3. Model atmospheres
The statistical equilibrium calculations are performed in horizontally homogeneous LTE model atmospheres in hydrostatic equilibrium. We account for metallic and molecular UV line absorption using Kurucz' (1992) opacity distribution functions interpolated for the proper solar mix of abundances. The solar abundances are adopted from Holweger (1979) with minor modifications for carbon (Stürenberg & Holweger 1990). For Fe and Mg we use values of 7.51 and 7.58, respectively. Four different solar temperature stratifications are explored for comparison. The first model in Fig. 3, labeled GRS88, is flux-conserving, convection is taken into account parametrically with a mixing-length of 0.5 pressure scale heights (Fuhrmann et al. 1993) and line blanketing included with Kurucz' ODFs. This is our standard solar model atmosphere although we are aware that it does not reproduce the centre-to-limb variation of the solar continuum intensities very well. The reason for adopting this model as a standard is that we can easily use the model as a differential stellar atmosphere with full physics included. This does not hold for the second type of solar model in Fig. 3, labeled HM74, which is the semi-empirical model of Holweger & Müller (1974). As was shown by Fuhrmann et al. (1993, 1994) scaled semi-empirical temperature stratifications should not be used for stars with parameters very different from the Sun; this is particularly important for metal-poor stars. Experience with the two solar temperature stratifications shows that they are both able to reproduce most of the line flux spectra provided that abundances and damping constants are properly adjusted. The third model in Fig. 3, labeled EAGLNT93, is a theoretical convective equilibrium model from the Uppsala group and their co-workers (Edvardsson et al. 1993). It was calculated with opacity sampling including millions of lines from the compilation of Kurucz (1990). The final model in Fig. 3, labeled MACKKL86, is the empirical model constructed at Harvard by Avrett and his co-workers (Maltby et al. 1986). They include more opacity from the ultraviolet lines that were later published by Kurucz (1990), which results in a less steep photospheric temperature gradient because the increase in the quasi-continuous line-haze opacity shifts the computed heights of formation of the observed ultraviolet continua outwards (Rutten 1988). Model MACKKL obviously differs from the other three models at greater heights by a chromosphere with a steep temperature rise.
In the context of fitting the Mg I lines considered here, we have made test calculations with the four models. Experience with these solar temperature stratifications shows that all of them are able to reproduce most of the line flux spectra provided that abundances and damping constants are properly adjusted. The abundance differences between the LTE and non-LTE calculations have nearly the same value for all four models even though we had to adjust some line parameters (e.g. , ) in order to obtain a better line fit.
3.1. Statistical equilibrium calculations
The statistical equilibrium is calculated using the DETAIL code (Giddings 1981; Butler & Giddings 1985) in a version based on the method of complete linearization as described by Auer & Heasley (1976). The calculation includes all radiative line transitions which are mostly represented by Doppler profiles; 99 lines were linearized. The Mg b lines are treated with full radiative and van der Waals damping. The linearized line transitions were selected from test calculations including different combinations with a preference for the stronger transitions including the levels. Adding more transitions did not change the results. The bound-free transitions of the lowest 22 levels were linearized, too.
3.2. Background opacities
In a star such as the Sun the flux in the ultraviolet spectral region is determined to a large part by opacities due to metal line absorption. We again use the opacity distribution functions of Kurucz (1992) to represent this opacity. In these ODF data single line opacities in small frequency intervals are represented by superlines; consequently, the ODF opacity is not identical to that required at a specific position in frequency space. For the calculation of a model atmosphere this simplification is a sufficient approximation, but for non-LTE line formation the actual radiation field across a line transition or an ionization continuum is important for the determination of the statistical equilibrium of an atom. For bound-free transitions the exact position of the absorbing lines is less important, and the use of the ODFs will be reliable, provided the intervals are small enough in the frequency region near the ionization edge. For a bound-bound transition with its narrow line width it can be important in which part of the broad synthetic ODF line it is formed. We include the additional ODF opacity in the UV for wavelengths between 1300 and 3860 Å to allow for a realistic behaviour of the ionization from the ground state and the first excited level without affecting most of the line transitions. Only a few Mg I lines are found in this region allowing us to omit additional line opacities as most of these transitions are of minor importance for the statistical equilibrium.
© European Southern Observatory (ESO) 1998
Online publication: April 15, 1998