On the Activity of Stars (Including Those with Planetary Systems) from Observations of the Kepler and K2 Missions

Abstract—In the first part of our work, we present the results of a comparative analysis of photometric observations of stars using the Kepler mission and its continuation K2. The comparability of the data was estimated by an indirect method based on the values of the stellar surface spottedness parameter $S$ calculated by us, which were determined by the amplitudes of the brightness variability of the objects ${{R}_{{{\text{var}}}}}$ from the observations of the main mission and K2. The main conclusion of our work is that the considered data form a single array that can be used in further research. Conclusions are drawn about changes in the activity of stars of different temperatures and how the activity is related to rotation. The conclusion that the data under consideration form a single array allowed us to conduct a generalized analysis of two independent samples of stars with planetary systems in the second part of the work. Using an estimate of the radii of stars (data from the Kepler telescope archive), we obtained the values $A$ of the area of spots on the surface of stars in millionths of the visible hemisphere of the Sun. We found a good agreement between the estimates of the characteristics of spots on the surface of stars with exoplanets based on the observations of the main mission Kepler and its continuation K2. From our list of more than 700 objects with planetary systems, 76 stars with effective temperatures ${{T}_{{{\text{eff}}}}}$ differing from the solar temperature by 100 K were selected for further consideration. For them, conclusions were made about the decrease in the activity of objects with rotation periods of more than 10–12 days. From the consideration of the ratio of the spottedness of objects $A$ and their age $t$ (established by the gyrochronological ratio), it is concluded that stars with an age of less than 1 billion years are the most active. For one set of observations (Q3) of the Kepler mission, a preliminary analysis of photometric observations of 9 objects with the largest values of the spottedness parameter $A$ (more than 10 000 in millionths of the solar hemisphere) has been performed. The variability of their light curves at times comparable to the periods of rotation and its noticeable amplitude, indicating an increased area of spots on the surface, are registered.