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

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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 ([FORMULA], [FORMULA], [FORMULA]), 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 [FORMULA] (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 [FORMULA] and a helium abundance [FORMULA], 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 [FORMULA] and [FORMULA]. The solar radius is taken as [FORMULA] cm according to Wittmann (1977), and the corresponding effective temperature of the Sun according to (Eq.  3) is [FORMULA] 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 [FORMULA] for modeling the present Sun.

[FIGURE] Fig. 1. The temperature dependence of different opacity tables at density [FORMULA]. 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 [FORMULA] will reproduce the solar radius, whereas for the representation of the Balmer lines a very low mixing-length parameter of [FORMULA] is required according to Fuhrmann et al.  (1993). The stellar evolution calculations with the theory of Böhm-Vitense have been carried out with [FORMULA], [FORMULA] and [FORMULA] (cf.  Eq. (5), Eq. (7)) in contrast to Fuhrmann et al. , who worked with the parameters of Mihalas (1978) [FORMULA], [FORMULA] and [FORMULA]. In principle a variation of y, w and [FORMULA] 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 [FORMULA] in Eq. (7) takes values higher than 10, and in case of [FORMULA] the expression [FORMULA] can be omitted and [FORMULA] 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 [FORMULA], [FORMULA] and [FORMULA] 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 [FORMULA]. Therefore, stellar evolution calculations with parameters that fit the Balmer lines always show too low effective temperatures for the present Sun. Reducing or enhancing [FORMULA] is inversely equivalent to a change of [FORMULA] (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 [FORMULA] in contrast to [FORMULA]. 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] Fig. 2. The temperature stratifications of different convection models. Full line: Canuto & Mazzitelli [FORMULA]. Dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Dot-dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA] (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 [FORMULA] 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 1. Results of the stellar evolution calculations with the convection theory of Canuto & Mazzitelli. [FORMULA] 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 [FORMULA] 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 [FORMULA]. In contrast connecting model atmospheres and recalibrating the stellar structure models to the present solar effective temperature requires a mixing-length parameter of [FORMULA].

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 [FORMULA] for structure and atmosphere is in the following referred to as unified model. The [FORMULA] 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 [FORMULA] 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 [FORMULA], 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] Fig. 3. Blue wing of theoretical [FORMULA] profiles. Full line: Canuto & Mazzitelli [FORMULA]. Dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Dot-dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA] (used by Fuhrmann et al.  1993)
[FIGURE] Fig. 4. Blue wing of theoretical [FORMULA] profiles. Full line: Canuto & Mazzitelli [FORMULA]. Dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Dot-dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA] (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 [FORMULA], [FORMULA], [FORMULA] and [FORMULA] 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] Fig. 5. Evolutionary tracks of stellar models with different convection models. Full line: Canuto & Mazzitelli [FORMULA]. Dot-dashed line: Böhm-Vitense [FORMULA], [FORMULA], [FORMULA] and [FORMULA]
[FIGURE] Fig. 6. Evolutionary tracks of solar models using the convection model of Canuto & Mazzitelli. Full line: Unified model with atmosphere and [FORMULA]. Dot-dashed line: Model with new low-temperature opacities and [FORMULA] 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

  1. The formulation [FORMULA] 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 [FORMULA] and a pressure scale height at the top of convection zone [FORMULA] in the equation [FORMULA] (Wagenhuber, private communications).
  2. 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 [FORMULA]. In contrast to this prediction the stellar models calculated above all show the bottom of the convection zone at [FORMULA] 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 [FORMULA], 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.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998