We present near-infrared (2.5-5.0 {mu}m) spectral studies of shocked H2 gas in the two supernova remnants IC 443 and HB 21, which are well known for their interactions with nearby molecular clouds. The observations were performed with Infrared Camera (IRC) aboard the AKARI satellite. At the energy range 7000 K <= E(v,J) <= 20000 K, the shocked H2 gas in IC 443 shows an ortho-to-para ratio (OPR) of 2.4+0.3-0.2, which is significantly lower than the equilibrium value 3, suggesting the existence of non-equilibrium OPR. The shocked gas in HB 21 also indicates a potential non-equilibrium OPR in the range of 1.8-2.0. The level populations are well described by the power-law thermal admixture model with a single OPR, where the temperature integration range is 1000-4000 K. We conclude that the obtained non-equilibrium OPR probably originates from the reformed H2 gas of dissociative J-shocks, considering several factors such as the shock combination requirement, the line ratios, and the possibility that H2 gas can form on grains with a non-equilibrium OPR. We also investigate C-shocks and partially-dissociative J-shocks for the origin of the non-equilibrium OPR. However, we find that they are incompatible with the observed ionic emission lines for which dissociative J-shocks are required to explain. The difference in the collision energy of H atoms on grain surfaces would make the observed difference between the OPRs of IC 443 and HB 21, if dissociative J-shocks are responsible for the H2 emission. Our study suggests that dissociative J-shocks can make shocked H2 gas with a non-equilibrium OPR.
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