stronger turbulence ($\sigma^2 \gtrsim 1$), or at smaller particle energies, $\Delta t$ is greater as particles spend more time within the turbulent region ($\Delta t \propto \kappa_\parallel^{-1}$, where $\kappa_\parallel$ grows with energy and decreases with $\delta B$, regardless of the assumed turbulence model) and stellar modulation effects need to be accounted for. We neglect also the energy loss due to curvature drifts of particles moving against the motional electric field ${\bf V}/c \times {\bf B}$.

We use simplifying assumptions for the radial and vertical structure of the protoplanetary disk. The total vertical column density typically decreases with radius to some power depending on the viscosity [e.g. @Pringle:81; @D'Alessio.etal:98], while the scale height increases due to the weakening vertical component of stellar gravity with radius. In our simulations, we are primarily interested in EP trajectories that intersect the disk; thus, any disk radial and vertical structure is [neglected]{}. The disk lies on the XY-plane and is assumed to have semi-thickness $0.1 R_{\star}$ independent of distance ($< 30 R_{\star}$ from the star) and an innermost radius of $2 R_{\star}$. The actual disk structure will depend on its temperature stratification and the equilibrium between gas, magnetic pressure and wind ram pressures at the 