Description
Treatments of particle acceleration in astrophysical plasmas generally treat the energetic particles separately from the background plasma. Ideal MHD suffices for the background, but it is essential to treat the anisotropy of the particles. The code Sapphire++ [1] does this by solving the Vlasov-Fokker-Planck (VFP) equation using a spherical harmonic expansion of the particle distribution function and taking the velocity and magnetic fields from a solution of the ideal MHD equations. However, if the flow contains a shock transition, i.e., a discontinuity in the MHD variables, an additional assumption is needed to connect the particle distribution across it.
Two possible assumptions are (1) energetic particles are assumed to traverse the shock without deflection or change in energy, i.e., Liouville’s theorem is exploited to match the distributions up- and downstream and (2) the discontinuous transition in plasma velocity and magnetic field is smoothed out on a length scale small compared to the gyroradii of the lowest energy particles and the VFP equation is then solved in the resulting continuous profiles. Here, we use a method based on 2-dimensional eigenfunctions [3] to implement assumption (1) and compare this with the Sapphire++ solution that implements assumption (2) [2]. We demonstrate that these approaches yield the same result for the spectral index of accelerated particles in the case of an oblique, non-relativistic shock front in a 1-dimensional flow. Since the eigenfunction method is, in this particular case, numerically much less resource intensive than the full Sapphire++ method, this result opens up the prospect of rapid, detailed modelling of the the spectral indices observed in astrophysical sources.
[1] Nils W. Schween, Florian Schulze and Brian Reville. “Sapphire++: A particle transport code combining a spherical harmonic expansion and the discontinuous Galerkin method”. In: Journal of Computational Physics 523
(2025), p. 113690. doi: https://doi.org/10.1016/j.
jcp.2024.113690.
[2] Asma Shirin T. et al. “Spectral curvature and breaks from Fermi acceleration at oblique shocks”. In: Monthly Notices of the Royal Astronomical Society: Letters 544.1 (Nov. 2025), pp. L160–L166.doi:10.1093/mnrasl/slaf113
[3] Makoto Takamoto and John G. Kirk. “Rapid Cosmic-ray Acceleration at Perpendicular Shocks in Supernova Remnants”. In: Astrophysical Journal
809.1, 29 (Aug. 2015), p. 29. doi: 10.1088/0004-637X/809/1/29.