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

Appendix: lower boundary condition

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

with the Eddington factors at any frequency . 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

since , the ratio , and the dimensionless absorption .

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

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

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

cf. Eq. (7-31) in Mihalas (1978).

Taylor expansion of unknown final values to the first order:

Therefore at a fixed running frequency we obtain, after discretization

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

which yields temperature corrections

with denoting the mean intensity of the external illumination.

Let us define auxiliary variables

then

Neglecting terms,

and, finally

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

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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