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

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4. Calibration data for solar-type stars

As for the Sun, we studied the behaviour of [FORMULA] and [FORMULA] over the HRD in the solar neighbourhood. This investigation demanded a large number of simulation runs since we had to cover many objects and had to clarify how densely the HRD should be sampled in order to record the relevant variations of [FORMULA]. The results are now based on 58 simulation runs for solar metallicity and [FORMULA], covering the range [FORMULA] and [FORMULA]. Corresponding results for metallicities down to [FORMULA] will be published in a subsequent paper.

Various aspects determined the region of the HRD which is covered by our models: Towards higher [FORMULA] the convective zones become shallow and hence insignificant for the overall stellar structure. Towards lower [FORMULA] molecular opacities become more and more important; these are not included in our opacities yet. Moreover, convection becomes very efficient, leading to small entropy jumps which cause problems in the numerical modeling. Towards lower [FORMULA] the large entropy jumps cause other numerical problems related to the steep temperature and velocity gradients in the surface cooling layer. And finally, no solar-type stars are located towards higher [FORMULA].

We present our findings in graphical as well as numerical form. Like in the solar case, several RHD simulation runs differing in numerical details were performed for certain atmospheric parameter combinations, thus allowing us to estimate the internal uncertainties associated with our models. In the figures we provide numerical values and show polynomial or exponential least-squares fits to the data points. The fits are constructed such that the residuals between fitted and actual data values show no remaining systematic dependence on the stellar parameters. In Appendix C we give explicit expressions for these fits. In this way our results can be easily utilized in other applications. We made efforts to produce fits with a simple and smooth functional dependence. Nevertheless we remind the reader to the fact that polynomial fits of higher order are not well suited for extrapolation. In other words: the fits quickly loose their meaning outside the region covered by the grid of RHD models and should not be extrapolated too far!

Fig. 3 displays [FORMULA] as a function of [FORMULA] and [FORMULA] in the HRD. In the region where [FORMULA], convection becomes inefficient and the convective zones are superadiabatic throughout. Entropies in that region refer to the value encountered at the base of the convective envelope and are still derived assuming [FORMULA]. Besides [FORMULA] itself, the entropy jumps are of interest since they measure the efficiency of the convective energy transport. The entropy jumps shown in Fig. 4 are the entropy differences between the photospheric entropy minimum on the optical depth scale and [FORMULA] (see Fig. 2).

[FIGURE] Fig. 3. [FORMULA] in units of [FORMULA] from the grid of RHD models for [FORMULA] and solar metallicity. Symbols indicate RHD models. Attached to the symbols the actual [FORMULA] values are given; the contour lines present a fit to them. The fitting function is given in Appendix C.

[FIGURE] Fig. 4. Entropy jumps in units of [FORMULA]. The entropy jump was calculated as difference between [FORMULA] and the entropy at the photospheric entropy minimum. See Fig. 3 for further explanations.

Fig. 4 shows that the RHD model grid covers more than an order of magnitude in the entropy jump, comprising stars with rather efficient envelope convection (with small entropy jumps) in the regions of K-dwarfs and stars with inefficient convection (large entropy jumps) in the region of F-dwarfs. The significant variation of the entropy jumps indicates that the simple dependence of [FORMULA] seen in Fig. 3 is non-trivial. It is not just reflecting the changes of the stellar surface conditions due to changes of opacity and ionization balance but carries information about changes in the efficiency of the convective energy transport.

With the aid of envelope models (as described before) we translated the [FORMULA] values given in Fig. 3 into equivalent mixing-length parameters which are shown in Fig. 5. We find a moderate, nevertheless significant variation of [FORMULA] between about 1.3 for F-dwarfs and 1.75 for K-subgiants. In the close neighborhood of the Sun we find a plateau where [FORMULA] remains almost constant. As discussed previously, the absolute values of [FORMULA] are probably less reliable than its scaling properties as displayed in Fig. 5.

[FIGURE] Fig. 5. [FORMULA] for standard mixing-length theory (Böhm-Vitense 1958) with [FORMULA] ([FORMULA]: mixing-length, [FORMULA]: local pressure scale height). The presentation of the data is analogous to Fig. 3.

The theory of Canuto & Mazzitelli (1991,1992; CMT) describes convection analytically within the picture of a turbulent medium. In view of the significant amount of work that went into the implementation and verification of this prescription by several groups, it appeared worthwhile to compare our results with CMT. Like MLT, CMT contains a free length-scale [FORMULA]. According to Canuto & Mazzitelli [FORMULA] is essentially the distance to the upper (Schwarzschild) boundary of the convective envelope; for purposes of fine tuning they add a small fraction of the pressure scale height at the upper boundary. We follow this recipe here by writing [FORMULA] and translated our [FORMULA] to [FORMULA] values, again with the help of envelope models, but now based on CMT. As shown in Fig. 6, we find [FORMULA] for the Sun. To assess the sensitivity of [FORMULA] to [FORMULA], we note that the overall entropy jump at fixed solar [FORMULA] and [FORMULA] increases by roughly a factor of 2 if one reduces [FORMULA] from 0.4 to zero. As in the MLT case, we find a region of rather constant [FORMULA] in vicinity of the Sun and a significant variation between zero in the region of K-dwarfs and almost 0.6 for F-dwarfs. Interestingly, for the model with the lowest [FORMULA], we find an [FORMULA] slightly smaller than zero. This value was formally derived by extrapolating the [FORMULA]-relation towards negative [FORMULA].

[FIGURE] Fig. 6. [FORMULA] for the Canuto & Mazzitelli convection theory with [FORMULA] (z: distance to the upper (Schwarzschild) boundary of the convective zone, [FORMULA]: pressure scale height at the upper boundary). The [FORMULA]-calibration reproduces the same underlying [FORMULA]-distribution as the [FORMULA]-calibration shown in Fig. 5. The presentation of the data is analogous to Fig. 3.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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