Determination of Starspot Covering Fraction as a function of Stellar Age from Observational Data

The association of starspots with magnetic fields leads to an expectation that quantities which correlate with magnetic field strength may also correlate with {starspot} coverage. Since younger stars spin faster and are more magnetically active, assessing whether {starspot} coverage correlates with shorter rotation periods and stellar youth tests these principles. Here we analyze the {starspot} covering fraction versus stellar age for M{-}, G{-}, K{-}, and F{-}type stars based on previously determined variability and rotation periods of over 30,000 {\textit{Kepler}} main-sequence stars. We determine the correlation between age and variability using single and dual power law best fits. We find that {starspot} coverage does indeed decrease with age. Only when the data {are} binned in an effort to remove the effects of activity cycles of individual stars, do statistically significant power law fits emerge for each stellar type. {Using bin averages,} we then find that the {starspot} covering fraction scales with the {X}-ray to bolometric ratio to the power $\lambda$ with {$0.22\pm 0.03 < \lambda < 0.32\pm 0.09$} for {G-type} stars of rotation period below 15 days and for the full range of F{-} and M{-type} stars. For K{-}type stars, we find two branches of $\lambda$ separated by variability bins, with the lower branch showing nearly constant starspot coverage and the upper branch {$\lambda \sim 0.35\pm 0.04$.} G{-}type stars with periods longer than $15$ days exhibit a transition to steeper power law of {$ \lambda \sim 2.4 \pm 1.0$.} The potential connection to previous rotation-age measurements suggesting a magnetic breaking transition at the solar age, corresponding to period of $24.5$ is also of interest.