to 4, so the solution appears pretty robust. Notice that this result depends only on the jump conditions across the shock, and it was necessary only to suppose that somehow plasma turbulence would isotropize the CRs on each side of the shock. Plasma shocks were long known to be highly turbulent downstream. The other key insight to the introduction of DSA was that unless the CRs returning to the upstream side of a shock where isotropized with respect to the upstream flow they would amplify local Alfvenic turbulence there by a streaming instability. Those waves should then, in turn scatter the subsequent CRs trying to escape in this way. This physics, even though it leads to a simple outcome, actually shows that DSA is a nonlinear aspect of shock formation, so was an early clue that the physics is actually more complex.

An important detail about DSA that we encountered in §3 was the finite amount of time required to accelerate particles up to a specified energy (or momentum). This is now easy to estimate from the fractional momentum gain per shock encounter expressed in equation 8 along with an estimate of the time interval between CR shock crossings. The 