3 Time-series light curve analysis

Figure 1: Photometric light variations of $\sigma$ Gem between 8 November 1996-21 January 1997 in the $\Delta y$ band and $\Delta (b-y)$colour. The upper panel shows the y data and the result of our time-series photometric spot modelling using either a two-spot (dashed line) or a three-spot assumption (solid line). The lower panel displays the colour curve and the fit obtained using a spot temperature of 600 K cooler than the photospheric background temperature.

Figure 2: Mercator plots of the time-series spot modelling at three different epochs. Shown are the times marked by the three arrows on the time-axis in Fig. 1. For more explanations see the text.

3.1 The light-curve modelling code

For the spot modelling, we apply the TISMO code developed by Bartus (1996) and sucessfully applied to the RS CVn binary HR 1099 in Strassmeier & Bartus (2000). The code is written for inverting the light curve variations in the time domain rather in the (rotational) phase domain. It is thus able to provide a continuous fit for the photometric light variations in consecutive rotational cycles caused by starspots with varying geometry. The output parameters are the time dependent spot coordinates. By assuming circular dark spots these are the spot longitudes ( $\lambda_i(t)$), the spot latitudes ( $\beta_i(t)$), and the spot radii ( $\gamma_i(t)$). Errors are estimated according to the photometric precision of the data and the procedure described in Bartus (1996).

Because the secondary component is not seen, the photometric variations are attributed entirely to the K1III primary component. For the modelling, two a priori assumptions are made. First, the surface temperature is fixed to $T_{\rm phot}=4630$ K (cf., Sect. 4.2) and second, the "unspotted'' brightness level remains fixed at the maximum brightness ever observed, i.e $V_{\rm max}=4\thinspace\hbox{$.\!\!^{\rm m}$ }137$ (cf. Strassmeier et al. 1988). The inclination of the stellar rotation axis was set to the most probable value of i=60$^\circ$ (cf., Sect. 4.2 and Eker 1986).

3.2 Results

We started the modelling with just two spots but after a few runs it became clear that the number of spots had to be increased to three to obtain a satisfactory fit to the light curve (Fig. 1). Having the spot temperature as a free parameter, we fit the b-ycurve to search for the most probable spot temperature. Fits are obtained for spot temperatures cooler than the surrounding photosphere of 200 K-900 K in steps of 100 K. The best fit was obtained using $\Delta T_{\rm spot}=600\pm100$ K, i.e., $T_{\rm spot}\approx 4030$ K. The upper panel of Fig. 1 shows the y-band data and the model fits with either two or three spots with this temperature, while the lower panel shows the model fit for b-y with just the three-spot case.

We find that the three spots are nearly equally spaced in longitude, as indicated in the upper bars in Fig. 1, and also illustrated in Fig. 2. The latter figure shows mercator plots of the three-spot solution at three different epochs following each other by one rotational cycle (these epochs coincide with the mid-epochs of the spectroscopic datasets SS2, SS4 and SS6, respectively, used later in Sect. 4). The spot areas are also comparable, between 20$^\circ$-30$^\circ$ in radii. The total spot coverage varies thus between 10-20% of a hemisphere during one rotation. Despite that the spot latitudes are the most uncertain parameters, the resulting latitudinal positions indicate spottedness predominantly at lower latitudes. In our solutions, cool regions do not reach latitudes higher than $\approx$60$^\circ$.

Considerable evolution of the spot parameters is not seen, however, a small systematic decrease of the longitudes of SPOT 1 and SPOT 2 is present, while SPOT 3 performs a mixed motion with replacements in both directions and also an increase in size. The spot migrations are of the order of 2-3$^\circ$ per rotational cycle, i.e. the order of the uncertainty of the spot modelling (see, e.g., Kovári & Bartus 1997). Thus, the detection of such small spot-motion patterns and their interpretation due to surface differential rotation requires longer baselines in time.