With the existing and upcoming large multi-fibre low-resolution spectrographs, the question arises how precise stellar parameters such as Teff and [Fe/H] can be obtained from low-resolution K-band spectra with respect to traditional photometric temperature measurements. Until now, most of the effective temperatures in galactic Bulge studies come directly from photometric techniques. Uncertainties in interstellar reddening and in the assumed extinction law could lead to large systematic errors. We aim to obtain and calibrate the relation between Teff and the $rm ^{12}CO$ first overtone bands for M giants in the galactic Bulge covering a wide range in metallicity. We use low-resolution spectra for 20 M giants with well-studied parameters from photometric measurements covering the temperature range 3200 < Teff < 4500 K and a metallicity range from 0.5 dex down to -1.2 dex and study the behaviour of Teff and [Fe/H] on the spectral indices. We find a tight relation between Teff and the $rm ^{12}CO(2-0)$ band with a dispersion of 95 K as well as between Teff and the $rm ^{12}CO(3-1)$ with a dispersion of 120 K. We do not find any dependence of these relations on the metallicity of the star, making them relation attractive for galactic Bulge studies. This relation is also not sensitive to the spectral resolution allowing to apply this relation in a more general way. We also found a correlation between the combination of the NaI, CaI and the $rm ^{12}CO$ band with the metallicity of the star. However this relation is only valid for sub-solar metallicities. We show that low-resolution spectra provide a powerful tool to obtain effective temperatures of M giants. We show that this relation does not depend on the metallicity of the star within the investigated range and is also applicable to different spectral resolution.
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