Effect of Acceleration and Escape of Energetic Particles on Spectral Steepening at Shocks

Energetic particles spectra at interplanetary shocks often exhibit a power law within a narrow momentum range softening at higher energy. We introduce a transport equation accounting for particle acceleration and escape with diffusion contributed by self-generated turbulence close to the shock and by preexisting turbulence far upstream. The upstream particle intensity steepens within one diffusion length from the shock as compared with diffusive shock acceleration rollover. The momentum spectrum, controlled by macroscopic parameters such as shock compression, speed, far-upstream diffusion coefficient, and escape time at the shock, can be reduced to a log-parabola and also to a broken power law. In the case of upstream uniform diffusion coefficient, the largely used power-law/exponential cutoff solution is retrieved.

Automated summary

The energetic particle spectra at interplanetary shocks often exhibit a power law with softening at higher energy levels. A transport equation has been introduced to account for particle acceleration and escape, with diffusion generated by turbulence close to the shock and preexisting turbulence far upstream. The momentum spectrum is controlled by macroscopic parameters and can be simplified to a log-parabola or a broken power law, depending on the conditions.