The Factory and The Beehive II. Activity and Rotation in Praesepe and the Hyades

Open clusters are collections of stars with a single, well-determined age, and can be used to investigate the connections between angular-momentum evolution and magnetic activity over a star's lifetime. We present the results of a comparative study of the relationship between stellar rotation and activity in two benchmark open clusters: Praesepe and the Hyades. As they have the same age and roughly solar metallicity, these clusters serve as an ideal laboratory for testing the agreement between theoretical and empirical rotation-activity relations at $\approx$600 Myr. We have compiled a sample of 720 spectra --- more than half of which are new observations --- for 516 high-confidence members of Praesepe; we have also obtained 139 new spectra for 130 high-confidence Hyads. We have collected rotation periods ($P_{rot}$) for 135 Praesepe members and 87 Hyads. To compare $H\alpha$ emission, an indicator of chromospheric activity, as a function of color, mass, and Rossby number $R_o$, we first calculate an expanded set of $\chi$ values, with which we can obtain the $H\alpha$ to bolometric luminosity ratio, $L_{H\alpha}/L_{bol}$, even when spectra are not flux-calibrated and/or stars lack reliable distances. Our $\chi$ values cover a broader range of stellar masses and colors (roughly equivalent to spectral types from K0 to M9), and exhibit better agreement between independent calculations, than existing values. We find no difference between the two clusters in their $H\alpha$ equivalent width or $L_{H\alpha}/L_{bol}$ distributions, and therefore take the merged $H\alpha$ and $P_{rot}$ data to be representative of 600-Myr-old stars. Our analysis shows that $H\alpha$ activity in these stars is saturated for $R_o\leq0.11^{+0.02}_{-0.03}$. Above that value activity declines as a power-law with slope $\beta=-0.73^{+0.16}_{-0.12}$, before dropping off rapidly at $R_o\approx0.4$...