M dwarf luminosity, radius, and $\alpha$-enrichment from $I$-band spectral features

Despite the ubiquity of M dwarfs and their growing importance to studies of exoplanets, Galactic evolution, and stellar structure, methods for precisely measuring their fundamental stellar properties remain elusive. Existing techniques for measuring M dwarf luminosity, mass, radius, or composition are calibrated over a limited range of stellar parameters or require expensive observations. We find a strong correlation between the $K_S$-band luminosity ($M_K$), the observed strength of the $I$-band sodium doublet absorption feature, and [Fe/H] in M dwarfs without strong H$\alpha$ emission. We show that the strength of this feature, coupled with [Fe/H] and spectral type, can be used to derive M dwarf $M_K$ and radius without requiring parallax. Additionally, we find promising evidence that the strengths of the $I$-band sodium doublet and the nearby $I$-band calcium triplet may jointly indicate $\alpha$-element enrichment. The use of these $I$-band features requires only moderate-resolution near-infrared spectroscopy to provide valuable information about the potential habitability of exoplanets around M dwarfs, and surface gravity and distance for M dwarfs throughout the Galaxy. This technique has immediate applicability for both target selection and candidate planet host system characterization for exoplanet missions such as \textit{TESS} and \textit{K2}.