## Appendix: lower boundary conditionAt 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 © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |