Cosmic ray transport and acceleration in an evolving shock landscape

¹ Ruhr-Universität Bochum, Fakultät für Physik und Astronomie, Institut für Theoretische Physik IV, Universitätsstraße 150 44780 Bochum, Germany
² Ruhr Astroparticle and Plasma Physics Center (RAPP Center), Bochum, Germany
³ Department of Astronomy, University of Wisconsin–Madison, 475 N. Charter Street, Madison, WI 53706, USA
⁴ Department of Physics, University of Wisconsin–Madison, 150 University Avenue, Madison, WI 53706-1390, USA
⁵ Chalmers University of Technology, Department of Space, Earth and Environment, 412 96 Gothenburg, Sweden

^⋆ Corresponding author: sophie.aerdker@rub.de

Received: 23 December 2024
Accepted: 4 September 2025

Abstract

Context. Cosmic rays are detected from 10⁹ eV up to 10²⁰ eV, with two distinct changes in the spectrum at ∼10¹⁵ eV and at 10^18.5 eV. Below the first break (i.e., cosmic-ray knee), sources of acceleration are believed to be located in the Milky Way. Above the second break (i.e., cosmic-ray ankle) the population is expected to be extragalactic. In between the two, there is a need to take into account a third population of sources with the capacity to accelerate particles to extreme energies. In addition to a Galactic wind and its termination shock, large-scale shock structures in the Galactic halo have been proposed to fill the gap.

Aims. In this paper, we investigate CR transport in a time-dependent landscape of shocks in the Galactic halo. These shocks could result from local outbursts, for instance, star-forming regions and superbubbles. CRs re-accelerated at such shocks can reach energies above the knee. Since the shocks are closer to the Galaxy than a termination shock and CRs escape downstream, they can propagate back more easily. With such outbursts happening frequently, shocks are bound to interact. This interaction could adjust the CR spectrum, particularly for the particles that are able to be accelerated at two shocks simultaneously.

Methods. The transport and acceleration of CRs at the shock is modeled by stochastic differential equations (SDEs) within the public CR propagation framework CRPropa. We developed extensions for time-dependent wind profiles and, for the first time, we connected the code to hydrodynamic simulations, which were run with the public Athena++ code.

Results. We find that depending on the concrete realization of the diffusion tensor, a significant fraction of CRs can make it back to the Galaxy. These could contribute to the observed spectrum around and above the CR knee (E ≳ 10 PeV). In contrast to simplified models, a simple power law does not describe the energy spectra well. Instead, for single shocks, we find a flat spectrum (E⁻²) at low energies, which steepens gradually until it reaches an exponential decline. When shocks collide, the energy spectra transiently become harder than E⁻² at high energies.