3 Starspot modeling

The photometric variation attributed to starspots can provide useful informations about physical characteristics (size, location, effective temperature etc.) of the spotted stars. Over the past few years, several attempts have been made to treat light curve of spotted stars with one or two-spot models (Bopp & Evans 1973; Budding 1977; Dorren 1987; Strassmeier 1988; Eker 1994; Henry et al. 1995; Kovári & Bartus 1997 and references therein). In order to account for long term variation analogous to sunspot cycle a time dependent formulation of the spot model has been introduced by Strassmeier & Bopp (1992), and later on used by others (Strassmeier et al. 1994; Oláh et al. 1997).

  

Table 1: Basic parameters of six RS CVn binary stars

1. Henry et al. (1995)
2. Raveendran & Mohin (1995)
3.  Eaton et al. (1983)
4. Poe & Eaton (1995), except for V711 Tau.

  

Table 2: Starspot parameters of six RS CVn binary stars

A new approach, considering multiple random spots, has recently been introduced by Eaton et al. (1996) to solve the erratic behaviour of light variation in active stars. To evaluate geometric parameters of starspots from observed light curves we have used our computer program based on the analytical formulation for the circular starspots given by Dorren (1987). For each star we have extracted the value of the maximum brightness ever observed from the literature (see Table 1). These values were further used as the unspotted light level of each star.

Polar spots as well as the spots uniformly distributed in longitudinal belts do not produce any rotational signature in the light output, and hence are indistinguishable as far as light modeling is concerned. In order to reproduce the light curve of a given star in the framework of a spot model either polar spots or spots distributed uniformly in a longitudinal belt are used to produce changes in the light level of photometric variation. The photometric spot modeling, Doppler imaging (Hatzes & Vogt 1992; Strassmeier et al. 1991) as well as theoretical models (Solanki et al. 1997) seem to support the existence of spots at high latitudes and/or polar spots in chromospherically active stars. Therefore, we have chosen one of the spots permanently located at the pole. To model the V as well as the colour light curves spot parameters, namely, the longitude ($\lambda$), latitude ($\beta$), radius ($\gamma$) and ($\Delta T$) were taken as free parameters and rest of the parameters i.e. inclination (i), limb darkening co-efficient (u) etc. were kept constant.

  

Figure 1: BVR light curves of RS CVn star V711 Tau during epoch 1996-97

We have adopted constant $\Delta T$ to evaluate flux ratio between the spot and photosphere using a simple black body approximation with $\Delta T$$\sim$ 1000 K for the stars showing insignificant colour variations. For the stars (IM Peg, II Peg and $\lambda$ And) having significant colour variation we have adopted the procedure of modeling (V-R) colour indices curve to extract $\Delta T$ and area as described by Poe & Eaton (1985). Using Dorren's (1987) spot model with three spots having one always at the pole, observed light curves were fitted and the best fit parameters ($\lambda$, $\beta$, $\gamma$, $\Delta T$) of the two spots plus the size of the polar spot were evaluated by simultaneously using gradient and grid search least square fitting method (Bevington 1969). We have also tested the modeling technique for various noise levels to check the reliability and stability of the results. To trace out absolute minima in $\chi^2$-parametric space we iterated the fitting procedure with different initial values of parameters, and the solutions with overlapping spots were discarded. The final best fit parameters with their uncertainties are listed in Table 2.

  

Figure 2: BVR light curves of RS CVn star UX Ari during epoch 1996-97