The document summarizes the results of a particle-in-cell simulation of a mildly relativistic quasi-parallel shock. The simulation shows: 1) Filamentation of ions at the shock front and magnetic field amplification beyond compression alone. 2) Electrons are accelerated nearly to equipartition with ions, reaching gamma factors of around 100. 3) Magnetic bubbles and peaks in magnetic pressure form downstream of the shock, driven by drift currents and gradient-B electric fields. The results suggest that significant synchrotron radiation could be produced by electrons accelerated in such relativistic shocks. Further simulations are planned to study transient shocks and fully 3D effects.

1. log(!), Time = 16000"t Ion Phase Space, Time = 16000!t
400
300 0.5
200 0.0
100 -0.5
0 -1.0
0 2000 4000 6000 8000 0 2000 4000 6000 8000
0.0 0.6 1.3 1.9 2.5 3.2 3.8 0.3 1.1 1.9 2.7 3.4 4.2 5.0
Magnetic field amplification and electron
acceleration to near ion energy equipartition
from a mildly quasi-parallel relativistic shock
Gareth Murphy(1), Mark Dieckmann(2), Luke Drury(1)
1. Dublin Inst for Advanced Studies, 2. Univ. of Linkoping

2. Motivation
• Explore the formation of mildly relativistic
shocks at kinetic level with PIC simulations
• How is non-thermal radiation produced?
• How does a quasi-parallel field affect shock
formation?

3. GRB internal shock
• When fast blobs meet
slow blobs
• Internal shock
• Emission of Gamma rays
• Synchrotron (jitter)
radiation implies fast
electrons and large
magnetic fields

4. Numerical Method
• Particle In Cell (PIC) Simulations
• Plasma Simulation Code (PSC; Ruhl et al
2003)
• MPI-Parallel

5. Problem setup
• 0.9c collision speed
• Density ratio of 10
• Ion/electron mass ratio of 250
• Quasi-parallel magnetic field
• 100 CP’s per cell
• 18,000 x400 cells in 2D

6. 1D structure
Electron Phase Space, Time = 25000!t
200
150
100
50
0
0 200 400 600 800 1000
0.3 0.9 1.5 2.1 2.7 3.3 3.9

7. Electron phase space
Electron Phase Space, Time = 16000!t
150
•
100
Gamma of ~ 100 for a
tenuous population of 50
electrons 0
0 2000 4000 6000 8000
0.3 1.2 2.1 3.0 3.9 4.8 5.7

8. Ion phase space
Ion Phase Space, Time = 16000!t
0.5
• Ions accelerated by 0.0
longitudinal electric field -0.5
• Shock drift acceleration
-1.0
0 2000 4000 6000 8000
0.3 1.1 1.9 2.7 3.4 4.2 5.0

9. Ion density -
400
filamentation
log(!), Time = 16000"t
300
200
Filaments
100
0
0 2000 4000 6000 8000
0.0 0.6 1.3 1.9 2.5 3.2 3.8
log(!), Time = 16000"t
400
Magnetic bubbles
300
200
100
0
0 2000 4000 6000 8000
0.0 0.8 1.6 2.3 3.1 3.9 4.7

10. Integrated density
structure
1000
Left-moving ions
Right-moving ions
800 Left-moving electrons
Right-moving electrons
600
400
200
0
0 2000 4000 6000 8000

11. Right Moving Ions

12. Left Moving Ions

13. Electron Density
log(!), Time = 16000"t
400
300
200
100
0
0 2000 4000 6000 8000
0.0 0.7 1.4 2.1 2.9 3.6 4.3
log(!), Time = 16000"t
400
300
200
Magnetic Bubbles
100
0
0 2000 4000 6000 8000
0.0 0.8 1.5 2.3 3.0 3.8 4.5

14. Magnetic Field
• Peak from gradient in convection (u x B)
electric field.
• Drift current
• Magnetic bubbles - formation observed in
lab plasmas but not in astrophysical context

15. 2D Magnetic Pressure
log(B2), Time = 16000!t
300
200
100
0
0 2000 4000 6000 8000
-3.1 -2.2 -1.2 -0.3 0.6 1.6 2.5

16. Integrated Magnetic
Pressure
B2
2.0
1.5
1.0
0.5
0 2000 4000 6000 8000
Peaks at shock front
Compression of magnetic field

17. Conclusions
• Filamentation at the shock front
• Field amplification in excess of that
expected from compression alone.
• Electron acceleration to near energy-
equipartition with ions.
• Expect significant synchotron (jitter)
radiation

18. Future Work
• 3D Simulations
• Transient Shock

19. Obrigado!
log(!), Time = 18000"t
log(B2), Time = 16000!t
300
300
200
200
100
100
0
0 0 2000 4000 6000 8000
0 2000 4000 6000 8000
-3.1 -2.2 -1.2 -0.3 0.6 1.6 2.5
0.0 0.6 1.3 1.9 2.5 3.2 3.8
Electron Phase Space, Time = 16000!t Ion Phase Space, Time = 16000!t
150 0.5
100 0.0
50 -0.5
0 -1.0
0 2000 4000 6000 8000 0 2000 4000 6000 8000
0.3 1.2 2.1 3.0 3.9 4.8 5.7 0.3 1.1 1.9 2.7 3.4 4.2 5.0