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

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4. Final remarks

Application of the `pure' shock acceleration mechanism in order to model UHE particle production in the discussed extragalactic jets meets a number of difficulties. Due to small spatial dimensions of hot spot regions, the diffusive particle escape may become a serious problem for particles, preventing them from reaching energies above [FORMULA] eV. We would like to note that the respective losses would be enhanced in the presence of oblique fields causing the drift motions along the shock surface. Therefore it is not clear if the speeding up of the acceleration process at oblique shocks will be accompanied by the spectrum cut-off energy increase. In the super-luminal magnetic field configuration, the acceleration to UHE can be completely suppressed if there is no other mechanism providing particles in the sub-UHE energy range (cf. Begelman & Kirk 1990, Ostrowski 1993a). Another difficulty may arise due to the high inclination of the spectrum, [FORMULA], expected in some cases in relativistic shocks (Ostrowski 1991, 1993a; Ballard & Heavens 1993). In realistic conditions it can substantially increase the shock wave energy conversion into lower energy cosmic rays. The total efficiency of cosmic ray generation must then be much larger to allow for the required power in UHE particles.

The presented hypothesis for the acceleration mechanism producing the most energetic cosmic ray particles at the jet side-boundary has a number of advantages over the `pure' shock hypothesis. At first, its efficient action is rather weakly dependent on detailed conditions in the acceleration site. The mean magnetic field elongated along the jet axis assumed in the present paper is in no way essential to the presented discussion. One may note, however, that such a field configuration can be produced in a natural way due to any form of viscosity near the jet boundary, including the viscous pull of magnetized plasma mediated by the accelerated particles (Eq. 2, cf. Paper I). Then, the field perturbations required near the boundary are created by growing short wave instabilities at the jet boundary surface, as well as by the anisotropically distributed cosmic rays streaming along the boundary. The required energy expense of the jet for the UHE cosmic ray production is quite reasonable in comparison to the energy available. Within the present model involving relativistic jets, as long as global instabilities do not disrupt the organized jet flow, the cut-off energy can be a factor of a few ones greater than the value obtained in the pure shock acceleration if seed particles for acceleration are injected all the way along the jet boundary. Then, the obtained spectrum can be very flat before the cut-off. One may note that no compression or de-compression is directly connected with this acceleration process and adiabatic losses seem not to be an obstacle here. A slow perpendicular expansion of the jet is in most cases controlled by the magnetic field structure and we do not expect it to play any noticeable role in particle deceleration within the UHE energy range. To recapitulate, we believe that because the respective conditions arise in a natural way, the process of particle acceleration at tangential discontinuity should be seriously considered as an important supplementary process to shock acceleration.

The main difficulty in analysing the details of the particle spectrum comes from an inadequate knowledge of local conditions in the acceleration region inside the jet and in the space surrounding the jet, including the magnetic field strength and configuration, the form and the amplitude of field perturbations, the distance of the `effective' escape boundary or, finally, at low particle energies, the velocity profile of the turbulent shear layer expected to occur at the interface between the jet and the surrounding medium. Of course, observations provide some information about magnetic field magnitude and the mean field structure, but, besides its fragmentary, projection and resolution dependent character, the measurements relate to regions where radiative losses of cosmic ray electrons take place, not necessarily strictly coinciding with the ion acceleration sites. The information to be borrowed from hydrodynamical jet simulations usually refers to the flows with limited values of the magnetic field (cf. Marti et al. 1995, 1997; Gómez et al. 1995). So, the available basic information required for the cosmic ray spectrum derivation is very limited. Fortunately, the rapid acceleration at the velocity discontinuity results in very flat spectra in all situations where particles near the discontinuity have a chance to cross this surface at least few times before the escape. When considering the energy budget of accelerated particles, the flat spectrum means that cosmic ray energy is contained in particles with highest energies. In the conditions of effective acceleration, the upper scale for the UHE particle energy density is provided by the magnetic field energy density near the jet boundary. Possessing the higher energy density, UHE cosmic rays would smear out the discontinuity into a wide shear layer with a much reduced acceleration efficiency.

Finally, let us comment on the particle spectrum in the presence of radiative loses decreasing the cut-off energy in the spectrum. If the loss process can be sufficiently effective to shift the cut-off energy substantially below the `geometric' cut-off present in our simulations, there will occur weaker mixing of the two acceleration processes in forming the total spectrum. No mixing means that there are two acceleration time scales present, the one for the shock acceleration and the other one for the tangential discontinuity acceleration. The spectra formed in these two processes - at the shock and at the jet boundary far from the shock - can be independent, with a different shape and an energy cut-off. For electrons the radiative cut-off can occur at such low energies that the acceleration process at the jet boundary involves only other processes (e.g. the second-order Fermi acceleration, highly oblique shocks, magnetic field reconnection) in the turbulent boundary layer, with the viscous shear acceleration playing only a secondary role (cf. Ostrowski 1997). Without a detailed consideration it is difficult to draw any conclusions about the resulting synchrotron spectra. We would like to note, however, that some observations of the synchrotron optical jets may require such mechanisms to operate (Ostrowski, in preparation). Till now the most detailed information available about the synchrotron jet structure is for the nearby M87 jet (e.g. Sparks et al. 1996, and references therein), but at least five more have been observed. However, the observations are not conclusive in the matter of the particular mechanism responsible for the energetic particle populations present in these sources.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998