An Analytical Theory of Diffusive Shock Acceleration for Gradual SEP

Events

Marty Lee

Durham, New Hampshire

USA

Acceleration at a CME-Driven Shock

Lee, 2005

First-Order Fermi Acceleration

vz

v

Upstream Waves I

Tsurutani et al., 1983

Upstream Waves II

Hoppe et al., 1981

Instability Mechanism

VA vz

v

January 20, 2005 GLE Event

Bieber et al., 2005

Large SEP Events

Reames and Ng, 1998

SEP Energy Spectra

Tylka et al., 2001

Shock Geometry

Cane et al., 1988

Shock Acceleration I

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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

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ζ(z)= [V/Kzz(z')]dz'0z∫

Shock Acceleration II

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−Kzz∂f∂zz→∞

=V(f0−f∞)e−ζ∞(1−e−ζ∞)−1

Shock Acceleration III

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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)

Shock Acceleration IV

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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)

⎧ ⎨ ⎩

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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∫ ⎫ ⎬ ⎭

⎧ ⎨ ⎩

Shock Acceleration V

Analytical and Numerical Calculations Proposed I

1. Calculation of with dependence on

2. Role of shock obliquity (Tylka and Lee, 2006)€

I (k,z)

€

ψ,K⊥,ξ p (ψ ),v0 (ψ )

€

(ψ )

Analytical and Numerical Calculations Proposed II

3. Form of exponential rollover with dependence on

4. Ion spatial profiles upstream of the shock with dependence on

5. Escaping ions and the streaming limit with dependence on

6. Inferring from ESP events

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E,ψ , A /Q,K⊥

€

E,ψ , A /Q,K⊥

€

E, A /Q

€

ξ ,β

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exp −βdv'

v'

e−ζ ∞'

1− e−ζ ∞'

v0

v

∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥→ exp −β

dv'

v'

1

ζ ∞'v0

v

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥€

I ∝ k−λ

€

K⊥= 0

Spectral Rollover:

€

ζ∞i ∝ (Qi / Ai)

2−λ v λ −3

€

→ exp −C (ψ )(Ai /Qi)2−λ v 3−λ

[ ]

Mewaldt et al.;

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E0i ∝ (Qi / Ai)

b

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b = 22 − λ

3− λ

Work Plan: Way Behind!

Year 1: Complete and publish analytical work

Year 2: Develop the numerical algorithm and test the analytical approximation.

Year 3: Systematic comparison of results with selected events