Astron. Astrophys. 332, 127-134 (1998)
4. Results and discussion
The stellar evolution code used for a reference model treats
the atmospheric boundary condition according to the rules described in
the introduction. Convection is included in terms of the standard
mixing-length theory ( , ,
), low-temperature opacities are taken from the
data of LAOL (see Sect. 2.3), and the metal abundance is set
equal to the meteoritic value (Holweger 1979,
Holweger et al. 1995). With this input data the reference model
is calibrated to fit the observed parameters of the present Sun.
Setting the mixing-length parameter and a
helium abundance , luminosity and effective
temperature of the model fit to the values of the Sun after an
evolution time that corresponds to its present age. Luminosity and age
of the present Sun have been adopted from Bahcall & Pinsonneault
(1995) as and . The solar
radius is taken as cm according to Wittmann
(1977), and the corresponding effective temperature of the Sun
according to (Eq. 3) is K. All
evolutionary tracks are calculated for one solar mass; they start at
the pre-main sequence and end at an age of 6 Gyr.
To evaluate the effects due to changes of the low-temperature
opacities in the atmospheric models (see Sect. 2.3) it is
necessary to investigate at which temperature the opacities of the new
table can be fitted smoothly to OPAL opacities.
As demonstrated in Fig. 1 the resulting temperature is around
10 000 K. This condition has been tested for different densities and
thus a switch to the new opacity table is installed for temperatures
lower than 10 000 K. Note that the new opacity tables have been
calculated with the 1979 ODFs and with the scaled 1992 ODFs.
The comparison between both data sets reveals that at temperatures
lower then 5 000 K only differences of less then 2% must be expected.
The calculation of evolutionary tracks with the new opacity tables
need a mixing-length parameter of for modeling
the present Sun.
![[FIGURE]](img88.gif) |
Fig. 1. The temperature dependence of different opacity tables at density . Full line: OPAL with old low-temperature opacities according to LAOL. Dashed line: New low-temperature opacities based on Kurucz' ODF statistics
|
The connection of atmospheric models to the stellar structure
calculations using the convection theory of Böhm-Vitense while
simultaneously fitting the present Sun and the Balmer lines failed.
With the new low-temperature opacity tables and connected atmospheric
models a mixing-length parameter of will
reproduce the solar radius, whereas for the representation of the
Balmer lines a very low mixing-length parameter of
is required according to Fuhrmann et al.
(1993). The stellar evolution calculations with the theory of
Böhm-Vitense have been carried out with ,
and (cf.
Eq. (5), Eq. (7)) in contrast to Fuhrmann et al. , who
worked with the parameters of Mihalas (1978) ,
and . In principle a
variation of y, w and could lead
to a solution of this problem but the following characteristics of the
Böhm-Vitense convection model are verified by calculations. The
optical thickness in Eq. (7) takes values
higher than 10, and in case of the expression
can be omitted and
becomes a single combined parameter. The enhancement of this combined
parameter can be compensated by reduction of the mixing-length
parameter.
Consequently the mixing-length parameter has to be reduced for
stellar evolution if the calculations are repeated with the parameters
, and
of Fuhrmann et al. . If the boundary
condition of fitting the present Sun is to be conserved, such a
reduction can never compensate a mixing-length as small as
. Therefore, stellar evolution calculations with
parameters that fit the Balmer lines always show too low effective
temperatures for the present Sun. Reducing or enhancing
is inversely equivalent to a change of
(see Eq. (5)); thus it is evident that the
variations of the different parameters nearly cancel, and the
reproduction of the observed Balmer lines and of the present Sun
cannot be obtained simultaneously using the convection model of
Böhm-Vitense.
A view at the temperature structure of the layers where the Balmer
lines are formed (Fig. 2) reveals that the Balmer lines require a
steeper temperature gradient such as is obtained by low mixing-length
parameter of in contrast to
. But through the steeper convective gradient
the whole convection zone is more extended, and therefore the
properties of the present Sun cannot be fitted.
![[FIGURE]](img103.gif) |
Fig. 2.
