## 3. The accelerated particle spectrumBelow, 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, , where the Lorentz factor and . An analogous estimate for the boundary acceleration (cf. Eq. 1) yields a somewhat larger energy gain, but for mildly relativistic flows the ratio 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 the shock acceleration becomes the first order process with 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 ## 3.1. Monte Carlo modellingIn 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 and is
advected downstream with the velocity, . In
present simulations we consider three values for the flow velocity,
, 0.5 and 0.9 of the light velocity. The shock
compression ratio 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 (
Below, we consider spectra of particles escaping through the
considered boundaries for the mono-energetic (
or ) seed particle injection either at the
shock () or at the jet side boundary far
upstream of the shock (), and for different
distances (, ) of the
boundaries. The particle distribution
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.
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 (, ) 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 . Therefore, the upper momentum cut-off cannot reach values above the scale .
In Fig. 5 we compare spectra of particles injected at the jet side boundary far upstream of the shock, in the distance , to the spectra of particles injected at the terminal shock. In the former case resulting distributions are very flat or even inverted (, 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.
We also performed simulations with the continuous injection process extended along the jet boundary from up to the shock at . As long as most particles were injected far from the shock the resulting spectra were very similar to the ones obtained with . 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 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 to values
allows the larger 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 and . 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
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
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 . 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,
, and the time required for these particles to
diffuse back to the jet boundary to be further accelerated is also
. 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
In Fig. 7, the spectra for different turbulence levels defined
by the respective values of or 0.0013 are
compared. Because in the simulations with smaller
© European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |