Power requirements for cosmic ray propagation models involving re-acceleration and a comment on second order Fermi acceleration theory

We derive an analytic expression for the power transferred from interstellar turbulence to the Galactic cosmic rays in propagation models which include re-acceleration. This is used to estimate the power required in such models and the relative importance of the primary acceleration as against re-acceleration. The analysis provides a formal mathematical justification for Fermi's heuristic account of second order acceleration in his classic 1949 paper.