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Particle Acceleration

0. The observation of high-energy g -rays from space implies that particles must be accelerated to very high energies (up to ~ 10 20 eV to explain highest-energy cosmic rays). Particle Acceleration.

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Particle Acceleration

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  1. 0 The observation of high-energy g-rays from space implies that particles must be accelerated to very high energies (up to ~ 1020 eV to explain highest-energy cosmic rays). Particle Acceleration Power-law spectra observed in many objects indicate that acceleration mechanisms produce power-law particle spectra, N(g) ~ g-p Generally, d(gmv) = 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. 0 General Idea: Particles bouncing back and forth across shock front: Particle Acceleration at Strong Shocks Shock rest frame Stationary frame of ISM v1r1 = v2r2 U v2 = (1/4)v1 v1 = U r1 / r2 = 1/4 p2, r2 p1, r1 Rest frame of shocked material v1‘ = (3/4) U v2 ‘ = (3/4)U At each pair of shock crossings, particles gain energy <DE / E> = (4/3) V/c = U/c Write E = b E0 = (1 + U/c) E0 ; b = 1 + U/c

  3. 0 Stationary frame of ISM Shock rest frame Particle Acceleration at Strong Shocks (cont.) U v1r1 = v2r2 v2 = (1/4)v1 v1 = U p2, r2 r1 / r2 = 1/4 p1, r1 Rest frame of shocked material v1‘ = (3/4) U v2 ‘ = (3/4)U Flux of particles crossing the shock front in either direction: Fcross = ¼ Nc Downstream, particles are swept away from the front at a rate : NV = ¼ NU Probability of particle to remain in the acceleration region: P = 1 - (¼ NU)/(¼ Nc) = 1 – (U/c)

  4. 0 Energy of a particle after k crossings: E = bk E0 Particle Acceleration at Strong Shocks (cont.) Number of particles remaining: N = Pk N0 => ln (N[>E]/N0) / ln (E/E0) = lnP / lnb = -1 or N(>E)/N0 = (E/E0)lnP/lnb => N(E)/N0 = (E/E0)-2

  5. 0 Weak nonrelativistic shocks p > 2 More General Cases Relativistic parallel shocks: p = 2.2 – 2.3 B U U Relativistic oblique shocks: p > 2.3 B U U

  6. 0 Effects of Cooling and Escape Evolution of particle spectra is governed by the Continuity Equation: ∂ne (g,t) . ______ ∂ __ ne (g,t) ______ = - (g ne) + Qe (g,t) - ∂t ∂g tesc,e Particle injection (acceleration on very short time scales) Escape Radiative and adiabatic losses

  7. 0 Effects of Cooling and Escape (cont.) Assume rapid particle acceleration: Q(g, t) = Q0g-q g1 < g < g2 Fast Cooling : tcool << tdyn, tesc for all particles Synchrotron or Compton spectrum: Particle spectrum: n-1/2 g-2 F(n) N(g) n-q/2 g-(q+1) n g n1 g1

  8. 0 Effects of Cooling and Escape (cont.) Assume rapid particle acceleration: Q(g, t) = Q0g-q g1 < g < g2 Slow Cooling : tcool << tdyn, tesc only for particles with g > gb Synchrotron or Compton spectrum: Particle spectrum: n-(q-1)/2 g-q F(n) N(g) n-q/2 g-(q+1) n g n1 nb g1 gb

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