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

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3. The accelerated particle spectrum

Below, with the use of the Monte Carlo simulations, we discuss the spectrum of particles accelerated at the jet if the acceleration process at the jet side-boundary is present. Let us note that in ultrarelativistic flows both processes - the acceleration at the terminal shock and at the considered here tangential discontinuity at the jet boundary - are in some way comparable. The mean relative energy gain of a particle at individual interaction with the relativistic shock, [FORMULA], where the Lorentz factor [FORMULA] and [FORMULA]. An analogous estimate for the boundary acceleration (cf. Eq. 1) [FORMULA] yields a somewhat larger energy gain, but for mildly relativistic flows the ratio [FORMULA] is quite close to 1. Therefore the resulting acceleration depends roughly on the number of particle interactions with the shock and the boundary discontinuity, respectively. These numbers depend on numerous factors including the injection site of cosmic ray particles and the transport properties for these particles. For smaller velocities with [FORMULA] the shock acceleration becomes the first order process with [FORMULA] and is expected to dominate over the second order tangential discontinuity acceleration (cf. Sect. 2.2).

As discussed in Sect. 2.1, the spectra of particles accelerated at relativistic shock waves depend in a large extent on the detailed conditions (magnetic field configuration, amplitude of field perturbations, etc.) near the shock. Unfortunately, these conditions are usually poorly known and any consideration of the process must be based on several rough assumptions. Therefore, in the present simulations we do not attempt to reproduce a detailed shape of the particle spectrum in any definite astrophysical object, but, rather, we consider the form of spectrum modifications introduced to the standard power-law with a cut-off shock spectrum by additional acceleration at the jet boundary. In order to limit the number of free parameters we decided to model stationary particle spectra without taking the radiative losses into account, i.e. the upper energy limit of accelerated particles (cf. Rachen & Biermann 1993, Sigl et al. 1995) is fixed by the boundary conditions allowing for the escape of the highest energy particles. Below, we consider the simplest parallel shock configuration. One should note, however, that the derivations of the acceleration time scales in relativistic shocks by Naito & Takahara (1995) and Bednarz & Ostrowski (1996) suggest the possibility of more rapid acceleration in shocks with oblique magnetic fields. The situation with losses being important is commented in Sect. 4.

3.1. Monte Carlo modelling

In the simulations, we simplify the jet structure near the terminal shock as depicted in Fig. 1. The shock is in rest with respect to the cocoon surrounding the jet. The upstream plasma hitting the shock moves with the relativistic velocity [FORMULA] and is advected downstream with the velocity, [FORMULA]. In present simulations we consider three values for the flow velocity, [FORMULA], 0.5 and 0.9 of the light velocity. The shock compression ratio [FORMULA] is derived with the use of approximate formulae presented by Heavens & Drury (1988) for shock propagating in cold electron-proton plasma and a negligible dynamical role of the magnetic field. The analysis of Celotti & Fabian (1993) suggests that (e, p) plasma can be a viable jet content in the strong FRII radio sources. The conditions occurring behind the jet terminal shock due to the flow divergence are modelled by imposing the particle free escape boundary a finite distance, [FORMULA], downstream of the shock and in the adjoining front side of the cocoon (`a front boundary', cf. Fig. 1). We use the jet radius, [FORMULA], as the unit to measure [FORMULA] and other spatial distances. However, one should note that the physical barrier `opposing' particle escape is defined by the downstream diffusive scale [FORMULA], where [FORMULA] is the downstream diffusion coefficient along the shock normal. Here the magnetic field is oriented along this normal and [FORMULA]. The value of [FORMULA] in units of [FORMULA] scales as [FORMULA], where [FORMULA] and [FORMULA] ([FORMULA]) is the particle diffusion coefficient along (across) the magnetic field. Since the finite extent of the actual cocoon and all realistic magnetic field structures therein allow for particle escape to the sides we introduce another, tube-like free escape boundary surrounding the jet (`a side boundary') in a distance of [FORMULA] from the jet axis. The unit for the particle momentum is defined by the `effective' (cf. Bednarz & Ostrowski 1996) magnetic field, [FORMULA], in such a way, that a particle with the momentum [FORMULA] has a gyroradius equal to the jet radius, [FORMULA]. A concept of the effective field is introduced to represent the additional magnetic field power contained in waves perturbing particle trajectories. The action of the effective field is observed in the simulations as bending of the particle trajectory at scales smaller than its gyroradius in the uniform component of the magnetic field, [FORMULA]. With a simplified scattering model applied, [FORMULA] is not uniquely defined (one is unable to evaluate the amount of turbulence at scales smaller than [FORMULA]), in the present simulations we use the value estimated by Bednarz & Ostrowski (1996) as [FORMULA], where [FORMULA] is the maximum angular momentum scattering amplitude and [FORMULA] is the mean scattering time multiplied by the angular gyration velocity (here [FORMULA]). In the present simulations, the values [FORMULA] and 1.05 arise for [FORMULA] and 0.0013, the strong and the weak scattering cases, respectively.

