A set of 27 evolutionary models of cool close binaries was computed under the assumption that their evolution is influenced by the magnetized winds. Initial periods of 1.5, 2.0 and 2.5 d were considered. For each period three values of 1.3, 1.1 and 0
.9 solar mass were taken as the initial masses of the more massive components. Here the results of the computations of the first evolutionary phase are presented, which starts from the initial conditions and ends when the more massive component reaches its critical Roche lobe. In all considered cases this phase lasts for several Gyr. For binaries with the higher total mass and/or longer initial periods this time is equal to, or longer than the main sequence life time of the more massive component. For the remaining binaries it amounts to a substantial fraction of this life time. From the statistical analysis of models, the predicted period distribution of detached binaries with periods shorter than 2 d was obtained and compared to the observed distribution from the ASAS data. An excellent agreement was obtained under the assumption that the period distribution in this range is determined solely by the mass and angular momentum loss due to the magnetized winds. This result indicates, in particular, that virtually all cool detached binaries with periods of a few tenths of a day, believed to be the immediate progenitors of W UMa-type stars, were formed from detached systems with periods around 2-3 d and that magnetic braking is the dominant formation mechanism of cool contact binaries. It operates on the time scale of several Gyr rendering them rather old, with age of 6-10 Gyr. The results of the present analysis will be used as input data to investigate the subsequent evolution of the binaries, through the mass exchange phase and contact or semi-detached configuration till the ultimate merging of the components.
Based on analysis of photometric observations of nearby M type stars obtained with ASAS, 31 periodic variables were detected. The determined periods are assumed to be related to rotation periods of the investigated stars. Among them 10 new variables
with periods longer than 10 days were found, which brings the total number of slowly rotating M stars with known rotation periods to 12 objects. X-ray activity and rotation evolution of M stars follows the trends observed in G-K type stars. Rapidly rotating stars are very active and activity decreases with increasing rotation period but the period-activity relation is mass-dependent which suggests that the rotation period alone is not a proper measure of activity. The investigated stars were grouped according to their mass and the empirical turnover time was determined for each group. It increases with decreasing mass more steeply than for K type stars for which a flat dependence had been found. The resulting Rossby number-activity relation shows an exponential decrease of activity with increasing Rossby number. The analysis of space motions of 27 single stars showed that all rapidly rotating and a few slowly rotating stars belong to young disk (YD) whereas all old disk (OD) stars are slowly rotating. The median rotation period of YD stars is about 2 days and that of OD stars is equal to 47 days, i.e. nearly 25 times longer. The average X-ray flux of OD stars is about 1.7 dex lower than YD stars in a good agreement with the derived Rossby number-activity formula supplemented with rotation-age relation and in a fair agreement with recent observations but in a disagreement with the Skumanich formula supplemented with the activity-rotation relation.
