A SIMPLE NONLINEAR MODEL FOR THE ROTATION OF MAIN-SEQUENCE COOL STARS. I. INTRODUCTION, IMPLICATIONS FOR GYROCHRONOLOGY, AND COLOR-PERIOD DIAGRAMS

We here introduce a simple nonlinear model to describe the rotational evolution of cool stars on the main sequence. It is formulated only in terms of the Rossby number (Ro = P/tau), its inverse, and two dimensionless constants which we specify using solar and open-cluster data. The model has two limiting cases of stellar rotation, previously called C and I, that correspond to two observed sequences of fast and slowly rotating stars in young open clusters. The model describes the evolution of stars from C-type, with particular mass and age dependencies, to I-type, with different mass and age dependencies, through the rotational gap, g, separating them. The proposed model explains various aspects of stellar rotation, and provides an exact expression for the age of a rotating cool star in terms of P and tau, thereby generalizing gyrochronology. Using it, we calculate the time interval required for stars to reach the rotational gap-a monotonically increasing, mildly nonlinear function of tau. Beginning with the range of initial periods indicated by observations, we show that the (mass-dependent) dispersion in rotation period initially increases, and then decreases rapidly with the passage of time. The initial dispersion in period contributes up to 128 Myr to the gyro-age errors of solar-mass field stars. Finally, we transform to color-period space, calculate appropriate isochrones, and show that this model explains some detailed features in the observed color-period diagrams of open clusters, including the positions and shapes of the sequences, and the observed density of stars across these diagrams.