The temperature stratifications of different convection models. Full line: Canuto & Mazzitelli . Dashed line: Böhm-Vitense , , and . Dot-dashed line: Böhm-Vitense , , and (used by Fuhrmann et al. (1993))
|
In order to escape this dilemma a high degree of overadiabaticity
should only occur at the stellar surface. Such a characteristic is
provided by the convection model of Canuto & Mazzitelli (see
Fig. 8 in Canuto & Mazzitelli 1991). Fig. 2 presents the
resulting temperature stratification between
and 10 for a convection model of Canuto & Mazzitelli that fits the
present Sun and the Balmer lines. Note that for deeper layers
the stratification is close to the one with the high mixing-length
parameter of Böhm-Vitense, which is a result of the calibration
to the present Sun. To distinguish between the mixing-length
parameters of Böhm-Vitense and Canuto & Mazzitelli an index
CM is introduced. Table 1 gives an overview of the models
calculated with the theory of Canuto & Mazzitelli.
![[TABLE]](img105.gif)
Table 1. Results of the stellar evolution calculations with the convection theory of Canuto & Mazzitelli. is the effective temperature found at the age of the present Sun; old and new opacities refer to the use of low-temperature opacities, and in the second column the connection of model atmospheres is indicated
All stellar structure models with convection theory of Canuto &
Mazzitelli have the same hydrogen, helium and metal abundances as the
models with Böhm-Vitense's theory. Calculating stellar
evolutionary tracks with the old low-temperature opacities and the
simplified treatment of the atmosphere as described in Sect. 1
requires a mixing-length parameter of to fit
the present Sun. Using the new low-temperature opacities and the grey
standard outer boundary condition recalibration to the present Sun can
be achieved with . In contrast connecting
model atmospheres and recalibrating the stellar structure
models to the present solar effective temperature requires a
mixing-length parameter of .
As can be seen in Fig. 3, 4 and 5 the last model fits not only
the position of the present Sun in the Hertzsprung-Russell diagram but
also the Balmer lines. This stellar evolution model with connected
atmospheres and a common mixing-length parameter of
for structure and atmosphere is in the
following referred to as unified model. The
profile of the unified model shows no
difference to the profile of Fuhrmann et al. (1993), who used
Böhm-Vitense's model for convection (see Fig. 3). The small
deviation between the profile of the unified
model and Fuhrmann et al. (cf. Fig. 4) corresponds to
an increase of only 10 K in the spectroscopic determination of the
effective temperature, which can be neglected in view of the larger
observational error for the Balmer lines. Both figures clearly show
that the Balmer line profiles produced by the Böhm-Vitense
mixing-length parameter , which is required when
stellar evolution should fit the present Sun, are too narrow compared
with the profiles of Fuhrmann et al. who have fitted their
profiles to spectra of the Sun and several cool stars.
![[FIGURE]](img111.gif) |
Fig. 3. Blue wing of theoretical profiles. Full line: Canuto & Mazzitelli . Dashed line: Böhm-Vitense , , and . Dot-dashed line: Böhm-Vitense , , and (used by Fuhrmann et al. 1993)
|
![[FIGURE]](img113.gif) |
Fig. 4. Blue wing of theoretical profiles. Full line: Canuto & Mazzitelli . Dashed line: Böhm-Vitense , , and . Dot-dashed line: Böhm-Vitense , , and (used by Fuhrmann et al. (1993)).
|
In Fig. 5 the evolutionary tracks of the unified model
compared to the Böhm-Vitense standard model with the parameters
, ,
and are presented. The
region of the early pre-main sequence shows no differences in the
track whereas at the position of the zero-age main sequence the
effective temperature of the model using Canuto & Mazzitelli is
about 20 K cooler. Larger deviations cannot be discovered since both
tracks are calibrated with the position of the present Sun as
indicated by the box. This calibration is also the reason for the good
agreement of the tracks in Fig. 6 where no significant difference
between the evolutionary tracks with and without connected atmospheres
can be revealed.
