and the Alfvén wave speed in the circumstellar medium (for $B_0 = 10$ G and $n=5\times 10^4$ cm$^{-3}$ at a distance $\sim 2R_{\star}$ from the star, the Alfvén speed is $\sim 2 \times10^4$ km$/$s and decreases outward as $\sim 1/R$). Thus, on the time-scale of EP propagation, the frame rotating with the star and the rest frame of the expanding plasma are, to good approximation, indistinguishable. As a consequence, it is reasonable to assume that the global EP anisotropy in the expanding wind frame is small. Thus, the wind advection has no effect on the EPs motion outward that is mainly of diffusive nature.

@Feigelson.etal:02 used observations of intense flaring in Orion T Tauri stars combined with a relation between X-ray and EP emission on the Sun to estimate that the circumstellar medium of a T Tauri star will be enriched in EPs by a factor of $\sim 10^5$ compared with the contemporary Sun. The sources of EPs closer to the central star are likely to be shocks preceding CMEs or flares in the stellar corona. Here, we are interested in the space-averaged effect of EPs on disk ionization. We mimic such sources by uniformly injecting EP populations on spheres 