Do some environments favor efficient conversion of molecular gas into stars? To answer this, we need to be able to estimate the H2 mass. Traditionally, this is done using CO observations and a few assumptions but the Herschel observations which cover the far-IR dust spectrum make it possible to estimate the molecular gas mass independently of CO and thus to investigate whether and how the CO traces H2. Previous attempts to derive gas masses from dust emission suffered from biases. Generally, dust surface densities, H I column densities, and CO intensities are used to derive a gas-to-dust ratio (GDR) and the local CO intensity to H2 column density ratio (XCO), sometimes allowing for an additional CO-dark gas component (Kdark). We tested earlier methods, revealing degeneracies among the parameters, and then used a sophisticated Bayesian formalism to derive the most likely values for each of the parameters mentioned above as a function of position in the nearby prototypical low metallicity (12 + log (O/H) ~ 8.4) spiral galaxy M 33. The data are from the IRAM Large Program mapping in the CO(2-1) line along with high-resolution H I and Herschel dust continuum observations. Solving for GDR, XCO, and Kdark in macropixels 500 pc in size, each containing many individual measurements of the CO, H I, and dust emission, we find that (i) allowing for CO dark gas (Kdark) significantly improves fits; (ii) Kdark decreases with galactocentric distance; (iii) GDR is slightly higher than initially expected and increases with galactocentric distance; (iv) the total amount of dark gas closely follows the radially decreasing CO emission, as might be expected if the dark gas is H2 where CO is photodissociated. The total amount of H2, including dark gas, yields an average XCO of twice the galactic value of 2 × 1020 cm-2/ K km s-1, with about 55% of this traced directly through CO. The rather constant fraction of dark gas suggests that there is no large population of diffuse H2 clouds (unrelated to GMCs) without CO emission. Unlike in large spirals, we detect no systematic radial trend in XCO, possibly linked to the absence of a radial decrease in CO line ratios.
