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Astron. Astrophys. 364, 543-551 (2000)

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2. Procedure of the light-curve analysis

In the analysis of the light curves, instead of the often used and somewhat questionable practice of forming normal points, we used the original observational data in order to avoid negative influences of such a normalization. To analyse these asymmetric light curves, probably deformed by the presence of spotted areas on the components, we used Djurasevic's (1992a) programme generalised to the case of an overcontact configuration (Djurasevic et al. 1998). The programme is based on the Roche model and the principles arising from the paper by Wilson & Devinney (1971). The light-curve analysis was made by applying the inverse-problem method (Djurasevic 1992b) based on Marquardt's (1963) algorithm.

According to this method, the stellar size in the model is described by the filling coefficients for the critical Roche lobes [FORMULA] of the primary and secondary component respectively, which tell us to what degree the stars in the system fill their corresponding critical lobes. For synchronous rotation of the components these coefficients are expressed via the ratio of the stellar polar radii, [FORMULA], and the corresponding polar radii of the critical Roche lobes, i.e., [FORMULA]. In the case of an overcontact configuration the potential [FORMULA] characterising the common photosphere is derived with the filling coefficient of the critical Roche lobe [FORMULA] of the primary, while the coefficient [FORMULA] may be excluded from further consideration. The degree of overcontact is defined in the classical way (Lucy & Wilson 1979) as:


where [FORMULA], [FORMULA], and [FORMULA] are the potentials of the common photosphere and of the inner and outer contact surfaces respectively. To achieve more reliable estimates of the model parameters in the programme for the light-curves analysis, we applied a quite dense coordinate grid having [FORMULA] elementary cells per each star. The intensity and angular distribution of the radiation of elementary cells are determined by the stellar effective temperature, limb-darkening, gravity-darkening and by the effect of reflection in the system.

The presence of spotted areas (dark or bright) enables one to explain the asymmetries, the light-curve anomalies and the O'Connell effect, as has been suggested by several investigators (e.g. Binnendijk 1960; Hilditch 1981; Maceroni et al. 1990and many others). In our programme these active regions are approximated by circular spots, characterised by the temperature contrast of the spot with respect to the surrounding photosphere ([FORMULA]), by the angular radius of the spot ([FORMULA]) and by the longitude ([FORMULA]) and latitude ([FORMULA]) of the spot's centre. The longitude ([FORMULA]) is measured clockwise (as viewed from the direction of the +Z-axis) from the +X-axis (line connecting star centres) in the range [FORMULA]-[FORMULA]. The latitude ([FORMULA]) is measured from [FORMULA] at the stellar equator (orbital plane) to [FORMULA] towards the "north" (+Z) and [FORMULA] towards the "south" (-Z) pole.

For a successful application of the model described above to the analysis of the observed light curves, the method proposed by Djurasevic (1992b) was used. By that optimum model parameters are obtained through the minimization of [FORMULA], where [FORMULA] is the residual between the observed (LCO) and synthetic (LCC) light curves for a given orbital phase. The minimization of S is achieved with an iterative cycle of corrections of the model parameters. In this way the inverse problem method gives us the estimates of system parameters and their standard errors.

The present light-curve analysis shows that during the deeper (primary) minimum the cooler (more massive and larger) component eclipsed the hotter (less massive and smaller) one. Since the primary minimum is an occultation, for the initial value of the mass ratio we used [FORMULA] ([FORMULA]), as was estimated by Hrivnak (1988), based on a new radial velocity study. The indices (h,c) refer to the less massive (hotter) and more massive (cooler) component, respectively. The temperature of the more massive component was fixed at the value [FORMULA], according to its spectral type (G5V) and colour index. Following Lucy (1967), Rucinski (1969) and Rafert & Twigg (1980), the gravity-darkening coefficients of the components, [FORMULA], and their albedos, [FORMULA], were set at the values of 0.08 and 0.5, respectively, appropriate for stars with convective envelopes. Moreover, in solving the inverse problem, the values of the limb-darkening coefficients were derived on the basis of the stellar effective temperatures of the two components, and the surface gravity according to their spectral types, using the polynomial proposed by Díaz-Cordovés et al. (1995).

