Time-dependent Acceleration and Escape of Charged Particles at Traveling Shocks in the near-Sun environment: the case of the Sept 5th 2022 Parker Solar Probe event

The Parker Solar Probe (PSP) approaches to the Sun in the past 6 years unveiled a broad variety of Traveling shocks (Ts) in the near-Sun environment, from the very weak Ts that would have been unlikely classified as shocks at 1 AU and are not associated with significant enhancement of energized particles, to the fastest ever Ts in-situ measured in the heliosphere, with unprecedented early-on signatures of particle (ions and electrons) acceleration. The interpretation of these measurements requires models for the time-evolution of particle acceleration and the escape from the source. We present the time-dependent version of a 1D transport model that incorporates particle escape at all supra- and non-thermal energies (not only the highest energies) into the diffusive shock acceleration (DSA) model via an energy- and position- dependent escape time. If the scattering is dominated by pre-existing solar wind turbulence the average time scale 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 leaking out into the upstream and never return. A simple scaling with time of the time-dependent energy spectrum is provided. Finally, we compare the ``nose'' structure at a few hundreds keV protons first measured in situ by PSP in crossing the very fast September 5th 2022 Ts 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 DSA version. A larger sample of such events by PSP will help identify the conditions in the Ts lifetime of an escape-dominated regime.