4 Discussion

The applications presented so far differ in their relative contributions from shear and centrifugal effects. We may estimate the relative strength of these two more transparently by comparing their corresponding acceleration coefficients: for the shear acceleration coefficient we have

$$<\dot{p}>_{\rm s}~=\frac{1}{p'^2}\frac{\partial}{\partial p'}... ...u_{\rm c}~\Gamma) =(4+\alpha)~p'~\frac{\lambda}{v'}~\Gamma~$$ (9)

where $\Gamma$ denotes the viscous energization coefficient given by Eq. (B.10) and $\tau_{\rm c} \propto \tau_0(r)~p'^{\alpha}$ with $\tau_{\rm c}=\lambda/v'$. On the other hand, centrifugal acceleration may be described by the coefficient

$$<\dot{p}>_{\rm c}~=\frac{1}{p'^2}\frac{\partial}{\partial p'}... ...ppa~~ (p'^0)^4~\gamma_f^4~ \frac{\Omega^4~r^2}{c^4}\right)~,$$ (10)

which for highly relativistic particles ( $p'\simeq p'^0$) and $\kappa=v'^2~\tau_{\rm c}/3$ reduces to

$$<\dot{p}>_{\rm c}~=\frac{4+\alpha}{3}~ p'\frac{\lambda}{v'}~ \gamma_f^4~\frac{\Omega^4~r^2}{c^4}\cdot$$ (11)

Hence, for the ratio $c_{\rm s}=~ <\dot{p}>_{\rm c}/<\dot{p}>_{\rm s}$ of centrifugal to shear acceleration we finally arrive at

$$c_{\rm s} \simeq \left\{ \begin{array}{ll} 2.2~ r_{\rm ms}/r & \m... ...\\ 5~ v_{\phi f}^2/(c^2-v_z^2) & \mbox{for flat rotation.} \end{array} \right.$$ (12)

For the case of Keplerian rotation and for the parameters chosen above, the relative strength becomes $c_{\rm s} \simeq 0.05$, i.e. the shear effects essentially dominate over those of centrifugal ones. On the other hand, in the case of galactic rotation we have $c_{\rm s} \simeq (0.3-2)$, which allows a complex interplay between centrifugal and shear acceleration with increasing azimuthal velocity.

Our approach utilises the relativistic diffusive particle transport theory as advanced by Webb (1989) and Webb et al. (1994) and thus assumes the diffusion approximation to be valid, i.e. the deviation of the particle distribution from isotropy to be small. In a strict sense, this requires the particle mean free path $\vert c~\tau_{\rm c}\vert$ to be much smaller than both, the typical length scale for the evolution of the mean (momentum-averaged) distribution function and the typical length scale of variation for the background flow. Our conclusions are therefore of restricted applicability if highly anisotropic distributions are expected as, for example, near ultra-relativistic shocks (cf. Kirk & Schneider 1987; Kirk & Webb 1988).

In the application presented here, energy changes as a result of radiative (e.g. synchrotron) losses or second-order Fermi effects due to stochastic motions of the scattering centres have not been considered. One expects the inclusion of radiative losses to introduce an upper bound to the possible particle energy at the point where acceleration is balanced by losses, thus leading to a cut-off in the particle momentum spectrum. On the other hand, the inclusion of second-order Fermi effects would give an additional diffusion flux in momentum space. It may formally be taken into account by a more careful treatment of the scattering term in the Boltzmann equation. For non-relativistic jet flows, a numerical study of second-order Fermi acceleration has recently been given by Manolakou et al. (1999). Our present omission of second-order Fermi acceleration in the relativistic transport equation appears justifiable for the cases where the typical random velocities of the scattering centres (as measured in the comoving frame, i.e. relative to the flow) are smaller than a product of the radial particle mean free path times the rotational flow velocity gradient. However, one should note, that estimating the effects of second-order Fermi acceleration for the case of flat rotation, for example, clearly show them to be of increasing relevance for decreasing azimuthal flow velocities. A more detailed analysis will thus be given in a subsequent publication, while the purpose of the present model is confined to the analysis of steady-state solutions and the essential physical features of shear and centrifugal acceleration.

So far, shear-type acceleration processes in the context of AGN jets have been investigated by Subramanian et al. (1999) and Ostrowski (1998, 2000): By following the road suggested by Katz (1991), who considered the particle acceleration in a low density corona due to flux tubes anchored in a keplerian accretion disk, Subramanian et al. (1999) investigated the acceleration of protons driven by the shear of the underlying Keplerian accretion disk. They demonstrated that the shear acceleration may transfer the energy required for powering the jet and showed the shear to dominate over second-order Fermi acceleration. However, their model does not deal with the acceleration of particles emitting high energy radiation, but is basically confined to the bulk acceleration of the jet flow up to asymptotic Lorentz factors of $\sim$10. On the other hand, Ostrowski (1998, 2000) examined the acceleration of cosmic ray particles at a sharp tangential flow discontinuity. He demonstrated that both, the acceleration to very high energies as well as the production of flat particle spectra are possible. In order for this model to apply efficiently, one requires several conditions to be satisfied, including the presence of a relativistic velocity difference and a thin (not extended) boundary, as well as a sufficient amount of turbulence on both sides of the boundary. It seems however that such conditions might be realized, for example, at the jet side boundary layer in powerful FR II radio sources. With respect to shear acceleration, our approach can thus be regarded as complementary to the one developed by Ostrowski. While our analysis considers the influence of a gradual (azimuthal) shear profile in the jet interior, the analysis by Ostrowski deals with a sharp tangential shear profile at the jet boundary layer. Generally, one expects both processes to occur and to contribute to the production of high energy particles, their relative contributions being dependent on the specific conditions realized in the jet interior and its boundary.