© ESO, 2016

1. Introduction

Isomers and isotopologs are excellent tools to probe physical and chemical properties, as well as evolutionary states, in the ISM. Cyclopropenylidene, the cyclic form of C₃H₂, is one of the most abundant and widespread molecules in our Galaxy (Matthews & Irvine 1985). Propadienylidene, H₂CCC, is a very polar carbene and is the less stable isomer of c-C₃H₂ by ~10 kcal mol^-1 (5000 K; Wu et al. 2010). Propadienylidene was first detected in the laboratory by Vrtilek et al. (1990) by means of millimetrer-wave absorption spectroscopy in a DC discharge. It has been observed in prestellar and protostellar cores (Cernicharo et al. 1991; Turner et al. 2000; Kawaguchi et al. 1991; Sakai & Yamamoto 2013), circumstellar envelopes (Cernicharo et al. 1991), translucent clouds (Turner et al. 2000), diffuse clouds (Cernicharo et al. 1999; Kulczak-Jastrzȩbska et al. 2012; Liszt et al. 2012), and photodissociation regions (PDRs; Teyssier et al. 2005; Cuadrado et al. 2015).

The cyclic-to-linear (hereafter c/l) ratio of C₃H₂ tends to increase with increasing A_V. Significant variations of this ratio are found in dense cores, ranging from 25 in TMC-1(CP) to 67 in TMC-1C (Spezzano et al. 2016). For completeness, we report in Table A.1 the values of the C₃H₂c/l ratio observed in several media, including dense cores. To be able to model the observed c/l ratios use this ratio as an astrochemical tool, the chemistry leading to the formation and destruction of c-C₃H₂ and H₂CCC (hereafter l-C₃H₂) needs to be better understood, with emphasis on the branching ratios of the reactions involved.

The main formation process of cyclic and linear C₃H₂ involves the radiative association of C₃H⁺ and H₂ to form cyclic and linear C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ and a subsequent dissociative recombination with electrons that leads to c-C₃H₂ and l-C₃H₂, respectively. C₃H⁺ exists in both cyclic and linear forms, but the cyclic isomer lies 17 kcal mol^-1 (8600 K) above the linear one (Ikuta 1997), hence for our purposes we can assume to deal with l-C₃H⁺ only. The radiative association of l-C₃H⁺ and H₂ will form cyclic and linear C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ in a 1:1 ratio (Maluendes et al. 1993). Talbi et al. (2009) suggested with their calculations that the dissociative recombination of c-C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ to form c-C₃H₂ is more efficient than its linear counterpart. Using an afterglow experiment at 300 K, Adams & Babcock (2005) observed that the cyclic C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ recombines faster than the linear isomer.

The chemistry of c and lC₃H₂ and that of other associated species has previously been modeled for instance by Turner et al. (2000) and Fossé et al. (2001). Both of these works pointed out that neutral-neutral reactions can also play a part in determining the C₃H₂c/l ratio. The rate coefficients of the reactions most relevant in the present context have changed a little since these studies.

A key point of the dissociative recombination of C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ is its branching ratio, which unfortunately has not yet been investigated either experimentally or theoretically. Chabot et al. (2013) studied the dissociative recombination of cyclic and linear C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{2}\end{array}$. Their results show that while c-C₃H is the main product of the dissociative recombination of c-C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{2}\end{array}$ with electrons, l-C₃H is not the main product of the recombination of l-C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{2}\end{array}$. In this work we study the effect of the branching ratios of the dissociative recombination of both cyclic and linear C₃H$\begin{array}{c}\mathrm{+}\\ \mathrm{3}\end{array}$ on the cyclic-to-linear ratio of C₃H₂. We use the well-studied prestellar core L1544 to compare the results of our model with the observational results.

2. Formation and destruction of C₃H₂

In physical conditions with high visual extinction (A_V > a few mag) where photoprocesses do not play a significant role, the formation and destruction of (cyclic and linear) C₃H₂ follow a relatively simple scheme. The main formation pathway is the dissociative recombination ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$. In this paper, we use the latest rate coefficient data from the KIDA database (Wakelam et al. 2015; see also below), where three different branches for this reaction are included. These are tabulated in Table 1. The third tabulated reaction for both forms of C₃H₂, whose rate coefficient we label here as k_DR,3, is the most important for the C₃H₂ formation because it produces C₃H₂ directly (see also Sect. 4).

