The Acceleration of Anomalous Cosmic Rays by the Heliospheric Termination Shock

J. A. le Roux, V. Florinski, N. V. Pogorelov, & G. P. ZankDept. of Physics & CSPARUniversity of Alabama in Huntsville, Huntsville, AL 35763

SHINE Meeting, Nova Scotia August 3-7, 2009

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1. The problem facing standard diffusive shock acceleration theory

• Near-isotropic distributions

• Distribution function continuous across the shock

• Distribution function forms a plateau downstream

• Power law spectra with a single slope

• Steady-state intensities

Standard diffusive shock acceleration (DSA) theory:

• Large field-aligned beams upstream directed away from shock– highly variable anisotropy – peak in anisotropy at ~0.4 MeV

• Highly anisotropic intensity spikes at shock

• Distribution function deviates from plateau downstream

• Power law spectra harder than predicted by DSA theory – multiple slopes-

spectrum concave?

• Upstream intensities highly variable

Energetic particle observations by Voyager contradict standard DSA:

A shock acceleration model that can handle large pitch-angle anistropies and includes the stochastic nature of the termination shock’s shock obliquity, the focused transport model

The solution:

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2. The Focused Transport Equation

jij

i

i

i

ii

x

f

x

p

fp

x

U

x

fU

t

f

3

1

PUI

iii

ij

iji

i

i

iii

ij

iji

i

i

i

i

iii

Qf

D

p

fpbE

p

q

dt

dUb

vx

Ubb

x

U

fbE

p

q

dt

dUbvx

Ubb

x

U

x

bv

x

fbvU

t

f

22

2

312

11

2

1

2231

2

1

Convection

Adiabatic energy changes

Diffusion

Focused transportStandard CR transport

1st order Fermi acceleration

Shock driftacceleration due to grad-B drift

Shock driftenergy loss due to curvature drift

FOCUSED TRANSPORT INCLUDE BOTH 1ST ORDER FERMI AND SHOCK DRIFT ACCELERATION – BUT NO LIMITATION ON PITCH-ANGLE ANISOTROPY

4

ct

ff

dt

d

M

f

dt

dMf

t

f

'

'

gg1 xV

bbBBbVbV

VE

VbV

Eg2

g2

Eg1

||2||||

||

,'

,0

,

vdt

d

qB

m

Bq

M

q

Mv

t

BMq

dt

ddt

dM

v

where

Grad-B drift Curvature driftElectric field drift

Conservation of magnetic moment

3. Drifts in the Focused Transport Equation

Grad-B and curvature drifts absent in convection

Shock drift included – with or without scattering

Guiding Center Kinetic Equation for f(xg, M,’,t)

5

•No cross-field diffusion – can be added or simulated by varying magnetic field angle

•Gradient and curvature drift effect on spatial convection ignored – might be negligible – or can be added – drift kinetic equation

•Magnetic moment conservation at shocks – reasonable assumption

•Gyrotropic distributions – reflection by shock potential at perpendicular shock not described

•No polarization drifts – can be added – higher order drift kinetic equation – only important at v~U

4. Possible Disadvantages of Focused Transport

Focused transport equation suitable for modeling anisotropic shock acceleration

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5. Results of Shock Acceleration of “core” Pickup Ions with a Time- dependent Focused Transport

v/Ue

100 101

f(v)

(s3 /

km6 )

10-1

100

MeV9sec 11 UvInjection speed if 1 = BN = 89.4o

Spiral angle

0 30 60 90 120 150 180 210 240 270 300 330 360 390

Eve

nts

0

20

40

60

80

100

120

140

160

180

200

220

Voyager 1 – 2004 – 1 hour averages

Mimics anomalous perpendicular diffusion

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|| cos

De Hoffman-Teller speed in SW frame is the injection speed

When including time variations in spiral angle (stochastic injection speed), shock acceleration of “core” pickup ions works

(i) Stochastic injection speed

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(ii) Multiple Power Law Slopes - Observations

Upstream spectra are volatileDownstream spectra more stableMultiple power law slopes

Cummings et al., [2006]

Decker et al. [2006]

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Both at V2 and V1, post-TS spectrum has multiple slopes

Exponential rollovers

Multiple power laws partly due to nonlinear shock acceleration?

