  (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.
