Temporal variation of the photometric magnetic activity for the Sun and Kepler solar-like stars

Context. The photometric time series of solar-like stars can exhibit rotational modulation, that is, brightness variations due to active regions co-rotating with the stellar surface. These signatures allow us to constrain properties of stellar rotation and magnetic activity.Aims. In this work we investigate the behavior, particularly the variability in terms of strength, of the photometric magnetic activity of Kepler solar-like stars and compare it with that of the Sun.Methods. We adopted the photometric magnetic activity proxy, S-ph, which was computed with a cadence of five times the rotation period (P-rot). The average S-ph was taken as the mean activity level, and the standard deviation was taken as a measure of the temporal variation of the magnetic activity over the Kepler observations. We also analyzed Sun-as-a-star photometric data from VIRGO (Variability of Solar Irradiance and Gravity Oscillations). Sun-like stars were selected from a very narrow parameter space around the solar properties, according to the recent Gaia-Kepler stellar properties catalog and the latest Kepler rotation catalog. We also looked into KIC 8006161 (HD 173701), a very active metal-rich G dwarf, and we compared its magnetic activity to that of stars with similar stellar fundamental parameters.Results. We find that the amplitude of S-ph variability is strongly correlated with its mean value, independent of spectral type. An equivalent relationship has previously been found for ground-based observations of chromospheric activity emission and magnetic field strength, but in this work we show that photometric Kepler data also present the same behavior. While, depending on the phase of the cycle, the Sun is among the less active stars, we find that the S-ph circle dot properties are consistent with those observed in Kepler Sun-like stars. KIC 8006161 is, however, among the most active of its peers, which tend to be metal-rich. This results from an underlying relationship between P-rot and metallicity and supports the following interpretation of the magnetic activity of KIC 8006161: its strong activity is a consequence of its high metallicity, which affects the depth of the convection zone and, consequently, the efficiency of the dynamo.