1 Introduction

Magnetic activity in late-type main-sequence stars is an observable manifestation of the stellar magnetic fields. The generation and amplification of surface magnetic fields in solar-like stars are commonly considered the end result of a complex dynamo mechanism, whose efficiency depends on the interaction between differential rotation and subphotosferic convection into the stellar interior. In this context, stellar rotation must play a very important role, and numerous studies have searched for relationships between magnetic activity and rotation. This connection has been explored ever since the pioneering work by Wilson (1966) and the study by Kraft (1967). At first, the rotation-activity relationship was considered to be an indirect one, through stellar age: in fact, Skumanich (1972), compared the rotation and the Ca II emission luminosity for Pleiades, Ursa Major, Hyades stars and the Sun, and showed that both the chromospheric emission and the rotation velocity decay roughly as the inverse square root of the age. According to Frazier's data (Frazier 1970), Ca II emission intensity (in a 1.1 Å band centered on the K-line) varies linearly with surface magnetic field strength, thus making appropriate to link the stellar Ca II emission luminosity with the average magnetic field strength. Thus, the Skumanich's work was the first to suggest the activity-rotation relationship as a causal consequence of the dynamo action, because of the indirect observation of the proportionality law between the average surface magnetic field and the stellar rotation.

When comparing any radiative measure of stellar magnetic activity with the stellar angular velocity, for stars of comparable effective temperature and gravity, one finds that activity increases monotonically with increasing angular velocity for rotation periods exceeding $\sim$1-2 days. The correlation between X-ray luminosity and rotation was first discovered for RS CVn systems by Walter & Bowyer (1981) and was generalized by Pallavicini et al. (1981) for late-type stars (F7 to M5), independently from the luminosity class, as $L_{\rm x} \approx 10^{27}(v_{\rm rot})^{2}$ erg s^-1, being  $v_{\rm rot}$ the rotational velocity in km s^-1. In the last 20 years many other authors have investigated the correlation between several chromospheric and coronal magnetic activity indicators and stellar rotation rate in both field stars and cluster members (Maggio et al. 1987; Dobson & Radick 1989; Stepien 1994; Hempelmann et al. 1995; Randich et al. 1996, to cite a few). For stars of a given spectral type, hence given radius, the X-ray vs. rotation relationship does not hold for very fast rotators (Micela et al. 1985). Obviously, if one were to increase $v_{\rm rot}$ into the Pallavicini's equation, extremely large X-ray luminosities are obtained. What is observed instead is that the most active stars reach a maximum X-ray luminosity such that $L_{\rm x}/L_{\rm bol}\approx 10^{-3}$ (Vilhu 1984; Vilhu & Walter 1987), where $L_{\rm bol}$ indicates the star's bolometric luminosity. The saturation limit appears to extend all the way along the main sequence from G-type stars to the latest M dwarfs (Fleming et al. 1993); using $L_{\rm x}/L_{\rm bol}$ as activity indicator for stars in the Pleiades and $\alpha$ Persei open clusters, Stauffer et al. (1994) and Randich et al. (1996), respectively, found a trend of increasing log $L_{\rm x}/L_{\rm bol}$ with the rotation velocity up to $\sim$15 km s^-1, while stars with higher velocities have $L_{\rm x}/L_{\rm bol}$ near the saturation level. It is yet unclear whether this saturation effect is caused by an intrinsic change in the behavior of the dynamo, or it is merely a sign of a limiting coronal emission reached because the star runs out of the available surface area to accommodate more active regions (Jardine & Unruh 1999).

In this context, we have studied the dependence of the X-ray emission on the stellar rotation in late-type main-sequence stars, using a sample of both field stars and members of open clusters, which includes more K and M-type dwarfs than in previous studies. In particular, we have investigated the relationships between X-ray luminosity or X-ray to bolometric luminosity ratio and the rotation period for stars grouped in selected mass or color ranges. In the first part of this paper we show that the rotation period by itself appears to be a good predictor of the stellar X-ray luminosity down to a critical value, below which the coronal emission saturates, and we explore how the transition from the non-saturated to the saturated X-ray emission regime depends on stellar properties, such as the spectral type and the mass.

The magnetic dynamo mechanism operates in a highly conductive fluid (plasma) subject to convective and rotational motions whose characteristic time scales are, respectively, the convective turnover time - i.e. the period of circulation within a convective cell - and the stellar rotation period. The Rossby number $R_{\rm o}=P_{\rm rot}/\tau_{\rm conv}$, i.e. the ratio between the measured rotation period, $P_{\rm rot}$, and the convective turnover time, $\tau_{\rm conv}$, was introduced in the context of stellar magnetic activity by Noyes et al. (1984), and since then, used by many authors (Micela et al. 1984; Schmitt et al. 1985; Maggio et al. 1987; Dobson & Radick 1989; Stepien 1994; Hempelmann et al. 1995; Randich 2000 - to cite a few) as the quantity best-suited to parameterize the level of stellar magnetic activity. Using the Rossby number, all these authors have investigated the dependence of coronal and chromospheric emission levels on the spectral type, stellar age and evolutionary stage for different samples of late-type stars. In all these works, however, the Rossby number was always calculated as the ratio of the rotation period, a measured quantity, with the convective turnover time estimated by means of some empirical color-dependent function $\tau_{\rm e}(B-V)$, usually the one provided by Noyes et al. (1984), which resembles the theoretical  $\tau_{\rm conv}$. Moreover, although this function was originally derived for main-sequence stars with 0.5<B-V<1.4 (but based on 5 points only redder than B-V=1.0), several authors have applied such an empirical Rossby number out of its range of applicability (for example, Maggio et al. 1987 have used this function to evaluate  $\tau _{\rm e}$ for stars of luminosity class IV, or Giampapa et al. 1998 and Randich 2000 for cluster stars younger and redder than the Noyes's dwarfs). To check the validity of this approach, in the second part of this paper we have empirically determined a new  $\tau_{\rm e}(M)$, as a function of stellar mass, based on the analysis of the same star sample introduced above, and we have investigated whether the use of our X-ray-based Rossby number effectively helps in our physical understanding of X-ray emission as a magnetic dynamo phenomenon.

In Sect. 2 we present the stellar sample used for the present work, and discuss some observational issues related to the completeness of the sample; in Sect. 3 we explore the dependence of the relationship between X-ray emission and rotation period on both stellar mass and color, while in Sect. 4 we determine an X-ray-derived empirical Rossby number and discuss its physical meaning. Section 5 is devoted to the conclusions.