During the solar magnetic activity cycle the emergence latitudes of sunspots change, leading to the well-known butterfly diagram. This phenomenon is poorly understood for other stars since starspot latitudes are generally unknown. The related changes in starspot rotation rates caused by latitudinal differential rotation can however be measured. Using the set of 3093 Kepler stars with activity cycles identified by Reinhold et al. (2017A&A...603A..52R, Cat. J/A+A/603/A52), we aim to study the temporal change in starspot rotation rates over magnetic activity cycles, and how this relates to the activity level, the mean rotation rate of the star, and its effective temperature. We measured the photometric variability as a proxy for the magnetic activity and the spot rotation rate in each quarter over the duration of the Kepler mission. We phase-fold these measurements with the cycle period. To reduce random errors we perform averages over stars with comparable mean rotation rates and effective temperature at fixed activity-cycle phases. We detect a clear correlation between the variation of activity level and the variation of the starspot rotation rate. The sign and amplitude of this correlation depends on the mean stellar rotation and -- to a lesser extent -- on the effective temperature. For slowly rotating stars (rotation periods between 15-28 days) the starspot rotation rates are clearly anti-correlated with the level of activity during the activity cycles. A transition is observed around rotation periods of 10-15 days, where stars with effective temperature above 4200K instead show positive correlation. Our measurements can be interpreted in terms of a stellar "butterfly diagram", but these appear different from the Sun's since the starspot rotation rates are either in phase or anti-phase with the activity level. Alternatively, the activity cycle periods observed by Kepler are short (around 2.5 years) and may therefore be secondary cycles, perhaps analogous to the solar quasi-biennial oscillations.

Cone search capability for table J/A+A/622/A85/cyclepar (Activity cycle parameters for 3093 stars)