1. A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004 Huntsville Modelling Workshop

2. McIlwain, 1966

3. Correlations Highest SW correlation for energetic particles in the radiation belts is: velocity. R=.7-.8 during high- speed streams) V is NOT an energy. Not a density. Nor a Force(mv) Multiplying by density  ram or mechanical energy, makes the correlation worse. Multiplying by Bz  Electrical energy, makes the correlation worse. Whatever the mechanism, it is not energy-starved (2 nd moment) or density starved (0 th moment).

4. Motivation The origin of radiation belt particles has been resistant to analysis and modelling for some 40 years since their discovery. Nevertheless, it is a crucial component of Space Weather. To paraphrase Jim Drake on reconnection, “We cannot hold up our heads until we solve this problem” Recent satellite discoveries and theories have converged that the cusp might hold the key. But, current radiation belt models are equatorial, they do not incorporate high latitude sources.

5. Outline I.Data support for high-latitude source a)POLAR data b)Simulations and invariants II.Comparison of accelerator efficiencies a)Definition of efficiency b)Dipole / Fermi / Quadrupole III.Transport from the cusp a) 4D Fokker-Planck

6. Sheldon et al., GRL 1998 POLAR/ CAMMICE data 1 MeV electron PSD in outer cusp

7. Maxwell 1880 Chapman 1930

8. Quadrupolar T87 Magnetosphere Since Chapman & Ferraro 1937, we’ve known the magnetosphere is a quadrupole. All trapping models have assumed a dipole. If Quadrupole is source, how does it change the transport?

9. The Simulated T96 Quadrupole Trap Lousy Trap Great Accelerator Can be made to trap better though.

10. e - Trap Regions of T96 Cusp

11. Cusp Provisional Invariants Minimum energy is defined by “separatrix” energy (ExB =  B) ~ 30 keV The max energy defined by rigidity.~ 4 MeV e - Mapping C-shell limits to the dipole give 5<L<∞.  very close to the PSD “bump” Mapping Quad Energy limits to the rad belts, give ~ 0-50 keV for protons, and 1-10 MeV for electrons. Mapping pitchangles give 0 o <  < 50 o  microburst? Cusp particles have all the right properties  ORBE

12. Accelerator Efficiency Why would the cusp accelerate at all? Why not just use standard well- known accelerators?

13. Plasma Thermodynamics A neglected field—so this discussion is very qualitative. (See Krall & Trivelpiece) In ideal gases the Maxwell equilibrium is extremely well maintained by collisions In plasmas, turbulence takes the place of collisions. But the long-range Coulomb interaction means that equilibrium is often attained very slowly. Therefore one must get accustomed to non- Maxwellian distributions, and non-standard measures of temperature and entropy. E.g. kappas

14. What is Acceleration? Entropy is often defined as S = Q/T. Thus highly energetic particles have low entropy. Often they are “bump-on-a-tail” distributions, or “power law” greater than Maxwellian. Such low-entropy conditions cannot happen spontaneously, so some other process must increase in entropy. Separating the two mechanisms, we have a classic heat-engine. Note: the entropy INCREASE of heat flow into the cold reservoir, is counterbalanced by the DECREASE of entropy into the Work.

15. Why does a Trap help? Single-stage acceleration mechanisms operate very far from equilibrium. They therefore have a huge difference in entropy between W and Q 1, and are therefore inherently inefficient. Likewise environmental conditions are far from equilibrium, and are thus inherently unlikely. The product of these two situations = low density of accelerated particles. Energy Source * Efficiency * Probability = Power A trap allows small impulses to be cumulative. The pulses are close to equilibrium and are also likely. Thus a trap = higher density of accelerated particles.

16. The Dipole Trap Accelerator The dipole trap has a positive B-gradient that causes particles to trap, by  B-drift in the equatorial plane. Three symmetries to the Dipole each with its own “constant of the motion” 1)Gyromotion around B-field  Magnetic moment, “  ”; 2) Reflection symmetry about equator  Bounce invariant “J”; 3) Cylindrical symmetry about z-axis  Drift invariant “L” Betatron acceleration by E ┴ compression, violation of 3 rd invariant, L-shell

17. The Fermi-Trap Accelerator Waves convecting with the solar wind, compress trapped ions between the local |B| enhancement and the planetary bow shock, resulting in 1-D compression, or E // enhancement. Pitchangle diffusion keeps it in.

