Recent studies of nearby spiral galaxies suggest that photodissociation regions (PDRs) are capable of producing much of the observed H Ⅰ in galaxy disks. In that case, measurements of the observed H Ⅰ column density and the far-ultraviolet (FUV) photon flux responsible for the photodissociation process provide a new probe of the volume density of the local underlying molecular hydrogen. We develop the method and apply it to the giant Scd spiral galaxy M101 (NGC 5457). The H Ⅰ column density and amount of FUV emission have been measured for a sample of 35 candidate PDRs located throughout the disk of M101 using the Very Large Array and the Ultraviolet Imaging Telescope. We find that, after correction for the best-estimate gradient of metallicity in the interstellar medium (ISM) of M101 and for the extinction of the ultraviolet emission, molecular gas with a narrow range of density from 30-1000 cm~(-3) is found near star-forming regions at all radii in the disk of M101 out to a distance of 12′ ≈ 26 kpc, close to the photometric limit of R_(25) ≈ 13.5′. In this picture, the ISM is virtually all molecular in the inner parts of M101. The strong decrease of the H Ⅰ column density in the inner disk of the galaxy at R_G < 10 kpc is a consequence of a strong increase in the dust-to-gas ratio there, resulting in an increase of the H_2 formation rate on grains and a corresponding disappearance of hydrogen in its atomic form.
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