Electron acceleration in parallel relativistic shocks with finite thickness

Virtanen and Vainio (2003a) performed test-particle simulations of electron acceleration in parallel relativistic shock waves with finite width. The simulations trace individual electrons under the 'guiding-center' approximation in a homogeneous background magnetic field with superposed (magnetic) scattering centers frozen-in to the plasma flow. Scatterings off the irregularities are simulated making small random displacements of the tip of the electron's momentum vector using a random generator (Vainio et al., 2000; Virtanen and Vainio, 2003b). The mean free path, X, of all charged particles is taken to be a power-law function of particle rigidity, consistent with the assumed magnetic nature of scattering. They consider relativistic particles with speeds close to that of light and characterized by

Y = E/mc2 >> 1. Such particles are efficiently scattered by Alfvén waves, and these wave-particle interactions can be, to the lowest approximation, described by quasi-linear theory. Thus, the scattering frequency of relativistic particles, v = c/X, of species i is f r \2-q

1 miY J

where vo and q are parameters depending on the spectrum of magnetic fluctuations, and r1 is the upstream bulk-speed Lorentz factor. The scatterings are performed in the local rest frame, denoted by primes, so the Lorentz factor is also measured in that frame. To simplify the numerical treatment there was neglected the dependence of v on pitch angle. In standard quasilinear theory, q is the spectral index of the magnetic fluctuations causing the scattering. Two values for this parameter are considered:

Q1. q = 2 giving an energy-independent mean free path;

Q2. q = 5/3 corresponding to the Kolmogorov spectrum of turbulence.

Proton rigidity at constant Lorentz factor is mpjme times the electron rigidity.

Thus the proton mean free path is (mpjme )-q times the electron mean free path. Thus a shock wave having a thickness lsh of about one thermal-proton mean free path, lsh= c/ Vp (n), (4.29.6)

seems like a thick structure to all electrons with Lorentz factors less than

and electron acceleration at these energies should be modest, resembling adiabatic compression.

Virtanen and Vainio (2003a) studied two velocity profiles, U1 and U2. U1. The tanh profile of Schneider and Kirk (1989):

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