundances differ by a factor of two and by 10% compared to the data previously available in the literature for TMC1 and L483, respectively. It is worth noting that the large discrepancy observed in the case of TMC1 is more likely due to an error in the early analysis of the observational spectra rather than an effect of the LTE assumption. The actual difference in the HCNH^{+} column density derived using LTE and LVG is approximately 5% for both TMC1 and L483. Therefore, we confirm that the use of LTE to model the abundance of HCNH^{+} in cold, dense regions is reasonable. However, we strongly recommend employing the rate coefficients reported in this work for multiline analysis and for observations toward regions of moderate densities.
Acknowledgments
The authors acknowledge the European Research Council (ERC) for funding the COLLEXISM project No 811363, the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of Centre National de la Recherche Scientifique(CNRS)/Institut National des Sciences de l’Univers (INSU) with Institut de Chimie (INC)/Institut de Physique (INP) cofunded by Commissariat a l’Energie Atomique (CEA) and Centre National d’Etudes Spatiales (CNES). F.L. acknowledges the Institut Universitaire de France. M.A. and J.C. acknowledge funding support from Spanish Ministerio de Ciencia e Innovación through grants PID2019107115GBC21 and PID2019106110GBI00. This work made use of ASAI “Astrochemical Surveys At IRAM”.
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All Tables
Column density of HCNH^{+} (in 10^{13} cm^{−2}) under the LTE assumption and the LVG formalism for TMC1 and L483.
All Figures
Fig. 1. Comparison of the HCNH^{+} cross sections for selected total energies. We note that σ_{J = 0 − 5} is the sum of partial cross sections over total angular momenta up to J = 5. The stars depict the impact of including j_{2} = 0 − 2 in comparison to the constraint of the H_{2} rotational basis to j_{2} = 0, while the empty circles estimate the influence of a more exhaustive H_{2} rotational manifold (j_{2} = 0 − 4) with respect to j_{2} = 0 − 2. The dashed diagonal lines delimit an agreement region of a factor of 1.5. 

In the text 
Fig. 2. HCNH^{+} hyperfine resolved rate coefficients for the (8, 9) → (${j}_{1}^{\prime}$, F′) transitions at 10 K. The blue (red) line stands for collisional data obtained using H_{2} (He) as a projectile. 

In the text 
Fig. 3. Lines of HCNH^{+} in TMC1 (j_{1} = 1 → 0 and j_{1} = 2 → 1 in the left panels) and L483 (j_{1} = 1 → 0 in the right panel). Black histograms correspond to the observed line profiles, while the red lines correspond to the synthetic line profile calculated with the LVG model (see text). 

In the text 
Fig. 4. Density dependence of the HCNH^{+} brightness temperature ratio for the j_{1} = 1 → 0, 2 → 1, and 3 → 2 lines. Solid and dashed lines were obtained using the collisional rate coefficients of H_{2} and He, respectively. The dashdotted line refers to thermalization. The line width was set to 1.0 km s^{−1}. Thanks to the optically thin regime, these ratios are valid for column densities lower than 5 × 10^{13} cm^{−2}. 

In the text 
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