Robert

Supernova remnants are known as sources of galactic cosmic rays for their non-thermal emission of radio waves, X-rays, and gamma-rays. However, the observed CR spectra are hard to reproduce within the standard acceleration theories based on the assumption of Bohm diffusion and steady-state calculations.

We point out that a time-dependent treatment of the acceleration process together with a self-consistent treatment of the scattering turbulence is necessary.

Therefore we numerically solve the coupled system of transport equations for cosmic rays and isotropic Alfv’enic turbulence. The equations are coupled through the growth rate of the turbulence determined by the cosmic-ray gradient and the spatial diffusion coefficient of cosmic rays given by the spectral energy density of the turbulence. The system is solved on a co-moving expanding grid extending upstream for dozens of shock radii, allowing for self-consistent study of cosmic-ray diffusion in the vicinity of their acceleration site. The transport equation for cosmic rays is solved in a test-particle approach based on pre-calculated hydro models.

We demonstrate that the system is typically not in a steady state. In fact, even after several thousand years of evolution, no equilibrium situation is reached. The resulting time-dependent particle spectra strongly differ from those derived assuming a steady state and Bohm diffusion. The turbulence spectra show that bohmlike diffusion is achieved only in a small energy band. Our results indicate that proper account for the evolution of scattering turbulence is crucial for the formation of the observed soft spectra.