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

We present a source-plane reconstruction of a ${\it Herschel}$ and ${\it Planck}$-detected gravitationally-lensed dusty star-forming galaxy (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.