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

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3. Results

The parameters derived from the light-curve analysis are listed in Table 1. The errors in the parameter's estimates arise from the nonlinear least-squares method, on which the inverse-problem method is based. The first three rows of the table present the number of observations n, the final sum of squares of residuals between observed (LCO) and synthetic (LCC) light curves


and the standard deviation of the observations


In Table 1, the indices [FORMULA] denote the B and V-filter observations, respectively.


Table 1. Results of the analysis of AB And (1968, 1970, 1978, 1982, 1990 and 1995) photometric observations obtained by solving the light-curve inversion problem for the Roche model with single spotted areas on both components.
Fixed (*) parameters:
[FORMULA] - temperature of the less massive (hotter) component,
[FORMULA] - temperature of the more massive (cooler) star
[FORMULA] - filling coefficient for the critical Roche lobe of the less massive (hotter) star,
[FORMULA] - orbit inclination (in arc degrees)
[FORMULA] - mass ratio of the components
[FORMULA] - nonsynchronous rotation coefficients of the components,
[FORMULA] - gravity-darkening coefficients of the components,
[FORMULA] - albedo coefficients of the components.
[FORMULA] - accepted metallicity of the components
n - number of observations, [FORMULA] - final sum of squares of residuals between observed (LCO) and synthetic (LCC) light curves, [FORMULA] - standard deviation of the observations, [FORMULA] - spots' temperature coefficients, [FORMULA], [FORMULA] and [FORMULA] - spots' angular dimensions, longitudes and latitudes (in arc degrees), [FORMULA] - limb-darkening coefficients of the components, [FORMULA] - common dimensionless surface potentials of the system's components, [FORMULA], [FORMULA] - the potentials of the inner and outer contact surfaces respectively, [FORMULA] - degree of overcontact, [FORMULA] - polar radii of the components in units of the distance between the component centres, [FORMULA] - luminosity of the hotter star (including spots), [FORMULA] - stellar masses in solar units, [FORMULA] - mean radii of stars in solar units, [FORMULA] - logarithm (base 10) of the mean surface acceleration (effective gravity) for the system's stars, [FORMULA] - absolute bolometric magnitudes of the system's components and [FORMULA] - orbital semi-major axis in units of solar radius.

In the same table the spot characteristics (spot temperature factor, [FORMULA]; longitude, [FORMULA]; latitude, [FORMULA]; angular radius, [FORMULA]) are also given. The determination of these parameters is based on a simultaneous fitting of the available light curves in the B and V photometric bands for the different epochs of observations with the same set of basic system parameters. Finally, in this table we present also some of the important absolute system parameters derived from this analysis. They are obtained from the mass ratio of the components ([FORMULA]), orbital period ([FORMULA]) and the semi-major orbital axis ([FORMULA]) estimated by Hrivnak (1988) on the bases of the radial velocity study.

The obtained solution for each individual light curve, is presented in Fig. 1, where the optimum fit of the observed light curves (LCO) by the synthetic ones (LCC) is shown. The reference light level of Bell's 1982 light curve at orbital phase 0.25, for the unspotted configuration of the system is denoted by the dashed line. The rest of the light curves are normalised to this light level. The O-C residuals between the observed (LCO) and optimum synthetic (LCC) light curves are given on the left-hand side of the Fig. 2. The right-hand side on these panels shows the Roche models of the system obtained with the parameters estimated by analysing the corresponding light curves. Thanks to such plots, one sees how a system would look at a given orbital phase, chosen so that the spots are visible.

[FIGURE] Fig. 1. Observed (LCO) and final synthetic (LCC) light curve of AB And. The reference light level of Bell's 1982 light curve, at orbital phase 0.25, for the unspotted configuration of the system is shown by the dashed line.

[FIGURE] Fig. 2. Left: Final O-C residuals obtained by solving the inverse problem within the framework of the Roche model with single spotted areas on both components; Right: The view of the Roche model for AB And at the noted orbital phase, obtained with parameters estimated by solving the inverse problem.

It is evident from Table 1 and from Fig. 1 & Fig. 2 (Left) that the Roche model with single spotted areas on both components gives a satisfying fit of the analysed light curves. The light curves from 1978, 1990 and 1995 were obtained from observations that exhibited night to night short-term intrinsic variation in the light of the system, and perhaps even some small, unaccounted variation due to atmospheric or instrumental effects. In order to obtain a light curve from a complete set of orbital phases the observations performed during several nights were coupled in yearly light curves. In the analysis we used the original observational data; consequently, the mentioned effects can produce a fit deviation shown in O-C residuals over some of the intervals of the orbital phase. Also, fit deviations could to some extent be a result of the applied model with the circular spot, which is only a rough approximation of the real shape of a spot. The contribution of the latter effect in O-C residuals is smaller than the contribution of the first two effects.

In the case of single spotted areas on both components, the system's basic parameters are constant within the whole set of the analysed light curves. This indicates that the complex nature of the light-curve variations during the examined period can be almost completely explained by the development and motion of the spotted areas on the components.

The size of a spotted area can be used as an indicator of the system's activity. On the 1968 light curves we have a small asymmetry, caused by spotted areas located on different stellar hemispheres, near the stellar polar regions. On those for 1970, the spots' location would hardly affect the symmetric shape of the light curves, but it would change the light level at the maxima. The larger spot is located on the hotter star at high latitude, and the small one is on the cooler star, under, but near, the stellar equator. The analysis of the light curves from 1978 gives large spots that are situated on the same stellar hemispheres. Spotted areas cover a significant part of the stellar surfaces. So, their presence leads to a conspicuous asymmetry of the light curves. It seems that the system's activity decreases after that year. The light curves obtained during 1982 are less asymmetrical and their analysis gives spots that are smaller in size and located on different hemispheres of the components. Later, the asymmetry increases again and the analysis of the observational material from 1990 shows the large spots located again on the same hemispheres of the stars. Finally, although the light curves from 1995 are less asymmetrical, the solutions indicate again a high activity level of the system. The model with a large spot located near the polar region of the hotter star and a small one on the cooler star on the opposite hemispheres of the stars, successfully fits the observations. It is possible that large spotted areas at high latitudes correspond to an enhanced activity of the system.

Finally, Fig. 3 presents observations of AB And combined for B and V filters respectively, and corresponding colour-index (B-V) drawn for each epoch separately.

[FIGURE] Fig. 3. Observations of AB And for B and V filters combined, and the corresponding B-V colour-index drawn for each year separately.

During the analysed period, the main variations of the light curves are in the height of both maxima, especially in the primary one, and in the part following the secondary maximum as well as in the depth of the primary minimum. The B-V colour-index for the whole set of the analysed light curves show some reddening around the light-curve minima. In 1995 the B-V colour-index shows the main reddening around the orbital phases 0.0 and 0.5, which is followed by a relatively slow increase of the colour index. The main changes of the colour index with the orbital phase for the whole set of the analysed observations may be attributed to the temperature differences between the system's components, and partially to the influence of spotted active regions.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001