3 Model photospheres and the generation of synthetic spectra

For the purpose of analysing our observations, synthetic spectra have been generated (Ryde et al. 1998) using model photospheres from the new grid of spherically-symmetric model photospheres of M giants, which is currently being calculated with the latest version of the MARCS code. This version is the final major update (of the code and its input data) in the suite of MARCS model-photosphere programs first developed by Gustafsson et al. (1975) and further improved in several steps, e.g. by Plez et al. (1992), Jørgensen et al. (1992), and Edvardsson et al. (1993).

These hydrostatic, spherical model photospheres are computed on the assumptions of Local Thermodynamic Equilibrium (LTE), homogeneity and the conservation of the total flux (radiative plus convective; the convective flux being computed using the mixing length formulation). The radiative field used in the model generation, is calculated with absorption from atoms and molecules by opacity sampling in approximately 84 000 wavelength points over the entire, relevant wavelength range considered for the star ( $2300~\mbox{\AA}$- $20~\mbox{$\mu$ m}$).

Data on the absorption by atomic species are collected from the VALD database (Piskunov et al. 1995) and Kurucz (1995, private communication). The opacity of CO, CN, CH, OH, NH, TiO, VO, ZrO, H₂O, FeH, CaH, C₂, MgH, SiH, and SiO are included and up-to-date dissociation energies and partition functions are used. The continuous absorption as well as the new models will be fully described in a series of forth-coming papers in A&A (Gustafsson et al.; Jørgensen et al.; Plez et al., all in preparation). The new models were used by, among others, Decin et al. (2000) in their modeling of the K5III-giant $\alpha$ Tau.

Using the computed model photospheres we calculated synthetic spectra by solving the radiative transfer at a high wavelength resolution in a spherical geometry through the model photospheres. Using extensive line lists (consisting of wavelengths, excitation energies of the lower state of the transition, and line strengths in the form of oscillator strengths), we produce synthetic spectra of wavelength regions around the water vapour bands ($\nu_1$ and $\nu_3$) at 2.60-3.66 $\mu$m. The line lists included in the generation of the synthetic spectra are H₂O (Partridge & Schwenke 1997) (including all lines stronger than given by the condition $gf \times 10^{-\chi \theta}/\max(gf \times 10^{-\chi \theta}) > 10^{-5}$, where $\theta=5040/3050$ and $\chi$ is the excitation energy in eV, leading to more than 2 million lines), CO (Goorvitch 1994), SiO (Langhoff & Bauschlicher 1993), CH (Jørgensen et al. 1996), CN (Jørgensen & Larsson 1990; Plez 1998, private communications), OH (Goldman et al. 1998), and C₂ (Querci et al. 1971; Jørgensen 2001, private communications). The accuracy and the completeness of these line lists are discussed in Decin (2000). We have also included CO₂ with data from the Hitran database (Rothman et al. 1987, 1992) and the Hitemp database (Rothman et al. in preparation).

In the generation of synthetic spectra, we calculate the radiative transfer for points in the spectrum separated by $\Delta \lambda \sim 1~\mbox{km~s$^{-1}$ }$ (corresponding to a resolution of $\lambda/\Delta \lambda \sim 330~000$) even though the final resolution is much less. With a microturbulence of $\xi\sim 2~\mbox{km~s$^{-1}$ }$ in the model photosphere, this means that we are sure of sampling all lines in our database in the generation of the synthetic spectrum. This is an important point since a statistical approach, by only taking fewer, opacity-sampled points, will give an uncertainty in the synthesised spectrum. There will be noticeable differences in the synthesised spectra calculated with different sets of points, unless one chooses the spacing between the points smaller than the physical width of the line broadening. This is especially important when dealing with molecular bands, since the separation between lines differs greatly with wavelength depending on whether the lines are close to a band head or not. For example, by choosing random points in the spectrum, at which the radiative transfer is calculated for the synthetic spectrum, with a larger separation than we have chosen, one will tend to overestimate the absorption at band heads (or in regions with a high line density) and underestimate it far from band heads (or in regions with a low line density).

The emergent model spectra are subsequently convolved with a Gaussian in the same manner and with the same resolution as the observed bands. This will allow a comparison with ISO observations.