We present the results of a detailed investigation into the physical conditions in interstellar material interacting with the supernova remnant IC 443. Our analysis is based on a comprehensive examination of high-resolution far-ultraviolet spectra obtained with the Space Telescope Imaging Spectrograph onboard the Hubble Space Telescope of two stars behind IC 443. One of our targets (HD 43582) probes gas along the entire line of sight through the supernova remnant, while the other (HD 254755) samples material located ahead of the primary supernova shock front. We identify low velocity quiescent gas in both directions and find that the densities and temperatures in these components are typical of diffuse atomic and molecular clouds. Numerous high velocity components are observed in the absorption profiles of neutral and singly-ionized atomic species toward HD 43582. These components exhibit a combination of greatly enhanced thermal pressures and significantly reduced dust-grain depletions. We interpret this material as cooling gas in a recombination zone far downstream from shocks driven into neutral gas clumps. The pressures derived for a group of ionized gas components at high positive velocity toward HD 43582 are lower than those of the other shocked components, pointing to pressure inhomogeneities across the remnant. A strong very high velocity component near -620 km/s is seen in the absorption profiles of highly-ionized species toward HD 43582. The velocity of this material is consistent with the range of shock velocities implied by observations of soft thermal X-ray emission from IC 443. Moderately high-velocity gas toward HD 254755 may represent shocked material from a separate foreground supernova remnant.
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