![[FIGURE]](img116.gif) |
Fig. 5. Evolutionary tracks of stellar models with different convection models. Full line: Canuto & Mazzitelli . Dot-dashed line: Böhm-Vitense , , and
|
![[FIGURE]](img119.gif) |
Fig. 6. Evolutionary tracks of solar models using the convection model of Canuto & Mazzitelli. Full line: Unified model with atmosphere and . Dot-dashed line: Model with new low-temperature opacities and but with old atmospheric treatment
|
To work out only the effect of connected atmospheres that use
Kurucz' opacity distribution functions, the track with the old
atmospheric treatment was recalculated with new low-temperature
opacities based on Kurucz' ODFs. The agreement in Fig. 6 suggests
that calculating stellar evolution with unified models provides no
significant change compared to the case without atmospheres. However,
the tracks have been calibrated with the present Sun, and thus their
mixing-length parameters are different. If the models for the
evolutionary tracks are calculated both with a common mixing-length
parameter one obtains for the present Sun in Table 1 an effective
temperature lower by about 50 K for the model with connected
atmospheres as compared to the model with low-temperature opacities
only.
Evolutionary tracks of different models in the Hertzsprung-Russell
diagram with changes in the input physics will not provide enormous
differences in the neighborhood of the present Sun if they have been
calibrated to fit the position of the Sun. However, differences
between the tracks may become more important in more advanced stages
of stellar evolution.
Before drawing the conclusions from the above investigations there
are two remarks
- The formulation
with z being the
distance from the top of the convection zone introduced by Canuto
& Mazzitelli (1991) was not tested here because in reality Canuto
& Mazzitelli did not use a completely parameter-free theory.
Instead they worked with a mixing-length parameter
and a pressure scale height at the top of
convection zone in the equation
(Wagenhuber, private communications).
- A further observational constraint for solar models are the
oscillation frequencies of the Sun. Christensen-Dalsgaard et al.
(1985) have shown that the observed p-mode frequencies can be inverted
to estimate the solar interior soundspeed. The transition between the
subadiabatic temperature gradient below the convection zone and the
adiabatic gradient in the convection zone can be seen as a distinct
feature in the soundspeed profile. The feature has been used for a
careful determination of the depth of the convection zone in the Sun
(Christensen-Dalsgaard et al. 1991), and the base of the solar
convection zone was found at
. In contrast to
this prediction the stellar models calculated above all show the
bottom of the convection zone at with no
changes between different convection models and the treatment of the
atmosphere. The fundamental test of calculating theoretical
frequencies emerging from the stellar models and comparing these
frequencies with the observed modes was not carried out here. Paternò
et al. (1993) investigated the influence of the different
convection models formulated by Böhm-Vitense and by Canuto &
Mazzitelli on the theoretical oscillation frequencies. Mainly the
frequencies are testing the properties of near-surface convection.
Paterno et al. found a better representation of the observed
frequencies using the model of Canuto & Mazzitelli. Nevertheless
the base of their convection zone was in both formulations at
, which is only marginally higher than in the
models presented here. The discrepancy to the depth of the convection
zone as obtained by inverting the observed p-modes can be removed by
incorporating gravitational element diffusion in the stellar evolution
code (e.g. Bahcall & Pinsonneault (1995), Proffitt (1994)).
The conclusion of the present investigation is that a simultaneous
fit of the observed temperature and luminosity of the present Sun by
stellar evolution according to the solar age and the observed Balmer
lines with corresponding model atmospheres cannot be achieved using
the convection theory of Böhm-Vitense. The replacement of the
simple atmospheric boundary condition in stellar evolution by more
realistic model atmospheres reveals that unified stellar models can be
created which use as a common convection theory the model of Canuto
& Mazzitelli, and which fulfill both observational constraints.
This can be seen as a strong argument in favor of the use of the
Canuto & Mazzitelli convection theory. It will be possible to test
the near surface effects of the unified model by comparison with the
observed solar oscillation frequencies. In order to remove the
discrepancies with the depth of convection zone and for further
improvement it will be most probably necessary to implement
gravitational diffusion in the stellar evolution code. This last
fine tuning by diffusion as much as improving the equation of
state or the opacities will not affect the fundamental advantages that
lie in the description of convection with a unified stellar model that
describes both the properties of the stellar atmosphere and its
interior.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998
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