The effect of temperature mixing on the observable (T,beta)-relation of
  interstellar dust clouds
  Detailed studies of the shape of dust emission spectra are possible thanks to
the current instruments capable of observations in several sub-millimetre bands
(e.g., Herschel and Planck). However, some controversy remains even on the
basic effects resulting from the mixing of temperatures along the
line-of-sight.
  Studies have suggested either a positive or a negative correlation between
the colour temperature T_C and the observed spectral index beta_Obs. Our aim is
to show that both cases are possible and to determine the factors leading to
either behaviour. We start by studying the sum of two or three modified black
bodies of different temperature. With radiative transfer modelling, we examine
the probability distributions of the dust mass as a function of the physical
dust temperature. With these results as a guideline, we examine the (T_C,
beta_Obs) relations for different sets of clouds.
  Even in the case of modified blackbodies at temperatures T_0 and T_0+ Delta
T_0, the correlation between T_C and beta_Obs can be either positive or
negative. If one compares models where Delta T_0 is varied, the correlation is
negative. If the models differ in their mean temperature T_0 rather than in
Delta T_0, the correlation remains positive. Radiative transfer models show
that externally heated clouds have different mean temperatures but the widths
of their temperature distributions are rather similar. Thus, the correlation
between T_C and beta_Obs is expected to be positive. The same result applies to
clouds illuminated by external radiation fields of different intensity. For
internally heated clouds a negative correlation is the more likely alternative.
  If the signal-to-noise ratio is high, the observed negative correlation could
be explained by the temperature dependence of the dust optical properties but
that intrinsic dependence could be even steeper than the observed one.
