## 2. Acceleration processes in relativistic jetsThe present considerations attempt to extend the discussion of
particle acceleration at shock waves formed in the end points of jets
by including an additional acceleration process acting at the jet
boundary layer. For particles with UHE energies both the shock
transition as well as the velocity transition layer between the jet
and the surrounding medium (`cocoon') can be approximated as surfaces
of discontinuous velocity change ## 2.1. Shock acceleration in a hot spotA review of the problems related to energetic particle acceleration
at relativistic shock waves is presented by Ostrowski (1996) and Kirk
(1997). In the present section we summarize some most important
findings in this subject. The main difficulty in considering cosmic
ray acceleration at relativistic shocks arises from the substantial
particle anisotropies involved. A consistent approach to the first
order Fermi acceleration at such a shock propagating along the
background magnetic field (`parallel shock') was conceived by Kirk
& Schneider (1987) who obtained solutions to a kinetic equation of
the Fokker-Planck type with a pitch-angle diffusion scattering term.
More general conditions at the parallel shock were considered by
Heavens & Drury (1988), who took into consideration the fluid
dynamics of relativistic shock waves. For a shock propagating in the
cold (e, p) plasma a trend was revealed to make the accelerated
particle spectrum slightly flatter () for the
shock velocity, To date, there was somewhat superficial information on the
acceleration time scales, , in relativistic
shocks as the applied approaches often neglected or underestimated a
significant factor controlling the acceleration process - the particle
anisotropy. The realistic particle distributions are considered in
Bednarz & Ostrowski (1996; see also Ellison et al. 1990 and Naito
& Takahara 1995 for specific cases) who considered shocks with
oblique, sub- and super-luminal magnetic field configurations and with
finite amplitude perturbations, . At parallel
shocks, diminishes with increasing perturbation
amplitude and the shock velocity . A new feature
discovered in oblique shocks is that due to the cross-field diffusion
can change with in a
non-monotonic way. The acceleration process at the super-luminal shock
leading to the power-law spectrum is possible only in the presence of
a large amplitude turbulence. Then, in contrast to the quasi-parallel
shocks, increases with the increasing wave
amplitude. For A discussion of UHE particle acceleration at mildly relativistic shocks formed at the powerful radio source hot spots was presented by Rachen & Bierman (1993) and Sigl et al. (1995). The partly qualitative considerations show a potential difficulty in accelerating particles to the highest required energies. Besides the difficulties with the acceleration time scale there are even more severe constraints for the particle energy due to the boundary conditions: the finite perpendicular extent of the jet and the finite extent of the shock's downstream region situated within the radio source hot spot. The considerations of the time dependent acceleration at shocks described above (Bednarz & Ostrowski 1996) allow for a very rapid acceleration only in some particular conditions. One should also be aware of the another problem. As discussed above, due to an anisotropic particle distribution at the shock, the different physical factors acting near it can substantially modify the particle energy distribution. Thus, let us stress again, the theory is not able to predict every particular spectral index for the accelerated particles (cf. Ostrowski 1994, 1996) or any particular acceleration time scale and any attempt to compare such existing predictions to the observations bears a substantial degree of arbitrariness. ## 2.2. Acceleration process at the jet side boundaryA tangential discontinuity of the velocity field (or a shear layer)
occurring at the jet side boundary can be an efficient cosmic ray
acceleration site if the considered velocity difference, where is the flow Lorentz factor. The
numerical factor depends on particle anisotropy
at discontinuity. It increases with the growing field perturbations'
amplitude but slowly decreases with growing flow velocity. Particle
simulations described in Paper I give values for
within the strong scattering limit as a
substantial fraction of unity. During the acceleration process,
particle scattering is accompanied by the jet's momentum transfer into
the medium surrounding it. On average, a single particle with momentum
where the value of ## 2.3. The upper energy limit for the tangential discontinuity accelerationFor acceleration at the considered tangential discontinuity the acceleration time scale - as long as the flow is mildly relativistic - can be approximately determined by the mean time between boundary crossings, , and the mean energy gain at the crossing, (Eq. 1; cf. Bednarz & Ostrowski (1996) for the case of non-vanishing correlations between and , when ). Within a simple diffusive model in Paper I, the time is expressed with the use of , where is the particle mean free path. depends on a number of physical factors, including jet velocity and magnetic field structure determining parameters of particle wandering at the acceleration region and the mean energy gain at individual boundary crossing. Thus we express the acceleration time scale as where is a numerical factor depending on the
local conditions at, and near the jet, including the escape boundary
distance, the turbulent magnetic field structure near the flow
discontinuity and the flow velocity. For the strong scattering limit
the estimates of Paper I give the values of
between, say, 10 and 1, if we require the
spectrum to be sufficiently flat, within the velocity range (0.5,
0.99) For numerical evaluations, let us consider the radio source jet on the kpc scale of interest here. To estimate the upper energy limit for accelerated particles, at first one should compare the time scale for energy losses due to radiation and inelastic collisions to the acceleration time scale. The discussion of possible loss processes is presented by Rachen & Biermann (1993), who provide the loss time scale for protons as where is the magnetic field in mGs units,
This equation can easily yield a large limiting with moderate jet parameters (e.g. with , and ). However, one should note that the particle gyroradius provides the minimum for the acceleration region's spatial extent allowing particles to reach the predicted value of . Thus, for the actual particle maximum energy the jet radius should be larger than the respective gyroradius (cf. simulations presented in Sect. 3). From the observations of 6 objects Meisenhaimer et al. (1989) derived the following `best-guess' parameters for the hot spots with emission extending up to the infrared or optical wavelengths: - mGs, hot spot diameters - kpc, jet velocities within , c and the shock compression ratios in the range , . If, with a bit of optimism, one deduces from these estimates the jet (i.e. upstream the shock) parameters mGs, kpc the maximum particle energy can reach eV (). This value is substantially smaller than the upper limit given in Eq. 6 and scales like . ## 2.4. The required efficiencyLet us consider an isotropic power-law phase-space cosmic ray
distribution with a cut-off at the momentum :
, where © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |