  Understanding how molecules and dust might have formed within a rapidly
expanding young supernova remnant is important because of the obvious
application to vigorous supernova activity at very high redshift. In previous
papers, we found that the H2 emission is often quite strong, correlates with
optical low-ionization emission lines, and has a surprisingly high excitation
temperature. Here we study Knot 51, a representative, bright example, for which
we have available long slit optical and NIR spectra covering emission lines
from ionized, neutral, and molecular gas, as well as HST visible and SOAR
Telescope NIR narrow-band images. We present a series of CLOUDY simulations to
probe the excitation mechanisms, formation processes and dust content in
environments that can produce the observed H2 emission. We do not try for an
exact match between model and observations given Knot 51's ambiguous geometry.
Rather, we aim to explain how the bright H2 emission lines can be formed from
within the volume of Knot 51 that also produces the observed optical emission
from ionized and neutral gas. Our models that are powered only by the Crab's
synchrotron radiation are ruled out because they cannot reproduce the strong,
thermal H2 emission. The simulations that come closest to fitting the
observations have the core of Knot 51 almost entirely atomic with the H2
emission coming from just a trace molecular component, and in which there is
extra heating. In this unusual environment, H2 forms primarily by associative
detachment rather than grain catalysis. In this picture, the 55 H2-emitting
cores that we have previously catalogued in the Crab have a total mass of about
0.1 M_sun, which is about 5% of the total mass of the system of filaments. We
also explore the effect of varying the dust abundance. We discuss possible
future observations that could further elucidate the nature of these H2 knots.

Astrophysics of Galaxies