[FIGURE] Fig. 1. A schematic representation of the terminal shock neighbourhood. The velocities and distances used in the text are indicated.

Below, we consider spectra of particles escaping through the considered boundaries for the mono-energetic ([FORMULA] or [FORMULA]) seed particle injection either at the shock ([FORMULA]) or at the jet side boundary far upstream of the shock ([FORMULA]), and for different distances ([FORMULA], [FORMULA]) of the boundaries. The particle distribution [FORMULA] 5, which gives the particle number per logarithmic momentum bandwidth, is derived for particles escaping through the boundaries. For simplicity, in order to limit the number of free parameters in the simulations, we assume the mean magnetic field to be parallel to the jet velocity both within the jet and in the cocoon (cf. Coleman & Bicknell 1988, Fedorenko & Courvoisier 1996). In the examples presented below we consider a (resulting from simulations) ratio [FORMULA] to be either 0.0013 (small ) or 0.97 (large cross field diffusion). We consider these values as the effective ones, representing the transport properties of the medium with realistic magnetic field structures. The geometric pattern of the introduced particle trajectory perturbations' is taken to be the same at all momenta (Ostrowski 1991). Thus both diffusion coefficients are proportional to the particle momentum and the above ratio of diffusion coefficients is constant. Further details of the simulations are described in Appendix A.

At Fig-s 2-4, we consider particle spectra for the seed particle injection at the shock. The fixed spatial distances to the escape boundaries are assumed, but the size of particle trajectory defined by its gyroradius, as well as the spatial diffusion coefficients, increase in proportion to the particle momentum. As a result, the escape probability grows with particle energy providing a cut-off in the spectrum. The energy scale of the cut-off is different for the front boundary spectrum and the side boundary spectrum, with the latter being often larger in our simulations. This difference can occur also in the case of numerical experiments with the jet boundary acceleration turned off. It is due to the fact that particles accelerated at the shock close to the jet boundary have the opportunity to diffuse back upstream across the static cocoon medium, to be accelerated again at the shock and then escape through the side boundary. The difference between these two scales increases for example, with the jet velocity, the extent of the diffusive cocoon, shifting the particle injection site upstream of the shock, increasing the effective particle radial diffusion coefficient. However, one should note that the role of the jet boundary acceleration is limited for the seed particle injection at the shock in the presence of the nearby front escape boundary (cf. Fig. 2, middle panel). As explained below the situation will change drastically for the injection at the jet boundary far upstream of the shock.

[FIGURE] Fig. 2. Particle spectra derived for the acceleration process involving the first-order Fermi acceleration at the jet terminal shock and the acceleration at the jet-cocoon interface. The case with [FORMULA] and a seed particle injection at the shock is considered. The spectrum of all escaping particles is presented with a full line, while with dashed lines the spectra at the respective escape boundaries: medium dashes for the front boundary and long dashes for the side boundary. We consider the case with the front boundary placed [FORMULA] downstream of the shock and the side boundary in the distance [FORMULA] from the jet axis. For the low energy part of the front boundary spectrum we provide a power-law fit representing the `pure' shock spectrum without the cut-off (short dashes). At the successive panels we present the cases for [FORMULA], 0.5 and 0.9 . For comparison, the spectrum formed at the shock with neglected energy changes at the jet side boundary is presented for the case of [FORMULA] (individual points).

In the spectra presented at Fig. 2 three parts can be clearly separated. The first one, with a wavy behaviour, reflects the initial conditions of the mono-energetic injected spectrum interacting with the accelerating surfaces - for the shock injection only the shock acceleration is important in this range. In the remaining part of the spectrum, at energies directly preceding the cut-off energy, the spectrum exhibits some flattening with respect to the inclination of the lower energy part. The low energy section of the spectrum - within computational accuracy - coincides with the analytically derived inclination of the spectrum formed at the infinitely extended shock (cf. Heavens & Drury 1988). The spectrum flattening at larger energies occurs due to additional particle transport from the shock's downstream region to the upstream one through the cocoon surrounding the jet (this effect occurs also if there is no side boundary acceleration !), and inclusion of a very flat spectral component resulting from the side boundary acceleration (see below).

A comparison of particle spectra generated at jets with different velocities is presented in Fig. 3. One may note a systematic shift of the spectrum cut-off toward higher energies with an increase of the jet velocity. Additionally, at the low energy portion of the spectrum, the expected spectral index change can be observed. The influence of varying distances ([FORMULA], [FORMULA]) at particle spectra can be evaluated by inspecting Fig. 4. Decreasing any boundary distance leads to decreasing the cut-off energy, however the actual changes depend in a substantial degree on the leading escape process removing particles from the acceleration region - either the diffusive/free-escape through the front boundary or the radial diffusion toward the side boundary. Let us finally stress, that even for an infinite diffusive volume surrounding the jet, the acceleration efficiency will decrease for particles with momenta [FORMULA]  . Therefore, the upper momentum cut-off cannot reach values above the scale [FORMULA]  .

