| Summary: | Stars form in cold molecular clouds. However, molecular gas is difficult to observe because the most abundant molecule (H_2) lacks a permanent dipole moment. Rotational transitions of CO are often used as a tracer of H_2, but CO is much less abundant and the conversion from CO intensity to H2 mass is often highly uncertain. Here we present a new method for estimating the column density of cold molecular gas (Σ_(gas)) using optical spectroscopy. We utilize the spatially resolved Hα maps of flux and velocity dispersion from the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey. We derive maps of Σ_(gas) by inverting the multi-freefall star formation relation, which connects the star formation rate surface density (Σ_(SFR)) with Σ_(gas) and the turbulent Mach number (M). Based on the measured range of Σ_(SFR) = 0.005-1.5M⊙ yr^(−1) kpc^(−2) and M=18–130, we predict Σ_(gas) = 7–200 M⊙ pc^(−2) in the star-forming regions of our sample of 260 SAMI galaxies. These values are close to previously measured Σ_(gas) obtained directly with unresolved CO observations of similar galaxies at low redshift. We classify each galaxy in our sample as 'star-forming' (219) or 'composite/AGN/shock' (41), and find that in 'composite/AGN/shock' galaxies the average Σ_(SFR), M and Σ_(gas) are enhanced by factors of 2.0, 1.6 and 1.3, respectively, compared to star-forming galaxies. We compare our predictions of Σ_(gas) with those obtained by inverting the Kennicutt–Schmidt relation and find that our new method is a factor of 2 more accurate in predicting Σ_(gas), with an average deviation of 32 per cent from the actual Σ_(gas). © 2017 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. Accepted 2017 March 22. Received 2017 March 19; in original form 2016 October 15. Published: 27 March 2017. We thank Mark Krumholz and the anonymous referee for their useful comments, which helped to improve this work. CF acknowledges funding provided by the ...
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