COSMIC-RAY-MODIFIED SHOCKS WITH INJECTION IN THE HYDRODYNAMIC APPROACH .1. INJECTION LINEAR IN THE THERMAL PRESSURE

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.