  Atmospheric temperatures are to be estimated from thermal emission spectra of
Earth-like exoplanets orbiting M-stars as observed by current and future
planned missions. To this end, a line-by-line radiative transfer code is used
to generate synthetic thermal infrared (TIR) observations. The range of
'observed' intensities provides a rough hint of the atmospheric temperature
range without any a priori knowledge. The equivalent brightness temperature
(related to intensities by Planck's function) at certain wavenumbers can be
used to estimate the atmospheric temperature at corresponding altitudes. To
exploit the full information provided by the measurement we generalize
Chahine's original approach and infer atmospheric temperatures from all
spectral data using the wavenumber-to-altitude mapping defined by the weighting
functions. Chahine relaxation allows an iterative refinement of this 'first
guess'. Analysis of the 4.3{\mu}m and 15{\mu}m carbon dioxide TIR bands enables
an estimate of atmospheric temperatures for rocky exoplanets even for low
signal to noise ratios of 10 and medium resolution. Inference of Trappist-1e
temperatures is, however, more challenging especially for CO2 dominated
atmospheres: the 'standard' 4.3{\mu}m and 15{\mu}m regions are optically thick
and an extension of the spectral range towards atmospheric window regions is
important. If atmospheric composition (essentially CO2 concentration) is known
temperatures can be estimated remarkably well, quality measures such as the
residual norm provide hints on incorrect abundances. In conclusion, temperature
in the mid atmosphere of Earth-like planets orbiting cooler stars can be
quickly estimated from thermal IR emission spectra with moderate resolution.