[FIGURE] Fig. 3. Comparison of the total spectra from Fig. 2 for different flow velocities. The velocities are given near the respective curves.

[FIGURE] Fig. 4. Comparison of particle spectra for different distances to the jet side boundary for [FORMULA]. The case with [FORMULA] and the seed particle injection at the shock is considered. Three spectra are presented for [FORMULA]: a.) [FORMULA], b.) [FORMULA], c.) [FORMULA], and two additional spectra for [FORMULA]: d.) [FORMULA] and e.) [FORMULA].

In Fig. 5 we compare spectra of particles injected at the jet side boundary far upstream of the shock, in the distance [FORMULA], to the spectra of particles injected at the terminal shock. In the former case resulting distributions are very flat or even inverted ([FORMULA], cf. Appendix B). This feature results from the character of the acceleration process with particles having an opportunity to hit the accelerating surface again and again due to an inefficient diffusive escape to the sides. The apparent deficiency of low-energy particles in these spectra results from the fact that most of these particles succeeded in crossing the discontinuity several times before they are able to diffusively escape through the side boundary. In other words, the escape due to particle energy increase (and the corresponding diffusion coefficient increase) is much more effective than the escape caused by diffusion of the low energy particles across the cocoon. If we neglect the tangential discontinuity acceleration, the particle advection toward the terminal shock appended with the side boundary diffusive escape determines an initial phase of the upstream transport. The final energy spectrum is formed at the shock.

[FIGURE] Fig. 5. The particle spectra formed with and without the jet boundary acceleration for [FORMULA]. Spectra formed due to acceleration both at the jet boundary and at the terminal shock are presented with full lines, while the spectra for the neglected boundary acceleration are given with dashed lines. The results are presented for [FORMULA], [FORMULA] and a.) [FORMULA] or b.) [FORMULA].

We also performed simulations with the continuous injection process extended along the jet boundary from [FORMULA] up to the shock at [FORMULA]. As long as most particles were injected far from the shock the resulting spectra were very similar to the ones obtained with [FORMULA].

The role of the cocoon spatial extension in the acceleration process can be evaluated from Figs 4,6. One may note that for the particle injection at the terminal shock in the presence of a nearby front boundary, any change in [FORMULA] is accompanied by only a minor modification to the spectrum seen at highest energies. This behaviour is determined by the efficient particle escape through the front boundary. Only increasing [FORMULA] to values [FORMULA] allows the larger [FORMULA] to increase the spectrum cut-off substantially. However, for the injection far upstream of the shock the main process removing particles from the acceleration is the diffusive escape through the side boundary. In such a case particles can reach larger energies with a more extended cocoon. For the shock injection, the spectra presented in Fig. 6 only insignificantly differ in the momentum range between [FORMULA] and [FORMULA]. This feature illustrates the fact that spectrum inclination depends on the local conditions near the shock if the particle energy is insufficient to allow for non-diffusive escape through the boundary. For the upstream injection the spectrum with smaller [FORMULA] is shifted up in this momentum range because the larger proportion of all particles have a chance to escape at a given momentum. However, the spectrum inclination does not significantly depend on [FORMULA] until a chance for particle escape becomes comparable to the probability of doubling its energy. The process can be described in the following way. After commencing the acceleration process at injection, the energized particles fill diffusively the volume near [FORMULA]. The normal to the jet boundary diffusion coefficient is proportional to particle momentum in our model. Thus the diffusing relativistic particles fill this volume in a time inversely proportional to the particle momentum, [FORMULA], and the time required for these particles to diffuse back to the jet boundary to be further accelerated is also [FORMULA]. As energy gains of particles interacting with the jet boundary - with the mean value depending on the particle anisotropy and the jet velocity - are proportional to p, the resulting spectrum inclination only slightly depends on the size [FORMULA] at energies much smaller than the cut-off energy. A simplified analytic approach to this acceleration process is presented in Appendix B.

[FIGURE] Fig. 6. Comparison of the particle spectra formed with wide ([FORMULA] ; dashed lines (a) and (c)) and narrow ([FORMULA] ; full lines (b) and (d)) cocoon. The results are presented for [FORMULA] and two possibilities for particle injection: [FORMULA] for cases (a) and (b), and [FORMULA] for the cases (c) and (d).

In Fig. 7, the spectra for different turbulence levels defined by the respective values of [FORMULA] or 0.0013 are compared. Because in the simulations with smaller D, particles were injected at larger initial momentum, in order to make the comparison more simple, we scaled (i.e. vertically shifted) the spectrum (a) to coincide in the power-law section with the spectrum (b). One can observe that the cross-field diffusion, changing as [FORMULA], has a substantial influence on the spectrum if the particle radial diffusion is the main process removing particles from the jet vicinity.

[FIGURE] Fig. 7. Particle spectra formed with [FORMULA] (full lines; [FORMULA]) versus the spectra for the small [FORMULA] (dashed lines; [FORMULA]). The spectra (a) and (b) are formed for the shock injection [FORMULA], while the spectra (c) and (d) for the jet boundary injection at [FORMULA]. All presented results were derived with [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998