Effect of adiabatic cooling on the fitted parallel mean free path of solar energetic particles

TLDR: In this article, the authors considered the adiabatic energy loss effect, provided by the divergence of the solar wind flows, into the focused transport equation and solved numerically using a time-backward stochastic integration method.

Abstract: The focused transport equation without adiabatic energy loss is widely used to model solar energetic particles' (SEP) interplanetary propagation by fitting spacecraft data. We incorporate the adiabatic energy loss effect, provided by the divergence of the solar wind flows, into the focused transport equation. The equation is then solved numerically using a time-backward stochastic integration method. We show the comparison between solutions of focused transport equations with and without energy loss. We found the effect of adiabatic cooling is significant on the time profile of the intensity of SEPs. It is also shown that without energy loss, for gradual events, we can only fit the initial phase of SEP events. However, with energy loss, we can fit the entire ( initial and decaying) phases. In addition, the values of the mean free path obtained by fitting the SEP events with energy loss is always smaller than that without. The results suggest that including adiabatic cooling effect is another way to partially fix the solar energetic particle mean free paths' "too small'' problem discussed by Bieber et al. ( 1994), i.e., the mean free paths obtained by fitting transport equation to observation data are much larger than the quasi-linear theory results.