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Appendix A: MADEX: a local, non-LTE LVG code

The physical conditions in ISM clouds are such that molecular excitation is usually far from LTE. MADEX solves the non-LTE level excitation and line radiative transfer in a 1D isothermal homogeneous medium assuming a large velocity gradient (LVG) and spherical geometry. In this approximation the statistical-equilibrium equations are solved assuming local excitation conditions and a geometrically averaged escape probability formalism for the emitted photons (see details in Sobolev 1960; Castor 1970). This description allows one to take into account radiative trapping and collisional excitation and deexcitation more easily and computationally faster than more sophisticated non-local codes in which the radiative coupling between different cloud positions is explicitly treated (Montecarlo simulations, ALI methods, etc.). As a small benchmark, and in order to place the conclusions of our work on a firm ground, here we compare MADEX results with those obtained with RADEX⁶, a publicly available escape probability code (see van der Tak et al. 2007 for the basic formulae).

We ran several models for CO (a low-dipole moment molecule with μ = 0.12 D) and for HCO⁺ (μ = 3.90 D). These are typical examples of molecules with low and high critical densities, respectively (e.g. n_cr(CO 2 → 1) of a few 10⁴ cm^-3 and n_cr(HCO⁺2 → 1) of a few 10⁶ cm^-3). We note that for optically thin emission lines and for densities n(H₂) ≫ n_cr, collisions dominate over radiative excitations and level populations get closer to LTE (T_ex → T_k) as the density increases. For optically thick lines, line trapping effectively reduces n_cr and lines can be thermalized at lower densities. In the low density limit (n(H₂) ≪ n_cr), level populations are subthermally excited and, as the density decreases, tend to thermalize to the background radiation temperature (T_k>T_ex → 2.7 K in the millimetre domain).

Fig. A.1 CO isothermal models carried out with MADEX (filled squares) and RADEX (empty squares) non-LTE radiative transfer codes. Two gas densities are considered: n(H2) = 3 × 103 cm-3 (upper panels) and n(H2) = 3 × 105 cm-3 (lower panels). Excitation temperatures and line centre opacities (left and right panels, respectively) are shown for several rotational transitions in the millimetre domain as a function of CO column density. Two gas temperatures are considered, 10 K (blue points) and 150 K (red points).

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Figure A.1 (upper panels) shows low density model results (n(H₂) = 3 × 10³ cm^-3) for N(CO) from 10¹⁵ to 10¹⁸ cm^-2 at two different gas temperatures (T_k = 10 and 150 K, blue and red points, respectively). The left and right figures show the computed excitation temperatures and line centre opacities, respectively, for the CO 2 → 1, 3 → 2, and 4 → 3 transitions. Figure A.1 (lower panels) shows higher density models (n(H₂) = 3 × 10⁵ cm^-3) close to thermalization (T_ex ≃ T_k). We note that the selected range of column densities represents a transition from optically thin to optically thick emission. A line width of 1 km s^-1 is adopted in all models.

	Fig. A.2 Same as Fig. A.1 but for HCO+.
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Figure A.2 shows the same kind of models for HCO⁺ (column densities from 10¹² to 10¹⁵ cm^-2). Owing to the much higher critical densities of HCO⁺ rotational transitions, their excitation is sub-thermal (T_ex<T_k) in most of the explored parameter space. In addition, their associated emission lines become optically thick for column densities smaller than those of CO.

The filled and empty square marks in Figs. A.1 and A.2 represent computations performed with MADEX and RADEX codes, respectively. We checked that for the considered models, the predicted excitation temperatures and line opacities agree within ~20% and ~40%, respectively. This translates into maximum brightness temperature differences of ~50% in the most extreme cases.

Appendix B: Identified hydrocarbon lines

© ESO, 2015