Let us now investigate the influence of limb darkening on the
line photometry. In the simple Milne-Eddington approximation of
a plane-parallel atmosphere the observed specific intensity for
direction cosine is given by
The source-function can be taken constant within the line-profile and approximately linear in a mean (e.g. Rosseland) optical depth ,
If the opacity both in the continuum () and in the line (, where is normalized line profile) can be taken approximately proportional to the mean opacity () through the optical depths where the line is formed, then
The limb-darkening is thus generally steeper in continuum than in a line. During an eclipse, the ratio z of visible and total (i.e. visible plus eclipsed) light is different at different (i.e. ). Consequently, the contribution of the eclipsed component to the total specific intensity is
are moments of , which are algebraically independent. Unlike the case of a homogeneous stellar disk investigated in the previous section, where both the continuum and line intensities were proportional to a single factor z, the contribution of the eclipsed component to the total intensity in continuum (i.e. at )
is now affected by both and , while its contribution to the line intensity
by only. The generalization of Eqs. (9 (click here)), (10 (click here)) for observed changes of line intensities thus reads
Figure 2: Preliminary results of relative line photometry of V436 Per. Magnitude differences between the primary and secondary component (calculated according to Eq. (14) for H line () and He I 6678 (+)) are plotted in dependence on the orbital phase (from periastron). The secondary minimum occurs close to phase 0.42 and the primary minimum at phase 0.98
The eclipse of an edge of the stellar disk, where , can have negligible effect on and the line intensity, but observable effect on and the continuum intensity. This can lead to enhancement of relative line intensity even of the eclipsed component at this stage of eclipse. This effect is also marginally observable in the example of V436 Per shown in Fig. 2 (click here) (see Harmanec et al. 1997 for details on this system). Moreover, there seems to be a systematically higher effect in the stronger H line than in the He I line. The reason for this may lie beyond the Milne-Eddington approximation and the assumption (18 (click here)).
Unlike the case simplified in the previous section by the assumption of homogeneous stellar disk, all free parameters (, for each exposure and , ) can not be directly calculated now. Instead of this, radii and orbital parameters which should match the geometry of eclipse to the orbital phase are to be fitted to the observed data together with appropriate models of stellar atmospheres. In such an application the relative line photometry can yield an additional test on model atmospheres.
The author is indebted to D. Holmgren, J. Kubát and R.E. Wilson for useful discussions and comments. The suggestions of an anonymous referee are also highly appreciated. This study was supported by the grant 205/96/0162 of the Grant Agency of the Czech Republic, project K1-003-601 and grant 303401 of the Grant Agency of Academy of Sciences.