# The interstellar chemistry of C3H and C3H2 isomers

Jean-Christophe Loison, Marcelino Marcelino, Valentine Wakelam, Evelyne Roueff, P. Gratier, Núria Marcelino, Dianailys Nuñez Reyes, José Cernicharo, Maryvonne Gerin, Marcelino Agúndez

► To cite this version:

Jean-Christophe Loison, Marcelino Marcelino, Valentine Wakelam, Evelyne Roueff, P. Gratier, et al. The interstellar chemistry of C3H and C3H2 isomers. Monthly Notices of the Royal Astronomical Society, 2017, 470 (4), pp.4075 - 4088. 10.1093/mnras/stx1265. hal-01583213

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# The interstellar chemistry of C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub> isomers

Jean-Christophe Loison,<sup>1*</sup> Marcelino Agúndez,<sup>2*</sup> Valentine Wakelam,<sup>3</sup>

Evelyne Roueff,<sup>4</sup> Pierre Gratier,<sup>3</sup> Núria Marcelino,<sup>2</sup> Dianailys Nuñez Reyes,<sup>1</sup>

José Cernicharo<sup>2</sup> and Maryvonne Gerin<sup>5</sup>

<sup>1</sup>Institut des Sciences Moléculaires (ISM), CNRS, Université Bordeaux, 351 cours de la Libération, F-33400, Talence, France

<sup>2</sup>Instituto de Ciencia de Materiales de Madrid, CSIC, C\ Sor Juana Inés de la Cruz 3, E-28049 Cantoblanco, Spain

<sup>3</sup>Laboratoire d'astrophysique de Bordeaux, Université Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, F-33615 Pessac, France

<sup>4</sup>LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Université Paris 06, F-92190 Meudon, France

<sup>5</sup>LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Université Paris 06, Ecole Normale Supérieure, F-75005 Paris, France

Accepted 2017 May 19. Received 2017 May 16; in original form 2016 December 24

## ABSTRACT

We report the detection of linear and cyclic isomers of C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub> towards various starless cores and review the corresponding chemical pathways involving neutral (C<sub>3</sub>H<sub>x</sub> with x = 1,2) and ionic (C<sub>3</sub>H<sub>x</sub><sup>+</sup> with x = 1,2,3) isomers. We highlight the role of the branching ratio of electronic dissociative recombination (DR) reactions of C<sub>3</sub>H<sub>2</sub><sup>+</sup> and C<sub>3</sub>H<sub>3</sub><sup>+</sup> isomers, showing that the statistical treatment of the relaxation of C<sub>3</sub>H<sup>*</sup> and C<sub>3</sub>H<sub>2</sub><sup>*</sup> produced in these DR reactions may explain the relative c,l-C<sub>3</sub>H and c,l-C<sub>3</sub>H<sub>2</sub> abundances. We have also introduced in the model the third isomer of C<sub>3</sub>H<sub>2</sub> (HCCCH). The observed cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio varies from 110 ± 30 for molecular clouds with a total density of about 1 × 10<sup>4</sup> molecules cm<sup>-3</sup> to 30 ± 10 for molecular clouds with a total density of about 4 × 10<sup>5</sup> molecules cm<sup>-3</sup>, a trend well reproduced with our updated model. The higher ratio for molecular clouds with low densities is determined mainly by the importance of the H + l-C<sub>3</sub>H<sub>2</sub> → H + c-C<sub>3</sub>H<sub>2</sub> and H + t-C<sub>3</sub>H<sub>2</sub> → H + c-C<sub>3</sub>H<sub>2</sub> isomerization reactions.

**Key words:** astrochemistry – ISM: abundances – ISM: clouds – ISM: molecules.

## 1 INTRODUCTION

C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub> species are observed in two isomeric forms, cyclic and linear, in various interstellar environments, with the cyclic isomers being the most stable in both cases. The ring molecule cyclopropynylidyne (c-C<sub>3</sub>H) has been widely observed in the Galaxy (Turner et al. 2000; Fossé et al. 2001; Pety et al. 2012, Liszt et al. 2014) since its discovery by Yamamoto et al. (1987). Its linear counterpart, propynylidyne (l-C<sub>3</sub>H), was discovered in TMC-1 and IRC+10216 by Thaddeus et al. (1985) and is also widely present in the Galaxy (Turner et al. 2000; Fossé et al. 2001; Pety et al. 2012; McGuire et al. 2013). It is associated with the cyclic isomer but in general with a notably smaller abundance. Cyclopropenylidene (c-C<sub>3</sub>H<sub>2</sub>) has also been observed in many environments (Turner et al. 2000; Fossé et al. 2001; Pety et al. 2012; McGuire et al. 2013; Liszt et al. 2014; Sakai et al. 2010) since its discovery by Thaddeus et al. (1985). The linear isomer, propadienylidene (l-C<sub>3</sub>H<sub>2</sub>), a cumulene carbon chain (H<sub>2</sub>C = C = C:), was discovered in TMC-1

by Cernicharo et al. (1991) and in other sources (Cernicharo et al. 1999; Turner et al. 2000; Fossé et al. 2001; Pety et al. 2012, Liszt et al. 2012), despite the fact that it is always much less abundant than the cyclic isomer. The linear l-C<sub>3</sub>H<sub>2</sub> isomer has been proposed as a potential carrier of the 4881- and 5450-Å diffuse interstellar bands (DIBs) (Maier et al. 2011). This attribution was finally abandoned, however, because the column densities required to reproduce the DIBs were two to three orders of magnitude higher than the values measured in the diffuse medium at radio frequencies in absorption (Krełowski et al. 2011; Liszt et al. 2012).

The presence of these isomers raises various key questions about their chemistry. In particular, because cyclic and linear C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub> are not well reproduced by the usual models (Agúndez & Wakelam 2013; Sipilä et al. 2016), are rings and chains formed from the same progenitors. Furthermore, is there a relationship between the abundance ratio of the most stable isomer to that of the other isomers and their energy differences, a relationship that is believed to follow the 'minimum energy principle' (Lattelais et al. 2009) despite some counter examples (Loomis et al. 2015; Loison et al. 2016). Moreover, can we explain the fact that, as already noted by Fossé et al. (2001), the abundance ratio of cyclic to linear isomers is not constant across the Galaxy. In addition to these questions, we

* E-mail: jean-christophe.loison@u-bordeaux.fr (J-CL);
marcelino.agundez@icmm.csic.es (MA)
<table><caption>**Table 1.** C<sub>3</sub>H<sub>x</sub> = 1-3 neutral and ionic isomer characteristics.</caption><thead><tr><th>Species</th><th>ΔH<sub>f</sub><sup>298</sup> kJ mol<sup>-1</sup></th><th>μ (D)</th></tr></thead><tbody><tr><td>c-C<sub>3</sub>H cyclo-propynylidyne</td><td>715 ± 8 kJ/mol (Costes et al. 2009)</td><td>2.30 (experimental) (Lovas et al. 1992)</td></tr><tr><td>l-C<sub>3</sub>H propynylidyne</td><td>727 ± 8 kJ/mol (Costes et al. 2009)</td><td>3.55 (theory) (Woon 1995)</td></tr><tr><td>l-C<sub>3</sub>H<sup>+</sup></td><td>1591 kJ/mol (Costes et al. 2009; Wang et al. 2007)</td><td>3.06 (theory) (Huang et al. 2013)</td></tr><tr><td>c-C<sub>3</sub>H<sup>+</sup></td><td>1661 kJ/mol (Costes et al. 2009; Wang et al. 2007)</td><td>1.49 (theory) (This work)</td></tr><tr><td>c-C<sub>3</sub>H<sub>2</sub> cyclo-propenylidene</td><td>497 ± 4 kJ/mol (Vazquez et al. 2009)</td><td>3.27 (experimental) (Lovas et al. 1992)</td></tr><tr><td>t-C<sub>3</sub>H<sub>2</sub> (HCCCH) Propynylidene</td><td>543 ± 8 kJ/mol (Aguilera-Iparraguirre et al. 2008)</td><td>0.51 (Nguyen et al. 2001)</td></tr><tr><td>l-C<sub>3</sub>H<sub>2</sub> (H<sub>2</sub>CCC) propadienylidene (vinylidencarbene)</td><td>557 ± 4 kJ/mol (Vazquez et al. 2009)</td><td>4.16 (theory) (Wu et al. 2010).</td></tr><tr><td>c-C<sub>3</sub>H<sub>2</sub><sup>+</sup></td><td>1382.8 ± 9.2 kJ/mol (C. Lau & Ng 2006)</td><td>1.2 (theory) (this work)</td></tr><tr><td>l-C<sub>3</sub>H<sub>2</sub><sup>+</sup> (HCCCH<sup>+</sup>)</td><td>1396 ± 16 kJ/mol (Prodnuk et al. 1990)</td><td>0 (theory) (Wong & Radom 1993; Taatjes et al. 2005)</td></tr><tr><td>l-C<sub>3</sub>H<sub>2</sub><sup>+</sup>, b (H<sub>2</sub>CCC<sup>+</sup>)</td><td>1565 kJ/mol (C. Lau & Ng 2006)</td><td>3.0 (theory) (this work)</td></tr><tr><td>C<sub>3</sub>H<sub>3</sub> (CH<sub>2</sub>CCH) 2-propynyl (propargyl)</td><td>352 ± 4 kJ/mol (Vazquez et al. 2009)</td><td>0.14 (theory) (this work)</td></tr><tr><td>c-C<sub>3</sub>H<sub>3</sub><sup>+</sup></td><td>1076 ± 6 kJ/mol (CCSD(T)-F12/aug-cc-pVQZ calculations by comparison with l-C<sub>3</sub>H<sub>3</sub><sup>+</sup>, this work)</td><td>0 (theory) (this work)</td></tr><tr><td>l-C<sub>3</sub>H<sub>3</sub><sup>+</sup></td><td>1191 ± 4 kJ/mol (ΔH<sub>f</sub><sup>298</sup>(C<sub>3</sub>H<sub>3</sub>) + IE(C<sub>3</sub>H<sub>3</sub>) = 839.5 ± 0.1 kJ/mol (Jacovella et al. 2013))</td><td>0.524 (theory) (Huang et al. 2011)</td></tr></tbody></table>

emphasize the specific behaviour of C<sub>3</sub>H<sup>+</sup>, which is found in only two photodissociation regions (PDRs) (Pety et al. 2012; McGuire et al. 2014), and nowhere else, despite thorough searches in other interstellar environments (McGuire et al. 2013, 2015). In order to gain a better picture of C<sub>3</sub>H<sub>x</sub> chemistry, we performed an extensive and systematic review and update of the KIDA chemical network (Wakelam et al. 2015b) using the methodology presented in Loison et al. (2014a) and in Appendix A of this paper, including specific theoretical calculations presented in Appendix B. We have in particular introduced the third isomer of C<sub>3</sub>H<sub>2</sub> (t-C<sub>3</sub>H<sub>2</sub>: H-CCC-H), a quasi-linear species (Nguyen et al. 2001; Aguilera-Iparraguirre et al. 2008), which may play a non-negligible role as it has a different chemistry from that of the other C<sub>3</sub>H<sub>2</sub> isomers. The thermochemical properties of the C<sub>3</sub>H<sub>x</sub> = 0-3 neutral and ionic isomers are summarized in Table 1. It should be noted that the geometry of l-C<sub>3</sub>H<sub>2</sub><sup>+</sup> corresponds to that of t-C<sub>3</sub>H<sub>2</sub>, namely HCCCH<sup>+</sup>. We did not introduce the cyclic isomer of C<sub>3</sub>H<sup>+</sup> nor the third isomeric form of C<sub>3</sub>H<sub>2</sub><sup>+</sup> (H<sub>2</sub>CCC<sup>+</sup>), which are notably less stable than the other

isomers and, considering the chemistry, are not supposed to play an important role.

