realization, especially since the CRs diffuse upstream from the subshock a characteristic length $x_d$ as part of the acceleration process. That means there will be a pressure gradient $\sim P_{cr}/x_d$ influencing the flow upstream, so that the flow into the subshock will in this “shock precursor” be fractionally reduced by an amount $\delta u/u_1 \sim P_{cr}/(\rho_1 u^2_1)$. The maximum postshock thermalized pressure is $\frac{3}{4}\rho_1 u^2_1$, so if the CR and thermal pressures expected at the subshock are comparable, the shock must be significantly modified from the classical hydrodynamic shock.

The CR feedback generally tends to enhance the dynamical role of the CRs, since the deceleration through the precursor weakens the subshock and thus reduces the postshock thermal pressure. At the same time it turns out that the total compression through the shock, including the precursor and subshock generally exceeds that for a purely hydrodynamical shock. Then the expected value of $q$ is reduced for higher energy CRs, which tend to have longer scattering lengths, so see a larger compression. That adds to the CR pressure as well, since the relative number of the most energetic particles is greater. At the same time, if the highest energy CRs escape upstream into 