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Astron. Astrophys. 326, 257-262 (1997)
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 ( ), atmospheres
in hydrostatic equilibrium using plane-parallel structure, local
thermodynamic equilibrium and modeling convection through mixing
length.
The effective temperature , the surface
gravity the stellar mass the mixing length parameter
, 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
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
to .
2.1. The method
Starting from a reference model ( =4.5) we
adjust for models having another
(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 by linear regression.
Taking the most distorted components of some high quality EB
solutions we find =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
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 over a
large interval of =1.5 to calculate
. Equation (1) proved to be valid to a very high
degree all over the interval, with both the
exponents and the linear regression correlation coefficients
practically equal no matter if we used =1.5 or
smaller intervals (e.g. 0.30). However, an uncertainty of
5 K in the effective temperatures of the models
could affect the values for small
intervals. Then, we decided to use in our study
of grey atmospheres (Sect. 2.2) an interval 5
times the one easily happening for the components of close EBS, in
order to minimize the effect of uncertainties in
on and to have our studies valid even for very
distorted stars. In Sect. 2.3 we give for both
the large and the small intervals, but adopt
the more realistic =0.30.
To have layers deep enough for the adiabatic regime to be fully
established, we extended the temperature structure from
=-4.2 to +7.3, going as deep in the atmosphere
as the program allowed. In these deep layers 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 from -4.2 to
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 with M and R, and
then have difficulties in generating a distorted star, where
is larger precisely where the curvature radius
is larger (but R is smaller). Both approximations, however,
should lead to similar results for with our
method, as we compare models with the same approximation (in a way,
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
and 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
(= ), which is a measure of the degree of
efficiency in the convective energy transport. However,
did not show a strong dependence on
, as shown in Table 1 and Fig. 2a. This was also noted by Sarna (1989). It can
be seen, also, that the change in with
is dependent on the model effective
temperature. For some models ( ,
) we calculated for very
small values (0.1, 0.01). The resulting values
show that does not change significantly from its
value for a normal atmosphere ( =1.5).
![[TABLE]](img52.gif)
Table 1. Values of for convective grey models. Models marked with an * are non-Main Sequence stars, used to investigate the influence of the mass and temperature on . All models have solar chemical abundance.
![[FIGURE]](img55.gif) |
Fig. 1. for convective grey (thin lines) and non-grey (thick lines) models with .
|
![[FIGURE]](img64.gif) |
Fig. 2. 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 very different from the one a
normal star with the same mass should have on this phase). The cold
1.65 model with =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
depends mostly on for
normal (i.e. non-illuminated, see Sect. 3.1) atmosphere models. Fig. 2a shows our theoretical values of
vs. listed in Table 1 and
the third order polynomial adjusted with all the
values for grey atmospheres, given by the equation:
![[EQUATION]](img58.gif)
with = and the values
of given in Table 2.
![[TABLE]](img62.gif)
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
we were forced to diminish the extension in
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.
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, .
The weak dependence of on
is confirmed for non-grey models (Table 3),
although some trends become evident: the maximum of
shifts slightly, but systematically, in
with (Fig. 2b). However, the influence of
on (same
) is small if compared with the normal errors
involved in the empirical determinations of (see
Sect. 3.1). Then, we used all values of
(calculated with =0.3) in fitting the third
order polynomial to the non-grey (Table 2, Fig. 2b) which should be the best approximation for
real stars.
![[TABLE]](img69.gif)
Table 3. Values of for convective non-grey models. The values for =1.5 were calculated using =1.5, while all other used =0.3. In all cases the reference model has =4.5.
The values calculated by Lucy (1967) are
shown in Fig. 2b, also. Although the procedure
used to compute 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 =0.3 in the interval
6 000 K 6 800 K is .
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998
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