A Detailed Gravitational Lens Model Based on Submillimeter Array and Keck Adaptive Optics Imaging of a Herschel-ATLAS Sub-millimeter Galaxy at z=4.243

We present high-spatial resolution imaging obtained with the Submillimeter Array (SMA) at 880um and the Keck Adaptive Optics (AO) system at Ks-band of a gravitationally lensed sub-millimeter galaxy (SMG) at z=4.243 discovered in the Herschel-Astrophysical Terahertz Large Area Survey. The SMA data (angular resolution ~0.6") resolve the dust emission into multiple lensed images, while the Keck AO Ks-band data (angular resolution ~0.1") resolve the lens into a pair of galaxies separated by 0.3". We present an optical spectrum of the foreground lens obtained with the Gemini-South telescope that provides a lens redshift of z_lens = 0.595 +/- 0.005. We develop and apply a new lens modeling technique in the visibility plane that shows that the SMG is magnified by a factor of mu = 4.1 +/- 0.2 and has an intrinsic infrared (IR) luminosity of L_IR = (2.1 +/- 0.2) x 10^13 Lsun. We measure a half-light radius of the background source of r_s = 4.4 +/- 0.5 kpc which implies an IR luminosity surface density of Sigma_IR = (3.4 +/- 0.9) x 10^11 Lsun kpc^-2, a value that is typical of z > 2 SMGs but significantly lower than IR luminous galaxies at z~0. The two lens galaxies are compact (r_lens ~ 0.9 kpc) early-types with Einstein radii of theta_E1 = 0.57 +/- 0.01 and theta_E2 = 0.40 +/- 0.01 that imply masses of M_lens1 = (7.4 +/- 0.5) x 10^10 Msun and M_lens2 = (3.7 +/- 0.3) x 10^10 Msun. The two lensing galaxies are likely about to undergo a dissipationless merger, and the mass and size of the resultant system should be similar to other early-type galaxies at z~0.6. This work highlights the importance of high spatial resolution imaging in developing models of strongly lensed galaxies discovered by Herschel.