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ALMA reveals a warm and compact starburst around a heavily obscured supermassive black hole at z=4.75

We report ALMA Cycle 0 observations at 1.3mm of LESS J033229.4-275619 (XID403), an Ultraluminous Infrared Galaxy at $z=4.75$ in the Chandra Deep Field South hosting a Compton-thick QSO. The source is not resolved in our data at a resolution of $\sim$0.75 arcsec, placing an upper-limit of 2.5 kpc to the half-light radius of the continuum emission from heated-dust. After deconvolving for the beam size, however, we found a $\sim3σ$ indication of an intrinsic source size of $0.27\pm0.08$ arcsec (Gaussian FWHM), which would correspond to $r_{half}\sim0.9\pm0.3$ kpc. We build the far-IR SED of XID403 by combining datapoints from both ALMA and Herschel and fit it with a modified blackbody spectrum. For the first time, we measure the dust temperature $T_d=58.5\pm5.3$ K in this system, which is comparable to what has been observed in other high-z submillimeter galaxies. The measured star formation rate is SFR=$1020\pm150$ $M_{\odot}$ yr$^{-1}$, in agreement with previous estimates at lower S/N. Based on the measured SFR and source size, we constrain the SFR surface density to be $Σ_{SFR}>26\;M_{\odot}$yr$^{-1}$kpc$^{-2}$ ($\sim200\;M_{\odot}$yr$^{-1}$kpc$^{-2}$ for $r_{half}\sim0.9$ kpc). The compactness of this starburst is comparable to what has been observed in other local and high-z starburst galaxies. If the gas mass measured from previous [CII] and CO(2-1) observations at low resolution is confined within the same dust region, assuming $r_{half}\sim0.9\pm0.3$ kpc, this would produce a column density of $N_H\sim0.3-1.1\times10^{24}$cm$^{-2}$ towards the central SMBH, similar to the column density of $\approx1.4\times10^{24}$cm$^{-2}$ measured from the X-rays. Then, in principle, if both gas and dust were confined on sub-kpc scales, this would be sufficient to produce the observed X-ray column density without any need of a pc-scale absorber [abridged].