Supernovae from core-collapse of massive stars drive shocks into the molecular clouds from which the stars formed. Such shocks affect future star formation from the molecular clouds, and the fast-moving, dense gas with compressed magnetic fields is associated with enhanced cosmic rays. This paper presents new theoretical modeling, using the Paris-Durham shock model, and new observations, using the Stratospheric Observatory for Infrared Astronomy (SOFIA), of the H$_2$ S(5) pure rotational line from molecular shocks in the supernova remnant IC443. We generate MHD models for non-steady-state shocks driven by the pressure of the IC443 blast wave into gas of densities $10^3$ to $10^5$ cm$^{-3}$. We present the first detailed derivation of the shape of the velocity profile for emission from H$_2$ lines behind such shocks, taking into account the shock age, preshock density, and magnetic field. For preshock densities $10^3$-$10^5$ cm$^{-3}$, the the predicted shifts of line centers, and the line widths, of the H$_2$ lines range from 20-2, and 30-4 km/s, respectively. The a priori models are compared to the observed line profiles, showing that clumps C and G can be explained by shocks into gas with density 10$^3$ to $2times 10^4$ cm$^{-3}$ and strong magnetic fields. For clump B2 (a fainter region near clump B), the H$_2$ spectrum requires a J-type shock into moderate density (~100 cm$^{-3}$) with the gas accelerated to 100 km/s from its pre-shock location. Clump B1 requires both a magnetic-dominated C-type shock (like for clumps C and G) and a J-type shock (like for clump B1) to explain the highest observed velocities. The J-type shocks that produce high-velocity molecules may be locations where the magnetic field is nearly parallel to the shock velocity, which makes it impossible for a C-type shock (with ions and neutrals separated) to form.
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