The selfconsistent steady state solution for a strong shock , significantly modified by accelerated particles is obtained on the level of a kinetic description , assuming Bohm-type diffusion .
The original problem that is commonly formulated in terms of the diffusion-convection equation for the distribution function of energetic particles , coupled with the thermal plasma through the momentum flux continuity equation , is reduced to a nonlinear integral equation in one variable .
The solution of this equation provides selfconsistently both the particle spectrum and the structure of the hydrodynamic flow .
A critical system parameter governing the acceleration process is found to be \Lambda \equiv M ^ { -3 / 4 } \Lambda _ { 1 } , where \Lambda _ { 1 } = \eta p _ { 1 } / mc , with a suitably normalized injection rate \eta , the Mach number M \gg 1 , and the cut-off momentum p _ { 1 } .

We are able to confirm in principle the often quoted hydrodynamic prediction of three different solutions .
We particularly focus on the most efficient of these solutions , in which almost all the energy of the flow is converted into a few energetic particles .
It was found that ( i ) for this efficient solution ( or , equivalently , for multiple solutions ) to exist , the parameter \zeta = \eta \sqrt { p _ { 0 } p _ { 1 } } / mc must exceed a critical value \zeta _ { cr } \sim 1 ( p _ { 0 } is some point in momentum space separating accelerated particles from the thermal plasma ) , and M must also be rather large ( ii ) somewhat surprisingly , there is also an upper limit to this parameter ( iii ) the total shock compression ratio r increases with M and saturates at a level that scales as r \propto \Lambda _ { 1 } ( iv ) despite the fact that r can markedly exceed r = 7 ( as for a purely thermal ultra-relativistic gas ) , the downstream power-law spectrum turns out to have the universal index q = 3 \onehalf over a broad momentum range .
This coincides formally with the test particle result for a shock of r = 7 ( v ) completely smooth shock transitions do not appear in the steady state kinetic description .
A finite subshock always remains .
It is even very strong , r _ { s } \simeq 4 for \Lambda \ll 1 , and it can be reduced noticeably if \Lambda \gtrsim 1 .