The main destruction partners of C₃H₂ change with time, but at the time when C₃H₂ is the most abundant (see Sects. 3 and 4), the main destruction reactions are ion-molecule reactions that convert C₃H₂ back to ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$: ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}\mathrm{+}\mathrm{HC}{\mathrm{O}}^{\mathrm{+}}\mathit{/}{\mathrm{H}}_{\mathrm{3}}{\mathrm{O}}^{\mathrm{+}}\mathit{/}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}\mathrm{CO}\mathit{/}{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\mathit{/}{\mathrm{H}}_{\mathrm{2}}$ with rate coefficients k₁, k₂, and k₃, respectively. Thus, the abundance of C₃H₂ in steady state would be given by $[{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}]\mathrm{=}\frac{{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}[{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}][{\mathrm{e}}^{\mathrm{-}}]}{{\mathit{k}}_{\mathrm{1}}[\mathrm{HC}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{2}}[{\mathrm{H}}_{\mathrm{3}}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{3}}[{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}]}\mathit{,}$(1)which leads to an expression of the c/l ratio of C₃H₂$\begin{array}{ccc}\frac{[\mathit{c}\mathrm{-}{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}]}{[\mathit{l}\mathrm{-}{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}]}& \mathrm{=}& \frac{{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}[\mathit{c}\mathrm{-}{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}][{\mathrm{e}}^{\mathrm{-}}]}{{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}[\mathit{l}\mathrm{-}{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}][{\mathrm{e}}^{\mathrm{-}}]}\\ & & \mathrm{\times }\frac{{\mathit{k}}_{\mathrm{1}}^{\mathit{l}}[\mathrm{HC}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{2}}^{\mathit{l}}[{\mathrm{H}}_{\mathrm{3}}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{3}}^{\mathit{l}}[{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}]}{{\mathit{k}}_{\mathrm{1}}^{\mathit{c}}[\mathrm{HC}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{2}}^{\mathit{c}}[{\mathrm{H}}_{\mathrm{3}}{\mathrm{O}}^{\mathrm{+}}]\mathrm{+}{\mathit{k}}_{\mathrm{3}}^{\mathit{c}}[{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}]}\mathrm{·}\end{array}$(2)Because of gas-grain chemical interaction, a steady-state solution is not attained in our models, but the above expressions turn out to give a rather accurate representation of the C₃H₂ peak abundances. Inspection of the KIDA data reveals that the rate coefficients of the three C₃H₂-destroying ion-molecule reactions discussed above are almost identical (slightly higher for l-C₃H₂ than for c-C₃H₂), so that the latter fraction in Eq. (2) is ~1. Furthermore, our chemical model (see below) indicates that the c-${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$/l-${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ ratio is usually also very close to unity, so that the c-C₃H₂/l-C₃H₂ ratio is determined to a good approximation by the rate coefficient ratio ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ at the time when HCO⁺, H₃O⁺, and ${\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ are abundant (i.e., before the onset of freeze-out of neutrals onto grain surfaces). This ratio is unity in the KIDA data, which implies c-C₃H₂/l-C₃H₂ ~ 1 according to the approximating formula presented above. However, the observed c/l abundance ratio toward the dust peak in L1544 is 32 ± 4 (Spezzano et al. 2016). We therefore set out to modify the various k_DR branching ratios to explain the observations. The assumption that the two isomers are formed in the same fashion and in similar conditions is strengthened by our recent maps of c and lC₃H₂ in the inner 2.5′ × 2.5′ of L1544, which demonstrate that c and lC₃H₂ trace the same region (Spezzano et al., in prep.). A spatial association of c and lC₃H₂ has also been observed in other sources (e.g., Fossé et al. 2001).

Fig. 1 Left panel: abundances of c-C3H2 (solid lines) and l-C3H2 (dotted lines) as functions of distance from the core center in the L1544 model at t = 105 yr. Middle panel: C3H2c/l ratio as a function of distance from the core center in the L1544 model at t = 105 yr. Right panel: density (green solid line), dust temperature (red solid line), and gas temperature (red dashed line) as functions of distance from the core center in the L1544 model of Keto et al. (2014). The different colors in the left and middle panels correspond to ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}\mathrm{=}\mathrm{1}$ (red), ~7 (blue), and 25 (green). The inset in the left panel shows a zoomed-in view of the innermost 10 000 AU in the model core. The black vertical line marks the position of the C3H2 abundance peak (the line is omitted in the left panel for clarity).

