Computation of Diffusive Shock Acceleration Using Stochastic Differential Equations

The present work considers diffusive shock acceleration at non-relativistic shocks using a system of stochastic differential equations (SDE) equivalent to the Fokker-Planck equation. We compute approximate solutions of the transport of cosmic particles at shock fronts with a SDE numerical scheme. Only the first order Fermi process is considered. The momentum gain is given by implicit calculations of the fluid velocity gradients using a linear interpolation between two consecutive time steps. We first validate our procedure in the case of single shock acceleration and retrieve previous analytical derivations of the spectral index for different values of the Péclet number. The spectral steepening effect by synchrotron losses is also reproduced. A comparative discussion of implicit and explicit schemes for different shock thickness shows that implicit calculations extend the range of applicability of SDE schemes to infinitely thin 1D shocks. The method is then applied to multiple shock acceleration that can be relevant for Blazar jets and accretion disks and for galactic centre sources. We only consider a system of identical shocks which free parameters are the distance between two consecutive shocks, the synchrotron losses time and the escape time of the particles. The stationary distribution reproduces quite well the flat differential logarithm energy distribution produced by multiple shock effect, and also the piling-up effect due synchrotron losses at a momentum where they equilibrate the acceleration rate. At higher momenta particle losses dominate and the spectrum drops. The competition between acceleration and loss effects leads to a pile-up shaped distribution which appears to be effective only in a restrict range of inter-shock distances of ∼ 10-100 diffusion lengths. We finally compute the optically thin synchrotron spectrum produced such periodic pattern which can explain flat and/or inverted spectra observed in Flat Radio spectrum Quasars and in the galactic centre. Send offprint requests to: A. Marcowith at e-mail address [email protected]