2. The spot model and the inverse problem method

For the purpose of analysing the asymmetric light curves deformed by the presence of spotted areas on the components a Roche model based on the principles originated in the paper by Wilson & Devinney (1971) is developed. Here only some basic elements of the model are indicated, whereas the details can be found in Djurasevic (1992a).

The stellar size in the model is described by the filling coefficients for the critical Roche lobes F_1,2 of the primary and secondary respectively. They indicate to what degree the stars of the system fill the corresponding critical lobes (see Fig. 1 (click here)). In the case of synchronous rotation of the components these coefficients are expressed through the ratio of the stellar polar radii, R_1,2, and the corresponding polar radii of the critical Roche lobes, i.e. . The stellar nonsynchronous rotation is described by coefficients , where is the angular rotation rate of the components and is the Keplerian orbital revolution rate. In this case, the critical Roche lobes belong to the critical nonsynchronous lobes, and the filling coefficients F_1,2 are defined with respect to their polar radii. For a given mass ratio of the components q=m₂/m₁, and the nonsynchronous rotation coefficients f_1,2, the stellar shape and size in a CB Roche model are unequivocally determined by the filling coefficients F_1,2 of the critical lobes (Djurasevic 1992a).

  
Figure 1: The Roche model of an active CB with spotted areas on the components

The presence of spotted areas (dark or bright) enables one to explain the asymmetry and the depressions on the light curves of active CBs. In the model these regions are approximated by circular spots (Fig. 1) characterised by the temperature contrast of the spot with respect to the surrounding photosphere , by the angular dimensions (radius) of the spot and by the longitude and latitude of the spot centre.

For a successful application of the model in analysing the observed light curves an efficient method unifying the best properties of the Steepest Descent and the Differential Corrections method into a single algorithm (Djurasevic 1992b) is proposed. This method is obtained by modifying the Marquardt's (1963) algorithm.

The interpretation of photometric observations is based on the choice of optimal model parameters yielding the best agreement between the observed light curve and the corresponding synthetic one. Some of these parameters can be determined a priori in an independent way, while the others are found by solving the inverse problem. Typical case of the inverse problem involves the estimate of the following parameters: mass ratio of the components (q=m₂/m₁), filling coefficients of the critical Roche lobes (F_1,2), orbit inclination (i), temperature of either component (T) and spotted areas parameters (, and ). The temperature contrast of the spotted regions with respect to the surrounding photosphere is usually given a priori.

The Roche model given above and the method for the inverse-problem solution allow us to obtain a direct analysis of the observed light curves.