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Astron. Astrophys. 323, 202-210 (1997)

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2. Models and spectrum synthesis

In order to study the behavior of the SiO bands and their impact on the atmospheric structure we have calculated a grid of 138 hydrostatic model atmospheres for cool oxygen-rich stars using an improved version (Jorgensen et al. 1992) of the MARCS code (Gustafsson et al. 1975) with spherical radiative transfer routines from Nordlund (1984). We included molecular opacities of CO, TiO, H2 O and CN (Jorgensen 1994aand references therein) by treating them in the opacity sampling approximation (Jorgensen 1992). Elemental abundances for C, N and O were taken from Grevesse & Sauval (1994) otherwise from Anders & Grevesse (1989). In addition we calculated all models with and without taking into account the effects of SiO. The corresponding molecular data originate from a linelist compiled by Langhoff & Bauschlicher (1993), which represents at the moment the most accurate source of SiO opacities. The list is complete for all transitions with [FORMULA] and [FORMULA] for the three isotopes 28 Si16 O, 29 Si16 O and 30 Si16 O. To check the completeness of the data with respect to the total SiO opacity we calculated the vibrational-rotational partition function as a sum of all the energy levels included in the linelist and compared it with a similar partition function for all levels up to [FORMULA] and [FORMULA]. The difference between the two results was less than 0.2% for [FORMULA]. This means that the linelist is complete (in the sense of inclusion of relevant energy levels) to [FORMULA] for all temperatures of interest for stellar atmosphere computations. We also compared the obtained partition functions with the semi-analytical summations of Sauval & Tatum (1984) and Rossi et al. (1985). We found that the agreement is again as good as it should be expected from a high degree of completeness. We therefore conclude that neither the atmospheric structures nor the low or medium resolution spectra will be affected by spectral lines due to transitions between higher energy levels than those included in our list.

First we have produced a systematic grid of 67 atmospheres by changing the effective temperatures and gravitational accelerations in the range [FORMULA] and [FORMULA] concentrating on objects with one solar mass and solar chemical abundances. The rest of the models were designed to show the effects of different masses (sphericity effects), metallicities, C/O ratios and Si abundances for several selected [FORMULA] and [FORMULA] values. The chosen parameters cover the following ranges: [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. For all our atmospheres we have adopted the solar system relative isotopic abundances of 0.9223 for 28 Si, 0.0467 for 29 Si and 0.0310 for 30 Si, which are taken from Anders & Grevesse (1989).

As a next step the model atmospheres were used to calculate synthetic rotation-vibration spectra for SiO. This was done with a simple radiation transport program based on the corresponding routines in the MARCS code. For observational reasons we focused on the following two items: At the beginning we worked on high and medium resolution spectra covering the short wavelength part of the first overtone transitions and then we calculated low resolution spectra (opacity sampling resolution) for the whole infrared range between 2.0 and 12.5 µm including all [FORMULA], 2 and 3 bands.

To study the behavior of the first overtone bands we produced synthetic spectra for the wavelength range between 3.96 and 4.14 µm. This region is interesting, because it covers the existing SiO observations in the L-band (e.g. Rinsland & Wing 1982, Aringer et al. 1995) and it can be easily accessed by future ground based measurements. It contains the short wavelength part of the first overtone transitions starting with the V(2,0) feature of 28 SiO at approximately 4 µm. An overview including all SiO bandheads in the selected range is given in Table 1.


Table 1. Positions of SiO bandheads in the wavelength range between 3.96 and 4.14 µm

The model spectra have been calculated with a very high resolution of [FORMULA], which is enough for a good description of the line shapes. The latter were assumed to be simple Doppler profiles, since we did not attempt any exact fit to results from high resolution FTS spectroscopy. In addition, especially close to the bandheads, the wings of even the strongest lines will often be weaker than the Doppler cores of the many overlapping neighboring ones. On the other hand many of the SiO lines are extremely saturated in the atmospheres of cooler giants causing especially their width, which is mainly determined by the microturbulence ([FORMULA]), to have a strong influence on medium and low resolution work. As a consequence the intensity of the SiO absorption depends much on the value of [FORMULA] (This is also the reason, why the resolution must be high enough for representing the line profiles well.). For our grid of synthetic spectra we have adopted [FORMULA] km/s, which is consistent with the opacity sampling data used for the model atmospheres. In addition we have varied [FORMULA] for several selected stellar parameters between 1.0 and 6.0 km/s to investigate the effect of different line widths.

In Fig. 1 we present a typical result of our calculations. This is a medium resolution SiO spectrum ([FORMULA]) of an AGB star, which was simply derived by averaging over the points from the synthetic data. It shows the wavelength range from 3.96 to 4.12 µm. The continuum is set to one and the corresponding stellar parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA], C/O = 0.48, [Si] = 7.55 and [FORMULA] (C/O and [Si] solar). One can clearly see most of the different bandheads mentioned in Table 1. It is also obvious that the absorption of the main isotope 28 SiO can be very strong in cool objects.

[FIGURE] Fig. 1. Medium resolution SiO spectrum ([FORMULA]) for the wavelength range from 3.96 to 4.12 µm. The stellar parameters are [FORMULA], [FORMULA], [FORMULA], solar mass and chemical abundances. The continuum is set to one and only SiO lines are included. For an identification of the bandheads see Tab. 1.

In order to compare the intensity of different band systems we created for all model atmospheres synthetic spectra covering the wavelength range between 2.0 and 12.5 µm, which includes the fundamental as well as the first and second overtone transitions. Because of very practical reasons like calculation time and disk space this could only be done at a rather low resolution. Although such an approach does not allow to resolve line profiles, which is necessary for describing the SiO absorption correctly, one can still obtain realistic results by using the opacity sampling approximation that has already been applied for the model atmospheres. Fig. 2 shows a spectrum derived by this method adopting the same stellar parameters as in Fig. 1. The continuum is again set to 1 and only SiO features are included. The original resolution of [FORMULA] (between [FORMULA] at 2 µm and [FORMULA] at 10 µm; approximately equidistant stepsize in wavenumber) was reduced to a value of [FORMULA], because due to its statistical nature the opacity sampling approximation only gives correct results, if one takes the average of a larger number of spectral points. But in the case of the SiO band systems this is still enough to determine their total intensity.

[FIGURE] Fig. 2. Low resolution SiO spectrum ([FORMULA]) for the wavelength range from 2.0 to 12.0 µm. The stellar parameters are [FORMULA], [FORMULA], [FORMULA], solar mass and chemical abundances. The continuum is set to one and only SiO opacities are included. The fundamental, first and second overtone bands can be seen.
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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998