Inferring the rotation period distribution of stars from their projected rotation velocities and radii: Application to late-F/early-G Kepler stars

While stellar rotation periods P_rot may be measured from broad-band photometry, the photometric modulation becomes harder to detect for slower rotators, which could bias measurements of the long-period tail of the P_rot distribution. Alternatively, the P_rot distribution of stars can be inferred from their projected rotation velocities vsin i and radii R, without being biased against photometrically quiet stars. We solve this inference problem using a hierarchical Bayesian framework, which (i) is applicable to heteroscedastic measurements of vsin i and R with non-Gaussian uncertainties and (ii) does not require a simple parametric form for the true P_rot distribution. We test the method on simulated data sets and show that the true P_rot distribution can be recovered from ≳ 100 sets of vsin i and R measured with precisions of $1\, \mathrm{km\, s}^{-1}$ and 4 per cent, respectively, unless the true distribution includes sharp discontinuities. We apply the method to a sample of 144 late-F/early-G dwarfs in the Kepler field with vsin i measured from Keck/HIRES spectra, and find that the typical rotation periods of these stars are similar to the photometric periods measured from Kepler light curves: we do not find a large population of slow rotators that are missed in the photometric sample, although we find evidence that the photometric sample is biased for young, rapidly rotating stars. Our results also agree with asteroseismic measurements of P_rot for Kepler stars with similar ages and effective temperatures, and show that $\approx 1.1\, \mathrm{M}_\odot$ stars beyond the middle of their main-sequence lifetimes rotate faster than predicted by standard magnetic braking laws.