A two-fluid model is established to study the effect of injection
of cosmic-ray-modified shocks. The model comprises a thermal plasma and
rays. The cosmic rays are considered to be a massless fluid with significant
pressure. Its behaviour is governed by the cosmic-ray energy equation,
can be derived
from the transport equation. A term in the equation responsible for injection
proportional to the negative divergence of the plasma velocity. The rest
dependence has to be introduced phenomenologically. It is expected that
of cosmic rays increases if the thermal plasma energy density increases.
fix ideas, we assume that the injection is linear in the thermal pressure.
By suitable redefinition of some quantities, this model can formally be
recast into a model without
injection, i.e. the injection is ‘transformed’ away mathematically.
Hence the results
are similar to those for the modified shocks without injection. Nonetheless,
are differences. The system with injection prefers a smooth shock transition.
reason is that the injection softens the equation of state of the plasma,
an extra cosmic-ray partial pressure. The dependence of the effficiency
acceleration on the injection is also discussed.