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Appendix A: Atmospheric parameters of TO stars in 47 Tuc and derived abundances of Li, O, and Na

In this section we provide the determined atmospheric parameters of the studied turn-off (TO) stars in 47 Tuc, as well as the abundances of lithium, oxygen, and sodium derived in their atmospheres. The abundances were derived assuming identical microturbulence velocity of ξ_micro = 1.0 km s^-1 for all stars. The contents of the Table A.1 are as follows: Col. 1: star ID; Cols. 2 and 3: right accension and declination; Cols. 4 and 5: B and V magnitudes; Cols. 6 and 7: effective temperature and surface gravity; Col. 8: 3D NLTE abundance of Li; Cols. 9 and 11: 1D NLTE abundances of O and Na, respectively; Cols. 10 and 12: 3D+NLTE abundances of O and Na, respectively (i.e., 1D NLTE abundances with 3D–1D abundance corrections taken into account).

Appendix B: On the relationship between 3D NLTE, 1D NLTE, and 3D LTE abundance corrections for the Li 670.8 nm line

In the derivation of oxygen and sodium abundances we corrected the 1D NLTE abundances for 3D hydrodynamical effects by adding Δ_{3D LTE − 1D LTE} abundance correction. It is obvious that such procedure does not account for the strong dependence of atomic level departure coefficients on the horizontal fluctuations of temperature in the 3D hydrodynamical models. Ideally, elemental abundances should be derived using full 3D NLTE approach where NLTE spectral synthesis computations are performed in the framework of 3D hydrodynamical models. This, however, is still rarely possible since the majority of current 3D spectral synthesis codes lack the capability of NLTE radiative transfer computations. One is then, therefore, forced to resort to different simplifications, e.g., such as applying Δ_{1D NLTE − 1D LTE} + Δ_{3D LTE − 1D LTE} abundance correction to take into account both 3D hydrodynamical and NLTE effects.

Fig. B.1 Top: temperature stratifications in the 3D hydrodynamical (grey scales indicating the temperature probability density), average ⟨3D⟩ (dashed red line), and 1D (solid red line) model atmospheres at [M/H] = 0.0. Horizontal bars show the optical depth intervals where 90% of the line equivalent width is formed (i.e., 5% to 95%): black and dashed blue bars correspond to the line forming regions in the full 3D and 1D model atmospheres, respectively (the equivalent widths of Li, O, and Na lines are 2 pm, 4 pm, and 19 pm, respectively). Bottom: RMS value of horizontal temperature fluctuations in the 3D model (black line) and temperature difference between the average ⟨3D⟩ and 1D models (dashed blue line).

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	Fig. B.2 Same as in Fig. B.1 but for models at [M/H] = −1.0.
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It would be therefore instructive to assess whether the Δ_{3D NLTE − 1D LTE} and Δ_{1D NLTE − 1D LTE} + Δ_{3D LTE − 1D LTE} corrections would lead to different final elemental abundances, and, if so, how large these differences may be expected to be. For this purpose, we used 3D hydrodynamical CO⁵BOLD and 1D hydrostatic LHD model atmospheres, together with the NLTE3D code, to compute full 3D, average ⟨3D⟩, and 1D synthetic profiles of the 670.8 nm resonance lithium doublet⁷. Spectral line synthesis was done with the Linfor3D code (see Sect. 5.2.4). Two 3D hydrodynamical models were used in this exercise (of the four utilised in Sect. 5.2.4), with the following atmospheric parameters: T_eff = 5930 K, log  g = 4.0, [M/H] = 0.0, and T_eff = 5850 K, log  g = 4.0, [M/H] = −1.0. The analysis was done for the cases of (i) weak (EW = 0.5 pm); and (ii) strong (EW = 7 − 9 pm) lithium lines. The obtained synthetic line profiles were used to compute Δ_{3D NLTE − 1D LTE}, Δ_{1D NLTE − 1D LTE}, and Δ_{3D LTE − 1D LTE} abundance corrections. The resulting abundance corrections are provided in Tables B.1 and B.2.

The obtained results clearly show that full Δ_{3D NLTE − 1D LTE} correction is always different from the sum Δ_{1D NLTE − 1D LTE} + Δ_{3D LTE − 1D LTE}, both at [M/H] = 0.0 and −1.0. It is nevertheless important to stress that while at solar metallicity the abundances obtained with both approaches agree to ≈0.03 dex, significant differences are seen at [M/H] = −1.0. In the latter case, the full 3D NLTE correction is positive and amounts to Δ_{3D NLTE − 1D LTE} = 0.065 dex for both weak and strong lines. The combined abundance correction, on the other hand, is negative and reaches to Δ_{1D NLTE − 1D LTE} + Δ_{3D LTE − 1D LTE} = −0.094 dex and −0.100 dex for weak and strong lines, respectively. Obviously, application of such combined correction in the case of lithium with the [M/H] = −1.0 models may lead to erroneous results.

The reason why the differences between the two corrections become different at [M/H] = −1.0 is that horizontal temperature fluctuations in the 3D hydrodynamical model and differences between the temperature profiles of the average ⟨3D⟩ and 1D models become larger at lower metallicities. Since atomic population numbers depend very sensitively on the local temperature, this leads to stronger deviations from NLTE in the 3D models compared to what would be expected in the 1D case. In this sense, situation may be somewhat safer in the case of oxygen, the lines of which form deeper in the atmosphere and thus experience smaller temperature variations.

Appendix C: 3D–1D LTE abundance corrections for oxygen and sodium

In this section we provide Δ_{3D LTE − 1D LTE} abundance corrections for the oxygen and sodium spectral lines used in this study. Abundance corrections are provided for all four models used in Sect. 5.2.4, and a range of line equivalent widths, EW.

The 3D-1D LTE abundance corrections provided in Tables C.1−C.4 for ξ_micro = 1.0 km s^-1 were interpolated to the observed equivalent widths and added to the 1D NLTE abundances to obtain the 3D+NLTE abundances for oxygen and sodium as given in Table A.1. As demonstrated in Appendix B, the 3D+NLTE abundances are not a good approximation to the real 3D NLTE abundances in the case of Li (the 3D–1D LTE corrections do not even yield the correct sign). For oxygen and sodium, the validity of this approximation is unclear, and the corrections given in Tables C.1– C.4 should be considered as an order of magnitude estimate at best.

The corrections given for ξ_micro = 0.5 and 1.5 km s^-1 serve to demonstrate that the dependence of the 3D–1D LTE abundance corrections on the adopted microturbulence parameter is weak (see also Table 5), even for stronger, partly saturated lines. This is explained by the fact that the thermal line broadening largely dominates over the turbulent broadening for these relatively light atoms. For heavier elements, like e.g. iron, the impact of microturbulence on the 3D abundance corrections would be significant, and a proper evaluation of the 3D corrections for stronger lines would require a more elaborate treatment than the simple assumption of a fixed microturbulence that was sufficient in the present work.

© ESO, 2014