Cm. Ko et al., COSMIC-RAY-MODIFIED SHOCKS WITH INJECTION IN THE HYDRODYNAMIC APPROACH .1. INJECTION LINEAR IN THE THERMAL PRESSURE, Journal of Plasma Physics, 57, 1997, pp. 677-694
A two-fluid model is established to study the effect of injection on t
he structure of cosmic-ray-modified shocks. The model comprises a ther
mal plasma and cosmic rays. The cosmic rays are considered to be a mas
sless fluid with significant pressure. Its behaviour is governed by th
e cosmic-ray energy equation, which can be derived from the transport
equation. A term in the equation responsible for injection is proporti
onal to the negative divergence of the plasma velocity The rest of the
dependence has to be introduced phenomenologically. It is expected th
at the injection of cosmic rays increases if the thermal plasma energy
density increases. To fix ideas, we assume that the injection is line
ar in the thermal pressure. By suitable redefinition of some quantitie
s, this model can formally be recast into a model without injection, i
.e. the injection is 'transformed' away mathematically Hence the resul
ts are similar to those for the modified shocks without injection. Non
etheless, there are differences. The system with injection prefers a s
mooth shock transition. The reason is that the injection softens the e
quation of state of the plasma, and induces an extra cosmic-ray partia
l pressure. The dependence of the efficiency of shock acceleration on
the injection is also discussed.