where $v_{acc}$ is a characteristic large scale speed (e.g., $\Omega \times R$). This argument leads to the expression $$E < \beta_{acc} ZeBR,$$ which differs from the relation (2) only by the velocity factor, $\beta_{acc}$.

A related argument is that the time necessary to accelerate a particle must be less than the lifetime of the accelerator. Let us apply this to the diffusive shock acceleration process to be outlined in §4. The mean time to accelerate a particle to energy $E$ is given by equation 9, so we have $$t_{acc} \sim \frac{C_d\kappa}{v^2_{shock}} < t_{dynamic} \sim \frac{R}{v_{shock}},$$ where $\kappa$ is a characteristic spatial diffusion coefficient for the highest energy particles to be accelerated, and $C_d \sim 10$ is a numerical factor. Under most circumstances the applicable diffusion coefficient is $\kappa \sim (1/3) \lambda c$, where $\lambda$ is the scattering length of the particles. If we require that the scattering length exceed the particle gyroradius we once again obtain relation (3) above within a numerical factor of order unity.

I will simply call relation (3) the “Hillas constraint”. As did Hillas, we can quickly filter phenomena from our discussion of viable UHECR candidate sources with a plot of this relation. The line in 