Multi-wavelength lens reconstruction of a $\mathit{\text{Planck}}$ $\mathrm{\&}$ $\mathit{\text{Herschel}}$-detected star-bursting galaxy

We present a source-plane reconstruction of a $Herschel$ and ${\it Planck}$-detectedgravitationally-lenseddustystar-forminggalaxy(DSFG)at$z=1.68$ using {\it Hubble}, Sub-millimeter Array (SMA), and Keck observations. The background sub-millimeter galaxy (SMG) is strongly lensed by a foreground galaxy cluster at $z=0.997$ and appears as an arc of length $\sim 15^{\prime \prime}$ in the optical images. The continuum dust emission, as seen by SMA, is limited to a single knot within this arc. We present a lens model with source plane reconstructions at several wavelengths to show the difference in magnification between the stars and dust, and highlight the importance of a multi-wavelength lens models for studies involving lensed DSFGs. We estimate the physical properties of the galaxy by fitting the flux densities to model SEDs leading to a magnification-corrected star formation rate of $390\pm 60$ M${}_{\odot }$ yr${}^{-1}$ and a stellar mass of $1.1\pm 0.4\times {10}^{11}$ M${}_{\odot }$. These values are consistent with high-redshift massive galaxies that have formed most of their stars already. The estimated gas-to-baryon fraction, molecular gas surface density, and SFR surface density have values of $0.43\pm 0.13$, $350\pm 200$ M${}_{\odot }$ pc${}^{-2}$, and $\sim 12 \pm 7~$M$_{\odot}$yr$^{-1}$kpc$^{-2}$, respectively. The ratio of star formation rate surface density to molecular gas surface density puts this among the most star-forming systems, similar to other measured SMGs and local ULIRGS.