We present our observations in Section 2 and our chemical model, including various updates, in Section 3. Our conclusions are presented in Section 4. This study is the last part of a large review of C<sub>3</sub>H<sub>x</sub> = 0-8 O<sub>y</sub> = 0-1 chemistry in cold interstellar media (Wakelam et al. 2015a; Loison et al. 2016; Hickson et al. 2016b).

## 2 OBSERVATIONS

We carried out observations of cyclic and linear C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub> towards various cold dark clouds. The observations were performed with the IRAM 30-m telescope using the frequency-switching technique and the EMIR 3-mm receiver connected to a fast Fourier transform spectrometer providing a spectral resolution of 50 kHz. At the IRAM 30-m telescope, spectra are calibrated by comparing the sky emissivity with that of hot and cold loads and using the ATM program (Cernicharo 1985; Pardo et al. 2001). The uncertainty in
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Sample of lines of c-C<sub>3</sub>H<sub>2</sub>, l-C<sub>3</sub>H<sub>2</sub>, c-C<sub>3</sub>H, and l-C<sub>3</sub>H observed in the cold dense clouds, where the lines of each source have been shifted by fixed amounts along the velocity axis for a better visualization. Dashed vertical lines indicate the position of the V<sub>LSR</sub> of each source. Note that for TMC-1 and B1-b, lines can be fitted to two velocity components. Transition frequencies and quantum numbers of the observed lines are listed in Tables 2–4.</figcaption></figure>

the calibration in the 3-mm band is ~10 per cent. Pointing was checked on nearby planets and quasars. The error in the pointing is 3 arcsec at most. Data reduction, carried out using the software CLASS within the package GILDAS<fnref n="1" />, consisted in the subtraction of a baseline generated by smoothing the spectral regions free of lines. The observations of TMC-1 and B1-b are part of a 3-mm line survey (Marcelino et al. 2009), a good part of which was observed between 2012 January and May (for details, see Cernicharo et al. 2012). In the cases of L483, Lupus-1A, L1495B, L1521F and Serpens South 1a, the observations were carried out from 2014 September to November in selected frequency ranges across the 3-mm band (Agúndez et al. 2015). Examples of the observed spectra are shown in Fig. 1. The line parameters given in Tables 2–4 were obtained by Gaussian fitting to the observed line profiles. In some sources, lines were simultaneously fitted to two Gaussian components because this resulted in a better fit. For example, in B1-b there are two velocity components at systemic velocity in the Local Standard of Rest frame (V<sub>LSR</sub>) + 6.5 and + 7.4 km s<sup>-1</sup>. In TMC-1 the existence of the two velocity components is less obvious, but still the lines of some molecules can be decomposed into two velocity components at V<sub>LSR</sub> + 5.7 and + 6.0 km s<sup>-1</sup>.

## 2.1 Observed abundances of c/l-C<sub>3</sub>H<sub>2</sub> and c/l-C<sub>3</sub>H

From the observed line intensities, we determined the column densities of c/l-C<sub>3</sub>H<sub>2</sub> and c/l-C<sub>3</sub>H in the various sources, averaged over

the beam of the IRAM 30-m telescope, which, at the observed frequencies, ranges from 23 to 29 arcsec. These column densities are given in Table 5, as are the physical conditions corresponding to the various sources.

### 2.1.1 Statistical equilibrium: c-C<sub>3</sub>H<sub>2</sub> and l-C<sub>3</sub>H<sub>2</sub>

In the cases of cyclic and linear C<sub>3</sub>H<sub>2</sub>, we carried out statistical equilibrium calculations using the Large Velocity Gradient (LVG) method and assuming a medium with a uniform volume density of H<sub>2</sub> and gas kinetic temperature. These two physical parameters were taken from the literature for each source, and, when possible (i.e. when multiple lines of c-C<sub>3</sub>H<sub>2</sub> or l-C<sub>3</sub>H<sub>2</sub> were available), additional constraints were provided by the line intensity ratios observed in this study. The adopted values of T<sub>k</sub> and n(H<sub>2</sub>) are given in Table 5. The gas kinetic temperature in most sources is about 10 K, whereas densities span over almost two orders of magnitude in the range 10<sup>4</sup>–10<sup>6</sup> cm<sup>-3</sup>. It is important to note that the derived column densities of cyclic and linear C<sub>3</sub>H<sub>2</sub> are very sensitive to the adopted volume density (the higher the density, the lower the column density needed to be in order to reproduce the observed line intensities), although the cyclic-to-linear ratios show little dependence on n(H<sub>2</sub>). The ortho- and para-species of both c-C<sub>3</sub>H<sub>2</sub> and l-C<sub>3</sub>H<sub>2</sub> were considered independently in the statistical equilibrium calculations, and a statistical ortho-to-para ratio of 3 was assumed for both isomers, which is consistent with the intensity ratios between the ortho- and para-lines observed. In order to derive the column densities, we used various LVG models in which the column density of C<sub>3</sub>H<sub>2</sub> was varied (for those sources for which

<footnote n="1">¹ See http://www.iram.fr/IRAMFR/GILDAS.</footnote>
<table><caption>**Table 2.** Observed line parameters of *c*-C<sub>3</sub>H<sub>2</sub> and *l*-C<sub>3</sub>H<sub>2</sub> in TMC-1 and B1-b.</caption><thead><tr><th>Molecule</th><th>Transition</th><th>Frequency (MHz)</th><th>E<sub>up</sub> (K)</th><th>A<sub>ul</sub> (s<sup>-1</sup>)</th><th>HPBW (arcsec)</th><th>B<sub>eff</sub>/F<sub>eff</sub></th><th>V<sub>LSR</sub> (km s<sup>-1</sup>)</th><th>&Delta;v (km s<sup>-1</sup>)</th><th>&int; T<sup>*</sup><sub>A</sub> dv (K km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td colspan="10" style="text-align: center; font-weight: bold;">TMC-1</td></tr><tr><td rowspan="4">c-C<sub>3</sub>H<sub>2</sub></td><td>3<sub>1,2</sub> &ndash; 3<sub>0,3</sub> (ortho)</td><td>82966.196</td><td>13.7</td><td>9.91 &times; 10<sup>-6</sup></td><td>29.3</td><td>0.86</td><td>+5.83(2)</td><td>0.46(1)</td><td>0.279(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.32 &times; 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+5.82(1)</td><td>0.56(1)</td><td>1.659(1)</td></tr><tr><td>4<sub>3,2</sub> &ndash; 4<sub>2,3</sub> (ortho)</td><td>85656.415</td><td>26.7</td><td>1.52 &times; 10<sup>-5</sup></td><td>28.4</td><td>0.86</td><td>+5.81(1)</td><td>0.49(5)</td><td>-0.013(1)<sup>a</sup></td></tr><tr><td>3<sub>2,2</sub> &ndash; 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 &times; 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+5.78(2)</td><td>0.50(2)</td><td>0.078(2)</td></tr><tr><td rowspan="5">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> &ndash; 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 &times; 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+5.65(2)</td><td>0.24(5)</td><td>0.033(4)</td></tr><tr><td>5<sub>1,5</sub> &ndash; 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 &times; 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+5.94(2)</td><td>0.27(6)</td><td>0.024(4)</td></tr><tr><td>5<sub>1,4</sub> &ndash; 4<sub>1,3</sub> (ortho)</td><td>104915.583</td><td>14.1</td><td>1.022 &times; 10<sup>-4</sup></td><td>23.2</td><td>0.84</td><td>+5.65(3)</td><td>0.27(5)</td><td>0.010(2)</td></tr><tr><td>4<sub>0,4</sub> &ndash; 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 &times; 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+5.93(6)</td><td>0.35(11)</td><td>0.009(2)</td></tr><tr><td>5<sub>0,5</sub> &ndash; 4<sub>0,4</sub> (para)</td><td>103952.926</td><td>15.0</td><td>1.032 &times; 10<sup>-4</sup></td><td>23.4</td><td>0.84</td><td>+5.69(2)</td><td>0.31(4)</td><td>0.018(2)</td></tr><tr><td colspan="10" style="text-align: center; font-weight: bold;">B1-b</td></tr><tr><td rowspan="4">c-C<sub>3</sub>H<sub>2</sub></td><td>3<sub>1,2</sub> &ndash; 3<sub>0,3</sub> (ortho)</td><td>82966.196</td><td>13.7</td><td>9.912 &times; 10<sup>-6</sup></td><td>29.3</td><td>0.86</td><td>+6.49(1)</td><td>0.88(2)</td><td>0.289(6)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 &times; 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+7.39(1)</td><td>0.58(3)</td><td>0.100(5)</td></tr><tr><td>4<sub>3,2</sub> &ndash; 4<sub>2,3</sub> (ortho)</td><td>85656.415</td><td>26.7</td><td>1.522 &times; 10<sup>-5</sup></td><td>28.4</td><td>0.86</td><td>+6.55(1)</td><td>0.92(1)</td><td>1.492(9)</td></tr><tr><td>3<sub>2,2</sub> &ndash; 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 &times; 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+7.41(1)</td><td>0.66(1)</td><td>0.597(9)</td></tr><tr><td rowspan="5">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> &ndash; 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 &times; 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+6.54(2)</td><td>0.67(5)</td><td>0.020(1)</td></tr><tr><td>5<sub>1,5</sub> &ndash; 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 &times; 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+7.45(7)</td><td>0.30(14)</td><td>0.002(1)</td></tr><tr><td>5<sub>1,4</sub> &ndash; 4<sub>1,3</sub> (ortho)</td><td>104915.583</td><td>14.1</td><td>1.022 &times; 10<sup>-4</sup></td><td>23.2</td><td>0.84</td><td>+6.50(2)</td><td>0.86(3)</td><td>0.087(3)</td></tr><tr><td>4<sub>0,4</sub> &ndash; 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 &times; 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+7.38(3)</td><td>0.55(5)</td><td>0.025(3)</td></tr><tr><td>5<sub>0,5</sub> &ndash; 4<sub>0,4</sub> (para)</td><td>103952.926</td><td>15.0</td><td>1.032 &times; 10<sup>-4</sup></td><td>23.4</td><td>0.84</td><td>+6.56(4)</td><td>0.95(7)</td><td>0.075(5)</td></tr></tbody></table>

Notes. Numbers in parentheses are $1\sigma$ uncertainties arising from the line fitting in units of the last digits. The values of $\int T_A^* dv$ have an additional uncertainty of ~10 per cent owing to calibration. The antenna temperature ($T_A^*$) can be converted to the main beam brightness temperature ($T_{MB}$) by dividing by ($B_{eff}/F_{eff}$).

