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

## 1. Introduction

The differential equations of stellar structure respond to boundary conditions both at the center and at the surface. In structure calculations the center conditions are easy to treat whereas an accurate solution taking account of the surface boundary conditions is more complicated. Approaching the outer layers of a star the gas is gradually becoming optically thin, and as a result the diffusion approximation of radiative energy transport will no longer be valid. Furthermore, the location of the stellar 'surface' is not as simple as it is for the center. Whereas the theory of stellar atmospheres provides the tools for accurately computing stellar envelopes such calculations are time-consuming. It is therefore customary to use some simplifying approximations.

The method described by Kippenhahn et al.  (1967, 1990) is widely used. The treatment of the atmosphere is separated from that of the interior structure at a certain cut or fit point. Integration of the hydrostatic equation

along the Rosseland optical depth using the well-known Eddington temperature law Eq. (2) and the ideal gas equation results in radius and pressure at the atmospheric fit point (see Sect.2.1). The Henyey algorithm needs two outer boundary conditions which are now available. Generally, the fit point is situated at optical depth , where the diffusion approximation is assumed to be a valid description of radiative energy transfer. But this method has two important disadvantages. First the diffusion approximation is definitely not valid at a fit point at . Second the Eddington temperature law, which is used for the integration of Eq. (1), is also derived using the diffusion approximation. The influence of these approximations on the solution of the stellar differential equations cannot be predicted in a simple way.

Morel et al.  (1994) postulate that only with a reference star model well constrained by observations the influence can be revealed, and only for the Sun one has such accurate values for age, radius, mass, effective temperature and luminosity. In their careful investigation of the Sun on her way from the zero-age main sequence to present age they conclude that the diffusion approximation is not valid outside Rosseland optical depth . They extract several temperature stratification laws from different model atmospheres and restore them in their stellar structure calculations. Finally, they compare the temperature stratifications from Eq. (2), Kurucz' (1992) ODF model atmosphere program ATLAS9, and the empirical Harvard-Smithsonian Reference Atmosphere (HSRA) of Gingerich et al.  (1971). The latter one suffers from the difficulty to establish the temperature distribution empirically at great optical depths. Gingerich et al.  admit that their temperature is possibly 200 K too hot at . The disadvantage of the Eddington stratification is already mentioned above, and therefore only the temperature stratifications of ODF models are sufficiently realistic. However, Morel et al.  did not use one important piece of information in their investigation which seems to be well constrained by observation.

The spectrum of a star is formed in the stellar atmosphere and therefore any model atmosphere connected to the stellar structure model should represent the observed spectrum. From spectroscopic observations of the Sun, Procyon, and the two metal-poor stars HD 140283 and G41-41 Fuhrmann et al.  (1993) demonstrated that a precise determination of a cool star's effective temperature is determined by fitting the observed Balmer lines to the theoretical line profiles. A consistent fit to all Balmer lines requires an adjustment of the temperature stratification in the inner photosphere between and 10, which in turn forces the convective mixing-length parameter to using the convection model of Vitense (1953) and Böhm-Vitense (1958). This has recently been confirmed by Steffen et al.  (1995), who carried out 2D hydrodynamic calculations and van't Veer-Menneret & Mégessier (1996) who used the Kurucz ATLAS9 program and derived a mixing-length parameter of for the Sun and for Procyon in full agreement with Fuhrmann et al.  (1993).

In contrast Morel et al.  worked with ATLAS9 for some temperature stratifications always with a mixing-length parameter which is at variance with the observation of the Balmer lines. Van't Veer-Menneret & Mégessier point at this discrepancy between the low value of compared with a mixing-length parameter of 1.8 derived by Morel et al.  for the Sun as calibration star, who found that the parameter is the optimal choice to connect the atmosphere and the thermodynamic quantities associated with the solar convection zone to a satisfying accuracy. This paper will overcome the dichotomy of two mixing-length parameters, and the conclusion will be that the problems cannot be solved with the convection model of Böhm-Vitense. Replacement of convective energy transfer by the model of Canuto & Mazzitelli (1991, 1992) will remove the outer boundary problem and lead to unified model of cool stars.

A basic description of the programs and the method connecting the model atmospheres to the stellar structure model is given in Sect. 2. Sect. 3 discusses the attempt to produce a consistent convective flux using the convection models of both Böhm-Vitense and Canuto & Mazzitelli. In Sect. 4 the results of the calculations and the conclusions are presented.

© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998