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Astron. Astrophys. 326, 257-262 (1997)

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2. The atmosphere model

We use the Uppsala Model Atmosphere (UMA, Gustafsson et al. 1975, Bell et al. 1976) code, in a version by Vaz & Nordlund (1985). The code is designed for cool stars ([FORMULA]), atmospheres in hydrostatic equilibrium using plane-parallel structure, local thermodynamic equilibrium and modeling convection through mixing length.

The effective temperature [FORMULA], the surface gravity [FORMULA] the stellar mass [FORMULA] the mixing length parameter [FORMULA], and the chemical composition (fixed at solar abundance in this work) define a model. We use stellar models (Claret 1995, X =0.70, Z =0.02) to associate [FORMULA] with M for undistorted stars at the ZAMS. We refer to models with and without line absorption as "non-grey" and "grey", respectively, including in the "grey" models, though, the continuum opacity variation with frequency (as in Nordlund & Vaz 1990). We study grey (i.e. "continuum only") and non-grey (line blanketed or ODF) atmospheres in convective equilibrium. The ODF tables limit the temperature range of this study to those corresponding to (ZAMS) stars with masses ranging from [FORMULA] to [FORMULA].

2.1. The method

Starting from a reference model ([FORMULA] =4.5) we adjust [FORMULA] for models having another [FORMULA] (distorted) until the adiabat at the bottom of the distorted model becomes equal to that of the reference one, i.e. these models must represent the same star by having the same entropy at the bottom. We then examine how the total flux varies with g, determining [FORMULA] by linear regression.

Taking the most distorted components of some high quality EB solutions we find [FORMULA] =0.28 for V Pup (detached, Andersen et al. 1983), 0.22 for LZ Cen (detached, Vaz et al. 1995) and 0.30 for RY Aqr (semi-detached, Helt 1987). Larger values for [FORMULA] will certainly be found amongst the most deformed components of contact EB. In the beginning of this study we used equally spaced steps of 0.1 in [FORMULA] over a large interval of [FORMULA] =1.5 to calculate [FORMULA]. Equation (1) proved to be valid to a very high degree all over the interval, with both the [FORMULA] exponents and the linear regression correlation coefficients practically equal no matter if we used [FORMULA] =1.5 or smaller intervals (e.g. 0.30). However, an uncertainty of [FORMULA] 5 K in the effective temperatures of the models could affect the [FORMULA] values for small [FORMULA] intervals. Then, we decided to use in our study of grey atmospheres (Sect. 2.2) an interval [FORMULA] 5 times the one easily happening for the components of close EBS, in order to minimize the effect of uncertainties in [FORMULA] on [FORMULA] and to have our studies valid even for very distorted stars. In Sect. 2.3 we give [FORMULA] for both the large and the small [FORMULA] intervals, but adopt the more realistic [FORMULA] =0.30.

To have layers deep enough for the adiabatic regime to be fully established, we extended the temperature structure from [FORMULA] =-4.2 to +7.3, going as deep in the atmosphere as the program allowed. In these deep layers [FORMULA] is in practice zero, but there is non-adiabaticity because of mixing. The opacity hence is of negligible importance, unless it becomes completely wrong, what does not happen: the continuum opacity is indeed quite reasonable at the temperatures reached. The equation of state is close to modern formulations, such as the one by Mihalas et al. (1988), and is reasonable as well (it lacks effects like Coulomb screening, but these should come into play only much further down). Other formulations for the equation of state exist now, driven by helioseismology (Guenter et al. 1996, Elliott 1996, Antia 1996), but the level of differences are not significant in the current context. Moreover, these differences are particularly small at the surface, where the traditional (ODF) model atmospheres are actually covering effects not included in these newer formulations. The thermodynamic variables are both internally and physically consistent, as they obey the thermodynamic relations in the code (i.e., they stem from the same thermodynamic potential) and are supported by reasonable values for the continuum opacity and by the use of a realistic equation of state. Besides, these deep models produce the same spectrum as models (same parameters) having [FORMULA] from -4.2 to [FORMULA] 1.0. We are, therefore, still generating physically valid models.

The plane-parallel approximation means that we generate models for small regions over the distorted atmosphere (in terms of the geometrical depth our deep models have less than 0.6% of the star radius). Spherical symmetry models, apparently a better approach, would connect [FORMULA] with M and R, and then have difficulties in generating a distorted star, where [FORMULA] is larger precisely where the curvature radius is larger (but R is smaller). Both approximations, however, should lead to similar results for [FORMULA] with our method, as we compare models with the same approximation (in a way, [FORMULA] is a "differential" effect).