<sup>a</sup>Line in absorption against the cosmic microwave background.

multiple lines were available, the volume density was also varied) and selected the model that resulted in the best overall agreement between the observed and calculated velocity-integrated intensities. When some of the lines were calculated to be optically thick, as occurs for the $2_{1,2} - 1_{0,1}$ line of c-C<sub>3</sub>H<sub>2</sub> in most sources, we prioritized in order to reproduce the intensities of those lines that are optically thin.

For c-C<sub>3</sub>H<sub>2</sub>, level energies were computed from the rotational constants derived by Bogey et al. (1987), and line strengths were computed from the dipole moment of 3.27 D measured by Lovas et al. (1992), which is slightly lower than the value of 3.43 D previously measured by Kanata et al. (1987). Rate coefficients for the de-excitation of c-C<sub>3</sub>H<sub>2</sub> through inelastic collisions with He have been calculated by Avery & Green (1989), including 16 and 17 levels for ortho- and para- c-C<sub>3</sub>H<sub>2</sub>, respectively and covering the temperature range 10–30 K, and by Chandra & Kegel (2000), accounting for 47 and 48 levels for ortho- and para- c-C<sub>3</sub>H<sub>2</sub>, respectively, and covering the temperature range 30–120 K. Unfortunately, Chandra & Kegel (2000) did not extend their calculations down to 10 K. Therefore, in this work we adopted the values of Avery & Green (1989). We note, however, that by comparing the rate coefficients calculated in both studies at 30 K the values of Avery & Green (1989) are systematically higher (by up to a factor of 2) than those of Chandra & Kegel (2000), and thus at low densities it may lead to column densities of c-C<sub>3</sub>H<sub>2</sub> significantly lower than those obtained using the coefficients of Chandra & Kegel (2000). In this

study, the collision rate coefficients of Avery & Green (1989) at 10 K were adopted, scaled up by a factor of 1.38 to account for H<sub>2</sub> instead of He as collider.

In the case of l-C<sub>3</sub>H<sub>2</sub>, level energies were computed from the CDMS archive<sup>2</sup> (Müller et al. 2005), which is based on the experimental data of Vrtilek et al. (1990), and a dipole moment of 4.16 D was adopted, based on the *ab initio* calculations of Wu et al. (2010). As rate coefficients for the de-excitation of l-C<sub>3</sub>H<sub>2</sub> through inelastic collisions with H<sub>2</sub> and He, we adopted those calculated for H<sub>2</sub>CO by Green (1991).

The density of H<sub>2</sub> at the position of the cyanopolyyne peak in TMC-1 has been estimated as $(3-8) \times 10^4 \text{ cm}^{-3}$ by Pratap et al. (1997), based on observations of three transitions of HC<sub>3</sub>N, and as about $3 \times 10^4 \text{ cm}^{-3}$ by Lique et al. (2006), based on the observations of multiple transitions of SO. Here we adopt this latter value, which was also adopted by Fossé et al. (2001) in their study of c/l-C<sub>3</sub>H<sub>2</sub> in TMC-1. The fact that the $4_{3,2} - 4_{2,3}$ transition at 85.6 GHz, with an upper level energy of 26.7 K, is observed in absorption against the cosmic microwave background (see Table 1) indicates that this transition has an excitation temperature below 2.7 K and that c-C<sub>3</sub>H<sub>2</sub> is not thermalized in TMC-1, which is consistent with the adopted value of $n(H_2)$. From the intensities of the four lines of c-C<sub>3</sub>H<sub>2</sub> observed in TMC-1 we derive a column density of $1.2 \times 10^{14} \text{ cm}^{-2}$.

<footnote n="2"><sup>2</sup>See http://www.astro.uni-koeln.de/cdms.</footnote>
<table><caption>**Table 3.** Observed line parameters of *c*-C<sub>3</sub>H<sub>2</sub> and *l*-C<sub>3</sub>H<sub>2</sub> in various cold dense clouds.</caption><thead><tr><th>Molecule</th><th>Transition</th><th>Frequency<br>(MHz)</th><th>E<sub>up</sub><br>(K)</th><th>A<sub>ul</sub><br>(s<sup>-1</sup>)</th><th>HPBW<br>(arcsec)</th><th><i>B</i><sub>eff</sub>/<i>F</i><sub>eff</sub></th><th><i>V</i><sub>LSR</sub><br>(km s<sup>-1</sup>)</th><th>Δ<i>v</i><br>(km s<sup>-1</sup>)</th><th>∫ <i>T</i><sup>*</sup><sub>A</sub> dv<br>(K km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td colspan="11" style="text-align: center; font-weight: bold;">L483</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+5.26(1)</td><td>0.55(1)</td><td>1.637(9)</td></tr><tr><td>4<sub>3,2</sub> – 4<sub>2,3</sub> (ortho)</td><td>85656.415</td><td>26.7</td><td>1.522 × 10<sup>-5</sup></td><td>28.4</td><td>0.86</td><td>+5.28(2)</td><td>0.58(6)</td><td>0.069(5)</td></tr><tr><td>3<sub>2,2</sub> – 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 × 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+5.29(1)</td><td>0.39(1)</td><td>0.165(2)</td></tr><tr><td rowspan="3">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> – 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 × 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+5.30(1)</td><td>0.36(1)</td><td>0.091(3)</td></tr><tr><td>5<sub>1,5</sub> – 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 × 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+5.25(1)</td><td>0.35(2)</td><td>0.055(2)</td></tr><tr><td>4<sub>0,4</sub> – 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 × 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+5.30(1)</td><td>0.34(2)</td><td>0.054(2)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">Lupus-1A</td></tr><tr><td rowspan="2">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+5.00(1)</td><td>0.47(1)</td><td>1.259(6)</td></tr><tr><td>3<sub>2,2</sub> – 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 × 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+5.06(1)</td><td>0.36(1)</td><td>0.112(2)</td></tr><tr><td rowspan="4">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> – 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 × 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+5.05(1)</td><td>0.33(1)</td><td>0.078(1)</td></tr><tr><td>5<sub>1,5</sub> – 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 × 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+5.01(1)</td><td>0.33(1)</td><td>0.044(1)</td></tr><tr><td>4<sub>0,4</sub> – 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>14.1</td><td>8.742 × 10<sup>-5</sup></td><td>23.2</td><td>0.84</td><td>+5.06(2)</td><td>0.27(4)</td><td>0.039(5)</td></tr><tr><td>5<sub>0,5</sub> – 4<sub>0,4</sub> (para)</td><td>103952.926</td><td>15.0</td><td>5.152 × 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+5.06(1)</td><td>0.35(2)</td><td>0.042(2)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1521F</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+6.36(1)</td><td>0.50(1)</td><td>1.491(9)</td></tr><tr><td>4<sub>3,2</sub> – 4<sub>2,3</sub> (ortho)</td><td>85656.415</td><td>26.7</td><td>1.522 × 10<sup>-5</sup></td><td>28.4</td><td>0.86</td><td>+6.56(3)</td><td>0.32(6)</td><td>0.017(3)</td></tr><tr><td>3<sub>2,2</sub> – 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 × 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+6.43(1)</td><td>0.39(1)</td><td>0.101(2)</td></tr><tr><td rowspan="3">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> – 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 × 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+6.44(1)</td><td>0.43(1)</td><td>0.071(1)</td></tr><tr><td>5<sub>1,5</sub> – 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 × 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+6.41(1)</td><td>0.40(2)</td><td>0.039(1)</td></tr><tr><td>4<sub>0,4</sub> – 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 × 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+6.46(1)</td><td>0.43(2)</td><td>0.039(1)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">Serpens South 1a</td></tr><tr><td rowspan="2">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+7.33(1)</td><td>0.73(1)</td><td>1.246(12)</td></tr><tr><td>3<sub>2,2</sub> – 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 × 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+7.56(1)</td><td>0.64(1)</td><td>0.146(3)</td></tr><tr><td rowspan="3">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> – 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 × 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+7.53(1)</td><td>0.61(2)</td><td>0.090(2)</td></tr><tr><td>5<sub>1,5</sub> – 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 × 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+7.51(1)</td><td>0.59(3)</td><td>0.050(2)</td></tr><tr><td>4<sub>0,4</sub> – 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 × 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+7.57(2)</td><td>0.71(4)</td><td>0.054(2)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1495B</td></tr><tr><td rowspan="2">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+7.57(1)</td><td>0.37(1)</td><td>0.975(3)</td></tr><tr><td>3<sub>2,2</sub> – 3<sub>1,3</sub> (para)</td><td>84727.687</td><td>16.1</td><td>1.042 × 10<sup>-5</sup></td><td>28.7</td><td>0.86</td><td>+7.63(1)</td><td>0.29(1)</td><td>0.033(1)</td></tr><tr><td rowspan="3">l-C<sub>3</sub>H<sub>2</sub></td><td>4<sub>1,3</sub> – 3<sub>1,2</sub> (ortho)</td><td>83933.699</td><td>9.1</td><td>4.962 × 10<sup>-5</sup></td><td>29.0</td><td>0.86</td><td>+7.62(1)</td><td>0.32(2)</td><td>0.019(1)</td></tr><tr><td>5<sub>1,5</sub> – 4<sub>1,4</sub> (ortho)</td><td>102992.379</td><td>13.8</td><td>9.602 × 10<sup>-5</sup></td><td>23.6</td><td>0.84</td><td>+7.60(2)</td><td>0.36(6)</td><td>0.010(1)</td></tr><tr><td>4<sub>0,4</sub> – 3<sub>0,3</sub> (para)</td><td>83165.345</td><td>10.0</td><td>5.152 × 10<sup>-5</sup></td><td>29.2</td><td>0.86</td><td>+7.68(1)</td><td>0.18(9)</td><td>0.007(1)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1172</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+2.68(1)</td><td>0.75(1)</td><td>1.358(6)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1251A</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>-4.02(1)</td><td>0.47(1)</td><td>0.753(7)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1389</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>-4.76(1)</td><td>0.47(1)</td><td>1.280(12)</td></tr><tr><td colspan="11" style="text-align: center; font-weight: bold;">L1512</td></tr><tr><td rowspan="3">c-C<sub>3</sub>H<sub>2</sub></td><td>2<sub>1,2</sub> – 1<sub>0,1</sub> (ortho)</td><td>85338.900</td><td>4.1</td><td>2.322 × 10<sup>-5</sup></td><td>28.5</td><td>0.86</td><td>+7.05(1)</td><td>0.28(1)</td><td>0.601(6)</td></tr></tbody></table>

