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Appendix A

Appendix A.1: Rotational diagram analysis for levels in LTE and optically thick lines

We adopted the rotational diagram analysis that takes optical depth effects into account (also called population diagram analysis in Goldsmith & Langer 1999). A similar approach was taken by Goto et al. (2011). The line intensity emitted by a slab at a single temperature reads (A.1)B_ν is the Planck function at the excitation T_ex = T_rot = T_vib. The second term in the right-hand side is an optical depth correction factor to the optically thin rotational diagram. The term β = (1 − exp( − τ_ν))/τ_ν is akin to the escape probability. The maximum optical depth occurs at E_rot ~ kT_rot, where E_rot is the rotational upper level energy of an emission line. In the Born-Oppenheimer approximation, E_u = E_rot + E_vib. The rotational energy for CO can be approximated by E_rot = B_rotJ_u(J_u + 1), with the rotational constant B_rot = 2.76 K. The maximum optical depth occurs when . The line optical depth is (A.2)where ν is the line frequency, A the Einstein spontaneous emission probability of the transition, c the speed of light, Δv the turbulent width, N(CO) the CO column density, i the disc inclination, and (A.3)is the CO fractional population in the initial level (v′,J′). For CO, Q_rot ≃ kT_rot/B_CO + 1/3 (B_CO = 2.76 K) and Q_vib = 1/(1 − exp( − 3122/T_vib) are the rotational and vibrational partition function respectively (Brittain et al. 2007).

The rotational diagram of AB Aur (Fig. A.1) shows three parts: a steep slope corresponding to increasing optical depth until E_rot ~ kT_ex, then a shallow slope where the line optical depth decreases because the higher the rotational level, the less they are populated. At very high J, the slope steepens again. This behaviour also appears in our theoretical rotational diagrams, but the second turning point occurs at higher J than in the observations. The second slope change corresponds to lines with τ < 1. Assuming that the population is in LTE, we fitted the AB Aur rotational diagram by a model that takes the optical depth effects into account (Goldsmith & Langer 1999). The model parameters are the CO column density N(CO) = 4.2 × 10¹⁷ cm^-2, the mean excitation temperature T_ex ≃ 600 K (we assume that the vibrational temperature is equal to the rotational temperature), the turbulent width Δv = 0.05, and the inclination i = 30 degree. In the upper panel of Fig. A.1, the analytical solid curve compares well with the observations. The dashed-line curve shows the same model where optical depth effects are not taken into account. The lower panel shows the derived optical depths, which reach ~47.

Appendix A.2: Excitation, radiation, and gas kinetic temperatures

We define the radiation (brightness) temperature T_rad at a given wavelength as the equivalent blackbody temperature that will match the specific intensity computed by the continuum radiative transfer at a given location in the disc (Figs. A.2 and A.3). We also show temperature ratios in Figs. A.4 and A.5: the excitation over the radiation and the excitation over the gas kinetic temperatures.

Fig. A.1 The upper panel shows the comparison between AB Aur and an analytical model rotational diagram for the 12CO v = 1 − 0 transitions observed by Brittain et al. (2003). The solid blue line shows the fit by an analytical model that takes optical depths into account. The results from the same model but without the effect of optical depth are shown in dashed blue line. The lower panel shows the derived line optical depth.

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	Fig. A.2 Radiation temperature Trad at 0.15 micron and 4.844 micron for the Mdisc = 10-2M⊙, Rin = 1 AU disc models (left panel) and for the Mdisc = 10-4M⊙, Rin = 1 AU disc models (right panel). The black contour shows the regions that emit 49% of the fluxes at 4.844 micron. The black dashed-line contours contain 70% of the fluxes in the vertical direction.
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	Fig. A.3 Radiation temperature Trad at 0.15 micron and 4.692 micron for the Mdisc = 10-2M⊙, Rin = 1 AU disc models (left panel) and for the Mdisc = 10-4M⊙, Rin = 1 AU disc models (right panel). The black contours show the regions that emit 49% of the fluxes at 4.692 micron. The black dashed-line contours contain 70% of the fluxes in the vertical direction.
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Fig. A.4 Tex(v = 1,J = 19)/Trad(4.844) (upper panels) and Tex(v = 1,J = 19)/Tgas (lower panels) structures for the Mdisc = 10-2M⊙, Rin = 1 AU disc models. The left panels correspond to the model with UV pumping and the right panels to models without UV pumping. The contours are labelled in red. The white contours encompass the regions that emit 49% of the fluxes at 4.844 micron. The dashed-line contours contain 70% of the fluxes in the vertical direction.

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Fig. A.5 Tex(v = 1,J = 19)/Trad(4.844) (upper panels) and Tex(v = 1,J = 19)/Tgas (lower panels) structures for the Mdisc = 10-4M⊙, Rin = 1 AU disc models. The left panels correspond to the model with UV pumping and the right panels to models without UV pumping. The contours are labelled in red. The white contours encompass the regions that emit 49% of the fluxes at 4.844 micron. The dashed-line contours contain 70% of the fluxes in the vertical direction.

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© ESO, 2013