2.2. Convective grey (continuum only) atmospheres

Some models obtained with this method are shown in Fig. 1. The relation between [FORMULA] and [FORMULA] is the same at the bottom of the atmosphere in all the models that must represent different parts of the same distorted star.

We calculated models varying the values of [FORMULA] (= [FORMULA]), which is a measure of the degree of efficiency in the convective energy transport. However, [FORMULA] did not show a strong dependence on [FORMULA], as shown in Table 1 and Fig. 2a. This was also noted by Sarna (1989). It can be seen, also, that the change in [FORMULA] with [FORMULA] is dependent on the model effective temperature. For some models ([FORMULA], [FORMULA]) we calculated [FORMULA] for very small [FORMULA] values (0.1, 0.01). The resulting values show that [FORMULA] does not change significantly from its value for a normal atmosphere ([FORMULA] =1.5).


Table 1. Values of [FORMULA] for convective grey models. Models marked with an * are non-Main Sequence stars, used to investigate the influence of the mass and temperature on [FORMULA]. All models have solar chemical abundance.

[FIGURE] Fig. 1. [FORMULA] for convective grey (thin lines) and non-grey (thick lines) models with [FORMULA].
[FIGURE] Fig. 2. [FORMULA] for convective (a) grey and (b) non-grey models (filled symbols are Lucy calculations for a similar chemical composition). The third order polynomials are shown.

The last lines of Table 1 list tests made with non-main sequence stars (i.e. with a [FORMULA] very different from the one a normal star with the same mass should have on this phase). The cold 1.65 [FORMULA] model with [FORMULA] =4800K does correspond to a real Pre-Main Sequence (PMS) star (the secondary of TY CrA, Casey et al. 1993, 1995, 1997, see Sect. 3.2), while the other tests correspond to some different evolutionary PMS phases of that same star. These non-main sequence models show, however, that [FORMULA] depends mostly on [FORMULA] for normal (i.e. non-illuminated, see Sect. 3.1) atmosphere models. Fig. 2a shows our theoretical values of [FORMULA] vs. [FORMULA] listed in Table 1 and the third order polynomial adjusted with all the [FORMULA] values for grey atmospheres, given by the equation:


with [FORMULA] = [FORMULA] and the values of [FORMULA] given in Table 2.


Table 2. Coefficients for Eq. (2) in convective models.

2.3. Convective non-grey (line blanketed) atmospheres

In Fig. 1 we have models calculated for convective non-grey atmospheres: the non-grey models differ from the grey ones being colder in the upper layers, and slightly hotter in the deep ones. For [FORMULA] [FORMULA] [FORMULA] we were forced to diminish the extension in [FORMULA] in order to be able to calculate for the same range of effective temperatures used for the grey models, due to limitations of the ODF files.

[FORMULA] turned out to be very close to those for grey models, showing that the phenomenon is more related to the constitutive equations than to the ODF table used or to the value of the mixing length parameter, [FORMULA].

The weak dependence of [FORMULA] on [FORMULA] is confirmed for non-grey models (Table 3), although some trends become evident: the maximum of [FORMULA] shifts slightly, but systematically, in [FORMULA] with [FORMULA] (Fig. 2b). However, the influence of [FORMULA] on [FORMULA] (same [FORMULA]) is small if compared with the normal errors involved in the empirical determinations of [FORMULA] (see Sect. 3.1). Then, we used all values of [FORMULA] (calculated with [FORMULA] =0.3) in fitting the third order polynomial to the non-grey [FORMULA] (Table 2, Fig. 2b) which should be the best approximation for real stars.


Table 3. Values of [FORMULA] for convective non-grey models. The values for [FORMULA] =1.5 [FORMULA] were calculated using [FORMULA] =1.5, while all other used [FORMULA] =0.3. In all cases the reference model has [FORMULA] =4.5.

The [FORMULA] values calculated by Lucy (1967) are shown in Fig. 2b, also. Although the procedure used to compute [FORMULA] was the same, our opacity tables and atmosphere model were different from those he used. Even then, our results are in very close agreement: the mean of the 16 values of Table 3 calculated with [FORMULA] =0.3 in the interval 6 000 K [FORMULA] 6 800 K is [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998