Notes. Numbers in parentheses are $1\sigma$ uncertainties resulting the line fitting in units of the last digits. The values of $\int T_A^* dv$ have an additional uncertainty of ~10 per cent owing to calibration. The antenna temperature ($T_A^*$) can be converted to the main beam brightness temperature ($T_{MB}$) by dividing by ($B_{eff}/F_{eff}$).

This value is twice as high as that derived by Fossé et al. (2001) based on the observation of the $1_{1,0} - 1_{0,1}$ transition at 18.3 GHz with the Effelsberg 100-m telescope, probably as a result of optical depth effects and of the different beam sizes of the Effelsberg 100-m and IRAM 30-m telescopes. Our analysis indicates that the $2_{1,2} - 1_{0,1}$ line at 85.3 GHz is optically thick ($\tau \approx 10$), and thus the derived c-C<sub>3</sub>H<sub>2</sub> column density relies heavily on the other, optically thinner, observed transitions. The column density of l-C<sub>3</sub>H<sub>2</sub>

derived in TMC-1 is $1.8 \times 10^{12} \text{ cm}^{-2}$, in agreement with values derived by Cernicharo et al. (1991) and Fossé et al. (2001), and the cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio is 67.

The source B1-b is a dense core with a steep density gradient, with values ranging from a few times $10^6 \text{ cm}^{-3}$ at the centre to a few times $10^5 \text{ cm}^{-3}$ at angular scales of the order of the IRAM 30-m beam, according to the modelling of the continuum emission at 350 µm and 1.2 mm by Daniel et al. (2014). Based on this latter study, here
<table><caption>**Table 4.** Observed line parameters of *c*-C<sub>3</sub>H and *l*-C<sub>3</sub>H in TMC-1 and B1-b.</caption><thead><tr><th>Molecule</th><th>Transition</th><th>Frequency (MHz)</th><th>E<sub>up</sub> (K)</th><th>A<sub>ul</sub> (s<sup>-1</sup>)</th><th>HPBW (arcsec)</th><th><i>B</i><sub>eff</sub>/<i>F</i><sub>eff</sub></th><th>V<sub>LSR</sub> (km s<sup>-1</sup>)</th><th>&Delta;<i>v</i> (km s<sup>-1</sup>)</th><th>&int; T<sub>A</sub><sup>*</sup> d<i>v</i> (K km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td colspan="10" style="text-align: center; font-weight: bold;">TMC-1</td></tr><tr><td rowspan="8">c-C<sub>3</sub>H</td><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 3&ndash;2</td><td>91494.349</td><td>4.4</td><td>1.462 &times; 10<sup>-5</sup></td><td>26.6</td><td>0.86</td><td>+5.70(6)<br>+5.96(16)</td><td>0.27(2)<br>0.41(3)</td><td>0.080(1)<br>0.110(1)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 2&ndash;1</td><td>91497.608</td><td>4.4</td><td>1.262 &times; 10<sup>-5</sup></td><td>26.6</td><td>0.86</td><td>+5.79(2)<br>+6.05(4)</td><td>0.28(2)<br>0.48(5)</td><td>0.057(1)<br>0.063(1)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 2&ndash;2</td><td>91512.969</td><td>4.4</td><td>1.612 &times; 10<sup>-6</sup></td><td>26.6</td><td>0.86</td><td>+5.74(10)<br>+5.95(20)</td><td>0.30(15)<br>0.46(13)</td><td>0.009(1)<br>0.010(1)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 1&ndash;1</td><td>91681.696</td><td>4.4</td><td>2.852 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+5.72(4)<br>+5.63(2)</td><td>0.44(4)<br>0.38(4)</td><td>0.019(1)<br>0.033(3)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 1&ndash;0</td><td>91692.752</td><td>4.4</td><td>8.152 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+5.92(7)<br>+6.05(2)</td><td>0.34(10)<br>0.31(20)</td><td>0.019(3)<br>0.032(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 2&ndash;1</td><td>91699.471</td><td>4.4</td><td>1.262 &times; 10<sup>-5</sup></td><td>26.5</td><td>0.85</td><td>+5.75(2)<br>+6.06(15)</td><td>0.34(9)<br>0.68(19)</td><td>0.085(2)<br>0.017(3)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;3/2 F = 1&ndash;1</td><td>91747.372</td><td>4.4</td><td>3.272 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+5.69(4)<br>+6.06(15)</td><td>0.43(9)<br>0.68(19)</td><td>0.016(3)<br>0.018(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;3/2 F = 2&ndash;2</td><td>91780.518</td><td>4.4</td><td>2.052 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+5.79(2)<br>+6.16(6)</td><td>0.34(5)<br>0.32(10)</td><td>0.005(1)<br>0.005(1)</td></tr><tr><td rowspan="4">l-C<sub>3</sub>H</td><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 b F = 5&ndash;4</td><td>97995.166</td><td>12.5</td><td>6.122 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+5.59(2)<br>+5.89(5)</td><td>0.30(2)<br>0.33(7)</td><td>0.030(3)<br>0.010(3)</td></tr><tr><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 b F = 4&ndash;3</td><td>97995.913</td><td>12.5</td><td>5.952 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+5.59(2)<br>+5.94(2)</td><td>0.34(2)<br>0.23(7)</td><td>0.028(3)<br>0.005(3)</td></tr><tr><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 a F = 5&ndash;4</td><td>98011.611</td><td>12.5</td><td>6.132 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+5.62(3)<br>+6.00(6)</td><td>0.36(2)<br>0.25(7)</td><td>0.036(2)<br>0.006(1)</td></tr><tr><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 a F = 4&ndash;3</td><td>98012.524</td><td>12.5</td><td>5.962 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+5.54(4)<br>+5.90(10)</td><td>0.36(6)<br>0.34(14)</td><td>0.028(2)<br>0.009(1)</td></tr><tr><td colspan="10" style="text-align: center; font-weight: bold;">B1-b</td></tr><tr><td rowspan="8">c-C<sub>3</sub>H</td><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 3&ndash;2</td><td>91494.349</td><td>4.4</td><td>1.462 &times; 10<sup>-5</sup></td><td>26.6</td><td>0.86</td><td>+6.46(2)<br>+7.31(2)</td><td>0.88(4)<br>0.60(5)</td><td>0.143(10)<br>0.060(6)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 2&ndash;1</td><td>91497.608</td><td>4.4</td><td>1.262 &times; 10<sup>-5</sup></td><td>26.6</td><td>0.86</td><td>+6.70(5)<br>+7.58(5)</td><td>0.90(10)<br>0.55(8)</td><td>0.104(10)<br>0.033(4)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 5/2&ndash;3/2 F = 2&ndash;2</td><td>91512.969</td><td>4.4</td><td>1.612 &times; 10<sup>-6</sup></td><td>26.6</td><td>0.86</td><td>+6.79(6)<br>+6.60(9)</td><td>0.43(14)<br>0.74(20)</td><td>0.006(2)<br>0.010(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 1&ndash;1</td><td>91681.696</td><td>4.4</td><td>2.852 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+6.64(5)<br>+6.57(3)</td><td>1.25(11)<br>0.70(5)</td><td>0.049(4)<br>0.076(6)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 1&ndash;0</td><td>91692.752</td><td>4.4</td><td>8.152 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+6.73(4)<br>+7.38(4)</td><td>0.72(8)<br>0.84(7)</td><td>0.048(6)<br>0.013(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;1/2 F = 2&ndash;1</td><td>91699.471</td><td>4.4</td><td>1.262 &times; 10<sup>-5</sup></td><td>26.5</td><td>0.85</td><td>+6.49(9)<br>+6.79(9)</td><td>0.60(15)<br>1.03(16)</td><td>0.008(2)<br>0.016(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;3/2 F = 1&ndash;1</td><td>91747.372</td><td>4.4</td><td>3.272 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+6.47(2)<br>+7.36(2)</td><td>0.82(5)<br>0.40(6)</td><td>0.040(2)<br>0.013(2)</td></tr><tr><td>2<sub>1,2</sub> &ndash; 1<sub>1,1</sub> J = 3/2&ndash;3/2 F = 2&ndash;2</td><td>91780.518</td><td>4.4</td><td>2.052 &times; 10<sup>-6</sup></td><td>26.5</td><td>0.85</td><td>+6.49(3)<br>+7.44(6)</td><td>0.84(7)<br>0.43(13)</td><td>0.030(2)<br>0.006(2)</td></tr><tr><td rowspan="3">l-C<sub>3</sub>H</td><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 b F = 5&ndash;4</td><td>97995.166</td><td>12.5</td><td>6.122 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+6.51(2)<br>+7.36(4)</td><td>0.73(5)<br>0.54(9)</td><td>0.036(2)<br>0.013(2)</td></tr><tr><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 b F = 4&ndash;3</td><td>97995.913</td><td>12.5</td><td>5.952 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+6.40(3)<br>+7.24(8)</td><td>0.69(7)<br>0.47(10)</td><td>0.027(2)<br>0.008(2)</td></tr><tr><td><sup>2</sup>&Pi;<sub>1/2</sub> J = 9/2&ndash;7/2 a F = 5&ndash;4</td><td>98011.611</td><td>12.5</td><td>6.132 &times; 10<sup>-5</sup></td><td>24.8</td><td>0.85</td><td>+6.40(3)<br>+7.24(8)</td><td>0.69(7)<br>0.47(10)</td><td>0.027(2)<br>0.008(2)</td></tr></tbody></table>

Notes. Numbers in parentheses are $1\sigma$ uncertainties resulting from the line fitting in units of the last digits. The values of $\int T_A^* dv$ have an additional uncertainty of ~10 per cent owing to calibration. The antenna temperature ($T_A^*$) can be converted to the main beam brightness temperature ($T_{MB}$) by dividing by ($B_{eff}/F_{eff}$).

we adopt a mean gas kinetic temperature of 12 K and $n(H_2) = 4 \times 10^5 \text{ cm}^{-3}$, physical parameters that result in line intensity ratios in good agreement with the observed ones. The column densities we derive in B1-b are $N(\text{c-C}_3\text{H}_2) = 1.8 \times 10^{13} \text{ cm}^{-2}$ and $N(\text{l-C}_3\text{H}_2) = 6 \times 10^{11} \text{ cm}^{-2}$, which results in a cyclic-to-linear C$_3$H$_2$ ratio of 30.

