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Shocks and Fermi-I Acceleration

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Non-Relativistic Shocks p 1, 1, T 1 p 0, 0, T 0 vsvs p 1, 1, T 1 p 0, 0, T 0 v 0 = -v s Stationary Frame Shock Rest Frame v1v1

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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 = v s 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

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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)

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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

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More General Cases Weak nonrelativistic shocks p > 2 Relativistic parallel shocks: p = 2.2 – 2.3 Relativistic oblique shocks: Almost any spectral index possible U B U U BU

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Diffusive Shock Acceleration b a f r

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Electron Spectra from Diffusive Shock Acceleration = *r g = Pitch-angle scattering mean free path Moderately sub-luminal ( 1HT = 1x /cos Bf1 < 1) Marginally sub-luminal ( 1HT = 1x /cos Bf1 ~ 1) (Summerlin & Baring 2012)

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Asymptotic Particle Spectral Index n (Summerlin & Baring 2012)

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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:

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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:

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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+1) b F( ) q q/2 b Particle spectrum: Synchrotron or Compton spectrum: 1 1

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