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Astron. Astrophys. 346, 111-124 (1999)

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3. The solar benchmark

The natural benchmark for the scenario described above is of course the Sun. Not only are its global stellar parameters known with exceptional accuracy, but helioseismology has provided us with detailed information about the structure of the solar interior. Recent astrophysical convection theories were developed in the context of stellar evolution and tests of these theories were devised within that framework, in particular by studying their effects on the evolution of the Sun (Lydon et al. 1992, 1993a; Canuto & Mazzitelli 1991, 1992; Canuto et al. 1996). In our opinion, a more direct way for a validation of such theories is to look at the present solar structure without taking recourse to evolutionary calculations. This avoids the possibility that evolutionary changes of the solar structure interfere with model properties that are used for the assessment of the accuracy of the convection theory under consideration.

In the following we shall compare our RHD model predictions for [FORMULA] with related helioseismic measurements. On an absolute scale any deviation from the actual physical situation, e.g. in composition, shows up as mismatch between observations and theoretical predictions without being necessarily related to flaws in the convection model. In our solar models we paid particular attention to the role of the helium content Y and details of the radiative transfer .

Table 1 summarizes the findings for the Sun. For interpreting the different entropies given in the table we note here that the entropy jump over the superadiabtic layers in the Sun amounts to [FORMULA]. The first block of data (4 entries) in Table 1 refers to RHD models. For solar effective temperature and gravitational acceleration we calculated RHD models with [FORMULA] in addition to models with our standard [FORMULA]. We considered RHD models where the radiative transfer was treated in grey approximation ([FORMULA]), and where its frequency dependence was approximately taken into account using 5 representative frequency bands ([FORMULA], for details about the radiative transfer scheme see Ludwig et al. 1994). The entropies [FORMULA] of the adiabatic part of the convective envelopes are based on the assumption [FORMULA]. The entropies derived from the RHD simulations were converted into an effective mixing-length parameter [FORMULA] as outlined above.


Table 1. Results from hydrodynamical (RHD) models and helioseismology for different helium abundances. Y: helium abundance, [FORMULA]: entropy in the adiabatically stratified layers in units of [FORMULA], [FORMULA]: number of frequency bands for modeling the radiative transfer, [FORMULA]: mixing-length parameter. The data for [FORMULA] are obtained by linear interpolation between the data for [FORMULA] and 0.28.

There is a noticeable dependence of [FORMULA] on the treatment of the radiative transfer and the helium content. The different treatment of the radiative transfer leads to a different atmospheric temperature structure, resulting in a change of the absolute entropy level in the deeper atmosphere. There is some further influence on the entropy jump itself since the radiative energy exchange in the surface layers is slightly modified. But despite significant differences in [FORMULA] for the two RHD models with [FORMULA], both models give the same [FORMULA] values within the uncertainty of [FORMULA] of [FORMULA]. This is achieved by comparing RHD models and envelope models on a strictly differential basis. The [FORMULA]-relation in the envelope models was selected to closely follow the relation found in the RHD models: RHD models basing on grey radiative transfer were compared to envelope models calculated with a grey [FORMULA]-relation in their atmospheric layers. For RHD models basing on frequency-dependent radiative transfer, a [FORMULA]-relation was derived by horizontally and temporally averaging their temperature structure on surfaces of constant optical depth. This [FORMULA]-relation was subsequently used in the calculation of the related envelope models. When mixing solar RHD and envelope models with incompatible [FORMULA]-relations, [FORMULA] changes by up to 0.1. We have performed the same tests for a hotter RHD ([FORMULA]) model with similar findings. The insensitivity of [FORMULA] with respect to the treatment of the radiative transfer allowed us to largely rely on computationally less demanding RHD models employing grey radiative transfer for studying the scaling of [FORMULA] across the HRD.

