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

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Appendix A: mixing-length formulations

Generically, the well-known equations of MLT (cf. Cox & Giuli 1968) for the convective efficiency

[EQUATION]

the convective velocity

[EQUATION]

and the convective flux

[EQUATION]

contain four dimensionless free parameters [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. In the equations, the optical thickness of a convective eddy [FORMULA] and the isobaric expansion coefficient [FORMULA] are defined as

[EQUATION]

Further quantities entering the equations are the mixing-length [FORMULA], specific heat at constant pressure [FORMULA], Stefan-Boltzmann's constant [FORMULA], density [FORMULA], temperature T, actual temperature gradient [FORMULA], temperature gradient of a convective eddy [FORMULA], and adiabatic gradient [FORMULA] ([FORMULA] stands for [FORMULA]). Table A1 provides the f-parameters for various MLT formulations. The values of [FORMULA] given in this paper refer to the MLT formulation by Böhm-Vitense (1958) as specified in the first row of the table.


[TABLE]

Table A1. Constants [FORMULA] for the MLT formulations of various authors. (ML1 and ML2 are commonly used terms in the white dwarf community.)


A fine point in our implementation of MLT concerns the evaluation of the radiative gradient [FORMULA]. We do not use the commonly adopted expression for the radiative gradient in diffusion approximation, but instead derive [FORMULA] by differentiation of the [FORMULA]-relation. As a starting point, consider the [FORMULA]-relation for a grey atmosphere in radiative equilibrium

[EQUATION]

While in the case of a grey atmosphere q is the Hopf-function, we can introduce a modified q such that Eq. A5 represents any prescribed [FORMULA]-relation. A reasonable q should have the property

[EQUATION]

which guarantees that [FORMULA] becomes independent of [FORMULA] and consistent with the diffusion approximation for large optical depth (provided the Rosseland scale is adopted). The advantage of this procedure is that we obtain a smooth and realistic [FORMULA] which is consistent with the specified [FORMULA]-relation irrespective of optical depth. We never have to distinguish between atmosphere and optically thick layers and can use MLT throughout, for the small price of having to compute the [FORMULA]-scale as well.

Appendix B: entropy scale

The entropy is not always available from an EOS, and if so it involves an arbitrary additive constant. In order to relate entropy values given in this paper to more common thermodynamic variables, we provide a small list of quantities as a function of entropy and pressure in Table B1, calculated from the EOS which was used in the RHD simulations. We have concentrated on higher temperatures characteristic of the deeper layers of solar-type stellar envelopes. The data should allow to interrelate values for [FORMULA] to pressures, temperatures, and densities, and provide the entropy zero point of our EOS. For interpolation purposes one might take advantage of the following differential thermodynamic relations

[EQUATION]

and

[EQUATION]

[FORMULA] is related to the thermodynamic derivatives given in Table B1 according to

[EQUATION]


[TABLE]

Table B1. Thermodynamic quantities as a function of entropy and pressure from the RHD EOS for solar metallicity and [FORMULA]. Unless noted otherwise the values are given in cgs-units.


Appendix C: fitting functions

The fits for [FORMULA], entropy jump [FORMULA], [FORMULA], and [FORMULA] shown in Figs. 3, 4, 5, and 6 were computed from expressions (C3), (C4), and (C5) (for [FORMULA] and [FORMULA]) in terms of the auxiliary variables [FORMULA] and [FORMULA] defined below. The coefficients [FORMULA] are listed in Table C1. The fitting functions were generated automatically from a prescribed set of basis functions with the mere intention of providing numerical fits to the RHD data points which then allow the RHD results to be utilized easily in other contexts. Physical considerations did not enter, and we present the formulae in a form more suitable for computing rather than for human interpretation. We warn the reader that the fits quickly loose their meaning outside the region covered by the RHD model grid and should not be extrapolated too far.

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]


[TABLE]

Table C1. Coefficients for fitting functions


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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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