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Appendix A: Tables

In Table A.1, we list results for the solar metallicity. The complete table is available at the CDS and also at http://www.stagger-stars.net

Appendix B: Functional fits

Similar to Ludwig et al. (1999), we performed functional fits of the mixing length parameters and the mass mixing length parameter with the T_eff and log g for the different metallicities individually. We transformed the stellar parameters with x = (T_eff − 5777)/1000 and y = log g − 4.44, and fitted the values with a least-squares minimization method for the functional basis (B.1)The resulting coefficients, a_i, are listed in Table B.1.

Appendix C: Addendum on MLT

Appendix C.1: Mixing length formulation

In the framework of MLT, the convective flux is determined by (C.1)with c_P being the heat capacity, Δ the superadiabatic energy excess, and α_MLT the adjustable mixing length parameter, giving the mean free path of convective elements in units of pressure scale height. The convective velocity is determined by (C.2)where H_P is the pressure scale height, δ = −(∂lnρ/∂lnT)_p the thermal expansion coefficient, and ν the energy dissipation by turbulent viscosity. The superadiabatic excess is given by (C.3)and the convective efficiency factor by (C.4)with the optical thickness τ_e, and temperature distribution y of the convective element. The turbulent pressure (C.5)can be included, but a depth-independent turbulent velocity, v_turb is assumed, which is the common approach for atmospheric modeling. The resulting photospheric temperature stratifications are very similar to the MARCS (Gustafsson et al. 2008) and ATLAS models (Kurucz 1979; Castelli & Kurucz 2004). In Paper I, we showed that below the surface, where convective energy transport starts to dominate, the 1D models are systematically cooler than the ⟨3D⟩ stratifications because of the fixed α_MLT with 1.5, in particular for hotter T_eff.

Appendix C.2: Influence of additional MLT parameters

Fig. C.1 Entropy and superadiabatic gradient vs. depth (left and right panel, respectively) illustrating the influence of the additional MLT parameters ν, y, and β (top, middle, and bottom panel, respectively), the latter with the depth-independent vturb = 1 km s-1. The mixing length is kept fixed at αMLT = 1.5. We also included the standard values of β = 0, ν = 8 and y = 0.076 (dashed lines). Shown is the case for solar parameters.

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In the formulation of Henyey et al. (1965) of MLT, there are at least three additional free parameters apart from α_MLT, which usually are not mentioned explicitly, but are compensated for by the value of α_MLT. These are the scaling factor of the turbulent pressure, β, the energy dissipation by turbulent viscosity, ν, and the temperature distribution of a convective element, y. The default values are usually β = 1/2, ν = 8 and y = 3/4π² = 0.076 (see Gustafsson et al. 2008). In many cases, the turbulent pressure is neglected (β = 0). In the notation of Ludwig et al. (1999), these parameters would yield f₁ = ν^-1 and f₄ = y^-1, f₂ = 1/2 and f₃ = 8 /y.

The turbulent pressure indirectly influences the T-stratification, gradients, and hydrostatic equilibrium by reducing the gas pressure. The parameter ν enters the convective velocity inverse proportionally, v_MLT ∝ ν^-1 (see Eq. (C.2)), and since , an increase in ν would have the same effect as a reduction in α_MLT, i.e. ν ∝ s_bot. On the other

hand, y enters in the (nonlinear) convective efficiency factor, Γ, for the superadiabatic excess (see Eq. (C.4)), and therefore y is correlated with α_MLT in a more complex way.

Considering a variation of the three additional parameters in the computation of the solar 1D model, we notice that the adiabatic entropy value of the deep convection zone is altered significantly (see Fig. C.1). Furthermore, the two parameters ν and y also change the entropy jump and the superadiabatic temperature gradient, ∇_sad, and in particular, its maximum of ∇_sad. The effect of the variation of y on the entropy stratification is similar to that by α_MLT (see Fig. 1). However, the entropy of the deep convection zone exhibits a more nonlinear dependence with the y parameter. The increasing turbulent pressure with higher β changes the stratification only slightly, but shifts the location of the maximum of ∇_sad to the deeper interior. Towards the optical surface the influence of the MLT parameters decreases, as expected because of decreasing convective flux. A fine-tuning of β, ν and y is only useful when these parameters introduce an independent influence on the mixing length, since otherwise its effects can be summarized in α_MLT alone.

© ESO, 2014