The entropy difference due to a change in the helium content is primarily related to the associated change of the mean molecular weight. To first order, the RHD models react to changes in Y by keeping the temperature-pressure structure invariant. The density scales in proportion to the change in molecular weight (here: [FORMULA]); entropy and internal energy per unit volume remain the same, allowing the model to transport the same energy flux without a change of the flow velocity. The simple scaling behaviour of the RHD models with Y and the invariance of [FORMULA] found in the case [FORMULA] lead us to derive the values for [FORMULA] from an envelope model assuming the same [FORMULA] as in the case [FORMULA]. The entry for this model in the table is flagged as "inferred".

The second block of data (2 entries) in Table 1 shows results derived from helioseismic considerations. Helioseismology provides the sound speed-density profile in the Sun with high accuracy. In the deeper, adiabatically stratified layers of the envelope this profile defines an adiabat in the sound speed-density plane. If chemical composition and the EOS are known one can label this adiabat with its actual entropy value. Note that only thermodynamic properties enter into this procedure; it is independent of opacity or results from stellar structure models as long as one regards the helioseismic measurements themselves as independent. The helioseismic entropy values of the second block of data in Table 1 are derived in this way by assuming a certain Y and adopting the OPAL EOS (Rogers et al. 1996) to establish the relation between sound speed, density, and entropy. The [FORMULA] values are again derived from envelope models that match [FORMULA]. In contrast to [FORMULA], the determination of [FORMULA] depends on low-temperature opacities and stellar structure considerations. The helioseismic data are due to Antia (1996) and were kindly made available to us by the author.

The third block of data (2 entries) finally summarizes our best estimates for [FORMULA] from RHD models and helioseismology. They were obtained by linear interpolation between the data for [FORMULA] and [FORMULA] to Antia's (1996) best helioseismic estimate for the helium content [FORMULA] based on the OPAL EOS. The uncertainties given in the helioseismic case are dominated by the uncertainty of the helium content. We have adopted an uncertainty of [FORMULA], roughly comprising the range of Y discussed in the literature. In the RHD case the uncertainties reflect the variations which we find between simulation runs for the same physical parameters that differ in numerical details (grid resolution, size of the computational box, explicit or implicit treatment of the energy equation, etc.).

Of course, all our error estimates do not include possible systematic effects. The helioseismic Y can be affected by systematic errors in the EOS. The RHD results are certainly affected by the two-dimensionality of the models. Indeed, we interpret the discrepancy between RHD and helioseismic results as systematic effect: our 2D RHD models overestimate [FORMULA] by [FORMULA] (10% of the overall entropy jump between the surface and the adiabatic layers) and correspondingly underestimate the solar [FORMULA] by 0.1. This conclusion is supported by first results from a differential comparison of 2D and 3D RHD simulations for the Sun. The 3D run predicts a smaller [FORMULA] by [FORMULA], corresponding to a larger [FORMULA] by 0.07. Taking this fact into consideration, the theoretical and observational determination of [FORMULA] become fully consistent.

We expect another systematic effect to influence the absolute value of [FORMULA] affecting both the observationally and the theoretically derived values in the same manner. There exist indications that our low-temperature opacities underestimate the real stellar opacity in the deeper, subphotospheric layers. Our differential approach for determining [FORMULA] cannot fully compensate for this deficiency since the opacities affect RHD and MLT models differently. In RHD models the opacity effects are confined to the very thin cooling layer on top of upflowing regions, while in MLT models the opacities are important all over the zone of high superadiabaticity. This biases our [FORMULA] values again towards lower values. In the region of high superadiabaticity our opacities are presumably too small by 10-20%, causing [FORMULA] to be underestimated by [FORMULA].

However, in view of the fact that the RHD models are essentially parameter-free we consider the match in 2D as satisfactory. Besides the zero point, the functional dependence of [FORMULA] and [FORMULA] on the fundamental stellar parameters is of major interest. We believe that for these scaling relations our error estimates based on purely intrinsic model uncertainties are well justified.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999