Cosmic-ray-modified supernova remnant shocks

The time-dependent, two-fluid model for diffusive shock acceleration has been employed to study the evolution of cosmic-ray-modified supernova remnants (SNRs) during the early Sedov phase. We consider the sensitivity of net acceleration efficiency to model assumptions regarding density structure of the external medium as well as time dependence in the diffusion coefficient and the cosmic-ray adiabatic index. Simulations have been carried out treating expansion into an interstellar medium of either a uniform density or a density distribution decreasing as r^-2. Time-dependent behavior of diffusive acceleration in SNR shocks depends sensitively upon the specific heat ratio for the cosmic rays, γ_c, and the mean diffusion coefficient, 〈κ〉, both of which should change as the cosmic-ray distribution evolves. In order to take account of important aspects of the time evolution of the cosmic-ray distribution function within the two-fluid model, we have explored here simple time-dependent models for γ_c and 〈κ〉. To simulate evolution of distributions originating through in situ particle injection from the thermal plasma, γ_c is modeled to decrease exponentially from 5/3 to 4/3 on a time scale t_y = τt_d, where t_d is the mean diffusion time and τ ≳ 10² according to diffusion-advection simulations. We vary the mean diffusion coefficient according to a form developed from the Bohm diffusion model. Of the model assumptions explored here we find greatest sensitivity to time variations in γ_c. As established in earlier calculations, cosmic-ray pressure builds more quickly when γ_c is larger. In these simulations we found that if τ ≳ few times 100, dynamically significant cosmic-ray pressures are produced early in the Sedov phase, before γ_c decreases to near 4/3. That is important, since SNR models with initial γ_c = 4/3 tend to be inefficient cosmic-ray accelerators. If large cosmic-ray pressures can be generated near the beginning of the Sedov phase, then the subsequent energy transfer is dominated by the continued enhancement of a portion of the early cosmic-ray pressure that diffuses upstream of the shock. In this event, such features at later times as the value of γ_c or the injection of fresh, low-energy cosmic rays are not very important to the net acceleration efficiency. If these developments take place, then, for example, rather high net acceleration efficiencies can be achieved even if γ_c ≈ 4/3 over most of the Sedov phase. For the SNR models considered here, the total energy channeled into the cosmic rays can be of order 10% of the initial blast energy for both the uniform density and inverse-square density interstellar medium models.