4.1 Light-curve modelling

Since the discovery of stellar photometric variations due to cool spots, generally two approaches are used to model such variations and to deduce starspot properties. One is based on a trial-and-error direct light-curves modelling (LCM) when assuming a number of circular (or of other pre-defined shape) spots causing the variations. Numerical techniques employing this approach have been developed by Budding (1977 ), Vogt (1981

), Rodonò et al. (1986

), Dorren (1987 ), Strassmeier (1988 ) and Kiurkchieva (1990 ). A technique taking into account time evolution of starspots has been proposed by Strassmeier and Bopp (1992

). It allowed for detecting spot appearance and disappearance on time scales from a few to a hundred days. A zonal model with (near-)equatorial inhomogeneous bands of spots was considered by Eaton and Hall (1979

) and Alekseev and Gershberg (1996 ).

A disadvantage of the above mentioned models is that they have many free parameters and the shape of spots or of spot distribution has to be assumed. Moreover, the solution is not unique. To avoid many assumptions, an alternative approach performing light-curve inversion (LCI) into an image of the stellar surface has been developed. An inversion technique is usually applied to the photometric light curves in the two temperature approximation. The model assumes that, because of low spatial resolution, the intensity of each pixel on the surface $Ii$ contains contributions from both temperature components, the hot photosphere $Ip$ , and cool spots $Is$ weighted by the fraction of the surface covered with spots $fi$ (spot filling factor):

with $0 < fi < 1$ . The instantly observed stellar flux is the integral of the intensity contributions from the surface elements seen at a given rotational phase. The inversion of the light curve results in a distribution of the spot filling factor over the stellar surface, i.e., stellar image. Figure 1

shows examples of light curves and maps of the spot filling factor obtained from them for the RS CVn type star $s$ Gem. Since a light curve represents a one-dimensional time series, the resulting stellar image contains information on the spot distribution only in one direction, in longitudes, while spot extents and locations in latitudes remain uncertain. As a result, maximum spot concentration always appears at the central latitude of the stellar disk whose value depends on the assumed inclination of the rotational axis. Because of the projection effect and limb darkening, images show more well defined structures than the light curves and, therefore, are very useful for determining longitudes of spot concentrations. Numerical techniques based on such an approach have been elaborated by Messina et al. (1999 ) and Berdyugina et al. (2002

Figure 1:

Light curve inversion results for the RS CVn star $s$ Gem (Berdyugina & Henry, in preparation). The first and third columns are maps of the spot filling factor. Darker regions indicate larger values. Observed and calculated $V$ -band light curves are shown in the second and fourth columns by crosses and lines, respectively. The images also illustrate a flip-flop event that occurred in 1988: the active region near the phase 0.5 has diminished for about one year, while the region near the phase 0.0 has increased in size (see discussion of the flip-flop phenomenon in Section 6.2).

More detailed information on the spot pattern from light curves can be obtained in the case of eclipsing binaries by the eclipse mapping technique. This method employs the opportunity to scan the stellar disk in finer detail by the eclipsing companion. The inversion techniques based on the Maximum Entropy and Tikhonov regularisation methods (see Section 4.2) were developed by Rodonò et al. (1995

), Collier Cameron (1997 ), and Lanza et al. (1998a

Inversion or modelling of light-curves is clearly less informative than techniques which are based on spectroscopic observations. Continuous and frequent photometric data allow though for conclusions on longitudinal spot patterns and their long-term evolution (Section 6).