### Speaker

Prof.
Reinhard Schlickeiser
(Ruhr University Bochum)

### Description

The analytical theory of diffusive cosmic ray acceleration at parallel shock waves is generalized to arbitrary shock speeds $V_s=\beta _1c$, 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 $\mu =p_{\parallel }/p$ are taken in the rest frame of the streaming plasma, whereas the time and space coordiantes 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 $D_{\mu \mu }(-\mu )=D_{\mu \mu }(\mu )$ 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 $N(p\ge p_c)\propto p^{-\xi }$ with the spectral index $\xi (\beta _1) =1+[3/(\Gamma _1\sqrt{r^2-\beta _1^2}-1)(1+3\beta _1^2)]$. For nonrelativistic ($\beta _1\ll 1$) shock speds this spectral index agrees with the known result $\xi (\beta _1\ll 1)\simeq (r+2)/(r-1)$, whereas for ultrarelativistic ($\Gamma _1\gg 1$) shock speeds the spectral index value is close to unity.

Registration number following "ICRC2015-I/" | 134 |
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Collaboration | -- not specified -- |

### Primary author

Prof.
Reinhard Schlickeiser
(Ruhr University Bochum)