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Astron. Astrophys. 332, 127-134 (1998)

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2. Model interface between stellar interior and atmosphere

The calculations described in this paper are based on the Henyey algorithm as adapted in the Kippenhahn et al.  (1967) code. The program has recently been modified and described by Wagenhuber & Weiss (1994), who used the traditional way of connecting atmospheres to stellar structure models at [FORMULA]. In this former version the integration of Eq. (1) starts at [FORMULA] where the total gas pressure p equals the radiation pressure. The density [FORMULA] and the opacity [FORMULA] are evaluated with the approximation of the well-known Eddington temperature law,

[EQUATION]

and the ideal gas equation as equation of state. The integration is followed out to a fit point at optical depth [FORMULA]. The local kinetic temperature at that depth is equal to the effective temperature [FORMULA]  . The radius [FORMULA] of the star is defined with the Stefan-Boltzmann law and the total luminosity of the star [FORMULA].

[EQUATION]

Now radius and pressure at the atmospheric fit point are available as functions of luminosity and effective temperature, [FORMULA], and [FORMULA]. The interior part of the stellar structure can be calculated by the well-known Henyey scheme, and for a consistent connection to the atmosphere the outermost grid point in that scheme must be placed at [FORMULA]. Furthermore the physical parameters of the atmosphere and of the inner solution must 'fit' at this point and the total mass of the stellar model should be the sum of the atmospheric and the interior mass. In principle the conservation of total mass can be achieved by integrating the atmospheric mass and varying the inner mass by inserting or deleting the outermost gridpoints in the Henyey scheme. But often the atmospheric mass can be ignored with respect to the total mass. The physical parameters then can be fitted as follows. Luminosity and temperature from the last Henyey iteration are taken at the outermost gridpoint, and thus the temperature is equal to the effective temperature. From atmospheric integration [FORMULA] and [FORMULA] are now calculated, and they must fit the radius [FORMULA] and the pressure [FORMULA] at the outermost gridpoint of the Henyey scheme. The Henyey algorithm needs two outer boundary conditions which are now available ([FORMULA]) and the fit will be achieved with convergence of the Henyey iteration scheme.

Because of the arguments stated in the introduction it is recommended to replace this boundary condition by a more appropriate one.

2.1. New outer boundary conditions of the Henyey scheme

The principal difference between the boundary condition as described in the part before and those reported here is that in the latter the fit point is moved to [FORMULA] which, according the examination of Morel et al.  , is deep enough to establish the validity of the diffusion approximation.

An optical depth of [FORMULA] relates to an atmospheric depth of only 100 - 200 km in the Sun. Therefore in quiet evolution of cool unevolved stars (i.e. neglecting pulsations) the atmosphere is neither a source nor a sink for the energy, and it follows that the luminosity at the fit point is equal to the luminosity of the atmosphere, [FORMULA]. The same argument of a negligible extension of the atmosphere is used to evaluate the radius of the star at the fit point such that [FORMULA]. The error of this approximation can be estimated with help of Eq. (3). Varying the radius of the present Sun by 300 km leads to a temperature change of less then 2 K.

The modified outer boundary conditions thus require the interesting physical variables of a grid of stellar atmosphere models to be available (see Sect.2.2). These variables are the atmospheric pressure [FORMULA] and temperature [FORMULA] at the fit point, the effective temperature [FORMULA]   , radius [FORMULA], luminosity [FORMULA] and the mass [FORMULA] of the stellar atmosphere. The Henyey algorithm provides its own four physical variables at the fit point: [FORMULA], [FORMULA], [FORMULA], [FORMULA]. Luminosity and temperature are chosen to be independent, which means that the atmospheric table entry must fit both variables, [FORMULA], and [FORMULA]. If that operation is successful one obtains from the tables the effective temperature of the star and the two atmospheric fit point variables [FORMULA] and [FORMULA]. Finally, the two outer boundary conditions for the Henyey scheme are available ([FORMULA], [FORMULA]), and the Henyey iterations will converge to these values. A two-dimensional Taylor expansion is used to interpolate the atmospheric tables data with high accuracy.

2.2. Atmospheric models

The models of the stellar atmospheres have been calculated using a plane-parallel stratification with energy transported by radiation and convection and conserved through the atmosphere (Gehren 1977). In addition the program models line-blanketing with opacity distribution functions (ODF) provided by Kurucz (1979, 1995, and references therein). The basic input parameters are the effective temperature [FORMULA]   , the gravitational surface acceleration [FORMULA]   , and a set of abundance ratios by number, X/H, where X refers to all elements except hydrogen. [FORMULA]   relates a given stellar mass to its radius, and with [FORMULA]   and Eq. (3) one obtains the luminosity. The atmospheric mass results from integration of the equation

[EQUATION]

where m is the column mass in [g cm-2 ]. The program and the synthesis of the Balmer lines has been described in detail by Fuhrmann et al.  (1993, 1996).

2.3. The opacity problem

Currently two sets of ODFs are available, an older set (Kurucz (1979) and a more recent one (Kurucz 1992). Compared with the 1979 data the new ODFs differ not only in their substantially extended line list but also in their abundance pattern. In particular the iron abundance is the subject of discussions. Whereas the 1979 data was computed with an iron abundance of [FORMULA], with hydrogen number densities normalized to [FORMULA], the new data uses [FORMULA] with reference to results published by Blackwell et al.  (1984). Holweger et al.  (1990) instead derived [FORMULA], while the meteoritic value is [FORMULA].

There is an extended list of publications that deal with the question which of these results should be accepted as the solar iron abundance (see Holweger et al.  1991; Grevesse 1991; Hannaford et al.  1992; Grevesse & Noels 1993; Milford et al.  1994; Anstee & O'Mara 1995; Blackwell et al.  1995 and references therein; Biémont et al.  1991; Kostik et al.  1996). Currently the meteoritic iron abundance seems to be preferred. One way to deal with the new ODFs of Kurucz (1992) is to interpolate the opacity tables using a 'metal' abundance about 0.16 dex smaller than that entering his standard solar abundance mixture. This is only an approximation because iron contributes most but not all of the atmospheric line blanketing. Thus for comparison the calculations have also been repeated using the old ODFs.

The opacities at low temperatures in the evolution code provide yet another problem: the OPAL opacities (Rogers & Iglesias 1992) do not include entries below 6000 K. Thus Wagenhuber & Weiss (1994) implemented LAOL opacities (Weiss et al.  1990) for low temperatures. Consequently, differences of results obtained with either a standard treatment of the outer boundary condition or models with stellar atmospheres tied to the Henyey solution at [FORMULA] may be smeared out due to the different opacities in the model atmosphere program and in the evolution code. To distinguish these effects a table of Rosseland opacities for low temperatures has been generated with the model atmosphere program. This new opacity table was used for temperatures below 10 000 K in both types of boundary conditions, with and without model atmospheres. In the final section it will be shown that at 10 000 K the OPAL opacities and the new opacity table fit smoothly.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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