Magnetic Field Amplification in Diffusive Shock Acceleration
Theory of Shock Acceleration of Hot Ion Populations
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Transcript of Theory of Shock Acceleration of Hot Ion Populations
Theory of Shock Acceleration of Hot Ion Populations
Marty Lee
Durham, New Hampshire
USA
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Planar Stationary DSA - I
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f(z>0,v)=f0−(f0−f∞)(1−e−ζ)(1−e−ζ∞)−1
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f0≡β d′ v′ v0
v∫ f∞( ′ v)(1−e−′ ζ ∞)−1v′ v ⎛ ⎝ ⎜ ⎞ ⎠ ⎟-βexp−β d′ ′ v
′ ′ ve−′ ′ ζ ∞(1−e−′ ′ ζ ∞)−1′ vv∫ ⎡ ⎣ ⎢ ⎤
⎦ ⎥
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Vz∂fi∂z−∂∂z(Ki,zz
∂fi∂z)−13dVzdzv
∂fi∂v=Qi
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ζ(z)= [V/Kzz(z')]dz'0z∫
Planar Stationary DSA - II
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−Kzz∂f∂zz→∞
=V(f0−f∞)e−ζ∞(1−e−ζ∞)−1
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fp,∞= np(4πvp,02 )−1δ(v−vp,0)+ Cv−γS(v− vp,0)
Planar Stationary DSA - III
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log f
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log f
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log v
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Ani
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f∞ (v)∝δ (v − v0)
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Kzz = K0 (v /v0)2
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β =5
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Z = Vz /K0
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Z
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Z
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Z = 0
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Z = 8
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v /v0 = 1
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v /v0 = 5
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v /v0 = 1€
v /v0 = 5
Planar Stationary DSA - IV
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log f
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log f
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log v
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Ani
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f∞ (v)∝δ (v − v0)
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Kzz = K0 (v /v0)2
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β =5€
Z = Vz /K0
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Z
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Z
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Vzesc /K0 = 10
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Z = 0
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Z = 8
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v /v0 = 1
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v /v0 = 1€
v /v0 = 5
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v /v0 = 5
Planar Stationary DSA - V
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log f
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log f
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log v
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Ani
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Kzz = K0 (v /v0)2
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f∞ = (v /v0)−8
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β =5
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Z = Vz /K0
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Z = 0
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Z = 20
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Z
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Z
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v /v0 = 1.5
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v /v0 = 1.5
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v /v0 = 5
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v /v0 = 5
Planar Stationary DSA - VI
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log f
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log f
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log v
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Ani
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Kzz = K0 (v /v0)2
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f∞ = (v /v0)−4.2
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β =5
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Z = Vz /K0
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Z
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Z
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Z = 0
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Z = 20
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v /v0 = 1.01
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v /v0 = 1.01
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v /v0 = 5
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v /v0 = 5
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Tylka et al., 2005
Injection and Shock Obliquity
Anisotropy Limitation
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|S |
vf=
V
v1+
KA2 sin2 θ + (K|| − K⊥)2 sin2 θ cos2 θ
(K|| cos2 θ + K⊥sin2 θ)2
⎡
⎣ ⎢
⎤
⎦ ⎥
1/ 2
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K|| >> K⊥,KA :
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⇒|S |
vf=
V
v cosθGiacalone and Jokipii, 1999
“Energetic Storm
Particle” Event
Complexity
Van Nes et al., 1984
Ion Features
Van Nes et al., 1984
ACE Shock Survey
Lario et al., 2005
Quasi-Perpendicular
Shock Simulation
Giacalone, 1999
Shock Energetic Particle
Correlation Lengths
Neugebauer et al., 2006
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Example Event:1999 / 47-49(Cold Seeds)
Density compression: d/ u ~ 2.9
Clear proton foreshock of appropriate scale.
Softening particle spectra on the same ESP scale.
Particle intensity index appropriate to shock strength.
Small proton anisotropy.
Wave foreshock appropriate scale.
Argued and confirmed to be cold ion seed population leading to shock acceleration.
Example Event:1998 / 217-219
(Hot Seeds)Density compression: d/ u ~ 1.8
Very modest or questionable proton foreshock if SEP event is properly recognized.
Softening particle spectra on SEP scale.
Moderate proton anisotropy.
Nonexistent wave foreshock.
Argued and confirmed to be hot ion seed population leading to shock acceleration.
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Wave Excitation - I
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−V∂I±/∂z=2γ±I±
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I≅I+=Io+(k)+4π2k2VAV|Ωp|mpcosψ dvv3(1−Ωp2
k2v2)(fp−fp,∞)|Ωp/k|∞∫
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fp,∞= np(4πvp,02 )−1δ(v−vp,0)+ Cv−γS(v− vp,0)
Wave Excitation - II
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I=Io+ +4π2k2VAV|Ωp|mpcosψ dvv3(1−Ωp2
k2v2)|Ωp/k|∞∫ ⋅
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⋅ β n4πv0,p3 (
vv0,p)
−βS(v−v0,p)− n4πv0,p2 δ(v−v0,p)
⎧ ⎨ ⎩
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+ C v0,p−γβ−γ γ(v v0,p)−γ−β(v v0,p)
−β ⎡ ⎣ ⎢
⎤ ⎦ ⎥S(v− v0,p)
⎫ ⎬ ⎭⋅
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⋅exp−V d′ zcos2ψv34π
B20Ωp2dμ|μ|(1−μ2)I(Ωpμ−1v−1)+sin2ψK⊥−1
1∫ ⎡ ⎣ ⎢
⎤ ⎦ ⎥−1
0z∫ ⎫ ⎬ ⎭
⎧ ⎨ ⎩
Wave Excitation - IIIβ = 7; I0(k) 0
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Ion Fractionation
Tylka et al., 1999
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Complications within DSAWave-frame compression ratio:
Van Nes et al., 1984
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Fe/O Versus Energy
Tylka et al., 2005
Time Evolution
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vt ≈ V secθ
A Simple Model for Composition
Tylka and Lee, 2006
Fe/O Versus Energy
Tylka and Lee, 2006
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
The Blunt Termination Shock
McComas and Schwadron, 2006
Stone et al., 2008
Bow Shock
Kis et al., 2007
Kis et al., 2007
Current Issues & Challenges• Theory of Planar Stationary Diffusive Shock Acceleration
• Injection Rates and Injection Thresholds
• “Hot” and “Cold” Seed Particle Populations
• Quasilinear Wave Excitation
• Ion Fractionation Upstream of the Shock
• Power-law Index: Observations Versus Theory
• Varying Magnetic Obliquity Within an Event
• Geometry: Earth’s Bow Shock & Solar Wind Termination Shock
• Nonlinear Waves & SLAMS
Upstream Waves
Hoppe et al., 1981
SLAMS
Lucek et al., 2008