A&A 394, 1141-1144 (2002)

DOI: 10.1051/0004-6361:20021173

**M. Ostrowski ^{1,2} - J. Bednarz^{1}**

1 - Obserwatorium Astronomiczne, Uniwersytet Jagiellonski,
ul. Orla 171, 30-244 Kraków, Poland

2 -
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und
Astrophysik, Ruhr-Universität Bochum, Germany

Received 17 April 2002 / Accepted 23 July 2002

**Abstract**

The first-order Fermi acceleration process at an ultra-relativistic
shock wave is expected to create a particle spectrum with the unique
asymptotic spectral index
.
Below,
we discuss this result and differences in its various derivations, which
- explicitly or implicitly - always require highly turbulent
conditions downstream of the shock. In the presence of medium amplitude
turbulence the generated particle spectrum can be much steeper than the
above asymptotic one. We also note problems with application of the
pitch angle diffusion model for particle transport near the
ultra-relativistic shocks.

**Key words: **acceleration of particles - shock waves - X-rays: bursts - gamma-rays:
bursts - ISM: cosmic rays

In the discussion below we neglect the strictly parallel shocks, where some of our objections can be invalid. However, such shocks are not expected to frequently occur in the universe.

The work of Bednarz & Ostrowski (1997, 1998) was based on Monte Carlo
simulations of particle transport governed by small amplitude pitch
angle scattering. Thus, depending on the scattering parameter (a mean time between successive scattering acts) and
(the maximum angular scattering amplitude), we were able
to model situations with different mean field configurations and
different amounts of turbulence. One should note that the mean field
configuration downstream of the shock was derived here from the mean
upstream field using the appropriate jump conditions and trajectories of
particles interacting with the shock discontinuity were derived exactly
for such fields. A particle trajectory was derived in the respective
local plasma rest frame, with the Lorentz transformation applied at each
particle shock crossing. The approach takes into account correlations in
the process due to the regular part of the magnetic field, but
irregularities responsible for pitch angle scattering are introduced at
random. In order to model particle pitch angle diffusion upstream of the
shock, with nearly a delta-like angular distribution
(
- a momentum vector inclination to the shock
normal), an extremely small scattering amplitude should be used^{},
.
Increasing the
shock Lorentz factor results in decreasing the momentum perturbation
required for its transmission downstream and leaves a shorter time for
this perturbation, *t*_{1}. In the applied pitch angle diffusion approach,
the momentum variation due to the regular component of the magnetic
field scales like *t*_{1}, whence the diffusive change scales like
*t*_{1}^{1/2}. Thus growing
leads to decreasing *t*_{1} and the
diffusive term has to dominate at sufficiently large .
It is
the reason why in our simulations the orientation of the regular
magnetic field ceases to play a role in the limit
,
resulting in the spectral index convergence to its asymptotic value.

However, one should note that with decreasing
and
,
when the interaction proceeds at the sub-resonance
()
spatial scale, a serious physical problem with the applied
approach appears. In order to scatter particle momentum *uniformly*
within a narrow cone centred on the initial momentum, it requires the
short wave turbulence to be non-linear at the shortest scales. In our
discussion of the "effective'' magnetic field, ,
in the pitch angle
diffusion simulations (Bednarz & Ostrowski 1996; cf. Appendix below) we
evaluate the lower limit of such an effective field from the curvature of
simulated particle trajectories as

(1) |

taking into account both the background uniform field and the turbulent component evaluated with the use of the scattering parameters and (in this expression is given in angular units, it stands for ). Assuming the constant pitch angle diffusion coefficient ( ) for a series of computations involving smaller and smaller we had to use , which scales like . As a result, to be consistent with the assumed scattering model, for large shock Lorentz factors the effective magnetic field increases to large values due to the required growing power being concentrated in the short wave turbulence, . Such conditions seem to be unrealistic at least upstream of the shock.

