1$, and ${{\mathbf{B}}} \simeq {{\mathbf{B}}}'$. The plasma is assumed to be infinitely conductive so that any large-scale background electric fields are negligible; also the rate of energy change is so small that any motional electric field associated with the turbulent field is neglected. We also neglect the electric field associated with the gradient/curvature drift. In order to calculate ${\mathbf{B}} ({\bf x})$ at the particle position, we do not perform a magnetic field interpolation within the cell of the MHD grid, but simply calculate the field in the cell closest to the particle position; using the former would be more time-consuming as all six values of the B-components per cell-grid need to be computed, one on each face, with little advantage in numerical accuracy.

Equation (\[lorentz\]) is integrated by using two encapsulated time-steps: a coarse time-step $\sim 65 \times 2\pi/\Omega_0({\bf x})$ that varies according to the particle energy and ${\mathbf{B}}_0$, i.e., with the distance from the star; and a fine time-step for which we use the Bulirsch-Stoer method with adjustable time-step [@Press.etal:86]. This allows us to efficiently deal with variations of $B_0$ by orders of magnitude along the EPs propagation. We verified that the particle energy is conserved to a relative 