The dense core L483 hosts a Class 0 source and shows evidence of infall motions and a density gradient (Tafalla et al. 2000). The gas kinetic temperature has been estimated as 10 K (Anglada et al. 1997). Densities of H$_2$ of about $3 \times 10^4 \text{ cm}^{-3}$ were derived by Anglada et al. (1997), based on NH$_3$ observations, and by Jørgensen et al. (2002), based on the modelling of continuum observations at

450 and 850 µm, while Tafalla et al. (2000) derived a mean density of H$_2$ of $3 \times 10^5 \text{ cm}^{-3}$ over an area ~20 arcsec radius, based on the analysis of CH$_3$OH emission. Here we adopt this latter value, because at a gas kinetic temperature of 10 K the $4_{3,2} - 4_{2,3}$ transition of c-C$_3$H$_2$ can be excited up to the level observed in L483 only at densities well above $10^5 \text{ cm}^{-3}$.

In the starless core Lupus-1A, the kinetic temperature has been estimated as $14 \pm 2$ K (Agúndez et al. 2015), the value adopted here, and as $12.6 \pm 1.5$ K at a position 2.8 arcmin from Lupus-1A (Sakai et al. 2009). The density of H$_2$ has not been accurately estimated, but Sakai et al. (2009) point to densities as high as $10^6 \text{ cm}^{-3}$, while
<table><caption>**Table 5.** Physical properties, and derived column densities and cyclic-to-linear column density ratios.</caption><thead><tr><th>Source</th><th>T<sub>k</sub><br>(K)</th><th>n(H<sub>2</sub>)<br>(cm<sup>-3</sup>)</th><th>N(c-C<sub>3</sub>H)<sup>a</sup><br>(cm<sup>-2</sup>)</th><th>N(l-C<sub>3</sub>H)<sup>a</sup><br>(cm<sup>-2</sup>)</th><th>N(c-C<sub>3</sub>H<sub>2</sub>)<sup>b</sup><br>(cm<sup>-2</sup>)</th><th>N(l-C<sub>3</sub>H<sub>2</sub>)<sup>a</sup><br>(cm<sup>-2</sup>)</th><th>R<sub>1</sub><sup>c</sup></th><th>R<sub>2</sub><sup>c</sup></th></tr></thead><tbody><tr><td>TMC-1</td><td>10</td><td>3.02 &times; 10<sup>4</sup></td><td>6.12 &times; 10<sup>12</sup></td><td>1.12 &times; 10<sup>12</sup></td><td>1.22 &times; 10<sup>14</sup></td><td>1.82 &times; 10<sup>12</sup></td><td>5.5</td><td>67</td></tr><tr><td>B1-b</td><td>12</td><td>4.02 &times; 10<sup>5</sup></td><td>6.22 &times; 10<sup>12</sup></td><td>1.22 &times; 10<sup>12</sup></td><td>1.82 &times; 10<sup>13</sup></td><td>62 &times; 10<sup>11</sup></td><td>5.2</td><td>30</td></tr><tr><td>L483</td><td>10</td><td>3.02 &times; 10<sup>5</sup></td><td></td><td></td><td>4.02 &times; 10<sup>13</sup></td><td>72 &times; 10<sup>11</sup></td><td></td><td>57</td></tr><tr><td>Lupus-1A</td><td>14</td><td>4.02 &times; 10<sup>5</sup></td><td></td><td></td><td>1.42 &times; 10<sup>13</sup></td><td>3.52 &times; 10<sup>11</sup></td><td></td><td>40</td></tr><tr><td>L1521F</td><td>10</td><td>4.02 &times; 10<sup>5</sup></td><td></td><td></td><td>1.52 &times; 10<sup>13</sup></td><td>42 &times; 10<sup>11</sup></td><td></td><td>38</td></tr><tr><td>Serpens South 1a</td><td>11</td><td>4.02 &times; 10<sup>5</sup></td><td></td><td></td><td>1.42 &times; 10<sup>13</sup></td><td>52 &times; 10<sup>11</sup></td><td></td><td>28</td></tr><tr><td>L1495B</td><td>10</td><td>1.12 &times; 10<sup>4</sup></td><td></td><td></td><td>2.02 &times; 10<sup>14</sup></td><td>1.82 &times; 10<sup>12</sup></td><td></td><td>111</td></tr><tr><td>L1172</td><td>10</td><td>7.52 &times; 10<sup>4</sup></td><td></td><td></td><td>1.42 &times; 10<sup>13</sup></td><td></td><td></td><td></td></tr><tr><td>L1251A</td><td>10</td><td>2.12 &times; 10<sup>4</sup></td><td></td><td></td><td>2.22 &times; 10<sup>13</sup></td><td></td><td></td><td></td></tr><tr><td>L1389</td><td>10</td><td>5.22 &times; 10<sup>4</sup></td><td></td><td></td><td>2.22 &times; 10<sup>13</sup></td><td></td><td></td><td></td></tr><tr><td>L1512</td><td>10</td><td>2.62 &times; 10<sup>4</sup></td><td></td><td></td><td>1.72 &times; 10<sup>13</sup></td><td></td><td></td><td></td></tr></tbody></table>

Notes. $^a$Estimated error is a factor of 2.

$^b$Estimated error is 50 per cent for the first seven sources and a factor of 2 for L1172, L1251A, L1389 and L1512.

$^c$R<sub>1</sub> = N(c-C<sub>3</sub>H)/N(l-C<sub>3</sub>H) and R<sub>2</sub> = N(c-C<sub>3</sub>H<sub>2</sub>)/N(l-C<sub>3</sub>H<sub>2</sub>). Estimated uncertainties in R<sub>1</sub> and R<sub>2</sub> are 30 per cent. The error estimates and the assumed parameters are described in the text.

the observed intensity ratio between the 2<sub>1,2</sub> − 1<sub>0,1</sub> and 3<sub>2,2</sub> − 3<sub>1,3</sub> transitions of c-C<sub>3</sub>H<sub>2</sub> also points to densities above 10<sup>5</sup> cm<sup>-3</sup>. The value of n(H<sub>2</sub>) adopted here, 4 × 10<sup>5</sup> cm<sup>-3</sup>, results in line intensity ratios for cyclic and linear C<sub>3</sub>H<sub>2</sub> in reasonable agreement with the observed ones. The column densities derived are N(c-C<sub>3</sub>H<sub>2</sub>) = 1.4 × 10<sup>13</sup> cm<sup>-2</sup> and N(l-C<sub>3</sub>H<sub>2</sub>) = 3.5 × 10<sup>11</sup> cm<sup>-2</sup>, which results in a cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio of 40.

In L1521F, the volume density of H<sub>2</sub> has been estimated as 1.1 × 10<sup>6</sup> cm<sup>-3</sup>, based on observations of the continuum at 1.2 mm, and as in the range (1.3 − 4.5) × 10<sup>5</sup> cm<sup>-3</sup> from observations of N<sub>2</sub>H<sup>+</sup> and N<sub>2</sub>D<sup>+</sup> by Crapsi et al. (2005). These authors derived a rotational temperature of 4.8 K for N<sub>2</sub>H<sup>+</sup>, which is consistent with a gas kinetic temperature of 10 K. We find that a density of H<sub>2</sub> of 4 × 10<sup>5</sup> cm<sup>-3</sup> results in line intensity ratios for cyclic and linear C<sub>3</sub>H<sub>2</sub> in agreement with the observed ones, in particular regarding the intensity ratio between the 2<sub>1,2</sub> − 1<sub>0,1</sub> and 4<sub>3,2</sub> − 4<sub>2,3</sub> transitions of c-C<sub>3</sub>H<sub>2</sub>. The column densities we derived are N(c-C<sub>3</sub>H<sub>2</sub>) = 1.5 × 10<sup>13</sup> cm<sup>-2</sup> and N(l-C<sub>3</sub>H<sub>2</sub>) = 4 × 10<sup>11</sup> cm<sup>-2</sup>, which results in a cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio of 38.

For clump 1a of the Serpens South complex, Friesen et al. (2013) derived a gas kinetic temperature of 11 K through observations of NH<sub>3</sub>. The density of H<sub>2</sub> has not been accurately determined, but Friesen et al. (2013) suggested a volume density of about 10<sup>4</sup> cm<sup>-3</sup> based on time-scale arguments. However, on the assumption that the ortho-to-para ratio is 3 for c-C<sub>3</sub>H<sub>2</sub>, the observed intensity ratio between the 2<sub>1,2</sub> − 1<sub>0,1</sub> and 3<sub>2,2</sub> − 3<sub>1,3</sub> transitions requires gas densities above 10<sup>5</sup> cm<sup>-3</sup>. Adopting n(H<sub>2</sub>) = 4 × 10<sup>5</sup> cm<sup>-3</sup>, we derived column densities of N(c-C<sub>3</sub>H<sub>2</sub>) = 1.4 × 10<sup>13</sup> cm<sup>-2</sup> and N(l-C<sub>3</sub>H<sub>2</sub>) = 5 × 10<sup>11</sup> cm<sup>-2</sup>, which results in a cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio of 28.

The physical conditions in the sources L1495B, L1172, L1251A, L1389 and L1512 have been taken from the study by Cordiner et al. (2013). These authors adopted a common gas kinetic temperature of 10 K and derived the volume density from the observation of a couple of lines of HC<sub>3</sub>N, which are in the range (1 − 10) × 10<sup>4</sup> cm<sup>-3</sup>. In four of these sources, observations are restricted to just one line, the 2<sub>1,2</sub> − 1<sub>0,1</sub> transition of c-C<sub>3</sub>H<sub>2</sub>, and thus it is not possible to put additional constraints on the volume density. Moreover, according to our statistical equilibrium calculations, this line is optically thick (τ = 2 − 6) in these sources, which introduces a higher degree of uncertainty in the derived column densities of c-C<sub>3</sub>H<sub>2</sub>.

