Using a combined PIC-MHD code to simulate particle acceleration Allard Jan van Marle (APC) Fabien Casse (APC), Alexandre Marcowith (LUPM)

Acceleration of cosmic particles We know astrophysical shocks accelerate particles This process also influences the local magnetic field and the behaviour of the shocks For astrophysical shocks: magneto- hydrodynamics (MHD) Fluid approach Computationally efficient Lacks particle physics For particle acceleration: particle-in-cell (PIC) Kinetic approach Can simulate particle physics Computationally expensive The challenge: To combine both in a single code X-ray: Nasa/CXC/Rutgers/K. Eriksen et al.; Optical: DSS

Introducing Particles in MHD cells We split the work: MHD code (MPI-AMRVAC, van der Holst et al. 2008, https://homes.esat.kuleuven.be/~keppens/) describes the thermal plasma, particles represent the supra-thermal component component The MHD grid serves as cells for the PIC Lorentz force gives us the effect of electromagnetic field on particles Effect of supra-thermal particles on thermal plasma can be treated through Ohm’s law (Bai et al. 2015) Advantages Can simulate particle acceleration and feedback (unlike MHD) No need for a huge particle population that represents the thermal gas (unlike PIC) We don’t solve all Maxwell equations, which reduces numerical noise (no Cherenkov waves) Disadvantages Limited regime: nthermal >> nnon-thermal Some restrictions owing to use of grid-MHD Need to determine a particle injection rate (obtained from PIC simulations)

Ohm’s law For MHD With supra-thermal particles (R = ncr/ntotal) The Lorentz force (Bai et al. 2015)

The new PI(MHD)C equations MHD conservation equations Induction equation Particle equation of motion (Bai et al. 2015)

PI[MHD]C Move particles Using Boris-pusher and the B and E fields Update MHD quantities through conservation equations, including charge and current from particles Interpolate from particles to determine charge and current in cell centres Constrained transport ensures div.B=0 MHD cell-centres function as PIC cell-corners

Parallel shock Based on tests shown in Bai et al. 2015 Non-relativistic shock with pre-existing magnetic field (MA=30) Injecting particles (proton equivalent) at the shock (VCR = 3 VS) Box: 30x240 Rl Grid: 30x240 (+ 4 levels) Recovered analytical predictions (Bell ´04) and earlier numerical results. (Caprioli & Spitkovski ´14, Bai et al. ´15) Upstream: non-resonant streaming instability Downstream: turbulence van Marle, Casse & Marcowith (submitted)

SED for parallel shock

Near-oblique shock hybrid-PIC simulations (Caprioli & Spitkovsky 2014) NO B-field amplification or particle acceleration occurs at angles > ~60o. However, Vlasov-Focker-Planck based models do show upstream instabilities very close to the shock (Reville & Bell 2013) Hybrid PIC simulations show only results during early phase Their domain is approx. six times smaller along vertical axis They suffer from relatively low number of particles. Approx. 2x108, or 4 per cell, with only a small fraction non-thermal (Caprioli & Spitkovsky 2014) Particles are accelerated by repeated shock crossings (Decker 1988) Small instabilities cause shock corrugation Shock corrugation causes variation in upstream current Upstream current disturbs the upstream magnetic field Disturbed field increases shock corugation van Marle, Casse & Marcowith (submitted)

Particle SED for oblique shock Diffusive shock acc. Shock drift acc.

Summary PI[MHD]C can reliably reproduce hybrid-PIC models and analytical predictions for the parallel shock case The increased computational efficiency of this method has allowed us to improve on previous results for the near-oblique shock situation Next step: Extend 2-D models to realistic astrophysical scenarios (Stellar wind shocks, supernova remnants, cluster shocks, etc.) 3-D models (again, possible because of computational efficiency) Combining relativistic MHD with PIC (early supernova explosions, GRBs, AGNs)