ROTATION ACTIVITY CORRELATIONS IN K AND M DWARFS. I. STELLAR PARAMETERS AND COMPILATIONS OF v sin i AND P/sin i FOR A LARGE SAMPLE OF LATE K AND M DWARFS*
Reliable determination of Rotation Activity Correlation (RAC's) depends on precise measurements of the following stellar parameters: $T_{eff} parallax, radius, metallicity, and rotational speed $v\sin i$ In this paper, our goal is to focus on the determination of these parameters for a sample of K and M dwarfs. In a future paper (Paper II) we will combine our rotational data with activity data in order to construct RAC's. Here, we report on a determination of effective temperatures based on the (R I)$_C$ color from the calibrations of Mann et al. (2015) and Kenyon Hartmann (1995) for four samples of late K, dM2, dM3 and dM4 stars. We also determine stellar parameters $T_{eff} $log(g) and [M/H] using the PCA based inversion technique for a sample of 105 late K dwarfs. We compile all effective temperatures from the literature for this sample. We determine empirical radius [M/H] correlations in our stellar samples. This allows us to propose new effective temperatures, stellar radii, and metallicities for a large sample of 612 late K and M dwarfs. Our mean radii agree well with those of Boyajian et al. (2012) We analyze HARPS and SOPHIE spectra of 105 late K dwarfs, and have detected $v\sin i$ in 92 stars. In combination with our previous $v\sin i$ measurements in M and K dwarfs, we now derive $P/sin i$ measures for a sample of 418 K and M dwarfs. We investigate the distributions of $P/sin i$ and we show that they are different from one spectral sub type to another at a 99.9\ confidence level.