A homogeneous temperature calibration for K and M giants with an extensionto the coolest stars

In this paper we present a new estimate of the effective temperatures of 32 giant stars in the spectral range K0 to M10. The sample includes also 4 Mi ra stars. The temperatures are based on a homogeneous set of angular diamet ers obtained by our group by the technique of lunar occultations, and using a photometric and spectroscopic coverage with a combination of original me asurements and literature data. Most of this basic material had been presen ted in previous papers, but in the present work we derive improved effectiv e temperatures by using for the first time a grid of numerical center-to-li mb darkening models. We use this revision to derive a new calibration of th e effective temperature of K and M giant stars, which has the advantage of being based on only one, highly homogeneous set of data (while previous cal ibrations often used mixed data sets). The resulting calibration is extende d to the: coolest stars, reaching for the first time M9 for the non-Mira st ars, and M10 for the Mira stars. In this latter case the calibration does n ot account for pulsation phase variations and is only tentative. In the reg ion of overlap with previously existing calibrations, we find a largely sat isfactory agreement, although some differences exist and are discussed. In particular, it appears that the calibration of the effective temperature of cool Mira stars requires additional observational as well as theoretical e ffort. Seven of the stars in our sample appear to exhibit an effective temp erature that departs significantly from the mean relation (approximate to 9 00 K cooler between K1 and M8), and no simple explanation in terms of possi ble bias or experimental error could be found.