Explaining the Observed Relation Between Stellar Activity and Rotation

Observations of late-type main-sequence stars have revealed empirical scalings of coronal activity versus rotation period or Rossby number $Ro$ (a ratio of rotation period to convective turnover time) which has hitherto lacked explanation. For $Ro >> 1$, the activity observed as X-ray to bolometric flux varies as $Ro^{-q}$ with $2\le q \le 3$, whilst $|q| < 0.12$ for $Ro << 1$. Here we explain the transition between these two regimes and the power law in the $Ro >> 1$ regime by constructing an expression for the coronal luminosity based on dynamo magnetic field generation and magnetic buoyancy. We explain the $Ro<<1$ behavior from the inference that observed rotation is correlated with internal differential rotation and argue that once the shear time scale is shorter than the convective turnover time, eddies will be shredded on the shear time scale and so the eddy correlation time actually becomes the shear time and the convection time drops out of the equations. We explain the $Ro >> 1$ behavior using a dynamo saturation theory based on magnetic helicity buildup and buoyant loss.