ALMA imaging of SDP. 81-I. A pixelated reconstruction of the far-infrared continuum emission

We present a sub-50 parsec scale analysis of the gravitational lens system SDP. 81 at redshift 3.042 using Atacama Large Millimetre/submillimetre Array science verification data. We model both the mass distribution of the gravitational lensing galaxy and the pixelated surface brightness distribution of the background source using a novel Bayesian technique that fits the data directly in visibility space. We find the 1 and 1.3 mm dust emission to be magnified by a factor of mu(tot) = 17.6 +/- 0.4, giving an intrinsic total star formation rate of 315 +/- 60 M-circle dot yr(-1) and a dust mass of 6.4 +/- 1.5 x 10(8) M-circle dot. The reconstructed dust emission is found to be non-uniform, but composed of multiple regions that are heated by both diffuse and strongly clumped star formation. The highest surface brightness region is a similar to 1.9 x 0.7 kpc disc-like structure, whose small extent is consistent with a potential size-bias in gravitationally lensed starbursts. Although surrounded by extended star formation, with a density of 20-30 +/- 10 M-circle dot yr(-1) kpc(-2), the disc contains three compact regions with densities that peak between 120 and 190 +/- 20 M-circle dot yr(-1) kpc(-2). Such star formation rate densities are below what is expected for Eddington-limited star formation by a radiation pressure supported starburst. There is also a tentative variation in the spectral slope of the different star-forming regions, which is likely due to a change in the dust temperature and/or opacity across the source.