The column densities derived for c-C<sub>3</sub>H<sub>2</sub> and l-C<sub>3</sub>H<sub>2</sub> have uncertainties resulting from the observations (line-fitting plus telescope calibration), although the main source of error arises in the conversion of line intensities into column densities, with uncertainties in the kinetic temperature, gas density, collisional rate coefficients, and a probable lack of uniformity of the physical conditions of the cloud within the telescope beam. We estimate that the column densities derived for c-C<sub>3</sub>H<sub>2</sub> have an uncertainty of 50 per cent in those sources in which the availability of multiple lines permit constraints to be set on the gas density (TMC-1, B1-b, L483, Lupus-1A, L1521F, Serpens South 1a, L1495B), and an error of a factor of 2 in those sources in which only the 2<sub>1,2</sub> − 1<sub>0,1</sub> line was observed (L1172, L1251A, L1389, L1512). In the case of l-C<sub>3</sub>H<sub>2</sub>, in those sources in which this isomer was observed, multiple lines were available. However, the adopted collision rate coefficients were not specifically computed for this molecule but for formaldehyde, and thus we estimate uncertainties of a factor of 2 for the column densities of l-C<sub>3</sub>H<sub>2</sub>. Systematic errors in the kinetic temperature and gas density tend to cancel when computing cyclic-to-linear column density ratios, and thus we estimate errors of 30 per cent for the derived cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratios.

### 2.2.1 Local thermodynamic equilibrium: c-C<sub>3</sub>H and l-C<sub>3</sub>H

For cyclic and linear C<sub>3</sub>H, local thermodynamic equilibrium was assumed (e.g. Turner 1991) owing to the lack of adequate collision rate coefficients for these two isomers. The spectroscopy of c-C<sub>3</sub>H was taken from the JPL database<sup>3</sup> (Pickett et al. 1998), which is based on the experimental study by Yamamoto & Saito (1994), and the adopted dipole moment is 2.30 D, the value measured by Lovas et al. (1992). For l-C<sub>3</sub>H, level energies were taken from the CDMS database (Müller et al. 2005), which is based on laboratory work by Yamamoto et al. (1990) plus some further studies, and we adopted a dipole moment of 3.55 D based on ab initio calculations by Woon (1995). In this study, observations of c-C<sub>3</sub>H and l-C<sub>3</sub>H were restricted to various hyperfine components of a single rotational transition in TMC-1 and B1-b. Column densities were derived by assuming a rotational temperature of 5 K for the two isomers in both TMC-1 and B1-b, which is consistent with values from the

<footnote n="3"><sup>3</sup>See http://spec.jpl.nasa.gov.</footnote>
literature (Mangum & Wootten 1990; Fossé et al. 2001). The column densities derived imply that the cyclic isomer of C₃H is ~5 times more abundant than the linear one in both TMC-1 and B1-b. We estimate uncertainties of a factor of 2 for the column densities of c-C₃H and l-C₃H in TMC-1 and B1-b, while the cyclic-to-linear C₃H ratios are estimated to have errors of 30 per cent.

# 3 THE CHEMICAL MODEL

## 3.1 Model description

In order to calculate the abundances we used the chemical model Nautilus in its three-phase version from Ruaud et al. (2016). The Nautilus code computes the gas phase and dust ice composition as a function of time, taking into account reactions in the gas phase, sticking on grains and desorption from grain surfaces, and reactions at the surface. The ice is decomposed into a surface (the two most external monolayers of molecules) and a bulk. Both the surface and the bulk are chemically active based on the Langmuir-Hinshelwood mechanism with the formalism of Hasegawa et al. (1992). In the bulk though, the diffusion of species is much less efficient compared with the surface. All the details and surface parameters for diffusion are the same as in Ruaud et al. (2016). For the desorption from the surface (only), we consider thermal desorption and desorption induced by cosmic rays (Hasegawa & Herbst 1993) as well as by exothermic chemical reactions (the exothermicity of surface chemical reactions allows for the species to be desorbed after their formation) (Garrod et al. 2007). The Garrod et al. (2007) chemical desorption mechanism leads to the result that ~1 per cent of the newly formed species desorb and 99 per cent remain on the grain surfaces (this corresponds to a factor of 0.01 in Garrod et al. 2007). The binding energies of the species to the surface have been updated from Wakelam et al. (2017), and the atomic oxygen is allowed to diffuse by tunnelling (see Wakelam et al. 2017).

The gas phase network is based on kida.uva.2014⁴ (Wakelam et al. 2015b), with the modifications described in Section 3. The surface network and parameters are the same as in Ruaud et al. (2015), with some updates from Wakelam et al. (2017). Following Hincelin et al. (2015), the encounter desorption mechanism is included in the code. This mechanism accounts for the fact that the H₂ binding energy on itself is much smaller than on water ices and prevents the formation of several H₂ monolayers on grain surfaces.

The chemical compositions of the gas phase and the grain surfaces are computed as a function of time. The gas and dust temperatures are equal to 10 K, the total H₂ density is equal to $3 \times 10^4 \text{ cm}^{-3}$ (various runs were performed with the total H density varying between $1 \times 10^4$ and $2 \times 10^5 \text{ cm}^{-3}$). The cosmic-ray ionization rate is equal to $1.3 \times 10^{-17} \text{ s}^{-1}$, and the total visual extinction is set equal to 10. All elements are assumed to be initially in atomic form (elements with an ionization potential below the maximum energy of ambient ultraviolet photons (13.6 eV, the ionization energy of H atoms) are initially in a singly ionized state, i.e. C, S, Si, Fe, Na, Mg, Cl and P), except for hydrogen, which is entirely molecular. The initial abundances are similar to those of table 1 in Hincelin et al. (2011), the C/O elemental ratio being equal to 0.7 in this study. The grains are considered to be spherical with a 0.1-µm radius, a density of $3 \text{ g cm}^{-3}$ and about $10^6$ surface sites, all chemically active. The dust-to-gas mass ratio is set to 0.01.

## 3.2 Update of the chemistry

In order to determine the unknown rate constant that will be used in the network, we used a methodology developed in previous articles (Loison et al. 2014a,b, 2015) and summarized in Appendix A. This methodology includes an extensive literature review, various Density Functional Theory (DFT) and ab initio calculations for critical gas phase reactions, namely H + l-C₃H₂, H + t-C₃H₂, O + c-C₃H₂, O + l-C₃H₂, N + C₃, N + c-C₃H₂, N + l-C₃H₂, H + C₃O, O + C₃O, O + c-C₃H₃⁺, OH + C₃, OH + c-C₃H₂, H₂ + l-C₃H and H₂ + c-C₃H, to determine the presence, or not, of a barrier. When there is no barrier in the entrance valley and exothermic bimolecular exit channels, we chose to use the capture rate constant or sometimes a fraction of the capture rate constant by comparison with similar reactions (the capture rate is the upper limit of the rate constants for barrierless reactions). The dissociative recombination (DR) of c,l-C₃H₂⁺ and c,l-C₃H₃⁺ is an important source of c,l-C₃H and is the main source c,l,t-C₃H₂ in our network. The first step of c,l-C₃H₂⁺ and c,l-C₃H₃⁺ DR is the formation of highly excited C₃H₂** and C₃H₃**, which leads to bond fragmentation. Angelova et al. (2004) have shown that the DR of c-C₃H₂⁺ leads to 87.5 per cent of C₃Hₓ and 12.5 per cent of C₂H₃ + CHₓ, and the DR of c-C₃H₃⁺ leads to 90.7 per cent of C₃Hₓ and 9.3 per cent of C₂H₃ + CHₓ. Moreover, in DR processes, the H ejection is in general favoured more than H₂ ejection (Plessis et al. 2010, 2012; Janev & Reiter 2004). Considering the exothermicity for the ejection of two hydrogen atoms (endothermic for c-C₃H₃⁺ DR and only slightly exothermic for l-C₃H₂⁺, c-C₃H₂⁺ and l-C₃H₃⁺ DR, see Appendix B), this process will have a low branching ratio. Then, the dissociation of C₃H₂** and C₃H₃** will produce mainly C₃H + H and C₃H₂ + H, both C₃H and C₃H₂ species being also excited considering the exothermicity of the DR and the fact that the hydrogen atom will carry only a limited part of the available energy through kinetic energy. Part of the excited C₃H* and C₃H₂* molecules will lead to dissociation when they are populated above the dissociation limit, but most of them will relax through radiative emission of an infrared photon. As noted by Herbst et al. (2000), the typical time-scale for isomeric conversion is much shorter than that for relaxation by one infrared photon. Thus, because radiative relaxation occurs slowly, isomeric conversion leads to equilibrated isomeric (c-C₃H ⇌ l-C₃H, c-C₃H₂ ⇌ l-C₃H₂) abundances at each internal energy. The final balance is determined at or near the effective barrier to isomerization, which corresponds to the energy of the transition state. The ratio between the isomeric forms are then *approximated* by the ratio of the rovibrational densities of states of the isomers at the barrier to isomerization calculated using the MESMER program (Glowacki et al. 2012). Fig. 2 shows the isomerization pathway calculated at the DFT level. The calculated geometries of the stationary point can be found in Appendix A. The t-C₃H₂ has a *triplet ground state* and its production in the excited singlet ground state is neglected here. The production of t-C₃H₂ from the DR of c,l-C₃H₃⁺ is supposed to come from c,l-C₃H₃⁺ + e⁻ → c,l-C₃H₃ → t-C₃H₂ + H, which is supposed to be a minor channel in comparison to c,l-C₃H₃⁺ + e⁻ → c,l-C₃H₃ → c,l-C₃H₂ + H (t-C₃H₂ has a ground triplet state, in contrast to c,l-C₃H₂, which has a singlet ground state).

