We derive total (atomic + molecular) hydrogen densities in giant molecular clouds (GMCs) in the nearby spiral galaxy M33 using a method that views the atomic hydrogen near regions of recent star formation as the product of photodissociation. Far-UV photons emanating from a nearby OB association produce a layer of atomic hydrogen on the surfaces of nearby GMCs. Our approach provides an estimate of the total hydrogen density in these GMCs from observations of the excess far-UV emission that reaches the GMC from the OB association, and the excess 21-cm radio HI emission produced after these far-UV photons convert H2 into HI on the GMC surface. The method provides an alternative approach to the use of CO emission as a tracer of H2 in GMCs, and is especially sensitive to a range of density well below the critical density for CO(1-0) emission. We describe our "PDR method" in more detail and apply it using GALEX far-UV and VLA 21-cm radio data to obtain volume densities in a selection of GMCs in the nearby spiral galaxy M33. We have also examined the sensitivity of the method to the linear resolution of the observations used; the results obtained at 20 pc are similar to those for the larger set of data at 80 pc resolution. The cloud densities we derive range from 1 to 500 cm-3, with no clear dependence on galactocentric radius; these results are generally similar to those obtained earlier in M81, M83, and M101 using the same method.Comment: Accepted for publication in MNRAS. 25 figures, 16 tables, including online-only material
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