Diffusive Shock Acceleration Nepomuk Otte MAGIC/EUSO Seminar 21.01.05

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

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

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

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

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

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

Fermi Acceleration I follow: Origin and Propagation of the highest energy cosmic rays R. J. Protheroe astro-ph/9612212 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, 147-156 The acceleration of cosmic rays in shock fronts II A. R. Bell MNRAS (1978) 182, 443-455

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

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)

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

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

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

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

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

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

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

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

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:

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

Shock Acceleration Rate need tcycle

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

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

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

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