  Shocks in astrophysical fluids can generate suprathermal particles by first
order (or diffusive) Fermi acceleration. In the test particle regime there is a
simple relation between the spectrum of the accelerated particles and the jump
conditions at the shock. This simple picture becomes complicated when the
pressure of the accelerated particles becomes comparable with the pressure of
the shocked fluid, so that the non-linear backreaction of the particles becomes
non negligible and the spectrum is affected in a substantial way. Though only
numerical simulations can provide a fully self-consistent approach, a more
direct and easily applicable method would be very useful, and would allow to
take into account the non-linear effects in particle acceleration in those
cases in which they are important and still neglected. We present here a simple
semi-analytical model that deals with these non-linear effects in a
quantitative way. This new method, while compatible with the previous
simplified results, also provides a satisfactory description of the results of
numerical simulations of shock acceleration.
