(simplified) Results on the properties of warm H2 in 57 normal galaxies are derived from H2 rotational transitions, obtained as part of SINGS. This study extends previous extragalactic surveys of H2, the most abundant constituent of the molecular ISM, to more common systems (L_FIR = e7 to 6e10 L_sun) of all morphological and nuclear types. The S(1) transition is securely detected in the nuclear regions of 86% of SINGS galaxies with stellar masses above 10^9.5 M_sun. The derived column densities of warm H2 (T > ~100 K), even though averaged over kiloparsec-scale areas, are commensurate with those of resolved PDRs; the median of the sample is 3e20 cm-2. They amount to between 1% and >30% of the total H2. The power emitted in the sum of the S(0) to S(2) transitions is on average 30% of the [SiII] line power, and ~4e-4 of the total infrared power (TIR) within the same area for star-forming galaxies, which is consistent with excitation in PDRs. The fact that H2 emission scales tightly with PAH emission, even though the average radiation field intensity varies by a factor ten, can also be understood if both tracers originate predominantly in PDRs, either dense or diffuse. A large fraction of the 25 LINER/Sy targets, however, strongly depart from the rest of the sample, in having warmer H2 in the excited states, and an excess of H2 emission with respect to PAHs, TIR and [SiII]. We propose a threshold in H2 to PAH power ratios, allowing the identification of low-luminosity AGNs by an excess H2 excitation. A dominant contribution from shock heating is favored in these objects. Finally, we detect, in nearly half the star-forming targets, non-equilibrium ortho to para ratios, consistent with FUV pumping combined with incomplete ortho-para thermalization by collisions, or possibly non-equilibrium PDR fronts advancing into cold gas.
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