2 Energy gain per shock crossing cycle

The essential assumption of Fermi-type shock acceleration is that particles are deflected elastically in the local fluid frame on either side of the shock, so that upon recrossing upstream after an excursion downstream, say, they will have a net average energy gain. For particles of Lorentz factor $\gamma \gg \Gamma_{\rm sh}\gg 1$, this energy gain can be expressed as  

$$\frac{E_{\rm f}}{E_{\rm i}} = \Gamma_{\rm rel}^2 \left( 1 - ... ...ight) \left( 1 + \beta_{\rm rel}\cos \theta_{\rm f}' \right) ,$$ (1)

where $\Gamma_{\rm rel}\approx \Gamma_{\rm sh}/ \sqrt{2}$ is the relative Lorentz factor of the upstream and downstream media, $\theta$ is the angle between the particle's velocity at shock crossing and the shock normal, and primed and unprimed quantities are respectively measured in the downstream and upstream frames.

Kinematics require that the second factor in parentheses in Eq. (1) be greater than 1. In the initial shock crossing, $\theta_{\rm i}$is isotropically distributed, and we do find $E_{\rm f}/E_{\rm i}\sim \Gamma_{\rm rel}^2$.For physically realistic deflection processes upstream, however, it can be shown that for all subsequent shock crossings, $\theta_{\rm i} \sim \Gamma_{\rm sh}^{-1}$, so that we typically have $E_{\rm f}/ E_{\rm i}\sim 2$(Gallant & Achterberg 1998, 1999).