Acceleration time scale at ultrarelativistic shock waves

The first-order cosmic ray acceleration at ultrarelativistic shocks is investigated using the Monte Carlo method. We apply a method of discrete particle momentum scattering as a model of particle pitch angle diffusion to reproduce highly anisotropic conditions at the shock wave. Shocks with Lorentz factors γ up to 320 and varying magnetic field inclinations ψ are considered. Values of diffusion coefficients upstream in the point where energy spectral indices stabilize to the limit 2.2 were calculated. The obtained acceleration time does not depend on shock conditions. Introduction. Till quite recently the discussion of first order Fermi acceleration at relativistic shocks was restricted to small Lorentz γ-factors (for review see Ostrowski[1] and Kirk[2]). Bednarz and Ostrowski[3] found a simple mechanism that allows particles to be accelerated at shock waves with large Lorentz factors. It implies that an inclination of formed spectrum does not depend on downstream but only on upstream conditions and a shock velocity. The mechanism is a natural proccess accelerating particles in astronomical objects where such shocks occur. Among these phenomena where ultrarelativistic shocks are anticipated are gammaray bursts. They are non-thermal bursts of low energy γ-rays. Typical GRB lasts for about 10 sec. Their broad spectra usually peak between a few 100 keV and a few MeV (for review see Piran [4]). The Lorentz factor of an expanding fireball larger then 300 is required in order to the pair production optical depth to be sufficiently small (Baring [5]). Recently the host galaxy with redshift 3.42 was indentified for GRB971214 (Kulkarni et al. [6]) which confirms the suggestion that they originate at cosmological sources (Meegan et al. [7]). Numerical simulations of the acceleration process. In simulations we follow the procedure used in Bednarz and Ostrowski [8]. Seed particles are injected downstream the shock with the same initial energy. Theirs trajectories are derived in homogenous magnetic field B0 perturbed by inhomogenities which are simulated by particle momentum scattering within a cone with small angular opening ∆Ω less than the particle anisotropy ∼ 1/γ (see Ostrowski [9]). A particle is excluded from the simulation either if it escapes through the free-escape boundary placed far downstream the shock or if it reaches a time or energy larger than the assumed upper limit. These particles are replaced with ones arising from splitting the remaining high-weight particles with preserving their physical parameters. They are not replaced if they reach upper limit of time. -2.5 -2.0 -1.5 -1.0 -0.5 log10 (κ⊥ /κ ) 2.1 2.2 2.3 2.4 2.5