**B. Particle Acceleration**

It is now generally accepted that Cosmic rays (more specifically
the lower energy component below 10^{15} eV) are accelerated
within shocks in SNRs is the Galaxy (see e.g. Gaisser
[117]).
A beautiful demonstration of this effect
arises in the observation of synchrotron emission from Supernova
remnants, which shows X-ray emission from these accelerated
particles within the shocks.

The common model for particle shock acceleration is the Diffuse
Shock Acceleration (DSA) model. According to this model the
particles are accelerated when they repeatedly cross a shock.
Magnetic field irregularities keep scattering the particles back
so that they keep crossing the same shock. The competition
[96]
between the average energy gain, *E*_{f} /
*E*_{i} per
shock crossing cycle (upstream-downstream and back) and the escape
probability per cycle, *P*_{esc} leads to a power-law spectrum
*N*(*E*)*dE*
*E*^{-p}
*dE* with

(13) |

Note that within the particle acceleration literature this index
*p* is usually denoted as *s*. Our notation follows the
common notation within the GRB literature.

Blandford and Eichler
[29]
review the theory of DSA in
non-relativistic shocks. However, in GRBs the shocks are
relativistic (mildly relativistic in internal shocks and extremely
relativistic in external shocks). Acceleration in ultra
relativistic shocks have been discussed by several groups
[1,
17,
121,
161,
190,
424].
In relativistic shocks the considerations are quite different from
those in non-relativistic ones. Using the relativistic shock jump
conditions (Eq. 11 and kinematic considerations one can
find (see Vietri
[422],
Gallant and Achterberg
[122],
Achterberg et al.
[1])
that the energy gain in the first shock crossing is of the order
_{sh}^{2}. However, subsequent shock
crossing are not as efficient and the energy gain is of order unity
<*E*_{f} / *E*_{i}>
2
[1,
122].

The deflection process in the upstream region is due to a large
scale smooth background magnetic field perturbed by MHD
fluctuations. A tiny change of the particle's momentum in the
upstream region is sufficient for the shock to overtake the
particle. Within the downstream region the momentum change should
have a large angle before the particle overtakes the shock and
reaches the upstream region. As the shock moves with a
sub-relativistic velocity
( *c* /
3^{1/2}) relative to this
frame it is easy for a relativistic particle to overtake the
shock. A finite fraction of the particles reach the upstream
region. Repeated cycles of this type (in each one the particles
gain a factor of ~ 2 in energy) lead to a power-law spectrum
with *p* 2.2 - 2.3
(for _{sh}
>> 1). Like in
non-relativistic shock this result it fairly robust and it does
not depend on specific assumptions on the scattering process. It
was obtained by several groups using different approaches,
including both numerical simulations and analytic considerations.
The insensitivity of this result arises, naturally from the
logarithmic dependence in equation 13 and from the fact
that both the denominator and the numerator are of order unity.
This result agrees nicely with what was inferred from GRB spectrum
[369]
or with the afterglow spectrum
[291].
Ostrowski and Bednarz
[283]
point out, however, that this result requires highly turbulent
conditions downstream of the shock. If the turbulence is weaker the
resulting energy spectrum could be much steeper. Additionally as
internal shocks are only mildly relativistic the conditions in these
shocks might be different.

The maximal energy that the shock accelerated particles can be
obtained by comparing the age of the shock *R* / *c* (in the
upstream frame) with the duration of an acceleration cycle. For a simple
magnetic deflection, this later time is just half of the Larmour time,
*E* / *Z q*_{e}*B* (in the same frame). The
combination yields:

(14) |

where the values that I have used in the last equality reflect
the conditions within the reverse external shocks where UHECRs
(Ultra High Energy Cosmic Rays) can be accelerated (see
Section VIIIC below). For particle diffusion
in a random
upstream field (with a diffusion length *l*) one finds that *R*
in the above equation is replaced by (*R l* / 3)^{1/2}.

The acceleration process has to compete with radiation losses of the accelerated particles. Synchrotron losses are inevitable as they occur within the same magnetic field that is essential for deflecting the particles. Comparing the energy loss rate with the energy gain one obtain a maximal energy of:

(15) |

The corresponding Lorentz factor is of the order of 10^{8} for
_{sh} = 100
and *B* = 1 Gauss. Note that this formula assumes
that the acceleration time is the Larmour time and hence that the
synchrotron cooling time is equal to the Larmour time. Obviously
it should be modified by a numerical factor which is mostly likely
of order unity.