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\section{Setup} \section{Methods}  \subsection{Hydrodynamic Simulation}  In order to assess the recovery of information using ALMA, we post-process a self-gravitating radiation-hydrodynamic simulation in which we have complete three-dimensional temperature, density and velocity information. The simulation, th0.1fw0.3, was previously presented in (\citealt{Offner14}, hereafter OA14). We briefly describe the simulation properties here and refer the reader to OA14 for additional details.   The calculation was performed with the {\sc orion} adaptive mesh refinement (AMR) code \cite{truelove98,klein99}. The simulation follows the collapse of an isolated, turbulent low-mass core. It begins with an initially uniform, cold $4\msun$ sphere of radius $R_c=2\times 10^{17}$cm, density $\rho_c=2\times 10^{-19}$ g cm$^{-3}$ and temperature $T_c=10$ K . This core is embedded in a warm, diffuse gas with $\rho=rho_c/100$ and $T=100 T_c$ K. The dense gas is initialized with a grid of random velocity perturbations such that the initial rms velocity dispersion is 0.5 km s$^{-1}$.   Additional levels of adaptive mesh refinement (AMR) are inserted as the core collapses under the influence of gravity. The core itself is resolved with a minimum cell size of $\Delta_{\rm min}\simeq 0.001$ pc, where the maximum level of refinement has $\Delta_{\rm min}\simeq 26 AU$. Once the central region exceeds the maximum grid resolution ($\rho_{\rm max}\simeq 6.5 \times 10^{-15}$ g cm$^{-3}$, e.g., \citealt{truelove98}), a ``star" forms. This star, which is represented by a Lagrangian sink particle, accretes, radiates and launches a collimated bipolar outflow \citep{krumholz04, Offner09, cunninham11}. The rate of mass loss due to the outflow is set to a fixed fraction of the instantaneous accretion rate: $\dot m_w = \f_w \dot m_*$, where $f_w = 0.3$ is the outflow launching rate given by the X-wind model \citep{shu88}. The distribution of outflow momentum is parameterized by a fixed collimation angle, $\theta_0=0.1$, which is empirically determined to be similar to that of observed outflows \citep{matzner99,cunningham11}. Although $\theta_0$ is constant in time, the outflow injection into the AMR grid occurs on such small scales that the outflow properties such as the opening angle and morphology evolve hydrodynamically; these appear to agree well with observed low-mass outflows \citep{Offner11,Offner14}.    By the end of the calculation ($t= 0.5$ Myr), the simulation contains a single star with a mass of $\sim$1.45 $\msun$.   %Over the evolution, the outflow unbinds and ejects a significant fraction of the gas from the domain, producing a star formation efficiency of 47\%, which is comparable to observational and theoretical estimates for dense cores (e.g., \citealt{matzner00,alves07}).   Most of the core mass has been ejected from the domain and the remaining gas has a rms mass-weighted velocity dispersion of $\sim$1 km s$^{-1}$. Since OA14 found that the final stellar mass and star formation efficiency did not depend strongly on $\theta_0$ and $f_w$, we analyze only a single run. However, we note that different initial core masses, rotations, and magnetic field strengths might produce qualitatively different results \citep{machida13}.  \subsection{Molecular Line Modeling}  %  We use the non-local thermodynamical equilibrium radiative transfer code {\sc radmc-3d}\footnote{http://www.ita.uni-heidelberg.de/~dullemond/software/radmc-3d/} to compute the line emission   in $^{12}$CO(1-0) and $^{13}$CO(1-0). We adopt the Large Velocity Gradient (LVG) approximation \citep{shetty11}, which solves for the rotational level populations by solving the equations for local radiative statistical equilllibrium. {\sc radmc-3d} requires 3D input gas densities, velocities and temperatures, which are produced as outputs by the hydrodynamic simulation. We perform the radiative transfer on a uniform $256^3$ grid, where we interpolate all the AMR data to the second refinement level ($\Delta x =0.001$pc). We include turbulent line broadening on scales at and below the grid resolution by adding a constant microturbulence of 0.05 km s$^{-1}$. For $^{12}$CO we smooth the velocity field by using a doppler parameter of 0.025, such that the velocity field is linearly interpolated between velocity jumps greater that 0.025$c_s$, where $c_s$ is the local sound speed. $^{13}$CO has a doppler parameter of 0.25. %This parameter mainly affects the emission in cells abutting the warm atomic gas; it has negligible impact on the emission of the cooler, denser gas.  To obtain the CO abundances from the total gas density, we assume that molecular Hydrogen dominates in all gas cooler than 1,000 K, where $n_{{\rm H}_2}=\rho/(2.8 m_p)$. We adopt constant CO abundances of [$^{12}$CO/H$_2$]=$8.6 \times 10^{-5}$ and [$^{12}$CO/$^{13}$CO]=62 for gas cooler than 900 K; otherwise the CO abundance is set to zero. Thus, line emission only originates in relatively cold gas in the dense core and gas entrained by the outflow; the warm, low-density ambient material and the hot, outflow gas, which is ionized by construction, do not emit. We adopt the molecular collisional coefficients from \citet{schoier05}.   \subsection{Inteferometry Modelling}  The simulation data provided were in data cubes with 256 pixels/channels on each dimension. This represented an area of 0.26pc by 0.26pc, with a 20 km/s wide spectrum centered at the 0-1 transition of ^{13}CO and ^{12}CO.  The Common Astronomy Software Applications (CASA) is a software package that allows us to simulate observations with various interferometers and single dish telescopes. It also has tools that allow us to analyse this data.