Galactic Magnetic Field Reconstruction II. Constraints from polarized thermal dust sky as seen by $Planck$

The polarized Galactic thermal dust emission constitutes a major probe for the study and the characterization of the Galactic Magnetic Field (GMF). In this paper, we apply the maximum-likelihood analysis that we established in our companion paper (Pelgrims, Mac\'ias-P\'erez & Ruppin) to model the large-scale regular-component of the GMF from the polarized diffuse emission from Galactic thermal dust as measured by $Planck$ at 353 GHz. As a first attempt, we consider three models to describe the dust density distribution across the whole Galaxy and four models for the GMF. All models are parametric and heuristics and leave us with twelve reconstructions of the GMF geometrical structure. These reconstructions are obtained at the $N_{\rm{side}} = 64$ and provide the first constraints on the GMF obtained through Markov Chain Monte Carlo analysis from thermal dust polarization data. This work demonstrates that competitive constraints on the GMF can be obtained from the polarized thermal dust sky as compared to other observational probes