Abstract
An analytical calculation is presented to derive the mean free paths of energetic protons in large western solar energetic particle (SEP) events during the first 12 hr after the event onset utilizing the differential spectra averaged over successive time intervals of 3 hr. The model assumes diffusive transport of energetic particles in a radial magnetic flux tube and neglects solar wind convection and adiabatic cooling. The model further assumes that particles over the considered energies (>10 MeV) are injected instantaneously near the Sun. Using the model, we reproduce the differential spectra averaged over successive time intervals after event onset and derive the radial mean free paths (λ
mfp) for protons at the energies where the two spectra over successive time intervals intersect. We then select eight western SEP events during the past solar cycle and apply the scheme. The derived λ
mfp ranges from 0.016 to 0.149 au. Our method finds smaller mean free paths than the lower limit of the “Palmer consensus” (0.04 au) in approximately 35% of the cases. We also combine differential intensity spectra and fluence to estimate the number of crossings (N
c) of particles passing through 1 au at applicable energies. The average N
c, excluding a twin-CME event, is 17.7, which indicates that diffusive transport of SEPs is capable of elevating the fluence observed at 1 au by one order of magnitude compared to if particles stream through 1 au nearly scatter-free.
