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 <= q <= 3, whilst vertical bar q vertical bar < 0.13 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 behaviour 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 behaviour using a dynamo saturation theory based on magnetic helicity buildup and buoyant loss.