Volume 541, May 2012
|Number of page(s)||11|
|Section||Interstellar and circumstellar matter|
|Published online||24 April 2012|
The degeneracy between the dust colour temperature and the spectral index
The problem of multiple χ2 minima
1 Department of Physics, PO Box 64, University of Helsinki, 00014 Helsinki, Finland
2 IAS, Université Paris-Sud, 91405 Orsay Cedex, France
Received: 6 December 2011
Accepted: 5 March 2012
Context. With the current Herschel and Planck satellite missions, there is strong interest in the interpretation of the details of the sub-millimetre dust emission spectra. The data contain information on the properties of the interstellar clouds and the physics of the dust grains. A lot of work has been done to understand the negative correlation observed between the spectral index βObs and the colour temperature TC that in the χ2 fits is partly caused by the observational noise.
Aims. In the (TC, βObs) plane, the confidence regions are elongated, banana-shaped structures. Previous studies have indicated that the errors may exhibit strongly asymmetric features that have important consequences for the investigation of individual objects and the interpretation of the relation between the TC and βObs parameters. We study under which conditions the confidence regions exhibit such anomalous, strongly non-Gaussian behaviour that could affect the interpretation of the observed (TC, βObs) relations.
Methods. We examined a set of modified black body spectra and spectra calculated from radiative transfer models of filamentary interstellar clouds. We analysed simulated observations at discrete wavelengths between 100 μm and 850 μm. We performed modified black body fits and examined the structure of the χ2(TC, βObs) function of the fits.
Results. We demonstrate cases where, when the signal-to-noise ratio is low, the χ2 has multiple local minima in the (TC, βObs) plane. A small change in the weighting of the data points can cause the solution to jump to completely different values. In particular, if there is noise, the analysis of spectra with T > 10 K and βObs ≲ 2 can lead to a separate population of solutions with much lower colour temperature and higher spectral indices. The anomalies are caused by the noise. However, the tendency to show multiple χ2 minima depends on the model (in part via the influence on the signal-to-noise ratios) and on the set of wavelengths included in the analysis. Deviations from the underlying assumption of a single modified black body spectrum are not significant.
Conclusions. The presence of several local minima implies that the results obtained from the χ2 minimisation depend on the starting point of the optimisation and may correspond to non-global minima. Because of the strongly non-Gaussian nature of the errors, the obtained (TC, βObs) distribution 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 βObs(TC) relation to be unduly affected by these measurements.
Key words: ISM: clouds / infrared: ISM / radiative transfer / submillimeter: ISM
© ESO, 2012
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