Diffusive cosmic-ray acceleration: Two-fluid models with in situ injection

We present an enhanced formulation of the two-fluid model for diffusive acceleration at shocks. The formulation explicitly includes in situ injection of cosmic rays within the shock and allows the slope of the cosmicray momentum distribution to change across the shock. In addition, we argue that the two-fluid model has applications even when the energy-weighted mean slope of the particle distribution function, q = - 〈d ln f/d ln p〉 ≤ 4/3 or ≥ 5/3, in contrast to the conclusion of Achterberg, Blandford and Periwal. With the enhanced formulation we compare in steady state shocks the efficiencies of acceleration in shocks with and without in situ injection. Using an injection model fashioned after one studied in fully time-dependent simulations by Falle and Giddings, we show that our two-fluid model gives results consistent with the more complete simulations. We are then able to confirm earlier conclusions by Ellison and Eichler that at moderate Mach numbers the postshock cosmic-ray pressure is relatively insensitive to the details of the injection.