  We used K-band spectra to measure the H2 excitation temperatures in six
molecular knots associated with the filaments in the Crab Nebula. The
temperatures are quite high - in the range T ~ 2000-3000K, just below the H2
dissociation temperature. This is the temperature range over which the H2 1-0
S(1) line at 2.121\mum has its maximum emissivity per unit mass, so there may
be many additional H2 cores with lower temperatures that are too faint to
detect. We also measured the electron density in adjacent ionized gas, which on
the assumption of gas pressure balance indicates densities in the molecular
region n_mol ~ 20,000 H baryons cm-3, although this really is just a lower
limit since the H2 gas may be confined by other means. The excited region may
be just a thin skin on a much more extensive blob of molecular gas that does
not have the correct temperature and density to be as easily detectable. At the
opposite extreme, the observed knots could consist of a fine mist of molecular
gas in which we are detecting essentially all of the H2. Future CO observations
could distinguish between these two cases. The Crab filaments serve as the
nearby laboratories for understanding the very much larger filamentary
structures that have formed in the intracluster medium of cool-core galaxy
clusters.