Decker et al., [2008] – 78 day averages

Breaking points at ~0.06 MeV & 0.3 MeV

Rollover at ~ 0.7 MeV

Breaking points at ~0.07 & 0.2 MeV

Rollover at ~ 1-2 MeV

Bump at ~0.1 MeV

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v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-1110-1010-910-810-710-610-510-410-310-210-1100101102103104

(ii) Multiple Power Law Slopes - Simulations

v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-1110-1010-910-810-710-610-510-410-310-210-1100101102103104

v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-1110-1010-910-810-710-610-510-410-310-210-1100101102103104

upstream downstream 101 AU

Breaking points at ~0.01 & 0.4 MeV

v-4.2 v-3.3

DSA predicts v-4.4 if s = 3.2

Pickup proton “core” distribution

Successes:•Multiple power laws – stochastic injection speed•Higher energy breaking point at realistic and fixed energies downstream•Bump feature - magnetic reflection•Volatility in upstream spectra damped out deeper in heliosheath•3rd power law harder than predicted by DSA theory – magnetic reflection

Rollover at ~3.5 MeV

Bump at ~0.02-0.04 MeV

le Roux & Webb [2009], ApJ

1

2

3

10

le Roux et & Fichtner [1997], JGR

The ACR spectrum calculated with a nonlinear DSA model – TS modified self-consistently by ACR pressure gradient

Multiple power law slopes

Breaking points at 0.01-0.02 MeV and at ~0.3-0.4 MeV

Exponential rollover

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(iii) Episodic Intensity Spikes - Observations

Decker et al. [2005] V1 observations at TS

intensity spike just upstream of TS along magnetic field

Factor of ~5-10 increase in counting rate

Anisotropy of ~ 92 % - highly anisotropic

No spikes seen at V2

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r(AU)93.00 94.00 95.00

f(r)

10-2

10-1

100

r(AU)93.00 94.00 95.00

f(r)

10-2

10-1

100

(iii) Episodic Intensity Spikes - Simulations

10 MeV

1 MeV

1 MeV

t2

t1

t3

Spikes only occur when injection speed is low enough (BN is small enough) so that particles can magnetically be reflected upstream

Episodic nature of spikes controlled by time variations in BN

Spikes caused by magnetic reflectionle Roux & Webb [2009], ApJ

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(iv) Episodic Upstream Field-aligned Particle Beams - Observations

Upstream Downstream

TS

Upstream: – pitch-angle anisotropy is highly volatile, can reach ~ 100%, and field-aligned Downstream: – anisotropy converge to zero with increasing distance and is very stable

Decker et al., [2006] – V1 observations from 2004 -2006.6 – daily averages

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-1 0 1

f( )

0.0

5.0

-1 0 1

f( )

0.0

5.0

-1 0 1

f( )

0.0

5.0

upstream downstream 101 AU

t1

t2

t3

= 72%

= 50%

Success:Large fluctuations in anisotropies upstreamdie out deeper in heliosheath

(iv) Episodic Upstream Field-aligned Particle Beams - Simulations

le Roux & Webb [2009], ApJ 1 MeV

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(v) Energy Dependence of Upstream Anisotropy - Observations

Decker et al. [2006] V1 observations ~ 6 month averages

Upstream 1st order pitch-angle anisotropy peaks at ~0.3 MeV - no continuing increase with decreasing particle energy

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(v) Energy Dependence of Upstream Anisotropy - Simulations

Kinetic Energy (MeV)

10-3 10-2 10-1 100 101 102 103

0.10.20.30.40.50.60.70.80.91.0

1 MeV

10 keV

10 MeV

Vinj = U1/cos1

Shock acceleration

If Einj = 1 MeV, 1= BN = 88o

Peak in upstream anisotropy is signature of a nearly-perpendicular shockPeak indicates injection threshold energy– shock obliquity

Florinski et al.,[2008]le Roux & Webb [2009], ApJ

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Summary and Conclusions

•Multiple power law slopes – stable break points downstream •Strong fluctuations in upstream intensities – die out in heliosheath•Strong episodic intensity spikes at termination shock •Strong fluctuations in upstream B-aligned pitch-angle anisotropy – damped out in heliosheath•Peak in upstream anisotropy at ~ 1 MeV – peak is signature of nearly perpendicular shock

•The role of nonlinear shock acceleration in contributing to multiple power law slopes•Explanation of observed spectral slopes and TS compression ratio at V2 within shock acceleration context

•Inclusion of time variations in De Hoffman-Teller velocity determined by upstream time variations in BN

•Just as standard cosmic ray transport equation - Focused transport equation contains both 1st order Fermi and shock drift acceleration •Advantage – no restriction on pitch-angle anisotropy- Ideal for modeling injection close to the injection threshold velocity (de Hoffman-Teller velocity)

Successes:

Problems still to be addressed:

Useful features of Focused Transport model:

Key element in model’s success: