Revisiting the connection between magnetic activity, rotation period, and convective turnover time for main-sequence stars

The connection between stellar rotation, stellar activity, and convective turnover time is revisited with a focus on the sole contribution of magnetic activity to the Ca II H&K emission, the so-called excess flux, and its dimensionless indicator R$^{+}_{rm{HK}}$ in relation to other stellar parameters and activity indicators. Our study is based on a sample of 169 main-sequence stars with directly measured Mount Wilson S-indices and rotation periods. The R$^{+}_{rm{HK}}$ values are derived from the respective S-indices and related to the rotation periods in various $B-V$-colour intervals. First, we show that stars with vanishing magnetic activity, i.e. stars whose excess flux index R$^{+}_{rm{HK}}$ approaches zero, have a well-defined, colour-dependent rotation period distribution; we also show that this rotation period distribution applies to large samples of cool stars for which rotation periods have recently become available. Second, we use empirical arguments to equate this rotation period distribution with the global convective turnover time, which is an approach that allows us to obtain clear relations between the magnetic activity related excess flux index R$^{+}_{rm{HK}}$, rotation periods, and Rossby numbers. Third, we show that the activity versus Rossby number relations are very similar in the different activity indicators. As a consequence of our study, we emphasize that our Rossby number based on the global convective turnover time approaches but does not exceed unity even for entirely inactive stars. Furthermore, the rotation-activity relations might be universal for different activity indicators once the proper scalings are used.