6 Conclusion

The line-depth ratio method has proved to be very effective in detecting low variations of the disk-averaged temperature in active stars. The simultaneous use of many line pairs remarkably enhances the precision of the temperature measurements. Though the precision of the absolute value of temperature is limited by the accuracy with which the temperature scale itself can be established, i.e. $\sim$100 K (e.g. Gray 1992), the resolution in temperature is much higher, with errors in the 5-15 K range.

A well-defined rotational modulation of the average temperature, with amplitudes ranging from 119 K to 177 K, has been detected in all three systems studied in the present work.

Though an accurate modelling of the temperature curves could give information on spot distribution, in the present work we have performed a simple analysis of these curves, based essentially on their variation amplitude interpreted in terms of a dark spot crossing the star disk during its rotation. We have derived the possible solutions for such a spot in the parameters space $\frac{T_{\rm sp}}{T_{\rm ph}}- A_{\rm rel}$ (fractional temperature, fractional area). The possible solutions define a parabolic-shaped locus in the plane $\frac{T_{\rm sp}}{T_{\rm ph}}- A_{\rm rel}$, providing a lower limit for the fractional area of the starspot (or starspots). The lower limits we found indicate large spot coverage in all stars, with relative areas of 41%, 32% and 34% of star disk for VY Ari, IM Peg, and HK Lac, respectively. The filling factors, in units of star surface, are of 11.7%, 8.5%, and 9.0%, respectively. The temperature difference between spot and photosphere corresponding to these filling factors is $\Delta T \simeq 890$ K for VY Ari, $\Delta T \simeq 750$ K for IM Peg, and $\Delta T \simeq 810$ K for HK Lac.

Values of $\Delta T$, obtained by means of light and color curve analysis, comparable with ours have been found by previous investigations into these systems.

For instance, Eaton & Poe (1986) found $\Delta T \simeq 760$ K for VY Ari from a spot modelling applied to their BVRI, while Strassmeier & Bopp (1992) found $\Delta T \simeq 1200$ K from their UBVRI photometry.

For IM Peg, values of $\Delta T \simeq 920$ K (Poe & Eaton 1985) and  $\Delta T \simeq 1130$ K (Padmakar & Pandey 1999) are reported.

For HK Lac, values of $\Delta T \simeq 950$ K (Vogt 1981), $\Delta T \simeq 1080$ K (Poe & Eaton 1985), and $\Delta T \simeq 1200$ K (Olàh et al. 1997) are reported.

We remark that, although the spot parameters we find are only indicative, because they are based only on a lower limit of the spot filling factor, they are well within the values found with other methods.

The simultaneous use of LDR and photometry, combined with a detailed spot-model, can lead to univocal solutions, since we expect a different behaviour of the locus of light-curve solutions in the $\frac{T_{\rm sp}}{T_{\rm ph}}- A_{\rm rel}$ plane that can solve the ambiguity.

This matter will be the subject of a forthcoming paper.

Acknowledgements

This work has been supported by the Italian Ministero dell'Istruzione, Università e Ricerca (MIUR) and by the Regione Sicilia which are gratefully acknowledged. We are grateful to prof. D. Gray for several useful suggestions. We would like to thank the referee, Dr. Artie Hatzes, for his helpful comments and suggestions.