## Elementary model of diffusive shockwave acceleration

Let us consider the motion of a particle with velocity v >> u1 (where u1 is the velocity of shock front) in neighborhood of the shock wave. Let us take into account that the relative velocities of magnetic irregularities in the space plasma are very small in comparison with the velocity of a strong shock wave. So in the first approximation we may neglect the particle's change of energy during their interactions with magnetic irregularities. The change of particle momentum p as a result of scattering by the shock wave front will be

where pf and pi are final and initial particle momentum. The change of particle momentum during double crossing of the shock wave front will be

The flux of particles in the case of isotropic distribution after scattering by magnetic inhomogeneities per the unit of shock wave front, expressed through the accelerated particle density n(p) and angle 6between momentum p and axis x, will be

The averaging of Eq. 4.21.2 over the flux J(p) in the angle intervals n/2 <6X < n, nl2 < 6f < n, and 0 < 6 < n/2 gives the average gain of particle momentum for one cycle (a double crossing of the shock wave front):

where Pc is the probability of realization of the next cycle of double crossing by particle of the shock wave front, we obtain the following equation for determining the integral spectrum of accelerated particles N (> p):

If Pc1 is the probability that a particle from the region 1 (see Fig. 4.21.1) before the shock wave front will come back to the front, and Pc2 is the same for a particle coming from the region 2 behind the shock wave front, the value of Pc = Pc1Pc2 . It is expected that Pc1 = 1 because all particles from the region 1 by the convective motion came to the front. On the other hand, the probability

where J12 is the particle flux from the region 1 to region 2 (in the case of isotropic distribution of accelerated particles behind the front J12 = nv/4) and J2 = nu2 is the convection flux from the front to the region 2. Therefore by taking into account Eq. 4.21.6 we obtain

On the basis of Eq. 4.21.3a, 4.21.5, and 4.21.7 we obtain the following equation for the differential density of accelerated particles (differential energy spectrum) n(p) = -dN(> p)/dp :

is the degree of plasma compressibility by the shock wave. On the other hand the gas-dynamical consideration (see Pikelner, M1966; Landau and Lifshitz, M1957; Zeldovich and Raizer, M1966; Longmair, M1966) shows that c -

where Yg is the gas adiabatic index and M1 = u^us1 is the Mach number, and the sound velocity is determined by us1 = (YgP1 /P1 ))2 . It is important to note that according to Eq. 4.21.9 with increasing a from 2 to 4 the value of Ydecreases from 4 to 2 in agreement with what is observed in galactic and solar CR.