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

2. Acceleration processes in relativistic jets

The 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 change¹. Basing on such approximation we compare the acceleration at the non-compressive tangential discontinuity at the jet side boundary and the compressive terminal shock discontinuity.

2.1. Shock acceleration in a hot spot

A 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, U, growing up to roughly , and, then, increasing the inclination with further growth of U up to at the highest considered velocity . However, the resulting varying spectral index could still be reasonably approximated with the non-relativistic expression , where R is the shock compression ratio. They also noted that the spectrum inclination depends on the perturbations' spectrum near the shock, in contrast to the non-relativistic case. A qualitatively new possibility was revealed by Kirk & Heavens (1989) who considered the acceleration process in shocks with magnetic fields oblique to the shock normal (see also Ballard & Heavens 1991 and Ostrowski 1991). They demonstrated, again in contrast to the non-relativistic results, that such shocks led to flatter spectra than do the parallel ones, with for the (subluminal) quasi-perpendicular shocks. Their work relied on the assumption of adiabatic invariant conservation for particles interacting with the shock and, thus, was limited to only slightly perturbed background magnetic fields. A different approach to particle acceleration was presented by Begelman & Kirk (1990), who noted that in relativistic shocks most field configurations lead to super-luminal conditions. Then particles can be energized in a single shock transmission only, accompanied with a limited energy gain, but the acceleration in relativistic conditions is more efficient than that predicted by a simple adiabatic theory. The acceleration process in the presence of finite amplitude perturbations of the magnetic field was considered by Ostrowski (1991; 1993a), Ballard & Heavens (1992) and Bednarz & Ostrowski (1996). The considerations involved the Monte Carlo particle simulations for shocks with oblique perturbed magnetic fields. It was noted that the spectral index was not a monotonic function of the perturbation amplitude, enabling for the steeper spectra at intermediate perturbation amplitudes than those for the limits of small and large amplitudes. It has also been revealed that the conditions leading to very flat spectra involve an energetic particle density jump at the shock and probably lead to instability. The acceleration process in the case of a perpendicular shock shows a transition between the compressive acceleration described by Begelman & Kirk (1990) and, at larger perturbations, the regime allowing for formation of a wide range power-law spectrum. As a conclusion of this short review one should note that the present theory is unable to predict the spectral index of particles accelerated at the relativistic shock wave, e.g., the possible range of indices arising in computations for the sub-luminal shocks propagating in the cold (e, p) plasma is .

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 mildly relativistic shocks, in some magnetic field configurations one discovers a possibility to have extremely short acceleration time scales, comparable, or even smaller than the particle gyroperiod in the magnetic field upstream of the shock. It is also noted that there exist a coupling between the acceleration time scale and the resulting particle spectral index. Again, the above variety of different results illustrates the difficulty in providing an accurate acceleration time scale estimate without a detailed knowledge of the conditions in the shock.

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 boundary

A 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, U, is relativistic and the sufficient amount of turbulence on its both sides is present² (Paper I, Ostrowski 1997). If near the jet boundary particles exist with gyroradii (or mean free paths normal to the boundary) comparable to the actual thickness of the shear-layer interface, the acceleration process can be very rapid. One may note, that the particles with energies EeV, of interest here, could satisfy the last condition naturally. Any high energy particle crossing the boundary from, say, region I (within the jet) to region II (off the jet), changes its energy, E, according to the respective Lorentz transformation. It can gain or loose energy. In the case of a uniform magnetic field in region II, the successive transformation at the next boundary crossing, II I, changes the particle energy back to the original value. However, in the presence of perturbations acting at particle orbits between the successive boundary crossings there is a positive mean energy change:

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 p transports the following amount of momentum across the jet's boundary:

where the value of p is given as the one after transmission and the z -axis of the reference frame is chosen along the flow velocity. The numerical factor depends on the scattering conditions near the discontinuity and it can reach values also being a substantial fraction of unity. As a result, there exists a drag force per unit surface of the jet boundary acting on the medium along the jet, of a magnitude order same as the accelerated particles' energy density. Independent of the exact value of , the acceleration process can proceed very fast in the case of the mean magnetic field being parallel to the boundary (cf. Coleman & Bicknell 1988, Fedorenko & Courvoisier 1996) because particles can be removed at a distance from the accelerating interface only in the inefficient process of cross-field diffusion. One may note that in the case of a non-relativistic velocity jump, , the acceleration process becomes of the second-order in and a rather slow one.

2.3. The upper energy limit for the tangential discontinuity acceleration

For 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)³. The actual conditions are different from the plane-parallel case considered in Paper I, but we expect the above estimates to be still valid, as long as the particle gyroradius is smaller than the jet radius . If the perturbations are due to growing instabilities near the jet boundary, one may expect them to reach substantial amplitudes up to and above the kiloparsec scales that are of interest here, leading to magnetic field perturbations⁴ and . Then, the acceleration time scale becomes

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, a is the ratio of the energy density of the ambient photon field relative to that of the magnetic field, X is a quantity for the relative strength of p interactions compared to synchrotron radiation and is the proton Lorentz factor. For cosmic ray protons the acceleration dominates over the losses (Eq-s 4,5) up to the maximum energy

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 efficiency

Let us consider an isotropic power-law phase-space cosmic ray distribution with a cut-off at the momentum : , where H is the Heaviside step function and is the spectral index. For a flat spectrum, , the cosmic ray energy density ( ) peaks near the cut-off. For example, for , 99% of the cosmic ray energy density falls at the last two energy decades. For powerful sources, the rate of energy extraction from the galactic nucleus in a form of jet kinetic energy is estimated at eV/s. If a fraction of this energy is transformed into UHE cosmic rays and the particles are transmitted spherically-symmetric around the source, then the flux at Earth reaches the value eV/s/cm², where is the source distance in units of 10 Mpc. Above we neglect losses between the particle source and the Earth. As the measurements in the range above 1 EeV give the energy flux eV/cm² /s, it is enough to assume a small value to explain the observations with a single nearby source, or the respectively smaller value for numerous sources. For somewhat steeper spectra with , the required particle production efficiency can be an order of magnitude larger. The analogous efficiency estimates are obtained by Rachen & Biermann (1993; see also Rachen et al. (1993) and further references listed in the first section) and for an alternative model, by Kang et al. (1997).

© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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