Starspot rotation rates versus activity cycle phase: Butterfly diagrams of Kepler stars are unlike that of the Sun

Context. 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. Aims. Using the set of 3093 Kepler stars with measured activity cycles, 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. Methods. 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-folded these measurements with the cycle period. To reduce random errors, we performed averages over stars with comparable mean rotation rates and effective temperature at fixed activity-cycle phases. Results. 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 an effective temperature above 4200 K instead show positive correlation. Conclusions. Our measurements can be interpreted in terms of a stellar butterfly diagram, but these appear different from that of the Sun 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.