The degeneracy between dust colour temperature and spectral index Comparison of methods for estimating the β(T) relation

Context. Submillimetre dust emission provides information on the physics of interstellar clouds and dust. However, noise can produce a spurious anticorrelation between the colour temperature T-C and the spectral index beta. These artefacts must be separated from the intrinsic beta(T) relation of dust emission. Aims. We compare methods that can be used to analyse the beta(T) relation. We wish to quantify their accuracy and bias, especially for observations similar to those made with Planck and Herschel. Methods. We examine submillimetre observations that are simulated either as simple, modified black body emission or using 3D radiative transfer modelling. We used different methods to recover the (T, beta) values of individual objects and the parameters of the beta(T) relation. In addition to chi(2) fitting, we examined the results of the SIMEX method, basic Bayesian model, hierarchical models, and a method that explicitly assumes a functional form for beta(T). The methods were also applied to one field observed by Herschel. Results. All methods exhibit some bias, even in the idealised case of white noise. The results of the Bayesian method show significantly lower bias than direct chi(2) fits. The same is true for hierarchical models that also result in a smaller scatter in the temperature and spectral index values. However, significant bias was observed in cases with high noise levels. When the signal-to-noise ratios are different for different sources, all beta and T estimates of the hierarchical model are biased towards the relation determined by the data with the highest signal-to-noise ratio. This can significantly alter the recovered beta(T) function. With the method where we explicitly assume a functional form for the beta(T) relation, the bias is similar to the Bayesian method. In the case of the actual Herschel field, all methods agree on some degree of anticorrelation between T and beta. Conclusions. The Bayesian method and the hierarchical models can both reduce the noise-induced parameter correlations. However, all methods can exhibit non-negligible bias. This is particularly true for hierarchical models and observations of varying signal-to-noise ratios and this must be taken into account when interpreting the results.