Dust Spectral Energy Distributions in the Era of Herschel and Planck: A Hierarchical Bayesian-Fitting Technique

Abstract

We present a hierarchical Bayesian method for fitting infrared spectral energy distributions (SEDs) of dust emission to observed fluxes. Under the standard assumption of optically thin single temperature (T) sources, the dust SED as represented by a power-law-modified blackbody is subject to a strong degeneracy between T and the spectral index \(\beta\). The traditional non-hierarchical approaches, typically based on \(\chi^2\) minimization, are severely limited by this degeneracy, as it produces an artificial anti-correlation between T and \(\beta\) even with modest levels of observational noise. The hierarchical Bayesian method rigorously and self-consistently treats measurement uncertainties, including calibration and noise, resulting in more precise SED fits. As a result, the Bayesian fits do not produce any spurious anti-correlations between the SED parameters due to measurement uncertainty. We demonstrate that the Bayesian method is substantially more accurate than the \(\chi^2\) fit in recovering the SED parameters, as well as the correlations between them. As an illustration, we apply our method to Herschel and submillimeter ground-based observations of the star-forming Bok globule CB244. This source is a small, nearby molecular cloud containing a single low-mass protostar and a starless core. We find that T and \(\beta\) are weakly positively correlated—in contradiction with the \(\chi^2\) fits, which indicate a T-\(\beta\) anti-correlation from the same data set. Additionally, in comparison to the \(\chi^2\) fits the Bayesian SED parameter estimates exhibit a reduced range in values.