Computation of cosmic-ray acceleration by Ito's stochastic differential equations

Details Computation of cosmic-ray acceleration by Ito's stochastic differential equations Computation of cosmic-ray acceleration by Ito's stochastic differential equations

Jun 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994a%26a...286..314k&link_type=abstract

Astronomy and Astrophysics 286, 314-327 (1994)

Astronomy and Astrophysics

Astrophysics

38

Acceleration Of Particles, Radiation Mechanisms: Cyclotron And Synchrotron, Shock Waves, Methods: Numerical, Cosmic Rays

Scientific paper

We describe here a method to solve the general transport equation of cosmic rays numerically, including diffusive shock acceleration, second-order Fermi acceleration, adiabatic gains or losses, and synchrotron losses. We use the equivalence between Fokker-Planck equations and stochastic (ordinary) differential equations (SDEs) to transform the transport equation to a set of SDEs which is easier to implement numerically than the original partial differential equation. We are able to compute the cosmic-ray distribution function in the vicinity of a shock, determining its power-law slopes and cutoff energies for an arbitrary dependence on momentum, spatial position, and time of the diffusion coefficients. We use this method to analyse the influence of Kolmogorov Alfven wave turbulence on the characteristic properties of the solutions of the cosmic-ray transport problem near a strong shock. We have obtained the following results: The momentum spectrum of the accelerated particles is divided into two different regions. At lower momenta the spectrum is dominated by the influence of the second-order Fermi acceleration by Alfven waves in the vicinity of the shock. There is a turnover to a second region at higher energy, which is governed by the first-order Fermi process of shock acceleration. This turnover results from the fact that the typical momentum diffusion-time scales increases faster with momentum than the mean residence time of the particles in a finite acceleration region near the shock. The "first-order part" at higher energies shows a negligible influence of the second-order process. We propose this part to be responsible for the synchrotron spectrum as observed in the radio lobes of extragalactic radio sources. Nevertheless, Alfven speeds in these objects could be large, so that the turnover in the spectrum could move to observable frequencies if the size of the acceleration region is small.

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