The degeneracy between the dust colour temperature and the spectral
  index. The problem of multiple chi^2 minima
  Because of the Herschel and Planck satellite missions, there is strong
interest in the interpretation the sub-millimetre dust spectra from
interstellar clouds. Much work has been done to understand the dependence
between the spectral index beta_Obs and the colour temperature T_C that is
partly caused by the noise. The (T_C, beta_Obs) confidence regions are
elongated, banana-shaped structures. We studied under which conditions these
exhibit anomalous, strongly non-Gaussian behaviour that could affect the
interpretation of the observed (T_C, beta_Obs) relations. We examined modified
black body spectra and spectra calculated from radiative transfer models of
filamentary clouds at wavelengths 100um-850um. We performed modified black body
fits and examined the structure of the chi^2(T_, beta_Obs) function. We show
cases where, when the signal-to-noise ratio is low, the chi^2 has multiple
local minima in the (T_C, beta_Obs) plane. A small change in the weighting of
the data points can cause the solution to jump to completely different values.
In particular, noise can lead to the appearance of a separate population of
solutions with low colour temperatures and high spectral indices. The anomalies
are caused by the noise but the tendency to show multiple chi^2 minima depends
on the model and the wavelengths analysed. Deviations from the assumed single
modified black body spectrum are not important. The presence of local minima
implies that the results obtained from the chi^2 minimisation depend on the
starting point of the optimisation and may correspond to non-global minima. The
(T_C,beta_Obs) distributions may be contaminated by a few solutions with
unrealistically low colour temperatures and high spectral indices. Proper
weighting must be applied to avoid the determination of the beta_Obs(T_C)
relation to be unduly affected by these measurements.
