1 Introduction

Photometric variations attributed to starspots clearly depend on a number of physical parameters, i.e. size, location, effective temperature of the spots or spot groups. The numerous attempts made to extract this information by modelling the photometric light curve unavoidably meet with the non-uniqueness of the solutions. While a certain spot configuration can reasonably reproduce the photometric variations, it is not possible to invert models to derive unique values for spot sizes, shapes and temperatures. There is a fundamental trade-off between spot area and spot temperature. Very precise data at least in two colors are required to resolve this ambiguity. Unfortunately the most used color index, the B-V, is not of great help because, as the spot becomes darker its contribution to the color of the visible stellar hemisphere decreases and the observed color variation associated with the rotational modulation is always small. The use of the V-R color is more effective for the contemporaneous determination of starspot temperature and area, as earlier shown by Vogt (1981). The method determines the V-R color difference between the star and spot, while the spot temperature is derived through the (V-R)- $T_{\rm eff}$ calibration.

The Doppler-imaging technique (e.g. Vogt et al. 1987; Piskunov 1991) based on a series of high resolution spectral line profiles allows us to produce an image of the stellar surface with high degree of sophistication and accuracy (for recent developments see Rice & Strassmeier 2000). However, the temperature scale remains model dependent, in the sense that with a given set of stellar atmosphere models, the stellar image can be considered as the distribution of effective temperature across the stellar surface. The temperature scale of the image can be fixed by simultaneous photometric observations.

Following the earlier suggestions that TiO bands at 8860 Å  could be used to measure starspot temperatures (e.g. Vogt 1979; Ramsey & Nations 1980), Huenemoerder & Ramsey (1987) attempted a quantitative study of the effect of spots on the TiO bands to derive spot temperatures and surface coverage. Recently Neff et al. (1995) and O'Neal et al. (1996) obtained an evaluation of spot temperature with an estimated error of $\pm$100-200 K and of the surface filling factor for a number of active stars using both bands of TiO at 7055 and 8860 Å. Although the method looks very effective for stars with spot temperatures lower than 3500 K, it fails for stars with effective temperature higher than 5000 K if the average temperature difference is $T_{\rm eff}-$ $T_{\rm sp} \leq$ 1500 K, as generally observed. Moreover, the method requires an independent determination of $T_{\rm eff}$.

In a series of recent works Gray and collaborators have demonstrated that line-depth ratios are a powerful temperature discriminant, capable of resolving differences $\leq$10 K, and have determined $T_{\rm eff}$ of several main sequence (Gray & Johanson 1991) and giant stars (Gray & Brown 2001). Although the effective temperature scale calibration can be set to few tens of a degree, temperature differences can be measured with a precision less than one degree. Temperature variation in the 5-15 K range over the rotational period and the activity cycle has been reported for $\epsilon$ Eri (Gray & Baliunas 1995), $\sigma$ Dra (Gray et al. 1992), as well as for the Sun along the 11-year cycle (Gray & Livingston 1997).

Application of this method to the study of temperature variations associated with stellar surface features, as discussed by Gray (1996), is particularly challenging because several physical variables interact simultaneously, each impressing their signature on the spectral lines. In particular, cool surface features (spots) cause bumps in a line profile, which migrate through the line profile allowing Doppler-imaging in fast rotating stars. Since the size of the bump, to the first order, is determined by the loss of continuum light at the Doppler position of the bump, in a slowly rotating star we cannot see a clear Doppler shift but can expect a slight rotational modulation of the central line depth. For example, following Gray (1996), if the passage of a dark spot at the central meridian produces a decrease of 10% in the stellar disk light, then the bump would have a height of 10% of the depth of the line. The line depth will be reduced by about this quantity.

Let us now consider the line-depth ratios in a slowly rotating star and select two lines, one insensitive and one very sensitive to temperature. Due to the presence of a dark spot that produces a decrease of 10% in the continuum, the depth of both lines would be equally affected, and reduced by 10% and their depth ratio would remain unchanged. But due to the lower temperature in the spot, the line sensitive to the temperature will change its intrinsic depth and consequently also the depth ratio of the two lines will change. The amount of depth ratio variation should depend on the sensitivity to the temperature variation of the specific lines considered, and the fraction of surface covered by the spot.

On the basis of these considerations we have made some test observations on rather active, slowly rotating RS CVn binaries, namely VY Ari, IM Peg and HK Lac to investigate the applicability of the method and its ability to determine the spot temperature. The paper develops as follows: in Sect. 2 we present the observations and data reduction; in Sect. 3 the temperature calibration of the line-depth ratios; in Sect. 4 the results of temperature modulation of the active stars will be discussed, and in Sect. 5 a method to determine the spot temperature and the filling factor will be discussed.