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

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1. Introduction

Jet-like outflows are observed in a number of astrophysical environments, starting from young stars embedded in their parent molecular clouds, up to active extragalactic objects. In the later case, a number of interesting observational phenomena are noted over orders of magnitude of linear sizes. In particular, at the smallest milli-arc-second scales, one often observes relativistic jet velocities, with flow Lorentz factors [FORMULA] reaching the values above 10 (cf. Ghisellini et al. 1996). At larger scales, velocity measurements are more difficult, but without entrainment of large amount of matter near the active galactic nuclear source the jet flow velocity must be also relativistic. A possible loading of a jet with matter is expected to be appended by a substantial amount of turbulence (Henriksen 1987) and related jet kinetic energy dissipation. However, in the FR II radio sources, there are often observed jets efficiently transporting energy to the far-away hot spots and any jet breaking mechanism cannot act too effectively near the central core. Also, the existing hydrodynamical simulations of relativistic jets show for possibility of extended stable jet structures (Marti et al. 1995, 1997; Gómez et al. 1995). Another argument suggesting the relativistic jet speed at all scales, may be based on the visible asymmetry of jets with respect to the nuclear source, if one believes the effect is caused by the high velocity of the essentially bi-symmetric outflow (cf. Bridle et al. 1994). Let us also note that the Meisenheimer et al. (1989) modelling of the shock acceleration process at extragalactic radio-source hot spots yields `the best-guess' jet velocities in the range (0.1, 0.6)

The relativistic movement of the jet leads to shock wave formation in places where an obstacle or perturbation of the flow creates a sudden velocity jump. For jets loaded with a cold plasma the highly oblique conical shocks are formed within the jet tube. These shocks can have a non-relativistic character, involving the velocity jump perpendicular to the shock surface much smaller than the overall jet velocity [FORMULA]. They lead to a limited kinetic energy dissipation and are usually claimed to be responsible for forming the so called `knots' along the jet. A much more powerful shock is formed at the final working surface of the jet. There, a substantial fraction of the jet energy is transferred into heating the jet's plasma, generating strong turbulence, boosting magnetic fields within the turbulent volume, and finally accelerating electrons and nuclei to cosmic ray energies. Rachen & Biermann (1993) considered the process of particle acceleration to ultra-high energies (UHE) at such shocks. They show that given the favourable conditions the UHE particles up to [FORMULA] eV can be formed. Then Rachen et al. (1993) show that assumption of UHE particle acceleration in extragalactic powerful radio sources is compatible with the current measurements of cosmic ray abundances and spectra at energies above [FORMULA] eV. Additionally, the arrival directions of cosmic ray particles observed above 10 EeV are correlated with the local galactic supercluster structure (Stanev et al. 1995; see, also, Medina Tanco et al. 1996, Sigl 1996, Sigl et al. 1995, 1997, Hayashida et al. 1996, Elbert & Sommers 1995, Geddes et al. 1996 and Medina Tanco 1998). An alternative model involving the several-Mpc-scale non-relativistic shocks in galaxy clusters is proposed by Kang et al. (1996; see also Kang et al. 1997).

As noted by us (Ostrowski 1990; henceforth Paper I) a tangential discontinuity of the velocity field can also provide an efficient cosmic ray acceleration site if the considered velocity difference U is relativistic and the sufficient amount of turbulence on both its' sides is present. The problem was extensively discussed in the early eighties by Berezhko with collaborators (see the review by Berezhko 1990) and in the diffusive limit by Earl et al. (1988) and Jokipii et al. (1989). In the present paper we consider the process of ultra high energy cosmic ray acceleration in relativistic jets including the possibility of such boundary layer acceleration. As the considerations of Rachen & Biermann (1993) treat the acceleration process at relativistic shock in a somewhat simplified way (see, also Sigl et al. 1995), in the first part of the next section (Sect. 2.1) we review this process in some detail in order to understand the inter-relations between the conditions existing near the shock, the accelerated particle spectrum and the particle's upper energy limit. Then, in Sect. 2.2, we present a short description of the basic physical model for the considered acceleration process acting at the jet boundary. We show (Sect. 2.3) that in the conditions characteristic for relativistic jets in extragalactic radio sources, particles with energies above [FORMULA] eV can be produced in this process without extreme parameter fitting. The required efficiency is discussed in Sect. 2.4. We confirm the estimates presented previously for the shock acceleration, showing that the UHE particle flux observed at the Earth can be reproduced as a result of acceleration processes in jets of nearby powerful radio sources. In Sect. 3 we discuss the problem of the particles' spectrum. With the use of Monte Carlo simulations, we consider the action of both processes acting near the terminal shock in a relativistic jet. Modification of the spectrum due to varying boundary conditions and jet velocity is discussed for the case of ([FORMULA], p) jets expected to occur in the powerful FRII radio sources (cf. Celotti & Fabian 1993). The derived particle's upper energy limits are above the shock acceleration estimates and the spectrum modification at highest energies can resemble the observed above 10 EeV `ankle' structure. A short summary and final remarks are presented in Sect. 4. A preliminary report about this work was presented in Ostrowski (1993b, 1996).

For the discussion that follows, we consider the jet propagating with the relativistic velocity, [FORMULA]. We use c = 1 units.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998