We have observed warm molecular hydrogen in two nearby edge-on disk galaxies, NGC 4565 and NGC 5907, using the Spitzer high-resolution infrared spectrograph. The 0-0 S(0) 28.2 micron and 0-0 S(1) 17.0 micron pure rotational lines were detected out to 10 kpc from the center of each galaxy on both sides of the major axis, and in NGC 4565 the S(0) line was detected at r = 15 kpc on one side. This location lies beyond a steep drop in the radio continuum emission from cosmic rays in the disk. Despite indications that star formation activity decreases with radius, the H2 excitation temperature and the ratio of the H2 line and the far-IR luminosity surface densities, Sigma_L(H2}/Sigma_L(TIR}, change very little as a function of radius, even into the diffuse outer region of the disk of NGC 4565. This suggests that the source of excitation of the H2 operates over a large range of radii, and is broadly independent of the strength and relative location of UV emission from young stars. Although excitation in photodissociation regions is the most common explanation for the widespread H2 emission, cosmic ray heating or shocks cannot be ruled out. The inferred mass surface densities of warm molecular hydrogen in both edge-on galaxies differ substantially, being 4(-60) M_solar/pc^2 and 3(-50) M_solar/pc^2 at r = 10 kpc for NGC 4565 and NGC 5907, respectively. The higher values represent very unlikely point-source upper limits. The point source case is not supported by the observed emission distribution in the spectral slits. These mass surface densities cannot support the observed rotation velocities in excess of 200 km/s. Therefore, warm molecular hydrogen cannot account for dark matter in these disk galaxies, contrary to what was implied by a previous ISO study of the nearby edge-on galaxy NGC 891.
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