The analysis yields [FORMULA] as the filling coefficient of the critical Roche lobe, i.e. an overcontact configuration. So, we expect mutual tidal effects of the components to cause synchronisation between the stellar rotational period and the system's orbital period. Therefore, in the inverse problem we adopted the values [FORMULA] for nonsynchronous rotation coefficients.

In previous versions of our programme, there were two different possibilities in the application of the model with respect to the treatment of the radiation law: the simple black-body theory, or the stellar atmosphere models by Carbon & Gingerich (1969) (CG). Our current version of the programme for the light-curve analysis employs the new promising Basle Stellar Library (BaSeL). We have explored the "corrected" BaSeL model flux distributions, consistent with extant empirical calibrations (Lejeune et al. 1997 , 1998), with a large range of effective temperature [FORMULA], surface gravities, [FORMULA] and metallicity, [FORMULA], where [Fe/H] is the logarithmic metal abundance. The surface gravities can be derived very accurately from the masses and radii of the close binary (CB) stars by solving the inverse problem of the light-curve analysis, but the temperature determination is related to the assumed metallicity and strongly depends on photometric calibration.

In the inverse problem the fluxes are calculated in each iteration for current values of temperatures and [FORMULA] by interpolating both of these quantities in atmosphere tables, as an input, for a given metallicity of the CB components. The metallicity of the CB components involved can be different. Because of that we can use individual, different tables as an input, for each star, and in that way, choose the best calculations for its particular atmospheric parameters. Compared to Vaz et al. (1995) our two-dimensional flux interpolation in [FORMULA] and [FORMULA] is based on the application of the bicubic spline interpolation (Press et al. 1992). This proved a good choice.

By choosing and fixing the particular input switch, the programme for the light-curve analysis can be simply redirected to the Planck or CG approximation, or to the more realistic BaSeL model atmospheres. A disagreement obtained between individual B and V solutions decreases if we introduce the "corrected" BaSeL model flux distributions. A change in the assumed metallicity causes a noticeable change in the predicted stellar effective temperature. The value of the chemical abundance of the components was obtained by checking several different values around solar metallicity. In the case of AB And, the best fit of the B and V light curves was obtained with [FORMULA] for the metallicity of the components. With this value, the individual B and V solutions are in good agreement. The results presented here are given within this stellar atmosphere approximation.

Bell's 1982 curve (Bell 1984) was used to estimate the basic system parameters. On this light curve the system is brighter in the maximum after a primary minimum than on the rest of the analysed light curves. The obtained solutions show that Bell's 1982 light level, in the orbital phase 0.25 is very probably clean from spot effects. In the analysis of this light curve, the optimum photometric mass ratio was estimated as [FORMULA] ([FORMULA]). The inclination of the orbit was estimated as [FORMULA], the filling coefficient for the critical Roche lobe as [FORMULA] and the temperature of the hotter (less massive) star as [FORMULA].

The basic parameters of the system, obtained in this way, are fixed in the inverse-problem solution for other, more or less deformed and asymmetrical, light curves. The entire set of analysed light curves is normalised to the reference light level of Bell's 1982 light curve at the orbital phase 0.25, and their analysis was made with optimisation in the spot parameters.

Moreover, since the results of the light-curve analysis depend on the choice of the adopted working hypothesis, the present analysis was carried out within the framework of several hypotheses with spotted areas on the components. Since both stars of the system have external convective envelopes, which can show magnetic activity, we started the "spotted solution" by assuming that the components of AB And have cool spots, of the same nature as solar magnetic spots.

Our analysis shows that the Roche model satisfactorily fits the observations within two hypotheses on spot location on the components: I - single spotted areas on both of the components and II - two spotted areas on the cooler star.

In the first case we obtained a very good fit of the observations for the whole set of the analysed light curves without any changes of the system's basic parameters. The obtained results indicate that the complex nature of the light-curve variations during the examined period could be almost entirely explained by the changes in the parameters of the spotted areas.

The second hypothesis, with two spotted areas on the cooler star, requires, beside the changes of the spot's parameters, also significant changes of the system's basic parameters during the analysed period. Since we have no firmly established physical argument supporting these short-term variations, we consider this case as less reliable, to the point of being possibly excluded.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001