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State-to-state vibrational kinetics of H$_2$ and H$_2^+$ in a post-shock cooling gas with primordial composition

The radiative cooling of shocked gas with primordial chemical composition is an important process relevant to the formation of the first stars and structures, as well as taking place also in high velocity cloud collisions and supernovae explosions. Among the different processes that need to be considered, the formation kinetics and cooling of molecular hydrogen are of prime interest, since they provide the only way to lower the gas temperature to values well below $\sim$10$^4$~K. In previous works, the internal energy level structure of H$_2$ and its cation has been treated in the approximation of rovibrational ground state at low densities, or trying to describe the dynamics using some arbitrary $v>0$ H$_2$ level that is considered representative of the excited vibrational manifold. In this study, we compute the vibrationally resolved kinetics for the time-dependent chemical and thermal evolution of the post-shock gas in a medium of primordial composition. The calculated non-equilibrium distributions are used to evaluate effects on the cooling function of the gas and on the cooling time. Finally, we discuss the dependence of the results to different initial values of the shock velocity and redshift.