We present a sub-50 pc-scale analysis of the gravitational lens system SDP.81 at redshift 3.042 using Atacama Large Millimetre/submillimetre Array ( ALMA ) 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 \pm 0.4 , giving an intrinsic total star-formation rate of 315 \pm 60 M _ { \odot } yr ^ { -1 } and a dust mass of 6.4 \pm 1.5 \times 10 ^ { 8 } M _ { \odot } .
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 \sim 1.9 \times 0.7 kpc disk-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 \pm 10 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } , the disk contains three compact regions with densities that peak between 120–190 \pm 20 M _ { \odot } 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 .