1. Diffusive Shock Acceleration
Nepomuk Otte
MAGIC/EUSO Seminar

2. Outline About Trucks and Tennis Balls Second Order Fermi
First Order Fermi
Energy spectrum
Maximum Energies

3. About Trucks and Tennis Balls
mass M=40t
velocity vtruck = 80km/h on German Highways (sometimes 110 km/h)

4. About Trucks and Tennis Balls
mass m = 56,7…58,7 g
velocity vball = 200 km/h

5. About Trucks and Tennis Balls
vtruck
vball
what happens in a head on collision?
M>>m and vball>>vtruck

6. About Trucks and Tennis Balls
vtruck
vball
negligible relative change in momentum and energy of truck
But the tennis ball:
momentum |p| and kinetic energy W is increasing
due to energy transfer from the truck onto the ball
ΔE/E ≈ 4 vtruck/vball
 Acceleration of the ball

7. Diffusive Shock Acceleration
Light particles gain energy in head-on elastic collisions on heavy slower moving objects
The same mechanism also works in the universe:
at much longer timescales
with much smaller efficiency

8. Fermi Acceleration I follow:
Origin and Propagation of the highest energy cosmic rays
R. J. Protheroe astro-ph/
Cosmic Rays and Particle Physics
T. K. Gaisser chapter 11
High Energy Astrophysics: Volume 2
M. S. Longair chapter 21
The acceleration of cosmic rays in shock fronts I
R. Bell MNRAS (1978) 182,
The acceleration of cosmic rays in shock fronts II
A. R. Bell MNRAS (1978) 182,

9. The Original one: 2nd Order Fermi Acceleration or Elastic scattering of cosmic rays in magnetized clouds
Inputs:
relativistic isotropic particle distribution
heavy, magnetized, non relativistic gas cloud with velocity v  b

10. 2nd Order Fermi Acceleration
due to cloud movement head on collisions are slightly more probable
particle is randomly scattered on the magnetic field in the cloud
(diffusion process)

11. 2nd Order Fermi Acceleration
Transforming into rest frame of cloud:
Particle is scattered into
No change of particle Energy in the cloud system
Energy change in the lab frame

12. 2nd Order Fermi Acceleration
Second order because of
and
particle can win and loose energy in a single encounter
very small gain in energy after many encounters

13. The More Efficient Version 1st Order Fermi
planar shock front instead of gas clouds

14. 1st Order Fermi Acceleration
In SN ejected material propagates with
VP~104 km/s >> speed of sound (~10 km/s)
Shock wave with speed Vs= 4/3 Vp
particles crossing the shock front generate Alfvén waves
Alfvén waves are low frequency hydromagnetic plasma oscillation
chaotic distribution of magnetic fields

15. 1st Order Fermi Acceleration
Average
for a particle entering the shock
same scenario when the shock has bee crossed
the plasma on the other shock side is approaching with velocity Vp
Average energy gain for a full cycle

16. 1st Order Fermi Acceleration
Is first order in b
More efficient
always gain in energy

17. Probability for one Shock Crossing
Net flow of particles in downstream direction
Rate at which particles are lost from the shock in downstream:

18. Probability for one Shock Crossing
Net flow of particles in downstream direction
Rate at which particles cross the shock from upstream to downstream:
assume isotropic particle distribution upstream and particle speed

19. Probability for one Shock Crossing
Probability Pe for crossing the shock once and then escaping it:
assuming:
Probability Pr for crossing the shock once and returning to it:

20. Energy Spectrum Probability to cross the shock at least k times
Number of particles N with energy ≥ E
Eliminate k
Integral spectrum

21. Shock Acceleration Rate
need tcycle

22. Shock Acceleration Rate
time spent downstream:
particles move with velocity u2 in the downstream region away from shock
In addition diffusion adds to this movement
define border beyond which particle escapes the shock
t=4/c*k/u

23. Shock Acceleration Rate
analogue to downstream works also for the upstream region
Several possibilities for the diffusion constant
lower limit given by
with

24. Maximum Energies Acceleration is limited by synchrotron radiation
 maximum Energy for Electrons
duration of shock ~1000yrs
pion production
Bethe Heitler pair production
finite acceleration size

25. Summary Fermi acceleration is elastic scattering on magnetic fields
Energy gain per cycle
in second order Fermi
in first order Fermi
1st order produces E-2 differential spectrum
Maximum energy ≈ 100 TeV/nucleon