The H$\alpha$ emission of nearby M dwarfs and its relation to stellar rotation

The high-energy emission from low-mass stars is mediated by the magnetic dynamo. Although the mechanisms by which fully convective stars generate large-scale magnetic fields are not well understood, it is clear that, as for solar-type stars, stellar rotation plays a pivotal role. We present 270 new optical spectra of low-mass stars in the Solar Neighborhood. Combining our observations with those from the literature, our sample comprises 2202 measurements or non-detections of H$\alpha$ emission in nearby M dwarfs. This includes 466 with photometric rotation periods. Stars with masses between 0.1 and 0.6 solar masses are well-represented in our sample, with fast and slow rotators of all masses. We observe a threshold in the mass-period plane that separates active and inactive M dwarfs. The threshold coincides with the fast-period edge of the slowly rotating population, at approximately the rotation period at which an era of rapid rotational evolution appears to cease. The well- defined active/inactive boundary indicates that H$\alpha$ activity is a useful diagnostic for stellar rotation period, e.g. for target selection for exoplanet surveys, and we present a mass-period relation for inactive M dwarfs. We also find a significant, moderate correlation between $L_{\mathrm{H}\alpha}/L_{\mathrm{bol}}$ and variability amplitude: more active stars display higher levels of photometric variability. Consistent with previous work, our data show that rapid rotators maintain a saturated value of $L_{\mathrm{H}\alpha}/L_{\mathrm{bol}}$. Our data also show a clear power-law decay in $L_{\mathrm{H}\alpha}/L_{\mathrm{bol}}$ with Rossby number for slow rotators, with an index of $-1.7 \pm 0.1$.