The passage of energetic charged particles through interplanetary space

Link to ADS entry

It seems as though equation 2 in page 15 should logically come before the preceding equation. The change in the volume of a fluid element is given by $\frac{1}{V} \frac{dV}{dt} = \nabla \cdot \mathbf{U}$. The energy and volume are related in adiabatic expansion by

$U \cdot V^{\gamma-1} = const \Rightarrow \frac{d}{dt} \ln E = -\left( \gamma - 1 \right) \frac{d}{dt} \ln V = - \left( \gamma -1 \right) \nabla \cdot \mathbf{U}$

For classical monatomic gasses, the relation between the energy and momentum is $E \propto p^2$ and $\gamma = \frac{5}{3}$ so

$\frac{d}{dt} \ln p = - \frac{1}{3} \nabla \cdot \mathbf{U}$

Incidently, in the ultra - relativistic case where $E \propto p$ and $\gamma = \frac{4}{3}$ we get the same result.