Diffusive cosmic ray acceleration at shock waves of arbitrary speed

Abstract

The analytical theory of diffusive acceleration of cosmic rays at parallel stationary shock waves of arbitrary speed with magnetostatic turbulence is developed from first principles. The theory is based on the diffusion approximation to the gyrotropic cosmic ray particle phase space distribution functions in the respective rest frames of the up- and downstream medium. We derive the correct cosmic ray jump conditions for the cosmic ray current and density, and match the up- and downstream distribution functions at the position of the shock. It is essential to account for the different particle momentum coordinates in the up- and downstream media. Analytical expressions for the momentum spectra of shock-accelerated cosmic rays are calculated. These are valid for arbitrary shock speeds including relativistic shocks. The correctly taken limit for nonrelativistic shock speeds leads at relativistic \kr momenta to the power-law momentum spectrum $F_1(y_1)\propto y_1^{-q(r)}$ and at nonrelativistic \kr momenta to the power-law momentum spectrum $F_1(y_1)\propto y_1^{-(1+q(r))}$, where the power-law spectral index $q(r)$ is a factor 2 greater than the standard spectral index from nonrelativistic shock acceleration theory. Moreover, for nonrelativistic shock speeds we calculate for the first time the injection velocity threshold $\beta _c\ge \sqrt{3}\beta_u $, settling the long-standing injection problem for nonrelativistic shock acceleration.