Chemical Evolution in the Milky Way: Rotation-based ages for APOGEE-Kepler cool dwarf stars

We use models of stellar angular momentum evolution to determine ages for $\sim500$ stars in the APOGEE-\textit{Kepler} Cool Dwarfs sample. We focus on lower main-sequence stars, where other age-dating tools become ineffective. Our age distributions are compared to those derived from asteroseismic and giant samples and solar analogs. We are able to recover gyrochronological ages for old, lower-main-sequence stars, a remarkable improvement over prior work in hotter stars. Under our model assumptions, our ages have a median relative uncertainty of $14\%$, comparable to the age precision inferred for more massive stars using traditional methods. We investigate trends of galactic $\alpha$-enhancement with age, finding evidence of a detection threshold between the age of the oldest $\alpha$-poor stars and that of the bulk $\alpha$-rich population. We argue that gyrochronology is an effective tool reaching ages of 10--12 Gyr in K- and early M-dwarfs. Finally, we present the first effort to quantify the impact of detailed abundance patterns on rotational evolution. We estimate a $\sim15\%$ bias in age for cool, $\alpha$-enhanced (+ 0.4 dex) stars when standard solar-abundance-pattern rotational models are used for age inference, rather than models that appropriately account for $\alpha$-enrichment.