Statistical Fitting of Evolution Models to Rotation Rates of Sun-Like Stars

We apply for the first time a two-dimensional fitting statistic, $\tau^2$, to rotational evolution models (REMs) of stars (0.1 to 1.3 $M_{\odot}$) on the period-mass plane. The $\tau^2$ statistic simultaneously considers all cluster rotation data to return a goodness of fit, allowing for data-driven improvement of REMs. We construct data sets for Upper Sco, the Pleiades and Praesepe, to which we tune our REMs. We use consistently determined stellar masses (calculated by matching $K_\textrm{s}$ magnitudes to isochrones) and literature rotation periods. As a first demonstration of the $\tau^2$ statistic, we find the best-fitting gyrochronology age for Praesepe, which is in good agreement with the literature. We then systematically vary three parameters which determine the dependence of our stellar wind torque law on Rossby number in the saturated and unsaturated regimes, and the location of the transition between the two. By minimising $\tau^2$, we find best-fit values for each parameter. These values vary slightly between clusters, mass determinations and initial conditions, highlighting the precision of $\tau^2$ and its potential for constraining REMs, gyrochronology, and understanding of stellar physics. Our resulting REMs, which implement the best-possible fitting form of a broken power-law torque, are statistically improved on previous REMs using similar formulations, but still do not simultaneously describe the observed rotation distributions of the lowest masses, which have both slow and fast rotators by the Praesepe age, and the shape of the converged sequence for higher masses. Further complexity in the REMs is thus required to accurately describe the data.