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Appendix A: C₂H–e⁻ rate coefficients

The electronic ground-state symmetry of the radical C₂H is . Each rotational level N is therefore split by the spin-rotation coupling between N and the electronic spin S = 1/2 so that each rotational level N has two sublevels given by j = N ± 1/2. In addition, owing to the non-zero nuclear spin of the hydrogen atom (I = 1/2), each fine-structure level is further split into two hyperfine levels F = j ± 1/2. The rotational constant of C₂H is 1.46 cm^-1. The fine- and hyperfine-splittings are typically 0.01 and 0.001 cm^-1, respectively. The dipole moment of C₂H is 0.77 D (Woon 1995).

There is, to our knowledge, no previous estimate for the electron-impact hyperfine excitation rate coefficients of the C₂H radical. Recent R-matrix calculations have considered electron scattering from C₂H (Harrison & Tennyson 2011), but they concentrated on bound states of the anion and electronic excitation of the neutral. Here, electron-impact hyperfine rate coefficients for C₂H were computed using a three-step procedure, similar to the one employed recently for the radicals OH⁺ (Van der Tak et al. 2013) and CN (Harrison et al. 2013): i) rotational excitation rate coefficients for the dipolar (ΔN = 1) transitions were first computed using the Born approximation; ii) fine-structure excitation rate coefficients were then obtained from the Born rotational rates using the (scaled) infinite-order sudden (IOS) approximation; and iii) hyperfine excitation rate coefficients were finally obtained again using the IOS approximation. In the case of polar molecules, the long-range electron dipole interaction is well known to control the rotational excitation, especially at low collision energies (Itikawa & Mason 2005). Although the C₂H dipole is only moderate, the Born approximation is expected to be reasonably accurate for dipolar transitions. In practice, Born cross-sections were computed for collision energies ranging from 1 meV to 1 eV, and rate coefficients were deduced for temperatures from 10 K to 1000 K. The IOS approximation was employed to derive the fine-structure rate coefficients in terms of the rotational rates for excitation out of the lowest rotational level N = 0, following the general procedure of Faure & Lique (2012) (see Eqs. (1)−(3) of Harrison et al. 2013). The IOS approximation was also applied to derive the hyperfine rate coefficients (Eqs. (6)−(8) in Harrison et al. 2013).

The above three-step procedure was applied to the first 26 rotational levels of C₂H, which is up to the level (N,j,F) = (25,24.5,24), which lies 945 cm^-1 above the ground state (0, 0.5, 0), resulting in 246 collisional transitions. The largest rate coefficients are on the order of 5 × 10^-7 cm³ s^-1, i.e. typically four orders of magnitude more than the C₂H-He rate coefficients, and they are expected to be accurate to within a factor of 2. The complete set of de-excitation rate coefficients is available online from the LAMDA (Schöier et al. 2005) and BASECOL (Dubernet et al. 2013) databases.

Appendix B: Extrapolation of collision rates

Spielfiedel et al. (2012) only provided collision rates for temperatures up to 100 K and for rotational levels with up to N = 8 corresponding to an upper level energy of about 150 K, as given in BASECOL. This is clearly insufficient for modeling our observations because we observed up to N = 10−9, the gas temperatures in the Orion Bar are around 150 K, and densities of more than 10⁶ cm^-3 allow for a collisional excitation of much higher levels that will eventually decay radiatively, feeding the rotational ladder from levels above N = 9. A solution of the rate equations for the Orion Bar conditions thus asks for the inclusion of levels with energies of a few times the kinetic temperature.

Thus we have to extrapolate the rate coefficients from Spielfiedel et al. (2012) in terms of both additional levels and higher temperatures. A very general, but numerically demanding approach to extrapolating collision rates was proposed by Neufeld (2010). It is applicable to any type of molecule, but needs some “training” through collision rates from similar species. For C₂H we have, however, the advantage that it is basically a linear rotor, only superimposed by the hyperfine interaction. Since our measurements show no variation in the hyperfine ratio, and the collision rates from Spielfiedel et al. (2012) at temperatures above 20 K are dominated by the pure rotational transitions of the linear configuration with ΔN = ΔJ = ΔF (see Fig. B.1), it seems justified to concentrate on the rates for these transitions and to treat C₂H as a simple linear rotor.