Then, in contrast to the usual astrochemical models, in which it is assumed that the cyclic ions lead to cyclic neutral, and linear ions lead only to linear neutral, we consider that c,l-C₃H₂⁺ DR leads mainly to cyclic c-C₃H and that c,l-C₃H₃⁺ DR leads mainly to cyclic c-C₃H₂, with, in both cases, the cyclic-to-linear branching ratio given by the rovibrational densities of states of the isomers at the

<footnote n="4">⁴ http://kidaobs.u-bordeaux1.fr/models.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Intrinsic reaction coordinate pathway for the isomerization of c,l-C<sub>3</sub>H and c,l-C<sub>3</sub>H<sub>2</sub> in their ground state calculated at the M06-2X/aug-cc-pVTZ level.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Schematic diagram highlighting the important pathways for C<sub>3</sub>H<sub>x</sub> chemistry in dense molecular cloud for a typical age (between 1 × 10<sup>5</sup> and 6 × 10<sup>5</sup> yr). The thickness of each arrow is proportional to the fluxes (the very minor fluxes are not represented).</figcaption></figure>

barrier to isomerization. The reactions involving C<sub>3</sub>H<sub>x</sub> chemistry, including c,l-C<sub>3</sub>H<sub>2</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR, are represented in Fig. 3. It should be noted that if c,l-t-C<sub>3</sub>H<sub>2</sub> is produced almost exclusively from c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR, then c,l-C<sub>3</sub>H is produced not only from c,l-C<sub>3</sub>H<sub>2</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR, but also through the C + C<sub>2</sub>H<sub>2</sub> reaction as well as the H + C<sub>3</sub><sup>-</sup> reaction.

## 4 RESULTS AND COMPARISON WITH OBSERVATIONS

With our network and the physical conditions relevant to the various molecular clouds, the main formation pathway for c,l-t-C<sub>3</sub>H<sub>2</sub> is through the DR of c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup>, these ions being formed mainly through C<sub>3</sub> + H<sub>3</sub><sup>+</sup>, HCO<sup>+</sup>, HCNH<sup>+</sup> → C<sub>3</sub>H<sup>+</sup> reactions followed by the C<sub>3</sub>H<sup>+</sup> + H<sub>2</sub> reaction, the C<sub>3</sub>H<sup>+</sup> being also produced through

the C<sup>+</sup> + C<sub>2</sub>H<sub>2</sub> reaction, and the main consumption is with atoms: O, C for the four isomers, H for c,l-C<sub>3</sub>H and t,t-C<sub>3</sub>H<sub>2</sub>, and N for c,l-C<sub>3</sub>H and t-C<sub>3</sub>H<sub>2</sub>. Most of these reactions are introduced for the first time in the astrochemical network thanks to new *ab initio* calculations. For a typical dense cloud age (between 10<sup>5</sup> and 10<sup>6</sup> yr), the c-C<sub>3</sub>H<sub>2</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR is an important source of c,l-C<sub>3</sub>H, but the main c-C<sub>3</sub>H formation pathway is the C + C<sub>2</sub>H<sub>2</sub> reaction. Considering the importance of c-C<sub>3</sub>H<sub>2</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR, the branching ratio of the C<sub>3</sub>H<sup>+</sup> + H<sub>2</sub> reaction, particularly between c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> and c-C<sub>3</sub>H<sub>2</sub><sup>+</sup> + H, may have some importance. Rate constant measurements at room temperature for the C<sub>3</sub>H<sup>+</sup> + H<sub>2</sub> reaction clearly show a rate constant function of the pressure leading only to C<sub>3</sub>H<sub>3</sub><sup>+</sup>, the nature of the isomer not being clearly identified (Raksit & Bohme 1983; Smith & Adams 1987). A more recent experimental study (Savic & Gerlich 2005) led to a
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Abundances of C<sub>3</sub>H<sub>x</sub> gas phase species studied in this work as a function of time predicted by our model [$n(H_2) = 3 \times 10^4$ cm<sup>-3</sup>, T = 10 K]. The dashed horizontal lines represent the abundances observed for TMC-1 (this work, see Table 5). The dotted lines correspond to the uncertainties (see text).</figcaption></figure>

high rate constant for the radiative association at low temperatures, with, however, a non-negligible branching ratio towards c-C<sub>3</sub>H<sub>2</sub><sup>+</sup>. A global theoretical analysis, including branching ratio calculation using Rice-Ramsperger-Kassel-Marcus (RRKM) theory, was performed by Maluendes et al. (1993a,b), and led to the formation of both c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> isomers in approximately equal abundances. We adopt their rate constant expressions for these channels and we also consider some c-C<sub>3</sub>H<sub>2</sub><sup>+</sup> production with a high uncertainty factor. In order to estimate the importance of the uncertainties in these branching ratios we performed various runs considering only c-C<sub>3</sub>H<sub>3</sub><sup>+</sup> or l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> production, showing that the nature of the C<sub>3</sub>H<sub>3</sub><sup>+</sup> isomer is not negligible (up to 20 per cent for the cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio) but not critical, the key point being the relaxation of the C<sub>3</sub>H<sub>3</sub><sup>**</sup> and C<sub>3</sub>H<sub>2</sub><sup>*</sup> produced. It should also be noted that the low branching ratio for the c-C<sub>3</sub>H<sub>2</sub><sup>+</sup> + H channel considered in this study leads to a negligible importance for this channel.

The results of the simulation for the various species studied here and for typical molecular cloud conditions ($n(H_2) = 3 \times 10^4$ cm<sup>-3</sup>, T = 10 K) are shown Fig. 4, as well as observations for TMC-1.

### c-C<sub>3</sub>H and l-C<sub>3</sub>H

C<sub>3</sub>H isomers have been detected in this study in only two molecular clouds, TMC-1 and B1-b. The agreement between observations in TMC-1 and the model is good for a cloud time age between 10<sup>5</sup> and 10<sup>6</sup> yr (see Fig. 4), the age not being critical in that case. For TMC-1 and B1-b, the c-C<sub>3</sub>H/l-C<sub>3</sub>H ratio is close to 5. It is worth noting that, as shown in Fig. 5, the c-C<sub>3</sub>H/l-C<sub>3</sub>H ratio given by our model is almost independent of the total density of the molecular cloud in the range of H<sub>2</sub> between 1 × 10<sup>4</sup> and 2 × 10<sup>5</sup> molecule cm<sup>-3</sup>, in good agreement with observations (see Table 5) (measured only for two molecular clouds, however). In our model, c-C<sub>3</sub>H and l-C<sub>3</sub>H are produced not only from c,l-C<sub>3</sub>H<sub>2</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> DR but also through the C + C<sub>2</sub>H<sub>2</sub> reaction as well as the H + C<sub>3</sub><sup>-</sup> reaction. The larger amount of cyclic isomer is because both the C + C<sub>2</sub>H<sub>2</sub> reaction and the DR of c,l-C<sub>3</sub>H<sub>3</sub><sup>+</sup> and c,l-C<sub>3</sub>H<sub>2</sub> favour c-C<sub>3</sub>H. In the Horsehead PDR, the cyclic isomer is also favoured, with a slightly smaller ratio equal to 1.92 (Pety et al. 2012). In IRC+10216, however, the linear isomer is more abundant, with a c-C<sub>3</sub>H/l-C<sub>3</sub>H ratio equal to 0.38. In his simulation, Agúndez (2009) reproduced the fact that l-C<sub>3</sub>H is more abundant than c-C<sub>3</sub>H using
<figure><img src="image_7.png" /><figcaption>**Figure 5.** c-C<sub>3</sub>H/l-C<sub>3</sub>H and c-C<sub>3</sub>H<sub>2</sub>/l-C<sub>3</sub>H<sub>2</sub> ratios given by our model as a function of the total density of the molecular cloud. Dotted rectangles correspond to the possible values at the chemical cloud age given by the so-called distance of disagreement (Wakelam et al. 2006) at the given total densities.</figcaption></figure>

the theoretical results from Buonomo & Clary (2001) for the C + C₂H₂ reaction favouring l-C₃H production, contrary to the more recent experimental results from Costes et al. (2009). However, the result of Costes et al. (2009) involves a complex fitting, leading to large uncertainties. Considering our chemical network and the photodissociation cross-sections from van Hemert & van Dishoeck (2008), there is no alternative to the C + C₂H₂ reaction to form l-C₃H in circumstellar envelopes. Note that this reaction is the most important formation channel to l-C₃H in IRC+10216, but it is of little importance in molecular clouds. It can be noted that c-C₃H is supposed to be produced mainly through c-C₃H₂ photodissociation in IRC+10216 (there is no experimental or theoretical data on the c-C₃H₂ photodissociation branching ratio). There is, then, a crucial need to study the products of the photodissociations as well as of the C + C₂H₂ reaction, despite the considerable amount of work already performed on this topic (Bergeat & Loison 2001;

Buonomo & Clary 2001; Clary et al. 2002; Leonori et al. 2008; Costes et al. 2009; Hickson et al. 2016a).

### c-C₃H₂ and l-C₃H₂

C₃H₂ isomers have been detected in this study in most of the molecular clouds. The agreement between observations in TMC-1 and the model is notably less good for c,l-C₃H₂ than for c,l-C₃H for an early cloud age (a few times 10⁵ yr). At late times (after 10⁶ yr) the grain chemistry leads to a relatively large s-CH₄ abundance (a few per cent, compatible with observations on ice in young stellar objects; Boogert et al. 2015). Because CH₄ is not strongly bonded to the ice (Raut et al. 2007), the release of CH₄ into the gas phase triggers an intensive hydrocarbon chemistry. Moreover, as for these clouds ages, O and N atoms are depleted on grain leading mainly to H₂O, CO₂, NH₃ and HCN, all strongly bond to the ice with low desorption rate. Then, the hydrocarbons produced in the gas phase do not
react with O and N atoms but with C and other hydrocarbon neutrals and ions, leading to a rich hydrocarbon chemistry and large C₃Hₓ abundances. Among the various reactions, the C + C₂H₂ and C⁺ + C₂H₂ reactions are of particular importance. The most efficient c-C₃H₂ destruction pathway in dense molecular clouds is the O + c-C₃H₂ reaction, which shows no barrier at various theoretical levels (DFT, MP2, CCSD(T), MRCI+Q, detailed in Appendix A). The products and branching ratio for this reaction are not fully studied here, but HC₃O should be a non-negligible product of the O + c-C₃H₂ reaction.

