Efficiency of Nonlinear Particle Acceleration at Cosmic Structure Shocks

We have calculated the evolution of cosmic-ray (CR) modified astrophysical shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of diffusive shock acceleration (DSA) in one-dimensional quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as preexisting, upstream CR populations. Bohm-like diffusion is assumed. We model shocks similar to those expected around cosmic structure pancakes, as well as other accretion shocks driven by flows with upstream gas temperatures in the range T₀=10⁴-10^7.6 K and shock Mach numbers spanning M_s=2.4-133. We show that CR-modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc>~1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We determine and compare the ``efficiencies'' of CR acceleration in our simulated shocks by calculating the ratio of the total CR energy generated at the shock to the total kinetic energy that would pass through the shock over time in its initial frame of reference. For these models the time-asymptotic value for this efficiency ratio is controlled mainly by shock Mach number, as expected from the aforementioned similarity in CR shocks. In the presence of a preexisting CR population, shock evolution proceeds similarly to that for higher thermal injection rates compared to thermal leakage CR sources alone. This added contribution has little or no impact on the postshock or CR properties of the high Mach number shocks simulated. The modeled high Mach number shocks all evolve toward efficiencies ~50%, regardless of the upstream CR pressure. On the other hand, the upstream CR pressure increases the overall CR energy in moderate strength shocks (M_s~afew), since it is a significant fraction of the shock ram pressure. These shocks have been shown to dominate dissipation during cosmic structure formation, so such enhanced efficiency could significantly increase their potential importance as sources of cosmic rays.