Time-dependent Acceleration and Escape of Charged Particles at Traveling Shocks in the Near-Sun Environment

Current multi-spacecraft in situ measurements allow for the investigation of the time evolution of energetic particles at interplanetary shocks (IPs) at small (≲0.1 au) heliocentric distances. The energy spectrum of accelerated particles at IPs was shown by a previous 1D transport model that includes both self-excited plus preexisting turbulence and a term representing the escape of particles from the system to gradually steepen as a result of a finite acceleration-to-escape timescales ratio; such a model was found in excellent agreement with the entire sample of the ground-level enhancement spectra of solar cycle 23. We solve the time-dependent case of such a model in the case of diffusion dominated by preexisting turbulence. The average timescale for particle acceleration at various heliocentric distances, from 1 au down to the inner heliosphere (<0.1 au), is shorter than in the no-escape case, as higher energy particles have a shorter time to accelerate before completely leaving the system into the upstream medium. A simple scaling with time of the time-dependent spectrum is provided. We compare the “nose” structure at a few ∼100s keV protons first measured in situ by Parker Solar Probe in crossing the very fast 2022 September 5 shock at 0.07 au; we find that the nose is reasonably well explained by a lack of the highest energy particles not yet produced by the young shock by both our model and the no-escape version.