Issue 
A&A
Volume 573, January 2015



Article Number  A89  
Number of page(s)  17  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201423760  
Published online  23 December 2014 
Online material
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.staggerstars.net
Stellar parameters: effective temperature, T_{eff}, and surface gravity, log g (Cols. 1 and 2 in [K] and [dex]).
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 leastsquares 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 depthindependent 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 depthindependent v_{turb} = 1 km s^{1}. The mixing length is kept fixed at α_{MLT} = 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 = 3/4π^{2} = 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 f_{1} = ν^{1} and f_{4} = y^{1}, f_{2} = 1/2 and f_{3} = 8 /y.
The turbulent pressure indirectly influences the Tstratification, 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 finetuning 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
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