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Appendix A: Computation of synthetic spectra using stellar atmosphere models

To derive the abundances of different chemical elements, we computed synthetic spectra with and without atomic and molecular lines. We used the software package SPECTR (Shimansky et al. 2003; see also Menzhevitski et al. 2014 for its latest version). Shimanskaya et al. (2011) used this program for deriving chemical abundances for the secondary in the close binary system FF Aqr. They show that lines of chemical elements in high-resolution spectra strongly broadened by rotation are reproduced well by the theoretical spectra.

Our synthetic spectra are based on plane-parallel, hydrostatic stellar atmosphere models for a given set of parameters (T_eff, logg, [M/H]) computed by interpolating a model grid of Castelli & Kurucz (2003) using the technique described by Suleimanov (1996). The solar chemical abundances and isotope compositions were specified using the data of Asplund et al. (2006) for Fe, C, N, and O and of Anders & Grevesse (1989) for all the other elements.

The radiative transfer equation at each frequency (wavelength) was solved using the Hermite method with the determination of specific radiation intensities for fixed angles. For a model atmosphere, we calculated the fluxes emerging in three basic directions, with inclinations to the surface of 62°, 30°, 8°. The stellar surface was then subdivided into sectors. The fluxes emerging from sectors towards the observer were derived by interpolating the radiation intensities at the three basic angles to the actual visibility angle.

The computation of synthetic spectra took about 600 000 atomic and more than 1 800 000 ¹²CH, ¹³CH, and SiH molecular lines into account from the lists of Kurucz (1994) and Castelli & Kurucz (2003), and 28 bands of 10 molecules (VO, TiO, SO, SiO, NO, MgO, MgH, CO, CN, AlO) computed in terms of the theory of Nersisyan et al. (1989) and kindly provided by Ya. V. Pavlenko. We used the empirical oscillator strengths from Shimanskaya et al. (2011) for 1350 strong optical lines in the λ3900 − 7000 Å wavelength range.

We computed the profiles of the HI lines according to the broadening theories of Vidal et al. (1973) and Griem (1960). The standard Voigt profiles for the remaining lines were calculated with broadening due to thermal motion and microturbulence, natural damping, Stark broadening in the approximation of Kurucz & Furenlid (1979), and van der Waals broadening with constants determined from the classical Gray formula (Unsold 1955) with scaling factors Δlog C₆ = 0.7 ÷ 1.2 (Shimanskaya et al. 2011). The synthetic spectra were binned to the resolution of 0.05 Å. We estimated the intensity uncertainties of the line profiles provided by this value to be within 0.005% of the continuum flux.

We modelled the line profiles HI, MgI, MgII, AlI, and CaII taking deviations from local thermodynamic equilibrium (LTE) into account. For each model atmosphere, the non-LTE populations were computed using the complete linearisation method of Auer & Heasley (1976) with the package NONLTE3 of Sakhibullin (1983).

Summing the radiation from all the aforementioned sectors of the stellar atmosphere considering their areas and local radial velocities due to stellar rotation and radial and tangential microturbulence yielded the integrated radiation from the atmosphere. The resulting stellar spectra were broadened in accordance with the instrumental function of the spectrograph.

Appendix A.1: Determining the chemical abundances

Since the full width at half maximum in our observed spectra is FWHM ~ 5 Å, almost all spectral lines are weak and blended. The most intense of them are dominant. At this resolution, it is possible to derive the abundance of a chemical element accurately (σ ≤ 0.15 dex) in spectra with high S/N per resolution bin (S/N ≥ 100), if this element 1) influences the whole spectrum or a large part of it (Δλ ~ 100 Å); and 2) has several strong dominant lines. If an element has just one dominant line and contributes ≥ 50% of the line intensity, the accuracy of the derived abundance is ~ 0.2 dex. The iron abundance and the microturbulent velocity (ξ_turb) influence the whole spectrum. If ξ_turb is incorrect, it is not possible to achieve an optimal agreement between strong and weak iron lines in the theoretical and observed spectra. The macroturbulent and stellar rotation velocities were not taken into account, because these effects are too weak to be detectable at our observational resolution (Smith & Dominy 1979; Gray & Toner 1986). The following elements have prominent atomic absorption lines that are dominant at our resolution: MgI 5167 Å, 5172 Å, 5183 Å, CaI 4226 Å, CaII 3933 Å, and CaII 3968 Å. The shapes of strong molecular bands, such as CH, MgH including hundreds of lines. are easily recognised and reproduced well by the synthetic spectrum.

Figure A.2 shows theoretical spectra calculated with the stellar atmosphere model corresponding to the following parameters: [M/H] = 0.0 dex, T_eff = 4494 K, and log g = 2.56. In each of the panels, abundances for one of the elements are varied, while the abundances of other elements have the fixed values listed in Table 2. The derivation of the iron abundance is illustrated in the top right-hand panel. In the course of fitting [Fe/H], the micro-turbulent velocity ξ_turb was estimated (see top left panel of Fig. A.2). These two values ([Fe/H] and ξ_turb) are closely related. A normalised summed spectrum of 31 RC stars (Table A.1) is shown by a solid line. The fitting of the whole spectrum is illustrated in Fig. A.3.

	Fig. A.1 Comparison of the observed Arcturus spectrum from the ELODIE database and our calculated synthetic spectrum for a wavelength region including iron peak elements and for three wavelength regions corresponding to several prominent spectroscopic features (CH and MgH molecules, CaII H, K lines, CaI 4227 Å). The adopted parameters are from Ramírez & Prieto (2011).
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	Fig. A.2 Illustration of fitting the abundances of different chemical elements (Sect. 3.2). The derived abundances are listed in Table 2.
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	Fig. A.3 Fitting the whole spectrum of the RC stars in BH176 by the synthetic one (see Appendix A) The derived abundances are listed in Table 2.
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© ESO, 2014