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Astron. Astrophys. 363, 1055-1064 (2000)

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Appendix: lower boundary condition

At the lower boundary [FORMULA] we assume, that the radiation field is thermalized at any considered frequency, [FORMULA]. Radiation field is isotropic there, and consequently the second moment equals:

[EQUATION]

with the Eddington factors [FORMULA] at any frequency [FORMULA]. Asterisk attached to the Planck function indicates, that its value strictly fulfils the condition of radiative equilibrium.

Differentiating Eq. A.1 one can involve the temperature gradient at [FORMULA]

[EQUATION]

since [FORMULA], the ratio [FORMULA], and the dimensionless absorption [FORMULA].

At the lower boundary we assume also that the diffusion approximation is valid. In this case the bolometric flux is determined by the well known relation

[EQUATION]

and is proportional to the gradient of temperature. Bolometric flux [FORMULA], where [FORMULA] (cgs units).

Extracting the gradient [FORMULA] from the above equation, and substituting to Eq. A.2, we get the lower boundary condition in the form

[EQUATION]

cf. Eq. (7-31) in Mihalas (1978).

Taylor expansion of unknown final values to the first order:

[EQUATION]

Therefore at a fixed running frequency [FORMULA] we obtain, after discretization

[EQUATION]

Constraint of radiative equilibrium requires that (cf. Eq. 12)

[EQUATION]

which yields temperature corrections

[EQUATION]

with [FORMULA] denoting the mean intensity of the external illumination.

Let us define auxiliary variables

[EQUATION]

[EQUATION]

then

[EQUATION]

Neglecting [FORMULA] terms,

[EQUATION]

and, finally

[EQUATION]

The above lower boundary condition can be applied also for the computation of nonilluminated model atmosphere, when we simply set [FORMULA] for all [FORMULA]. In case of standard model atmosphere with coherent Thomson scattering, both functions [FORMULA] and [FORMULA], and therefore L and [FORMULA], are also set to zero.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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