The reflected ions with a gyro-radius exceeding the width
of the shock transition region can then be efficiently accelerated,
via the Fermi mechanism, by converging plasma flows carrying
magnetic inhomogeneities and MHD waves. In perpendicular shock a net
transverse particle momentum gain is due to the work of the electric
field on the particle drift motion. The electric field
perpendicular to the shock normal exists in all the reference frames
for the perpendicular shock. The particle of a momentum **p**
crossing back and forth the shock front and being scattered by MHD
waves carried with a flow of velocity **u** would undergo a
momentum increment
*p*
**p** ⋅ **u** / *v* + *O*((*u* /
*v*)^{2}) per scattering. A
velocity profile in the plane shock is illustrated by the dashed
line in Fig. 5 in the test particle case where one
neglects the back reaction effect of accelerated particles on the
shock. One way to calculate the accelerated particle spectra in a
scattering medium is to use the kinetic equation in the diffusion
approximation. The scattering medium in that approach is
characterised by momentum dependent particle diffusion coefficients
*k*_{1}(*p*) and *k*_{2}(*p*). The
shock is considered as a bulk velocity jump (see the dashed line in Fig. 5) assuming that the
test particles are injected at *p* = *p*_{0} and the
gyroradii of the
particles are larger than the shock width. Therefore, in a test
particle case particles must be injected at some super-thermal
energy to be accelerated by the shock. A solution to the kinetic
equation for a nearly isotropic test particle distribution in the
phase space is a power-law momentum distribution *f*(*p*,
*x*)
(*p* / *p*_{0})^{ - b}, *p*
*p*_{0} where
the index

(14) |

depends on the shock compression ratio *r*
(Axford et
al. 1977,
Krymskii 1977,
Bell 1978,
Blandford &
Ostriker 1978).
The CR spatial distribution in the model is
illustrated in Fig. 5. For a strong shock of
_{s} >> 1 and
_{a} >> 1 the
compression ratio given by Eqs. 6 and 8 is close to 4 if
_{g}
= 5/3 (or even larger if relativistic gas dominates the equation of
state). The pressure of the accelerated particles is

(15) |

Then for *b* = 4 one may see that *P*_{CR}
ln(*p*_{max} / *p*_{0})
indicating a potentially large cosmic ray (CR) pressure, if CRs are
accelerated to *p*_{max} >> *p*_{0}. The
maximal energy of accelerated
test particles depends on the diffusion coefficients, bulk velocity
and scale-size of the system. The finite scale-size of the shock is
usually accounted for by an energy dependent free escape boundary
located either in the upstream or in the downstream. For electrons
*p*_{max} can also be limited by synchrotron (or
inverse-Compton) losses of relativistic particles.

The test particle shock acceleration time τ_{a}(*p*)
can be estimated from the equation

(16) |

where the normal components of the shock upstream and downstream
bulk velocities *u*_{1n},
*u*_{2n} are measured in the shock rest
frame. Estimations based on a more rigorous approach distinguishing
between the mean acceleration time and the variance does not change
the results substantially, given the uncertainties in the diffusion
model.