Stellar parameters of early M dwarfs from ratios of spectral features at optical wavelengths

(Abridged) Low-mass stars have been recognised as promising targets in the search for rocky, small planets with the potential of supporting life. Doppler search programmes using high-resolution spectrographs like HARPS or HARPS-N are providing huge quantities of optical spectra of M dwarfs. We aim to calibrate empirical relationships to determine stellar parameters for early M dwarfs (spectral types M0-M4.5) using the same spectra that are used for the radial velocity determinations. Our methodology consists in the use of ratios of pseudo equivalent widths of spectral features as a temperature diagnostic. Stars with effective temperatures obtained from interferometric estimates of their radii are used as calibrators. Empirical calibrations for the spectral type are also provided. Combinations of features and ratios of features are used to derive calibrations for the stellar metallicity. Our methods are then applied to a large sample of M dwarfs that are being observed in the framework of the HARPS search for extrasolar planets.The derived temperatures and metallicities are used together with photometric estimates of mass, radius, and surface gravity to calibrate empirical relationships for these parameters. A total of 112 temperature sensitive ratios have been calibrated over the range 3100-3950 K, providing Teff values with typical uncertainties of the order of 70 K. Eighty-two ratios of pseudo equivalent widths of features were calibrated to derive spectral types. Regarding stellar metallicity, 696 combinations of pseudo equivalent widths of individual features and temperature-sensitive ratios have been calibrated, over the metallicity range from -0.54 to +0.24 dex, with estimated uncertainties in the range of 0.07-0.10 dex. We provide our own empirical calibrations for stellar mass, radius, and surface gravity.