The cyclic-to-linear C₃H₂ ratio observed here in seven molecular clouds varies between 111 and 28, with decreasing values when the total density of the cloud increases. Because the uncertainty for the observed cyclic-to-linear C₃H₂ ratios is estimated to be about 30 per cent, these uncertainties cannot explain the difference between the various molecular clouds. In Fig. 5, we have plotted the c-C₃H₂/l-C₃H₂ ratio evolution given by our model as a function of the total density. The trend of the c-C₃H₂/l-C₃H₂ ratio is in good agreement with observations (see Table 5), for a chemical cloud age given by the so-called distance of disagreement (Wakelam et al. 2006) (minimum average difference, in magnitude, between modelled and observed abundances), represented by the dotted rectangles in Fig. 5. Because C₃H₂ isomers are produced mostly from the DR of c,l-C₃H₃⁺, a DR independent of the total density of the molecular clouds, the main reaction impacting the cyclic-to-linear C₃H₂ ratio is supposed to be the l-C₃H₂ + H → c-C₃H₂ + H reaction. We performed a run suppressing the l-C₃H₂ + H → c-C₃H₂ + H reaction (presented in Appendix A). This suppression strongly reduces the cyclic-to-linear C₃H₂ ratio, which becomes close to the ratio given by the cyclic-to-linear C₃H₂ ratio of the c,l-C₃H₃⁺ DR and also close to the cyclic-to-linear C₃H₂ ratio of the most dense molecular clouds. This clearly shows that the high value of the cyclic-to-linear C₃H₂ ratio of low-density molecular clouds such as TMC-1 and L1495B (see Table 5) is due to the l-C₃H₂ + H → c-C₃H₂ + H reaction, which efficiently converts the linear l-C₃H₂ into the cyclic c-C₃H₂. This reaction becomes less efficient when the total density increases because the H atom density is almost constant with the total density of the cloud, in contrast to the fluxes of the C₃Hₓ and C₃Hₓ⁺ chemistry (for example, the C₃ + H₃⁺ flux increases with the total density for a typical molecular cloud age).

In the Horsehead PDR, the cyclic isomer is also favoured, but with a smaller ratio equal to 3.4 (Pety et al. 2012). In IRC+10216, the cyclic-to-linear C₃H₂ ratio is ~19 (Agúndez 2009), that is, close to the values found in dense molecular clouds. It seems clear that, in contrast to the case for C₃H, the chemistry is comparable in all these objects, the formation of C₃H₂ isomers being mainly due to the DR of C₃H₃⁺, with the photodissociation of C₃H₃ and CH₃CCH playing a minor role. In PDRs, the lower cyclic-to-linear C₃H₂ ratio may result from the fact that c-C₃H₂ is supposed to be notably more photodissociated than l-C₃H₂ (van Hemert & van Dishoeck 2008). In diffuse clouds, where atomic H can be significant, the H + l-C₃H₂ → H + c-C₃H₂ reaction (introduced in this study) may strongly enhance the cyclic-to-linear C₃H₂ ratio and compensate for the photodissociation effects.

### t-C₃H₂

We have introduced the third isomer of C₃H₂: t-C₃H₂ (HCCCH). This isomer, which is almost symmetric, has a stability between that of the cyclic and that of the linear isomer. It has a triplet ground state and it is reactive with H, C, O and also with the N atom (cyclic and linear isomers show a barrier for reaction with

N atoms). It should be noted that there is no measurement of the dipole moment of t-C₃H₂ and that the structure is not well defined, the C₂, C₈ and C₂ᵥ structures being very close in energy (Aguilera-Iparraguirre et al. 2008). The dipole moment calculated by Nguyen et al. (2001) is 0.51 D and corresponds to the C₂ structure compatible with electron paramagnetic resonance spectroscopy (Seburg et al. 2009). Its introduction does not notably change the C₃H and C₃H₂ chemistry, but the reaction with the N atom is a new pathway towards HC₃N production; it is minor but non-negligible.

### C₃H⁺

C₃H⁺ has been detected in only three objects: the Orion Bar PDR, the Horsehead PDR, and Sgr B2(N) (Pety et al. 2012; McGuire et al. 2013, 2015). The lack of detection of l-C₃H⁺ in cold dense molecular clouds (upper limit of 6 × 10⁻¹¹×[H₂] in TMC-1 from McGuire et al. 2013) is in agreement with our calculations, where the l-C₃H⁺ abundance is about a few times 10⁻¹³ to 10⁻¹² relative to H₂ for typical ages of dense molecular clouds (a few times 10⁵ yr). The formation of l-C₃H⁺ is relatively well known through the reaction of C⁺ with C₂H₂ and the reaction of C₃ with H₃⁺, HCO⁺ and HCNH⁺. However, its main destruction rate via reaction with molecular hydrogen is not so well known, despite the fact that the rate constant has been measured several times at various temperatures and pressures (Savic & Gerlich 2005; Raksit & Bohme 1983; Smith & Adams 1987). We use the global analysis performed by Maluendes et al. (1993a,b), leading to a high rate constant at low temperature, which strongly limits the l-C₃H⁺ abundance but has no, or a very small, effect on c,l-C₃H₃⁺ abundance and then almost no effect on c,l-C₃H and c,l-C₃H₂, as long as the l-C₃H⁺ consumption is dominated by reaction with H₂ instead of DR. The amount of l-C₃H⁺ in the gas phase can be notably higher than the one given in our model if l-C₃H⁺ reacts slightly more slowly with H₂ at low temperatures, which will have no, or a small, effect on c,l-C₃H and c,l-C₃H₂ abundances.

### l-C₃H₃⁺, c-C₃H₂⁺

Both l-C₃H₃⁺ and c-C₃H₂⁺ reach a non-negligible abundance in our calculation for typical dense molecular clouds (around a few times 10⁻¹¹ and up to 10⁻¹⁰ versus H₂) and both have non-zero dipole moments (see Table 1). As already pointed out by Huang & Lee (2011) for l-C₃H₃⁺, it may be detectable. However, it should be noted that the main formation pathway is the C₃H⁺ + H₂ reaction, for which the branching ratios towards l-C₃H₃⁺ and c-C₃H₂⁺ are not well known.

### The C₃ + O reaction

As already highlighted in Hickson et al. (2016b), C₃ is abundant in the gas phase as a result of various efficient neutral pathways producing C₃ and very few destruction mechanisms. The low reactivity of C₃ means that it can reach high abundance levels, which is at the origin of the rich C₃H₄, C₃H₆ and C₃H₈ production on grains through H addition reactions. In order to evaluate the role of the O + C₃ reaction, which has been studied only theoretically by Woon & Herbst (1996), who found a small barrier in the entrance valley, we performed a run considering no barrier for this reaction and a rate constant equal to 2.0 × 10⁻¹⁰ cm³ s⁻¹ molecule⁻¹ (close to the capture rate constant), which considerably reduces the C₃ abundance. The introduction of the O + C₃ reaction decreases all C₃Hₓ abundances, the agreement being notably less good, for typical dense cloud ages, than considering a barrier for the O + C₃ reaction, particularly for l-C₃H₂ and c-C₃H₂. At late times (after
10⁶ yr) the release of CH₄, produced on grains, into the gas phase
triggers an intensive hydrocarbon chemistry, allowing most of the
observations to be reproduced. Considering the large uncertainty of
grain reactions (most of the reactions on the grain surface are not
well characterized) and the large variability of the results with the
description of the physics of the grains (for example induced by
the new three-phase model introduced by Ruaud et al. 2016), the
good agreement between observations and calculations at late age,
considering no barrier for the O + C₃ reaction, may be fortuitous.
It clearly shows the need to perform experimental measurements of
the O + C₃ rate constant, which will be a way to constraint the CH₄
formation on grains and its release in gas phase.

## 5 CONCLUSION

We have detected both linear and cyclic isomers of C<sub>3</sub>H and C<sub>3</sub>H<sub>2</sub>
towards several molecular clouds using the IRAM 30-m telescope.
We have introduced in our gas-grain chemical model NAUTILUS a
new branching ratio of DR deduced from statistical theory, with the
relative amounts of c-C<sub>3</sub>H/l-C<sub>3</sub>H and c-C<sub>3</sub>H<sub>2</sub>/l-C<sub>3</sub>H<sub>2</sub> being propor-
tional to the density of the vibrational states of each isomer near the
effective barrier to isomerization. This is in contrast to the case for
the usual astrochemical models, which assume that cyclic ions lead
only to cyclic neutral, and linear ions lead only to linear neutral. We
have also introduced the third isomer of C<sub>3</sub>H<sub>2</sub>, t-C<sub>3</sub>H<sub>2</sub>, which plays
a minor but non-negligible role owing to specific reactivity. Our
model allows us to reproduce the observations for c-C<sub>3</sub>H, l-C<sub>3</sub>H
and the c-C<sub>3</sub>H<sub>2</sub>/l-C<sub>3</sub>H<sub>2</sub> ratio in dense molecular clouds despite a
notable underestimation for c-C<sub>3</sub>H<sub>2</sub> and l-C<sub>3</sub>H<sub>2</sub>. In particular, the
cyclic-to-linear C<sub>3</sub>H<sub>2</sub> ratio given by our model is in good agreement
with observations of the seven molecular clouds studied in this pa-
per, with the ratio value decreasing when the total density of the
cloud increases. This decrease is due to the l-C<sub>3</sub>H<sub>2</sub> + H → c-C<sub>3</sub>H<sub>2</sub>
+ H reaction, which is important for low-density molecular clouds.

We also highlight the role of the O + C₃ reaction, which is likely
to possess a substantial barrier, in good agreement with the calcu-
lations of Woon & Herbst (1996), leading to a high C₃ abundance
in dense molecular clouds.

Another important point is the branching ratio of the C + C₂H₂ re-
action, the amount of l-C₃H produced being critical for reproducing
the linear-to-cyclic ratio in IRC+10216.

## ACKNOWLEDGEMENTS

This work was supported by the program 'Physique et Chimie du Milieu Interstellaire' (PCMI) funded by Centre National de la Recherche Scientifique (CNRS) and Centre National des Etudes Spatiales (CNES). VW's research is funded by the European Research Grant (ERC) Starting Grant (3DICE, grant agreement 336474).

MA, NM and JC thank the ERC for support under grant
ERC-2013-Syg-610256 NANOCOSMOS. They also thank Span-
ish MINECO for funding support under grants AYA2012-32032,
and from the CONSOLIDER Ingenio program 'ASTROMOL'
CSD 2009-00038. IRAM is supported by Institut des Sciences de
l'Univers (INSU)/CNRS (France), Max Planck Gesellschaft (MPG)
(Germany) and Instituto Geografico Nacional (IGN) (Spain).

This work received financial support from the French Agence
Nationale de la Recherche (ANR) under grant ANR-13-BS05-0008
(IMOLABS: Molecules interstellaires: spectroscopie et synthèse en
laboratoire).

We also thank the anonymous reviewer for their useful comments,
which led to improvements in this paper, particularly the suggestion
to study in more detail the relationship between the c-C<sub>3</sub>H<sub>2</sub>/l-C<sub>3</sub>H<sub>2</sub>
ratio and the total H<sub>2</sub> density of the molecular cloud.

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