For such molecules, the IOS approximation is a handy approach to computing the collision rates for any series of ΔJ (see Appendix A) as long as the energy difference of the states is small compared to the kinetic energy of the collision partners. For higher levels, a correction is possible, as summarized by Schöier et al. (2005). We use here their Eq. (13) derived originally by de Jong et al. (1975), Bieging et al. (1998) to extrapolate the collision coefficients to higher levels based on the energy-level difference and the behavior of the collision rates for lower J levels in the series. Owing to the additional hyperfine split in C₂H, we have to perform the fit individually for the different F − J combinations in a pure rotational series. This assumes that we have no further interaction of the states, as justified by the behavior of the collision rates at temperatures between 20 K and 100 K. The result is shown in Fig. B.1. This figure shows the collision-rate behavior within the dominant series for the levels computed by Spielfiedel et al. (2012) and the extrapolation of the rates up to N = 18. All other collision rates are much lower so that they are not important for the population of the upper levels.

Fig. B.1 Rates for the de-excitation of C2H through collision with H2 at 100 K. Black crosses mark all collision rates from Spielfiedel et al. (2012) above 10-12 cm3s-1. The curves indicate the different transition series for pure rotational transitions of the molecule, i.e., ΔN = ΔJ = ΔF. Each series consists of four curves according to the fine and hyperfine structure split of the four N states with approximately the same energy: J = N ± /1/2 and F = J ± 1/2. The continuations of those curves into the colored symbols show the extrapolation of the rates to the higher level transitions using the formalism from Schöier et al. (2005) based on the IOS approximation.

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Fig. B.2 Temperature dependence for the main collision rates of C2H with H2. The curves are sorted by ΔJ but cover all transitions of the corresponding series. Every connected dot in Fig. B.1 produces one temperature-dependence graph in this plot. To improve the readability, we have omitted the ΔN = ΔJ = ΔF = 3 series here. Up to 100 K, the lines show the rates computed by Spielfiedel et al. (2012), from 100 to 400 K, and we extrapolated as a power law with the same slope as measured between 90 and 100 K.

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Besides the extrapolation to higher level energies, we also need to extrapolate the rates from Spielfiedel et al. (2012) to higher temperatures. To obtain the rate at higher temperatures, we have to compute the integral over collision cross sections for energies of about ten times the kinetic temperature of the gas (Flower 2001). This makes the high-temperature computations demanding, in principle. However, since we only want to expand the temperature range by a small extent, from 100 K to 400 K, we can use the temperature behavior measured up to 100 K to estimate the rates above this temperature. The simplest approach would be an extrapolation with the square root of temperature, reflecting the velocity of the collision partners. When

inspecting the rate coefficients at temperatures up to 100 K, we see, however, that the majority of the rates show a temperature dependence that is very different from an exponent of 0.5 (see Fig. B.2). But the double-logarithmic plot shows straight lines for basically all rate coefficients at high temperatures, indicating that a power-law extrapolation with a free exponent is a good description of the temperature dependence. The curvature in the curves is very small.

We extrapolated from 100 K to 400 K using the power law determined either between 80 K and 100 K or between 90 K and 100 K. When comparing the two extrapolations, we found only 11 transitions with a deviation of about 10%. All other transitions showed better agreement of the rates, proving that the power-law description is quite accurate. One should take into account, however, that this approach should not be continued over wide dynamic ranges because we clearly expect the coupling to change when dealing with factors of ten or more.

To check the sensitivity of our overall results to the extrapolations, we also ran the RADEX fits performed in Sect. 5 with an alternative collision rate file created just by a constant continuation of the rates from 100 K toward higher temperatures and higher levels. For the same input parameters, the average excitation temperature then drops from 26.9 K to 24.6 K, eventually indicating an overall relatively minor sensitivity of our results to the details of the extrapolations.

We finally note that rate coefficients for the fine-structure excitation of C₂H by para-H₂(j = 0) have been published recently by Najar et al. (2014). The largest rate coefficients, which correspond to transitions with Δj = ΔN = 2, were found to differ from those of C₂H-He by a factor of ~1.4, as assumed here. Larger differences were, however, observed for other transitions, especially those with Δj = ΔN = 1 (see Fig. 6 in Najar et al. 2014). Hyperfine-resolved rate coefficients for C₂H-H₂, for high levels and high temperatures, are therefore urgently required in order to avoid both collider-mass scaling and extrapolations.

© ESO, 2015