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4. Influence of limb darkening

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 tex2html_wrap_inline1283 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 tex2html_wrap_inline1285,
If the opacity both in the continuum (tex2html_wrap_inline1287) and in the line (tex2html_wrap_inline1289, where tex2html_wrap_inline1291 is normalized line profile) can be taken approximately proportional to the mean opacity (tex2html_wrap_inline1293) 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 tex2html_wrap_inline1297 (i.e. tex2html_wrap_inline1299). Consequently, the contribution of the eclipsed component to the total specific intensity is
are moments of tex2html_wrap_inline1301, 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 tex2html_wrap_inline1305)
is now affected by both tex2html_wrap_inline1307 and tex2html_wrap_inline1309, while its contribution to the line intensity
by tex2html_wrap_inline1311 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 Htex2html_wrap_inline1313 line (tex2html_wrap_inline1315) 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 tex2html_wrap_inline1317, can have negligible effect on tex2html_wrap_inline1319 and the line intensity, but observable effect on tex2html_wrap_inline1321 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 Htex2html_wrap_inline1323 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 (tex2html_wrap_inline1325, tex2html_wrap_inline1327 for each exposure and tex2html_wrap_inline1329, tex2html_wrap_inline1331) 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.

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