An analogous pitch angle diffusion modelling appended the considerations of
Gallant et al. (1999; for a more detailed description see Achterberg et al. 2001). They considered the highly turbulent conditions near the shock
leading to the particle pitch angle diffusion *with respect to the
shock normal*, i.e. the regular part of the magnetic field was
neglected. These computations gave essentially the same spectral indices
as the asymptotic one derived by Bednarz & Ostrowski (1998). Also, in a
variant of this model with uniform magnetic field upstream of the shock
and fully chaotic turbulent field downstream, the resulting spectral
index did not vary substantially. The physical content of the
discussed model is substantially different from the Bednarz &
Ostrowski one because it neglects the influence of the uniform field (or
long wavelength perturbations with
)
resulting in
magnetic field correlations at both sides of the shock. Thus, it
provides spectra with the asymptotic spectral index at quite moderate
,
a feature also present in the Bednarz & Ostrowski
simulations for parallel shocks. However, if the amplitude of the
magnetic field turbulence is limited, these simulations cannot
reproduce spectrum steepening (or flattening at intermediate Lorentz
factors) in the presence of oblique magnetic fields (cf. Ostrowski 1993;
Bednarz & Ostrowski 1998; Begelman & Kirk 1990). Both the above models
describe essentially the same physical situation for shocks
propagating in the highly turbulent medium and, of course, in rarely -
if ever - occurring parallel relativistic shocks.

An alternative discussion of the acceleration process presented by Gallant & Achterberg (1999) was based on a simple turbulence model. In their approach a highly turbulent magnetic field configuration was assumed upstream and downstream of the shock, idealized as cells filled with randomly oriented, uniform (within a cell) magnetic fields. With such an approach, particles crossing the shock enter a new cell with a randomly selected magnetic field configuration. Thus, there always occur configurations allowing some particles crossing downstream to reach the shock again and again. As a result of successive energy increases of the same finite fraction of accelerated particles, the power law spectrum is formed. In this model there is no need for upstream magnetic field perturbations if the considered oblique magnetic field configuration can turn all upstream particles back to the shock.

Two quasi-analytic approaches to the considered acceleration process
were presented by Kirk et al. (2000) and Vietri (2002). Both provide
methods to solve the Fokker-Planck equation describing particle
advection with the general plasma flow and the small amplitude
scattering of particle pitch angle as measured with respect *to the
shock normal*. The important work of Kirk et al. modified the Kirk &
Schneider (1987) series expansion approach to treat the delta-like
angular distribution upstream of the shock. An analytically more simple
Vietri approach applies convenient ansatz'es for the anisotropic
upstream and downstream particle distributions, resembling the Peacock
(1981) approach to acceleration at "ordinary'' relativistic shocks. Both
methods confirm the results of the earlier numerical modelling. A
deficiency of the above semi-analytic approaches is its inability to
treat situations with mildly perturbed magnetic fields, on average
oblique to the shock normal. If considered valid for different
configurations of the mean magnetic field, these models require the large
amplitude short wave turbulence to remove signatures in particle
trajectories of the uniform background field or of the long wave
perturbations. Thus it provides an alternative description of the same
physical situation discussed earlier with numerical methods by Bednarz
& Ostrowski (1998) in the
limit or their parallel
shock results and all other authors applying small amplitude pitch angle
scattering simulations at parallel shock waves.

The discussed approaches to the cosmic ray first-order Fermi acceleration at relativistic shocks yield consistent estimates of the asymptotic spectral index (2.2, 2.3). However, the result is not as universal as one could infer from the convergent conclusions of different authors, because all presented derivations require (explicitly or implicitly) large amplitude MHD turbulence near the shock. Only the Bednarz & Ostrowski (1998) modelling allows one to treat - in a simplified way - conditions with medium amplitude perturbations of the magnetic field. In such conditions particle spectra are expected to be very steep at high shock Lorentz factors. The spectra considered by Bednarz & Ostrowski flatten at large due to an implicit increase of the short wave turbulence in their model, approaching closer and closer the parallel shock configuration considered by the other authors. Until now the situation with the medium amplitude turbulence, , has not been studied in the limit of large , however, from comparison with the results of Begelman & Kirk (1990), Ostrowski (1993) and of Bednarz & Ostrowski (1998) for intermediate shock Lorentz factors, we expect very steep spectra to be formed in such conditions. Thus, if the conditions with limited turbulence are met at a large shock, it can be unable to accelerate particles to very high energies in the first-order Fermi mechanism. On the other hand the "low'' energy electrons radiating from ultra-relativistic shocks could be accelerated by the non-first-order processes, analogous to the ones discussed by Hoshino et al. (1992) or Pohl et al. (2001).

