An eigenfunction method for particle acceleration at ultra-relativistic shocks

We adapt and modify the eigenfunction method of computing the powerlaw spectrum of particles accelerated at a relativistic shock front via the first-order Fermi process [6] to apply to shocks of arbitrarily high Lorentz factor. The power-law index of accelerated particles undergoing isotropic small-angle scattering at an ultrarelativistic, unmagnetized shock is found to be s = 4.23± 0.2 (where s = d ln f/d ln p, with f the Lorentz-invariant phase-space density and p the momentum), in agreement with the results of Monte-Carlo simulations. We present results for shocks in plasmas with different equations of state and for Lorentz factors ranging from 5 to infinity. THE METHOD We study a stationary shock front in the x− y−plane. The accelerated particles are assumed to be test-particles without influence on the dynamics of the plasma or the jump conditions at the shock-front. The plasma flows along the z-axis, with constant velocities u − in the upstream (z < 0) region and u+ downstream (z > 0), the velocities are related by the Rankine-Hugoniot jump conditions. Test-particles are injected into the acceleration process and their interaction with the plasma flow is assumed to give rise to diffusion in the angle cos μ between a particle’s velocity and the shock normal. In the frame of the shock front this leads to a stationary transport equation valid for the local plasma rest frame and given in mixed coordinates as [6] Γ(u+ μ) ∂f ∂z = ∂ ∂μ Dμμ(1− μ ) ∂f ∂μ (1) where the plasma speed u is measured in units of the speed of light, Γ = (1− u) −1/2 is the Lorentz-factor, f(p, μ, z) is the (Lorentz invariant) phase-space density as 1) homepage: http://www.mpi-hd.mpg.de/theory/ a function of the particle momentum p, direction μ and position. p and μ are measured in the local rest frame of the plasma, whereas z is measured in the rest frame of the shock front. Equation (1) is solved using the separation Ansatz [6]