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Simulations of the spatial and temporal invariance in the spectra of gradual solar energetic particle events

The spatial and temporal invariance in the spectra of energetic particles in the gradual solar events is reproduced in the simulations. Based on a numerical solution of the focused transport equation, we obtain the intensity time profiles of solar energetic particles (SEPs) accelerated by an interplanetary shock in the three-dimensional interplanetary space. The shock is treated as a moving source of energetic particles with a distribution function. The time profiles of particle flux with different energies are calculated in the ecliptic at $1$ AU. According to our model, we find that shock acceleration strength, parallel diffusion and adiabatic cooling are the main factors in forming the spatial invariance in SEP spectra, and perpendicular diffusion is a secondary factor. In addition, the temporal invariance in SEP spectra is mainly due to the effect of adiabatic cooling. Furthermore, a spectra invariant region, which agrees with observations but is different than the one suggested by Reames and co-workers, is proposed based on our simulations.