The main deficiency of the approaches applying the pitch angle diffusion equation, in particular of our own attempt to discuss cases with oblique background magnetic fields, is their limitation to particular, highly turbulent conditions near the shock. This limitation may be significant or non-significant, depending on whether such conditions exist downstream of the ultra-relativistic shock. In the process discussed by Medvedev & Loeb (1999) such short-wave non-linear turbulence is created downstream of the shock, in the non-resonant wave-vector range for the shock accelerated particles.

MO is grateful to Reinhard Schlickeiser for the invitation to the Institute for Theoretical Physics of the Ruhr University, where this work was partly done, and to Yves Gallant and Bohdan Hnatyk for valuable discussions. The work was supported by theKomitet Badan Naukowychthrough the grant PB 258/P03/99/17.

Let us evaluate the magnetic field components responsible for regular,
,
and turbulent,
,
angular deviations
of the particle momentum during a single particle propagation time-step
.
For the regular deviation due to the mean magnetic field

(A.1) |

where ,

(A.2) |

For the considered model involving uniform scattering within a narrow cone of the opening angle , the mean scattering angle equals . If the magnetic field components and are oriented randomly with respect to each other, then the effective field modifying particle trajectory, , can be evaluated as . With the estimate one obtains and he can evaluate as

(A.3) |

In this rough estimate we give the lower contribution from the irregular magnetic field by assuming that trajectory perturbations are due to structures of the wavelength . In the case of turbulence power concentrated at shorter waves the required wave power is even higher to cause the considered scattering. The turbulence power with would provide correlations of successive scatterings, excluded in the considered model.

- Achterberg, A., Gallant, Y. A., Kirk, J. G., & Guthmann, A. W. 2001, MNRAS, 328, 393 In the text NASA ADS
- Bednarz, J. 2000a, Ph.D. Thesis, Jagiellonian University [
`astro-ph/0005075`] In the text - Bednarz, J. 2000b, MNRAS, 315, L37 NASA ADS
- Bednarz, J., & Ostrowski, M. 1996, MNRAS, 283, 447 In the text NASA ADS
- Bednarz, J., & Ostrowski, M. 1997, in Proc. 25th Int. Cosmic Ray Conf. (Durban), OG 9.1.6 In the text
- Bednarz, J., & Ostrowski, M. 1998, Phys. Rev. Lett., 80, 3911 In the text NASA ADS
- Bednarz, J., & Ostrowski, M. 1999, MNRAS, 310, L11 NASA ADS
- Begelman, M., & Kirk, J. G. 1990, ApJ, 353, 66 In the text NASA ADS
- Gallant, Y. A, & Achterberg, A. 1999, MNRAS, 305, L6 In the text NASA ADS
- Gallant, Y. A, Achterberg, A., & Kirk, J. G. 1999, A&AS, 138, 549 In the text NASA ADS
- Hoshino, M., Arons, J., Gallant, Y. A., & Langdon, A. B. 1992, ApJ, 390, 454 In the text NASA ADS
- Kirk, J. G., & Schneider, P. 1987, ApJ, 315, 425 In the text NASA ADS
- Kirk, J. G., Guthmann, A. W., Gallant, Y. A, & Achterberg, A. 2000, ApJ, 542, 235 In the text
- Medvedev, M. V., & Loeb, A. 1999, ApJ, 526, 697 In the text NASA ADS
- Ostrowski, M. 1993, MNRAS, 264, 248 In the text NASA ADS
- Peacock, J. A. 1981, MNRAS, 196, 135 In the text NASA ADS
- Pohl, M., Lerche, I., & Schlickeiser, R. 2001, in Proc. 27th Int. Cosmic Ray Conf., OG 2.3, 2713 In the text
- Vietri, M. 2002, submitted In the text

Copyright ESO 2002