1. Particle Acceleration The observation of high-energy  -rays from space implies that particles must be accelerated to very high energies (up to ~ 10 20 eV to explain highest-energy cosmic rays). Power-law spectra observed in many objects indicate that acceleration mechanisms produce power-law particle spectra, N(  ) ~  -p Generally, d(  mv) = e (E + v x B) No strong, static electric fields (most of the regular matter in the Universe is completely ionized) => Acceleration may be related to induced electric fields (time-varying or moving magnetic fields)

2. Particle Acceleration at Strong Shocks General Idea: Particles bouncing back and forth across shock front: At each pair of shock crossings, particles gain energy = (4/3) V/c = U/c p 2,  2 p 1,  1 U Stationary frame of ISM v 2 = (1/4) v 1 v 1 = U Shock rest frame v 1  1 = v 2  2  1 /  2 = 1/4 Rest frame of shocked material v 2 ‘ = (3/4) U v 1 ‘ = (3/4) U Write E =  E 0 = (1 + U/c) E 0 ;  = 1 + U/c

3. Particle Acceleration at Strong Shocks (cont.) Flux of particles crossing the shock front in either direction: F cross = ¼ Nc Downstream, particles are swept away from the front at a rate : NV = ¼ NU p 2,  2 p 1,  1 U Stationary frame of ISM v 2 = (1/4) v 1 v 1 = U Shock rest frame v 1  1 = v 2  2  1 /  2 = 1/4 Rest frame of shocked material v 2 ‘ = (3/4) U v 1 ‘ = (3/4) U Probability of particle to remain in the acceleration region: P = 1 - (¼ NU)/(¼ Nc) = 1 – (U/c)

4. Particle Acceleration at Strong Shocks (cont.) Energy of a particle after k crossings: E =  k E 0 => ln (N[>E]/N 0 ) / ln (E/E 0 ) = lnP / ln  or N(>E)/N 0 = (E/E 0 ) lnP/ln  Number of particles remaining: N = P k N 0 => N(E)/N 0 = (E/E 0 ) -2

5. More General Cases Weak nonrelativistic shocks p > 2 Relativistic parallel shocks: p = 2.2 – 2.3 Relativistic oblique shocks: p > 2.3 U B U U BU

6. Effects of Cooling and Escape = - (  n e ) + Q e ( ,t) - ________ ∂n e ( ,t) ∂t ∂ ∂∂. Radiative and adiabatic losses Escape ______ n e ( ,t) t esc,e Particle injection (acceleration on very short time scales) Evolution of particle spectra is governed by the Continuity Equation:

7. Effects of Cooling and Escape (cont.) Assume rapid particle acceleration: Q( , t) = Q 0  -q  1 <  <  2 Fast Cooling : t cool << t dyn, t esc for all particles N(  )      (q+1)  F( )   q/2  Particle spectrum: Synchrotron or Compton spectrum:

8. Effects of Cooling and Escape (cont.) Assume rapid particle acceleration: Q( , t) = Q 0  -q  1 <  <  2 Slow Cooling : t cool  b N(  )  qq   (q+1) bb F( )  q   q/2 b Particle spectrum: Synchrotron or Compton spectrum: 11 1