Fig. 2 Column densities (convolved to a 30′′ beam) of c-C3H2 (left panel) and l-C3H2 (middle panel), and the C3H2c/l column density ratio (right panel) as functions of time toward the center of the L1544 model. The different colors correspond to ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}\mathrm{=}\mathrm{1}$ (red), ~7 (blue), and 25 (green). The horizontal lines and bars in each panel correspond to the observed values toward the L1544 dust peak (Spezzano et al. 2016).

As a guide to the modifications, we analyzed the branching ratios calculated for the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ reaction by Chabot et al. (2013). These branching ratios, which are included in the KIDA database, are shown in Table 2. Calculations for the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ system are planned, but it is unclear when the results will be available (M. Chabot, priv. comm.). Therefore, we used the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ system as a template for the modifications. In the dissociation of ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}$, there are pathways that remove either one or both protons from ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}$ and pathways that break C-C bonds, that is, there are three groups of pathways. Inspection of the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ pathways (Table 1) reveals a similar grouping. As a first approximation we combined the branching ratios belonging to each group in the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}^{\mathrm{+}}$ data and translated them into the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$ system. This leads to the branching ratios given in Table 3 (mod_ch). In this modified model, the ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ ratio is ~5, which is still not high enough in light of the observations. We then proceeded to vary the branching ratios to attain a fit to the observed C₃H₂c/l ratio in L1544. The results of our analysis are presented in Sect. 3.

To simulate the C₃H₂ abundances in L1544, we used the physical model for L1544 presented by Keto & Caselli (2010) and updated by Keto et al. (2014), which gives us the density, (gas and dust) temperature, and A_V profiles as functions of distance away from the dust peak. We separated the model core to concentric shells and calculated the chemical evolution separately in each shell. In this way, we can produce simulated chemical abundance gradients and study how the column densities of the various molecules change as functions of time. A similar procedure has been used to successfully model the emission and absorption of the ortho and para H₂D⁺ ground-state rotational lines toward IRAS 16293A (Brünken et al. 2014). The (gas-grain) chemical code used here is described in detail in Sipilä et al. (2015a). The gas-phase chemical reaction set used in the present work is based on the latest KIDA reaction file (Wakelam et al. 2015), which was deuterated and spin-state separated according to the prescriptions of Sipilä et al. (2015a) and Sipilä et al. (2015b). However, in this paper we do not explicitly consider spin states or deuteration. The physical parameters of the model and the initial chemical abundances are adopted from Tables 1 and 3 in Sipilä et al. (2015a) except for the ratio of diffusion to binding energy (E_d/E_b), for which we adopt here a value of 0.60, in line with the recent results of Minissale et al. (2016).

3. Results

We calculated the abundances and column densities of c-C₃H₂ and l-C₃H₂ in the L1544 model with a grid of values for the ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ ratio, ranging from 1/1 (the KIDA value) up to 50/2 (mod_hr, Table 3). We plot in Fig. 1 the abundances of c-C₃H₂ and l-C₃H₂ and the C₃H₂c/l ratio at t = 10⁵ yr assuming three different values for ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$. We also show the density and temperature structures of the L1544 core model (Keto et al. 2014) for completeness. Additional calculations (not shown) indicate a simple increasing trend of the C₃H₂c/l ratio with ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$. Evidently, neither the KIDA model nor our initial modified model (mod_ch) produce a high enough c/l ratio, and a high ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ is needed to produce a c/l ratio comparable to the observed value.

To supplement the results shown in Fig. 1, we show in Fig. 2 the column densities of cyclic and linear C₃H₂ and their ratio as functions of time toward the center of the L1544 model. We also show the observed values (Spezzano et al. 2016). The simulated column densities were convolved to a 30′′ beam so that they correspond to the observations. The peak value of the C₃H₂c/l ratio is ~32, which corresponds to 8 × 10⁴ yr of chemical evolution. The best fit to the observations is attained at t ~ 10⁵ yr, when the column densities of both species and their ratio are all within the observed limits. The timescale required to attain the best fit depends on the initial chemical abundances, and so our results do not necessarily reflect the “absolute” age of the core. However, it is clear that we can only attain a high enough C₃H₂c/l ratio with the model if the production of cyclic C₃H₂ is strongly boosted over that of linear C₃H₂ in the dissociation of ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$.

