Towards understanding the Planck thermal dust models

Understanding the properties of dust emission in the microwave domain is an important premise for the next generation of cosmic microwave background (CMB) experiments, devoted to the measurement of the primordial $B$-modes of polarization. In this paper, we compare three solutions to thermal dust emission by the Planck Collaboration \cite{PlanckDust03,planck_16,planck_com} to point out significant differences between their respective parameters (the spectral index $\beta$, the optical depth $\tau$ and the dust temperature $T_d$). These differences originate from e.g. the priors on the parameters or the contribution of the Cosmic infrared background (CIB).