Gamma-ray emission during Long-Duration Gamma-Ray Flare (LDGRF) events is thought to be caused mainly by $>$300 MeV protons interacting with the ambient plasma at or near the photosphere. Prolonged periods of the gamma-ray emission have prompted the suggestion that the source of the energetic protons is acceleration at a Coronal Mass Ejection (CME)-driven shock, followed by particle back-precipitation onto the solar atmosphere over extended times. We study the latter phenomenon using test particle simulations, which allow us to investigate whether scattering associated with turbulence aids particles in overcoming the effect of magnetic mirroring, which reflects particles as they travel sunwards. The instantaneous precipitation fraction, $P$, the proportion of protons that successfully precipitate for injection at a fixed height, $r_i$, is studied as a function of scattering mean free path, $lambda$ and $r_i$. We find that the presence of scattering helps back-precipitation compared to the scatter-free case, although at very low $lambda$ values outward convection with the solar wind ultimately dominates. Upper limits to the total precipitation fraction, $overline{P}$, are calculated for 8 LDGRF events for moderate scattering conditions ($lambda$=0.1 AU). Due to strong mirroring, $overline{P}$ is very small for these events, between 0.56 and 0.93% even in the presence of scattering. Time-extended acceleration and large total precipitation fractions, as seen in the observations, cannot be reconciled for a moving shock source according to our simulations. These results challenge the CME-shock source scenario as the main mechanism for gamma-ray production in LDGRFs.
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