Cosmic ray-modified shocks: appearance of an isothermal jump

Abstract

We point out that for sufficiently strong shocks, with Mach number $M_1>\sqrt{\frac{3\gamma -1}{(3-\gamma )\gamma }}=1.34$ ($\gamma =5/3$), the solutions for cosmic ray-modified shocks experiences a bifurcation. As a result, for super-critical flows an isothermal jump forms (which is not a shock). The isothermal jump forms due to the energy diffusion of fast, but energetically subdominant cosmic rays. For super-critical flows the isothermal jump appears regardless of a particular feed-back mechanism from the CRs. The compression ratio at the isothermal jump is $2/(\gamma -1)=3$, so that in the test particle regime the expected spectrum of low energy CRs experiencing first-order Fermi process is $p=2\gamma /(3-\gamma )=5/2$, steeper than conventional $p=2$.