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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. v . v z. Upstream Waves I. Tsurutani et al., 1983. Upstream Waves II. Hoppe et al., 1981. - PowerPoint PPT Presentation
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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
€
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∫
Shock Acceleration II
€
−Kzz∂f∂zz→∞
=V(f0−f∞)e−ζ∞(1−e−ζ∞)−1
Shock Acceleration III
€
−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)
Shock Acceleration IV
€
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∫ ⎫ ⎬ ⎭
⎧ ⎨ ⎩
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
€
E,ψ , A /Q,K⊥
€
E,ψ , A /Q,K⊥
€
E, A /Q
€
ξ ,β
€
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.;
€
E0i ∝ (Qi / Ai)
b
€
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