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 SHINE Meeting, Nova ScotiaAugust 3-7, 2009

2 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:

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

4 where Grad-B driftCurvature drift Electric 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(x g, 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

6 5. Results of Shock Acceleration of “core” Pickup Ions with a Time- dependent Focused Transport Injection speed if  1 =  BN = 89.4 o Voyager 1 – 2004 – 1 hour averages Mimics anomalous perpendicular diffusion 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

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

8 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

9 (ii) Multiple Power Law Slopes - Simulations upstreamdownstream101 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 ~ 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 MeV and at ~ MeV Exponential rollover

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 10 MeV 1 MeV t2t2 t1t1 t3t3 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 reflection le Roux & Webb [2009], ApJ

13 (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 – daily averages

14 upstream downstream 101 AU t1t1 t2t2 t3t3  = 72%  = 50% Success: Large fluctuations in anisotropies upstream die out deeper in heliosheath (iv) Episodic Upstream Field-aligned Particle Beams - Simulations le Roux & Webb [2009], ApJ 1 MeV

15 (v) Energy Dependence of Upstream Anisotropy - Observations Decker et al. [2006] V1 observations ~ 6 month averages Upstream 1 st order pitch-angle anisotropy peaks at ~0.3 MeV - no continuing increase with decreasing particle energy

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

17 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 1 st 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: