All Sky dust modelling with Planck, IRAS and WISE Observations

We study the performance of the physical dust model presented by Draine \& Li (2007, DL07) in reproducing the high-resolution Planck, IRAS and \WISE infrared observations.The present work extends to the full sky the dust modelling done on nearby galaxies with Herschel and Spitzer data. We employ the DL07 dust model to generate maps of the dust mass surface density, the dust optical extinction $A_V$, the polycyclic aromatic hydrocarbon (PAH) molecules abundance $q_{PAH}$, and the starlight intensity heating the bulk of the dust, parametrized by Umin.We test the model by comparing these maps with independent estimates.The DL07 reproduces the observed spectral energy distribution (SED) satisfactorily in most of the sky, with small deviations in the Galactic disk near the Galactic centre, and in low Ecliptic latitude areas, presumably due to zodiacal light contamination.In the Andromeda galaxy (M31), the present dust mass estimates agree remarkably well (within 10%) with DL07 estimates based on independent Spitzer and Herschel data.The DL07 $A_V$ estimates agree with those generated by Planck Collaboration XI (2014)using a modified black body (MBB) approach, although DL07 suffers a global normalization discrepancy.In molecular clouds, we compare the DL07 $A_{V}$ estimates with maps generated from stellar optical observations from the 2MASS survey.The DL07 $A_{V}$ estimates are a factor 3  larger than values estimated from 2MASS observations.In the diffuse interstellar medium (ISM), we compare the DL07 optical extinction $A_V$ estimates with optical estimates from $\sim 2\times 10^5$ quasi stellar objects (QSOs) observed in the Sloan digital sky survey (SDSS).The DL07  $A_V$ estimates are larger than those estimated from the QSOs,   and this discrepancy depends strongly on Umin.The shape of the optical extinction curve appears to be independent of Umin.We investigate the dependence of the $A_V$ mismatch as a function of the DL07 fitted parameters.We propose an empirical ``renormalization" of the DL07 $A_V$ estimate, dependent of Umin, that compensates for the systematic differences found here.The renormalization also brings into agreement the  $A_V$ estimates on QSO and molecular clouds.In the diffuse ISM, the DL07 fitting parameter Umin,  effectively determined by the wavelength where the SED peaks, appears to trace variations in the far-infrared opacity of the dust grains.  We provide a family of SEDs per unit optical reddening, parameterized by Umin; these will be the constraints for a next generation of dust models.