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Astron. Astrophys. 335, 134-144 (1998)

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Appendix A: modelling of the particle acceleration process

We consider the particle diffusion in the jet-cocoon region in the case of the mean magnetic field being parallel to the jet velocity both inside and outside the jet. Modelling of the diffusive particle trajectories is based on the small-amplitude pitch-angle scattering approach (Ostrowski 1991). It assumes that the particle diffusion coefficients parallel to the mean field, [FORMULA], and the one perpendicular to the mean field, [FORMULA], are proportional to the particle momentum. Due to this fact, the amount of computations required for reproducing particle diffusive trajectories for low energy particles is much larger than the respective amount for particles with higher energies. Therefore, in order to deal with these low energy particles, we introduced a hybrid approach to derive particle trajectories. For particles near the surfaces of the flow velocity change at the jet boundary and at the shock, and not further than 2 particle gyroradii away in the former case and 1 diffusive scale [FORMULA] ([FORMULA], 2) from the shock surface, we use the exact form of the mentioned pitch-angle scattering method. When the particle diffuses further away from the respective discontinuity, we use the spatial diffusion model involving large time steps. However, the step length is always limited in order to prevent particles from crossing any discontinuity in an individual `diffusive' step - the mean diffusive step allows only for moving a particle 0.25 of the distance to its closest boundary. In these computations we use the values of [FORMULA] and [FORMULA] derived in independent simulations involving the `exact' pitch-angle diffusion procedure. This way we are able to consider particles with momenta which differ in a few orders of magnitude. During the simulations we use a variant of the trajectory splitting procedure described by Ostrowski (1991). A good test for this simulation procedure is provided by the agreement of our shock accelerated spectra fitted at low energies with the values derived analytically by Heavens & Drury (1988).

Appendix B: on the spectrum of particles accelerated at the tangential discontinuity

Let us consider a simple model for particles accelerated at the plane tangential discontinuity surrounded with infinite regions for particle diffusion. We derive the stationary spectrum at the discontinuity for the perpendicular (to discontinuity) diffusion coefficient being proportional to particle energy, [FORMULA]. For relativistic particles with [FORMULA] and the same velocity [FORMULA], the mean time between the successive particle interactions with the discontinuity is also proportional to the particle's energy. Thus the mean rate of particle energy gain is constant, independent of energy, [FORMULA] = const. The transport equation for the phase-space distribution function [FORMULA] has the following form,

[EQUATION]

In the above equation one assumes a continuous particle injection at [FORMULA]. For a general power-law form for [FORMULA] it yields the solution,

[EQUATION]

where [FORMULA] is the Heaviside step function. As long as there are no specific energy scales introduced into the acceleration process the obtained form for [FORMULA] is independent of the velocity difference at the discontinuity (particle anisotropy), and of possible correlation between the interaction time and the energy gain. In the case considered in the present paper [FORMULA] and [FORMULA].

For the discontinuity formed at the jet boundary, the jet radius and the escape boundary radius provide energy scales to the process. As a result, a cut-off occurs at large energies in the spectrum. However, at small energies, where the jet boundary curvature is insignificant and the diffusive regions very extended, the solution should be close to the one given in (B2). In fact, numerical fits for low energy sections of cases (c) and (d) at Fig. 6 give the respective fits [FORMULA] and [FORMULA] (the difference in the obtained spectral indices is not significant, depending on the momentum range used for fitting).

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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