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Diffusive cosmic ray acceleration at relativistic shock waves with magnetostatic turbulence

The analytical theory of diffusive cosmic ray acceleration at parallel stationary shock waves with magnetostatic turbulence is generalized to arbitrary shock speeds $V_s=β_1c$, including in particular relativistic speeds. This is achieved by applying the diffusion approximation to the relevant Fokker-Planck particle transport equation formulated in the mixed comoving coordinate system. In this coordinate system the particle's momentum coordinates $p$ and $μ=p_{\parallel }/p$ are taken in the rest frame of the streaming plasma, whereas the time and space coordinates are taken in the observer's system. For magnetostatic slab turbulence the diffusion-convection transport equation for the isotropic (in the rest frame of the streaming plasma) part of the particle's phase space density is derived. For a step-wise shock velocity profile the steady-state diffusion-convection transport equation is solved. For a symmetric pitch-angle scattering Fokker-Planck coefficient $D_{μμ}(-μ)=D_{μμ}(μ)$ the steady-state solution is independent of the microphysical scattering details. For nonrelativistic mono-momentum particle injection at the shock the differential number density of accelerated particles is a Lorentzian-type distribution function which at large momenta approaches a power law distribution function $N(p\ge p_c)\propto p^{-ξ}$ with the spectral index $ξ(β_1) =1+[3/(Γ_1\sqrt{r^2-β_1^2}-1)(1+3β_1^2)]$. For nonrelativistic ($β_1\ll 1$) shock speeds this spectral index agrees with the known result $ξ(β_1\ll 1)\simeq (r+2)/(r-1)$, whereas for ultrarelativistic ($Γ_1\gg 1$) shock speeds the spectral index value is close to unity.