The Acceleration of Anomalous Cosmic Rays by the

1. SHINE Meeting, Nova Scotia August 3-7, 2009 The Acceleration of Anomalous Cosmic Rays by the Heliospheric Termination Shock J. A. le Roux, V. Florinski, N. V. Pogorelov, & G. P. Zank Dept. of Physics & CSPAR University of Alabama in Huntsville, Huntsville, AL 35763

2. 1. The problem facing standard diffusive shock acceleration theory Standard diffusive shock acceleration (DSA) theory: • Near-isotropicdistributions • Distribution functioncontinuousacross the shock • Distribution function forms aplateaudownstream • Power law spectra with asingleslope • Steady-stateintensities Energetic particle observations by Voyager contradict standard DSA: • Largefield-aligned beamsupstreamdirectedawayfrom shock– highlyvariableanisotropy – peak in anisotropy at ~0.4 MeV • Highly anisotropicintensity spikesat shock • Distribution functiondeviatesfromplateaudownstream • Power law spectra harder than predicted by DSA theory–multipleslopes- • spectrum concave? • Upstreamintensitieshighly variable The solution: 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

3. 2. The Focused Transport Equation Standard CR transport Focused transport Shock drift energy loss due to curvature drift Convection Adiabatic energy changes Diffusion 1st order Fermi acceleration FOCUSED TRANSPORT INCLUDE BOTH 1ST ORDER FERMI AND SHOCK DRIFT ACCELERATION– BUT NO LIMITATION ON PITCH-ANGLE ANISOTROPY Shock drift acceleration due to grad-B drift

4. 3. Drifts in the Focused Transport Equation Guiding Center Kinetic Equation for f(xg, M,’,t) where Grad-B and curvature driftsabsent in convection Conservation of magnetic moment Shock drift included – with or without scattering Electric field drift Grad-B drift Curvature drift

5. 4. Possible Disadvantages of Focused Transport • 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 atv~U Focused transport equation suitable for modeling anisotropic shock acceleration

6. 5. Results of Shock Acceleration of “core” Pickup Ions with a Time- dependent Focused Transport (i) Stochastic injection speed Injection speed if 1 = BN = 89.4o De Hoffman-Teller speed in SW frame is the injection speed Voyager 1 – 2004 – 1 hour averages When including time variations in spiral angle (stochastic injection speed), shock acceleration of “core” pickup ions works Mimics anomalous perpendicular diffusion

7. (ii) Multiple Power Law Slopes - Observations Cummings et al., [2006] Decker et al. [2006] Upstreamspectra arevolatile Downstream spectra more stable Multiple power law slopes

8. Decker et al., [2008] – 78 day averages Breaking points at ~0.06 MeV & 0.3 MeV Rollover at ~ 0.7 MeV Bump at ~0.1 MeV Breaking points at ~0.07 & 0.2 MeV Rollover at ~ 1-2 MeV Both at V2 and V1, post-TS spectrum has multipleslopes Exponential rollovers Multiple power laws partly due to nonlinear shock acceleration?

9. le Roux & Webb [2009], ApJ (ii) Multiple Power Law Slopes - Simulations Bump at ~0.02-0.04 MeV Pickup proton “core” distribution upstream downstream 101 AU 1 2 3 Rollover at ~3.5 MeV v-4.2 v-3.3 Breaking points at ~0.01 & 0.4 MeV DSA predicts v-4.4 ifs = 3.2 • 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

10. le Roux et & Fichtner [1997], JGR Breaking points at 0.01-0.02 MeV and at ~0.3-0.4 MeV Exponential rollover The ACR spectrum calculated with a nonlinear DSA model – TS modified self-consistently by ACR pressure gradient Multiple power law slopes

11. (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

12. (iii) Episodic Intensity Spikes - Simulations Spikes caused by magnetic reflection le Roux & Webb [2009], ApJ t3 1 MeV t1 10 MeV 1 MeV t2 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

13. (iv) Episodic Upstream Field-aligned Particle Beams - Observations Decker et al., [2006] – V1 observations from 2004 -2006.6 – daily averages TS Upstream Downstream 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

14. (iv) Episodic Upstream Field-aligned Particle Beams - Simulations le Roux & Webb [2009], ApJ 1 MeV downstream  = 72% upstream 101 AU t3  = 50% t2 t1 Success: Large fluctuations in anisotropies upstream die out deeper in heliosheath

15. (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

16. (v) Energy Dependence of Upstream Anisotropy - Simulations Vinj = U1/cos1 le Roux & Webb [2009], ApJ Shock acceleration Florinski et al.,[2008] 1 MeV 10 MeV 10 keV If Einj = 1 MeV, 1= BN = 88o Peak in upstream anisotropy is signature of a nearly-perpendicular shock Peak indicates injection threshold energy– shock obliquity

17. Summary and Conclusions Useful features of Focused Transport model: • Just as standard cosmic ray transport equation - Focused transport equation contains • both1st 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) Key element in model’s success: • Inclusion of time variations in De Hoffman-Teller velocity • determined by upstream time variations in BN Successes: • 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 Problems still to be addressed: • 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