arXiv:astro-ph/0311310v1 13 Nov 2003 Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 6 September 2007 (MN L A T E X style file v2.2) Spot Sizes on Sun-like Stars S. K. Solanki, 1 and Y. C. Unruh 2 1 Max-Planck-Institut f¨ ur Aeronomie, D-37191 Katlenburg-Lindau, Germany 2 Astrophysics Group, Blackett Laboratory, Imperial College for Science, Technology and Medicine, London, SW7 2BW, United Kingdom Accepted 4th Nov 2003; Received Aug 2003; in original form May 2003 ABSTRACT The total area coverage by starspots is of interest for a variety of reasons, but direct techniques only provide estimates of this important quantity. Sunspot areas exhibit a lognormal size distribution irrespective of the phase of the activity cycle, implying that most sunspots are small. Here we explore the consequences if starspot areas were similarly distributed. The solar data allow for an increase in the fraction of larger sunspots with increasing activity. Taking this difference between the size distribution at sunspot maximum and minimum, we extrapolate to higher activity levels, assum- ing different dependencies of the parameters of the lognormal distribution on total spot coverage. We find that even for very heavily spotted (hypothetical) stars a large fraction of the spots are smaller than the current resolution limit of Doppler images and might hence be missed on traditional Doppler maps. Key words: sunspots, stars: spots, stars: late-type, stars: activity 1 INTRODUCTION In recent years, an ever increasing number of spotted stars have been mapped using Doppler imaging. The maps reveal the surface distribution of starspots, which in general are large compared to even the largest sunspots. While Doppler images do a good job of catching starspots that modulate the line profile, it is extremely difficult to detect a background of small starspots more or less homogeneously distributed over the stellar surface. While TiO-band mapping still suf- fers from notable uncertainties (see Sec. 2), it should in prin- ciple be able to pick up non-modulating and homogeneous spot distributions. Typically, techniques using TiO bands to determine spot temperatures and surface areas tend to find larger covering fractions than Doppler imaging tech- niques (though see also Berdyugina (2002)). Such differences in apparent spot coverage only provide a hint for unresolved starspots. One star for which the distribution of spot sizes is known in great detail is the Sun. Bogdan et al. (1988) found the size distribution to be well represented by a lognormal function. This implies that the number of small sunspots is much larger than that of large spots. This supports the idea that there could be additional small, i.e. unresolved, starspots on more active stars as well. The total starspot coverage is of interest as a measure of stellar magnetic activ- ity, in order to establish the proper ratio of starspot to stellar plage (Radick et al. 1998) and to obtain improved estimates of the total magnetic flux carried by the star (with the pos- sible exception of Zeeman Doppler imaging where most of the magnetic signal appears to come from penumbral-type structures, techniques of stellar magnetic field measurement mainly sample plage fields (Saar 1986; Solanki 1992)). Here we explore hypothetical scenarios for extrapolat- ing the solar spot-size distribution to activity levels typical for much more active stars. The basic assumption is that the size distribution of star spots can be described by a lognor- mal function, as in the case of sunspots. This assumption is not unreasonable since the magnetic fields on both the Sun and on more active cool stars are thought to be produced by a dynamo residing at the base of the convection zone (Petrovay 2001; Sch¨ ussler & Schmitt 2002). From there flux tubes carry the field to the solar surface. The fragmenta- tion of these tubes during their passage through the convec- tion zone is thought to give rise to the observed lognormal distribution (Bogdan et al. 1988). Lognormal distributions can, however, differ significantly from each other in their parameters. In order to constrain these parameters for active stars we investigate the possible range of behaviour between so- lar activity minimum and maximum and use these to ex- trapolate to larger levels of activity. Hence we assume that the processes which lead to the flux-tube size distribution do not change qualitatively with increasing activity. Such an assumption has in the past helped to reproduce, e.g., the high latitudes of starspots (Sch¨ ussler & Solanki 1992; Sch¨ ussler et al. 1996; Schrijver & Title 2001), or the pres- ence of active longitudes on the Sun and Sun-like stars Berdyugina & Usoskin (2003).