4. Discussion and conclusions

The ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ branches (Tables 1 and 3) show that the branch listed second (${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}\mathrm{H}\mathrm{+}{\mathrm{H}}_{\mathrm{2}}$) forms C₃H, which can be converted back into C₃H₂ through the sequence C₃H + H⁺ ➝ C₃H⁺ + H; ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}^{\mathrm{+}}\mathrm{+}{\mathrm{H}}_{\mathrm{2}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}$; ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}\mathrm{+}\mathrm{H}$. Therefore it seems reasonable to assume that increasing ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{l}}$ would further increase the C₃H₂c/l ratio. We tested this by modifying the ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{l}}$ and ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{1}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{1}}^{\mathit{l}}$ ratios, keeping ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ fixed at 50/2. We found that increasing ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{c}}$ and decreasing ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{l}}$ by 10%¹ further increases the peak C₃H₂c/l ratio to ~40, while decreasing ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{c}}$ and increasing ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{2}}^{\mathit{l}}$ by 10% leads to a peak C₃H₂c/l ratio of ~26. Therefore we can attain a good fit to the observations with several combinations of the k_DR,2 and k_DR,3 values, and it is difficult to derive strict upper limits to acceptable values of the branching ratios based on the present analysis. However, our calculations indicate that the effect of ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}\mathrm{+}\mathrm{H}$ is greater than that of ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}\mathrm{H}\mathrm{+}{\mathrm{H}}_{\mathrm{2}}$ in determining the C₃H₂c/l ratio. This is because the former pathway is a direct source of C₃H₂ while the latter pathway depends on multiple reactions (see above) that can also form species other than C₃H₂. We require ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ clearly higher than unity. We note that the C₃Hc/l ratio is in our models nearly independent of the various k_DR,n values, and is ~3 at the time of the C₃H₂ peak, increasing to ~10 at later times (as the C₃H₂c/l ratio decreases). These values agree with observations (Turner et al. 2000; Fossé et al. 2001; Cuadrado et al. 2015). However, because of its weak dependence on k_DR,n, the C₃Hc/l ratio cannot be used to constrain the branching ratios of the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ process.

Figure 2 demonstrates that the C₃H₂c/l ratio is not constant in time, meaning that the approximation of Eq. (2) does not hold universally. For the highest ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ ratio tested here (50/2, Table 3), the C₃H₂c/l ratio presents a maximum for a relatively brief period of time and finally settles to ~15. Our chemical model shows that at late times the slow neutral-neutral reaction H + CH₂CCH ➝ H₂ + C₃H₂ is a significant source of C₃H₂. Because the ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ ratio is so high, this neutral-neutral reaction is the most important formation pathway for l-C₃H₂ (and the second most important pathway for c-C₃H₂), so that the C₃H₂c/l ratio is no longer controlled solely by ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$. This finding is in line with previous discussions by Turner et al. (2000) and Fossé et al. (2001).

Figure 1 shows that the C₃H₂c/l abundance ratio presents a minimum about 10 000 AU away from the core center and increases again at larger radii. This behavior is a result of changing physical conditions, particularly the visual extinction. A detailed analysis of our results reveals that around 10 000 AU at t = 10⁵ yr, the formation of linear C₃H₂ is governed by reactions other than ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ because of its low rate coefficient, while at still higher radii photochemistry comes into play and the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ reaction is again the dominant formation pathway for both cyclic and linear C₃H₂. However, the abundances are very low in the outer core; by far the largest part of the cyclic and linear C₃H₂ column densities comes from the interval ~3000–7000 AU.

The total rate coefficient for ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ assumed in the earlier modeling works of Turner et al. (2000) and Fossé et al. (2001) is equal to that adopted here (Table 1) within a factor of two. Although only a limited amount of rate coefficient data is quoted in Turner et al. (2000) and Fossé et al. (2001), it seems that the rate coefficients for the main reactions in the present context have not significantly changed in the past two decades. Still, it is clear from all the models that the observed c/l ratios can only be reproduced if the relative rate coefficients for the formation and destruction of the various c and l species have different values.

