A critical shock Mach number for particle acceleration in the absence of
  pre-existing cosmic rays: $M=\sqrt 5$
  It is shown that, under some generic assumptions, shocks cannot accelerate
particles unless the overall shock Mach number exceeds a critical value M >
sqrt(5). The reason is that for M <= sqrt(5) the work done to compress the flow
in a particle precursor requires more enthalpy flux than the system can
sustain. This lower limit applies to situations without significant magnetic
field pressure. In case that the magnetic field pressure dominates the pressure
in the unshocked medium, i.e. for low plasma beta, the resistivity of the
magnetic field makes it even more difficult to fulfil the energetic
requirements for the formation of shock with an accelerated particle precursor
and associated compression of the upstream plasma. We illustrate the effects of
magnetic fields for the extreme situation of a purely perpendicular magnetic
field configuration with plasma beta = 0, which gives a minimum Mach number of
M = 5/2. The situation becomes more complex, if we incorporate the effects of
pre-existing cosmic rays, indicating that the additional degree of freedom
allows for less strict Mach number limits on acceleration. We discuss the
implications of this result for low Mach number shock acceleration as found in
solar system shocks, and shocks in clusters of galaxies.
