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

€

f(z>0,v)=f0−(f0−f∞)(1−e−ζ)(1−e−ζ∞)−1

€

f0≡β d′ v′ v0

v∫ f∞( ′ v)(1−e−′ ζ ∞)−1v′ v ⎛ ⎝ ⎜ ⎞ ⎠ ⎟-βexp−β d′ ′ v

′ ′ ve−′ ′ ζ ∞(1−e−′ ′ ζ ∞)−1′ vv∫ ⎡ ⎣ ⎢ ⎤

⎦ ⎥

€

Vz∂fi∂z−∂∂z(Ki,zz

∂fi∂z)−13dVzdzv

∂fi∂v=Qi

€

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

€

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

€

Z

€

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

€

log f

€

log v

€

Ani

€

f∞ (v)∝δ (v − v0)

€

Kzz = K0 (v /v0)2

€

β =5€

Z = Vz /K0

€

Z

€

Z

€

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

€

log f

€

log f

€

log v

€

Ani

€

Kzz = K0 (v /v0)2

€

f∞ = (v /v0)−8

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β =5

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Z = Vz /K0

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Z = 0

€

Z = 20

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Z

€

Z

€

v /v0 = 1.5

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v /v0 = 1.5

€

v /v0 = 5

€

v /v0 = 5

Planar Stationary DSA - VI

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log f

€

log f

€

log v

€

Ani

€

Kzz = K0 (v /v0)2

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f∞ = (v /v0)−4.2

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β =5

€

Z = Vz /K0

€

Z

€

Z

€

Z = 0

€

Z = 20

€

v /v0 = 1.01

€

v /v0 = 1.01

€

v /v0 = 5

€

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

€

|S |

vf=

V

v1+

KA2 sin2 θ + (K|| − K⊥)2 sin2 θ cos2 θ

(K|| cos2 θ + K⊥sin2 θ)2

⎡

⎣ ⎢

⎤

⎦ ⎥

1/ 2

€

K|| >> K⊥,KA :

€

⇒|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±

€

I≅I+=Io+(k)+4π2k2VAV|Ωp|mpcosψ dvv3(1−Ωp2

k2v2)(fp−fp,∞)|Ωp/k|∞∫

€

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|∞∫ ⋅

€

⋅ β n4πv0,p3 (

vv0,p)

−βS(v−v0,p)− n4πv0,p2 δ(v−v0,p)

⎧ ⎨ ⎩

€

+ C v0,p−γβ−γ γ(v v0,p)−γ−β(v v0,p)

−β ⎡ ⎣ ⎢

⎤ ⎦ ⎥S(v− v0,p)

⎫ ⎬ ⎭⋅

€

⋅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

€

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