Two-fluid models of cosmic-ray modified radiative shocks including the effects of an acoustic instability

Aims. We extend the study of two-fluid models of cosmic-ray modified radiative shocks by including energy transfer from the cosmic rays to the thermal gas due to a cosmic-ray driven acoustic instability. Methods. We perform hydrodynamic simulations to find solutions to the time-dependent, one-dimensional two-fluid equations governing the thermal fluid and cosmic-ray fluid. Source terms transferring energy between the two fluids are included. Results. The magnitudes of the source terms determine the degree to which the cosmic rays modify the radiative shock. We found a range of steady solutions characterised by moderate cosmic-ray acceleration and a compression ratio significantly greater than 7, which is the limit for cosmic-ray dominated shocks. The post-shock temperature and overall compression increase as the source term in the thermal fluid's energy equation increases. For sufficiently strong coupling, the solutions are thermally overstable.