As stated in the Introduction and expanded upon in Appendix A, the C₃H₂c/l ratio has been observed toward various environments (e.g., protostellar cores and diffuse clouds) with values ranging from 3 to ~70. The observations show a trend of increasing C₃H₂c/l ratio at increasing A_V. The analysis presented above demonstrates that various reactions can be responsible for the formation of C₃H₂ (and hydrocarbons in general; see Alata et al. 2014, 2015; Duley et al. 2015). We cannot, based on the present models, draw quantitative conclusions on the expected C₃H₂c/l ratio in objects other than L1544 without detailed (physical) modeling, particularly toward environments with low gas density and low A_V. Such an undertaking is beyond the scope of this Letter. As an example of the interpretational difficulty, we note that the C₃H₂c/l ratio in the outer parts of the L1544 model, which is roughly consistent with a PDR in terms of physical conditions, follows the ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}$ ratio (Fig. 1), and we checked that the C₃H₂c/l column density ratio is indeed high (~17–35 depending on the time) even 100′′ away from the core center. This agrees with the observations (Spezzano et al., in prep.).

We conclude that to explain the C₃H₂c/l ratio observed toward L1544 (32 ± 4; Spezzano et al. 2016), the branching ratios of the various outcomes of the ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}$ reaction should be adjusted to favor cyclic C₃H₂ over linear C₃H₂. In physical conditions with moderate visual extinction, the most important reaction determining the C₃H₂c/l ratio is ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{2}}\mathrm{+}\mathrm{H}$, followed by ${\mathrm{C}}_{\mathrm{3}}{\mathrm{H}}_{\mathrm{3}}^{\mathrm{+}}\mathrm{+}{\mathrm{e}}^{\mathrm{-}}\mathrm{-}\mathrm{\to }{\mathrm{C}}_{\mathrm{3}}\mathrm{H}\mathrm{+}{\mathrm{H}}_{\mathrm{2}}$. It is currently assumed in the KIDA database that c and lC₃H₂ are created in equal proportions in both the former and the latter reaction. However, our modeling results suggest that the ratio of the rate coefficients in the former reaction for cyclic and linear C₃H₂ should be of the order of ~20 depending on what is assumed for the latter reaction. Unfortunately, we cannot set strict limits on the branching ratios based on our current analysis. More theoretical and/or laboratory work is clearly needed to understand the chemistry of C₃H₂.

Acknowledgments

We thank the anonymous referee for helpful comments and suggestions that improved the manuscript. We acknowledge the financial support of the European Research Council (ERC; project PALs 320620).

References

Appendix A: Additional data

As noted in the introduction, the C₃H₂c/l ratio tends to increase with increasing A_V. Table A.1 reports the values of the c/l ratio observed in several media. The values range from 3 in diffuse clouds to over 50 in dense cores. However, the density of the medium is not the only quantity that has an effect on this ratio. Teyssier et al. (2005) showed that inside the Horsehead Nebula PDR the C₃H₂c/l ratio does not exceed 15 even though the extinction is on the order of 10−20, which is comparable with what is found in TMC-1(CP), where the c/l ratio has been observed to be 25 (Turner et al. 2000). In the Orion Bar a c/l ratio of 34 has been recently observed (Cuadrado et al. 2015), which is higher than the value of ~4 observed toward the Horsehead PDR (Teyssier et al. 2005). Furthermore, the c/l ratio is also different among dense cores, ranging from 25 in TMC-1(CP) to 67 in TMC-1C (Spezzano et al. 2016).

All Tables

All Figures

Fig. 1 Left panel: abundances of c-C3H2 (solid lines) and l-C3H2 (dotted lines) as functions of distance from the core center in the L1544 model at t = 105 yr. Middle panel: C3H2c/l ratio as a function of distance from the core center in the L1544 model at t = 105 yr. Right panel: density (green solid line), dust temperature (red solid line), and gas temperature (red dashed line) as functions of distance from the core center in the L1544 model of Keto et al. (2014). The different colors in the left and middle panels correspond to ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}\mathrm{=}\mathrm{1}$ (red), ~7 (blue), and 25 (green). The inset in the left panel shows a zoomed-in view of the innermost 10 000 AU in the model core. The black vertical line marks the position of the C3H2 abundance peak (the line is omitted in the left panel for clarity).

In the text

Fig. 2 Column densities (convolved to a 30′′ beam) of c-C3H2 (left panel) and l-C3H2 (middle panel), and the C3H2c/l column density ratio (right panel) as functions of time toward the center of the L1544 model. The different colors correspond to ${\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{c}}\mathit{/}{\mathit{k}}_{\mathrm{DR}\mathit{,}\mathrm{3}}^{\mathit{l}}\mathrm{=}\mathrm{1}$ (red), ~7 (blue), and 25 (green). The horizontal lines and bars in each panel correspond to the observed values toward the L1544 dust peak (Spezzano et al. 2016).

In the text