Context. The structures of the outer atmospheres of red giants are very complex. Recent interpretations of a range of different observations have led to contradictory views of these regions. It is clear, however, that classical model photospheres are inadequate to describe the nature of the outer atmospheres. The notion of large optically thick molecular spheres around the stars (MOLspheres) has been invoked in order to explain spectro-interferometric observations and low-and high-resolution spectra. On the other hand high-resolution spectra in the mid-IR do not easily fit into this picture because they rule out any large sphere of water vapour in LTE surrounding red giants. Aims. In order to approach a unified scenario for these outer regions of red giants, more empirical evidence from different diagnostics are needed. Our aim here is to investigate high-resolution, mid-IR spectra for a range of red giants, spanning spectral types from early K to mid M. We want to study how the pure rotational lines of water vapour change with effective temperature, and whether we can find common properties that can put new constraints on the modelling of these regions, so that we can gain new insights. Methods. We have recorded mid-IR spectra at 12.2−12.4 μm at high spectral resolution of ten well-studied bright red giants, with TEXES mounted on the IRTF on Mauna Kea. These stars span effective temperatures from 3450 K to 4850 K. Results. We find that all red giants in our study cooler than 4300 K, spanning a wide range of effective temperatures (down to 3450 K), show water absorption lines stronger than expected and none are detected in emission, in line with what has been previously observed for a few stars. The strengths of the lines vary smoothly with spectral type. We identify several spectral features in the wavelength region that are undoubtedly formed in the photosphere. From a study of water-line ratios of the stars, we find that the excitation temperatures, in the line-forming regions, are several hundred Kelvin lower than expected from a classical photospheric model. Conclusions. All stars in our sample show several photospheric features in their 12 μm spectra, which can be modelled with a classical model photosphere. However, in all stars showing water-vapour lines (stars cooler than ∼4300 K), the water lines are found to be much deeper than expected. The line ratios of these pure-rotational lines reveal low excitation temperatures. This could either be due to lower temperatures than expected in the outer regions of the photospheres caused by for example extra cooling, or due to non-LTE level populations, affecting the source function and line opacities, but this needs further investigation. We have demonstrated that these diagnostically interesting water lines are a general feature of red giants across spectral types, and we argue for a general explanation of their formation rather than explanations requiring specific properties, such as dust. Since the water lines are neither weak (filled in by emission) ...

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