Interpreting the Observed Positive Correlation between the Event-integrated Fluence and the Rollover Energy of Solar Energetic Particle Events by the PAMELA Mission with Coupled Hydromagnetic Wave Excitation and Proton Acceleration at Shocks in the Low Corona

An analytical model for diffusive shock acceleration (DSA) at one-dimensional stationary planar shocks in the lower corona is presented. The model introduces an upstream escape boundary through which a constant flux of protons streaming upstream out of the system is allowed. The nonvanishing flux of streaming protons out of the system limits the maximum attainable energy of DSA and produces a rollover in the high-energy spectra of the shock-accelerated protons. The condition for the rollover energy derived from the model can account for the approximately linear relation between the natural logarithm of event-integrated fluences and the natural logarithm of rollover energies as demonstrated in Bruno et al. Solar energetic particle (SEP) events with higher integrated fluences in principle exhibit higher rollover energies since proton-excited hydromagnetic waves in the turbulent sheath reduce the proton diffusion coefficient and throttle the upstream streaming of protons. The consistency between the observation and the theory of DSA at shocks in the lower corona serves as evidence for the shock origin of protons of the highest energies in large SEP events.