5 Summary and conclusions

Our study has confirmed that a single relationship between  $L_{\rm x}/L_{\rm bol}$ and an empirical Rossby number $R_{\rm e}$ can be obtained for main sequence late-type stars having different masses or spectral types (i.e. different properties of the convection zone). The importance of our result rests on the inclusion - for the first time with respect to past works by Noyes et al. (1984) and by Stepien (1994) - of very late-type stars with $M<0.5~M_{\odot}$, comprising some fully convective stars. For the first time we have also determined the functional dependence on the stellar mass of an empirically X-ray-derived characteristic time scale, $\tau _{\rm e}$, and, independently, we have carried out our analysis also with the B-V color.

However, the clear observational link between $L_{\rm x}/L_{\rm bol}$and the empirical Rossby number does not imply by itself the interpretation of the mass-dependent quantity  $\tau_{\rm e}(M)$ as a convective time. For this reason, we have performed a comparison between our empirical time scale and the theoretical values of the convective turnover time, $\tau_{\rm c}$. We have confirmed that the function  $\tau_{\rm e}(M)$ is very similar to  $\tau_{\rm c}$ for stars in the entire mass range considered (Fig. 10). For fully-convective late-type stars with $M<0.3~M_{\odot}$, we have shown that $\tau_{\rm e}(M)$ is still consistent with the corresponding values predicted by theoretical models. This is a new and significant result, because of insufficient data in previous works for such a comparison.

Another important point is that the function $\tau_{\rm e}(B-V)$ turns out to have the same functional shape as $L_{\rm bol}^{-1/2}$, in the range 0.5<B-V<1.5. For this reason we can assert that for non-saturated stars the $L_{\rm x}$ vs. $P_{\rm rot}^{-2}$relationship is equivalent to the relationship $L_{\rm x}/L_{\rm bol}\propto (P_{\rm rot}/\tau_{\rm e})^{-2}$.

The analysis developed to investigate the physical meaning of an empirical X-ray-derived Rossby number has forced us to revisit the relationship between X-ray emission and rotation. Our results show that the coronal X-ray emission increases roughly as a power law of the stellar rotation period independently of the stellar mass or spectral class, thus demonstrating that the stellar rotation dominates over convection in the magnetic dynamo operating in non-saturated late-type dwarfs, including stars with $M<0.5~M_{\odot}$.

On the other hand, the X-ray emission level from saturated stars depends only on  $L_{\rm bol}$, and hence it appears to be dependent on the characteristics of the stellar structure.

In this contest, the most important result is that, independently of the X-ray emission indicator used in our study, the X-ray emission saturation occurs below a critical rotation period, $P_{\rm rot}^{\rm sat}$, increasing with decreasing stellar mass, in a way that we have quantitatively estimated.

Equivalently, we have also demonstrated that $P_{\rm rot}^{\rm sat}$scales as  $L_{\rm bol}^{-1/2}$: this link suggests that the saturation of the X-ray emission in late-type main sequence stars can be considered as an effect depending on the properties of the convective region as well as an effect depending on the total energy budget available in the star. We believe that in the near future the study of the activity-rotation connection in samples of stars having different evolutionary stages will enable us to test on a more solid basis the relationship between the characteristic time scale  $\tau _{\rm e}$ and the bolometric luminosity, thus contributing to complete our current picture of the stellar magnetic activity.

Acknowledgements

The authors acknowledge partial support for this work from Agenzia Spaziale Italiana, Ministero dell'Università e della Ricerca Scientifica e Tecnologica and CNAA. F. D'Antona is also gratefully acknowledged for clarifying discussions, and we also thank the referee S. Randich for her useful comments and suggestions.