Reconstructing three-dimensional densities from two-dimensional observations of molecular gas

Star formation has long been known to be an inefficient process, in the sense that only a small fraction ϵff of the mass of any given gas cloud is converted to stars per cloud free-fall time. However, developing a successful theory of star formation will require measurements of both the mean value of ϵff and its scatter from one molecular cloud to another. Because ϵff is measured relative to the free-fall time, such measurements require accurate determinations of cloud volume densities. Efforts to measure the volume density from two-dimensional projected data, however, have thus far relied on treating molecular clouds as simple uniform spheres, while their real shapes are likely to be filamentary and their density distributions far from uniform. The resulting uncertainty in the true volume density is likely to be one of the major sources of error in observational estimates of ϵff. In this paper, we use a suite of simulations of turbulent, magnetized, radiative, self-gravitating star-forming clouds in order to examine whether it is possible to obtain more accurate volume density estimates and thereby reduce this error. We create mock observations from the simulations, and show that current analysis methods relying on the spherical assumption likely yield ∼0.26 dex underestimations and ∼0.51 dex errors in volume density estimates, corresponding to a ∼0.13 dex overestimation and a ∼0.25 dex scatter in ϵff, comparable to the scatter in observed cloud samples. We build a predictive model that uses information accessible in two-dimensional measurements - most significantly, the Gini coefficient of the surface density distribution - to produce estimates of the volume density with ∼0.3 dex less scatter. We test our method on a recent observation of the Ophiuchus cloud, and show that it successfully reduces the ϵff scatter.