Precise and accurate parameters for late-type (late K and M) dwarf stars are important for characterization of any orbiting planets, but such determinations have been hampered by these stars complex spectra and dissimilarity to the Sun. We exploit an
empirically calibrated method to estimate spectroscopic effective temperature ($T_{rm{eff}}$) and the Stefan-Boltzmann law to determine radii of 183 nearby K7-M7 single stars with a precision of 2-5%. Our improved stellar parameters enable us to develop model-independent relations between $T_{rm{eff}}$ or absolute magnitude and radius, as well as between color and $T_{rm{eff}}$. The derived $T_{rm{eff}}$-radius relation depends strongly on [Fe/H], as predicted by theory. The relation between absolute $K_S$ magnitude and radius can predict radii accurate to $simeq$3%. We derive bolometric corrections to the $VR_CI_CgrizJHK_S$ and Gaia passbands as a function of color, accurate to 1-3%. We confront the reliability of predictions from Dartmouth stellar evolution models using a Markov Chain Monte Carlo to find the values of unobservable model parameters (mass, age) that best reproduce the observed effective temperature and bolometric flux while satisfying constraints on distance and metallicity as Bayesian priors. With the inferred masses we derive a semi-empirical mass-absolute magnitude relation with a scatter of 2% in mass. The best-agreement models over-predict stellar $T_{rm{eff}}$s by an average of 2.2% and under-predict stellar radii by 4.6%, similar to differences with values from low-mass eclipsing binaries. These differences are not correlated with metallicity, mass, or indicators of activity, suggesting issues with the underlying model assumptions e.g., opacities or convective mixing length.
We confirm and characterize the exoplanetary systems Kepler-445 and Kepler-446: two mid-M dwarf stars, each with multiple, small, short-period transiting planets. Kepler-445 is a metal-rich ([Fe/H]=+0.25 $pm$ 0.10) M4 dwarf with three transiting plan
ets, and Kepler-446 is a metal-poor ([Fe/H]=-0.30 $pm$ 0.10) M4 dwarf also with three transiting planets. Kepler-445c is similar to GJ 1214b: both in planetary radius and the properties of the host star. The Kepler-446 system is similar to the Kepler-42 system: both are metal-poor with large galactic space velocities and three short-period, likely-rocky transiting planets that were initially assigned erroneously large planet-to-star radius ratios. We independently determined stellar parameters from spectroscopy and searched for and fitted the transit light curves for the planets, imposing a strict prior on stellar density in order to remove correlations between the fitted impact parameter and planet-to-star radius ratio for short-duration transits. Combining Kepler-445, Kepler-446 and Kepler-42, and isolating all mid-M dwarf stars observed by Kepler with the precision necessary to detect similar systems, we calculate that 21 $^{+7}_{-5}$ % of mid-M dwarf stars host compact multiples (multiple planets with periods of less than 10 days) for a wide range of metallicities. We suggest that the inferred planet masses for these systems support highly efficient accretion of protoplanetary disk metals by mid-M dwarf protoplanets.
