Физика космических лучей / HEA12
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HIGH ENERGY ASTROPHYSICS - Lecture 12
PD Frank Rieger
ITA & MPIK Heidelberg
Wednesday
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Fermi Particle Acceleration
1Overview
Scattering agent (plasma waves)
1st order Fermi acceleration
Shock Dynamics
Particle Spectra
Acceleration Timescale
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2Scattering Agents
Modern picture: Scattering/de ection mediated by plasma waves/magnetic
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turbulence B.
Resonant interaction when particle gryo-radius comparable to wavelength of perturbation.
rg >>
rg <<
rg ~
Figure 1: Example: Pitch angle scattering by B=B0, B B0, when gyro-radius becomes comparable to wavelength of perturbation, resulting in guiding centre drift by r rgyro .
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Charged particle in magnetic eld describes circular (cyclotron) motion about m.f. (=gyro-motion), v?, plus translational motion, vk along it:
When particle senses electromagnetic wave Doppler-shifted to its cyclotron frequency (or its harmonics), it can interact strongly with wave:
) ! kkvk = s
~ s = 0; 1; 2:::, !=wave frequency, kk=component of wave vector along B.
) When this is satis ed, particles remain in phase, leading to energy and momentum exchange.
General gryo-resonance ("Doppler") condition in magnetized plasma:
! s kkvk = 0
j~ j
where = q B = mc=relativistic gyro-frequency of particle with charge q
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and vk = velocity vector parallel to B.
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Example: for low-frequency Alfven waves, ' kkvk, i.e., rg = |
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kk |
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3Waves in magnetized plasmas - Example: Alfven wave (MHD)
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Mean background eld: B = B0x^ |
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{ Wave properties/perturbations: B;~ |
E;~ |
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~v / eik ~r i!t |
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{ Dispersion relation: ! = !(k) |
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Alfven waves:
{ low-frequency (large wavelength), transverse plasma waves, circularly polarised
{ ! = kkvA (dispersion-less) when ! i = |
ZeB |
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; with vA |
= B0 |
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{ion inertia vs. magnetic eld line tension as resorting force
~k~
{propagates along magnetic eld lines (k B0), like a wave on a string
~? ~
{B B0, so to 1st order no perturbation to eld strength
jB0 |
+ Bj = q |
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+ 2B B + ( B) |
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= pB0 |
+ ( B) |
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' B0 h1 + 2B02 |
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(B)2 |
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Field line
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4Resonance condition
helical trajectory
B0
Magnetic Field
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Consider elm. wave E = E0 cos(k ~x !t) |
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let ~x = ~x |
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+ vk b t, where b = B0=B0 unit vector. |
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) Primed frame moves with particle guiding centre. |
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~x |
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Then E = E0 cos[k |
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(! kkvk)t] where kk = k |
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! kkvk = Doppler-shifted frequency in guiding center frame.
("Frequency of wave as seen in guiding center frame")
Wave-particle resonance when ! kkvk = s .
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Pitch angle scattering of a particle due to interaction with uctuating
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B- eld of electromagnetic wave: |
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) Lorentz force FL = |
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~v B in same direction at same phase. |
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) interaction with perpendicular motion, v?, increases pk.
) interaction with parallel motion of particle, vk, slows down gyration (p?). ) change of pitch angle = arccos(pk=p).
) for di erent phases, di erent ) di usion in pitch angle.
Bw |
Bw |
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FL |
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B0 |
B0 |
Interaction between a positively charged particle and a wave. LEFT: interaction with v? velocity com-
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ponent. RIGHT: interaction with vk component. Lorentz force due to the particles motion and wave eld B can change particle momentum (and pitch angle). If the relative phase between particle and wave were shifted by , the situation would be reversed.
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Stochastic energy changes due to interaction with uctuating E- eld of electromagnetic wave:
Consider circularly polarised wave:
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(note: k E = |
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B for plane waves, i.e. j Ej = |
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j Bj = |
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j Bj for Alfven waves). |
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) Particle is in resonance if wave frequency equals gyration frequency. ) Particle can gyrate parallel or anti-parallel to electric wave eld.
) Acceleration or deceleration depending on random phase between wave and particle.
E |
E |
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E |
B0
v| 
E v|
Figure 3: ~
Resonant interactions with E- eld of a circularly polarised wave. Upper panel shows particle gyration
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along B0 together with uctuating electric eld. Lower panel gives two extreme cases: depending on phase between wave and gyro-orbit, particles moves either parallel or anti-parallel to electric eld, i.e. experiences acceleration or deceleration.
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Note: As shown by Achterberg (1981), 2nd order Fermi-type acceleration results from resonant damping (energy
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exchange) of/with magnetosonic waves (has component k |
B0) in the presence of e cient pitch-angle scattering |
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(by Alfven waves, k B0). Cherenkov resonance (s = 0; ! kkvk = 0) is possible for waves with a component of
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their electric eld along B0, i.e., for magnetosonic but not for Alfven waves (! = kjjvA). Resonant interactions cause particles to di use in pk, with p? remaining constant. Thus, without e cient pitch-angle scattering, distribution of fast particles would become increasingly anisotropic, favouring pk over p?, leading to a suppression of the acceleration.
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5Magnetic Turbulence
Magnetic elds in space can be highly turbulent - expect spectrum of waves:
Magnetic eld as sum of a mean eld plus turbulent component:
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+ B with < B >=< B0 |
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Isotropic < B >= 0, but non-vanishing power: |
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kmax |
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< ( B~ )2 >= Zkmin |
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E^(k) = E |
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k power spectrum of turbulence, |
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Kolmogorov turbulence: |
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< ( B~ )2 >= Z kmax E(k)dk |
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E(k) / k 5=3 |
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kmin |
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) most power at low k , large .
Charged particle transport dependent on
) intensity of turbulence = |
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, spatial di usion coe cient D = |
v rg |
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B2+B02 |
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( B)2 |
) kmin and rL, i.e. coherence length and particle Larmor radius.
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