Structure and Scale of Cosmic Ray Modified Shocks

Astrophysical shocks, diffusively accelerating cosmic rays (CR) ought to develop CR precursors. The length of the precursor $L_{p}$ is believed to be set by the ratio of the CR mean free path $\lambda$ to the shock speed, $L_{p}\sim c\lambda/V_{sh}\sim cr_{g}/V_{sh}$, which is independent of the CR pressure $P_{c}$. However, the X-ray observations of supernova remnant shocks suggest that the precursor scale may be significantly shorter than $L_{p}$ which would question the above estimate unless the magnetic field is strongly amplified and the gyroradius $r_{g}$ is strongly reduced. We argue that while the CR pressure builds up ahead of the shock, the acceleration enters into a strongly nonlinear phase in which an acoustic instability, driven by the CR pressure gradient, dominates other instabilities (for $\beta < 1$ ). In this regime the precursor steepens into a strongly nonlinear front whose size scales with \emph{the CR pressure }as $L_{f}\sim L_{p}\cdot\left(L_{s}/L_{p}\right)^{2}\left(P_{c}/P_{g}\right)^{2 }$, where $L_{s}$ is the scale of the developed acoustic turbulence, and $P_{c}/P_{g}$ is the ratio of CR to gas pressure. Since $L_{s}\ll L_{p}$, the precursor scale reduction may be strong in the case of even a moderate gas heating by the CRs through the acoustic and (possibly also) the other instabilities driven by the CRs.