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3. Relative line photometry

  The above described method for calculation of line-intensity variations yields the possibility of finding differential magnitude changes between the components and also to determine the ratio of component continua in the case that the intensity variations are caused by some overall darkening of a component e.g. by an eclipse. Let in the "normal'' state of a binary the intensities tex2html_wrap_inline1241 of components continua be normalized
The intensities tex2html_wrap_inline1243 of lines of the components found by solution of Eq. (6 (click here)) are expressed in ratio to this common continuum. If in another exposure the spectrum of component "1'' is tex2html_wrap_inline1245 increased (or decreased if z<1, see Fig. 1 (click here)), then the decomposed line intensities of both components referred to the instantaneous common continuum will be changed by factors tex2html_wrap_inline1249 to values
Specially, in our example shown in Fig. 1 (click here), both components have central line intensities 50% of their individual continua. In the maximum (left panel) the primary is twice as bright as the secondary, hence the lines of primary and secondary have depths of tex2html_wrap_inline1251 and tex2html_wrap_inline1253 of the total observed continuum, respectively. Let in the primary eclipse (right panel) the intensity of the primary be decreased to tex2html_wrap_inline1255 -times its normal value. The absolute depth of the line of primary is then 4-times lower while that of the secondary is unchanged. However, in ratio to the total continuum, which is now one half of its maximum value, the depth of primary line is only twice lower, while the intensity of the secondary line is twice higher.

Solving Eqs. (9 (click here)) and (10 (click here)), the factor z can be simply expressed as
and the ratio of continua intensities
Obviously, if z<1, then tex2html_wrap_inline1261 and tex2html_wrap_inline1263. This behavior can help to distinguish the variations caused by the "geometrical'' reasons (or their equivalent) from intrinsic variations of line intensities of a component or from the observational noise.

In the usual case of k exposures, the factors tex2html_wrap_inline1267 of the darkenings of the component "1'' can be calculated independently for each exposure from tex2html_wrap_inline1269 according to Eq. (11 (click here)). The ratio of continua intensities can be then obtained by least square fit of Eqs. (9 (click here)) and (10 (click here)) e.g. on a logarithmic (i.e. magnitude) scale, i.e. by solving the condition
for the variation tex2html_wrap_inline1271, i.e. simply by numerical minimization of function tex2html_wrap_inline1273. The differences of magnitudes of both components at the time of chosen exposure can be then expressed by
If the secondary component is supposed to be constant during the primary eclipse, then the magnitude of the whole system is given by
Vice versa, the total magnitude of the system in the course of secondary eclipse can be estimated by

The applicability of the method of relative line photometry in the above given approximation is certainly limited by the implicitly included assumptions that (i) the change of intensity is the same for the line and the continuum of the eclipsed component, (ii) the intensities of both component are constant outside their eclipses and (iii) that the shape of line profile is constant. The first assumption can be violated even in the case of a pure eclipse due to the limb darkening, which is different in the continuum and in the line. This problem will be studied in the next section. The second assumption is crucial to have well defined the "normal'' state (in which tex2html_wrap_inline1275 are directly observed), so that the line-intensity of the non-eclipsed component can be used as a "comparison''. This assumption can be violated e.g. for ellipsoidal variables. Obviously, a more sophisticated procedure (analogous to the light-curves solution) of fitting the observed values of tex2html_wrap_inline1277 (related to a reasonably chosen "reference'' state of tex2html_wrap_inline1279) by a theoretical model can yield an additional information in such a case. Finally, the third assumption can be violated e.g. by Rossiter effect when different equatorial parts of a rotating star are eclipsed. This assumption is crucial for the disentangling according to Eq. (1 (click here)). However, as far as the rotational broadening (even for a partially eclipsed component) is also given by a convolution in x, it is, in principle, possible to generalize the method of the disentangling and line photometry even to this case. On the other hand, such effects can be negligible in many cases not only for a pure eclipse, but in some approximation also for elliptic, reflecting or pulsating components of binaries.

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