18. The Quadrupole Trap A quadrupole is simply the sum of two dipoles. –Dipole moving through a magnetized plasma, heliosphere, magnetosphere, galactosphere. Maxwell showed how conducting plane (plasma) “reflects” image –A binary system of magnetized objects –binary stars –A distributed current system-Earth’s core, supernovae Quadrupoles have “null-points” which stably trap charged particles (eg. Paul trap used in atomic physics produced 3 mass-spectrometer designs and a several Nobel prizes.) Betatron acceleration by E ┴ compression, violation of 1 st & 3 rd invariants

19. Fermi I,II vs Alfven I,II 1-D compression, E // Upstream Alfven waves impinging on barrier Scattering inside trap due to waves Max Energy due to scale size of barrier (curvature) becoming ~ gyroradius Acceleration time exponentially increasing Critically depends on angle of SW B-field with shock 2-D compression, E ┴ Compressional waves impinging on cusp Scattering inside trap due to quadrupole null point Max energy due to gyroradius larger than cusp radius (rigidity). Acceleration time exponentially increasing Critically depends on cusp angle with SW

20. PROPERTYDIPOLEFERMI QUADRUPOLE Stochasticity.001:1:1000 s.001:>10 3 :>10 4 s0.1:1:10 s Process Flow rim>ctr>blockedend>side>diffusctr>rim>open Wave Coupling hi E weakall E samehi E best Accel. in trap TrapsDetrapsTrap/Release Diffusion EssentialHelpfulNeutral Adiabatic Heat 2D pancake1D cigar2D pancake Energy Source SW compressSW AlfvenSW+internal e - Max Energy 900MeV@10Re1.8 MeV@.1Re280 MeV@3Re e - Min Energy 45 keV2.5 keV30 keV Trap Volume 10 24 m 3 10 20 m 3 10 22 m 3 Trap Lifetime >10 13 s10 4 s10 9 :10 5 s Accel. Time >300,000s8,000s25,000s Trap Power <5x10 8 W10 6 W5x10 7 W

21. Cusp Feedback But the cusp is turbulent! How can the REAL cusp trap anything? It doesn’t. Usually.

22. Quadrupole Trap in the Laboratory ( Two 1-T magnets, -400V, 50mTorr)

23. Cusp Diamagnetic Cavities

24. Diamagnetic Effect of Cold Plasma

25. Time-Energy Dispersed Events

26. Cusp ORBE Scaling Laws Maximum energy from rigidity cutoffs, scaled by distance of planetary cusp to surface of planet. Assuming: –B rad ~ B surface = B 0 –B cusp ~ B 0 /R stag 3 –E rad = 5 MeV for Earth –E cusp ~ v 2 perp ~ (B cusp  ) 2 ~ [(B 0 /R stag 3 )R stag ]  E/B is constant E rad-planet ~(R stag-Earth /R stag-planet )(B 0-planet /B 0-Earth ) 2 E rad-Earth

27. Scaled Planetary ORBE Planet Mercury Earth Mars Jupiter Saturn Uranus Neptune E RAD 4 keV 5 MeV < 1.5 eV 150 MeV 1.2 MeV 1.4 MeV 0.42 MeV R STAG 1.4 10.4 1.25 65 20 25 B 0 (nT) 330 31,000 < 6 430,000 21,000 23,000 14,000

28. High Latitude Transport If energetic particles are trapped in the cusp, how do they get to the dipole radiation belts?

29. Warm PlasmasheetTransport Why is it important whether particles are transported or in situ accelerated, do empirical models care? In a word: prediction. Example: Williams, Frank & Shelley peak seen in ISEE-3 data (35 keV protons). Local acceleration or transport?

30. Fokker-Planck Equation df k /dt= ∂f k /∂t+ C∂f/∂y+ D z ∂ 2 f k /∂z 2 + ∂/∂x i (D ij ∂f k /∂x j ) –i,j= index for (M,K) or (E,  ); k = quad or dipole trap –1 st term is the source + loss terms –2 nd term is convective velocity in (U,B m )-space –3 rd term is diffusion in (U,B m )-space –4 th term is remaining diffusion in (E,a)-space Mixed parabolic & elliptic PDE solved with standard finite element techniques. Operator splitting, mapping between ƒ 1  ƒ 2 (Fast, efficient, not numerically diffusive).

31. 31 If coordinates are not separable, ==>new physics: –Vortices, convection –Anomalous, or enhanced diffusion, “migration” Convection + diffusion = confusion Diffusive-Convective Transport If coordinates are separable, then D = D1 * D2 ==> diagonally dominant, easily generalized from 1-D. D LL, D  D  ==> radial diffusion models (Schulz 74) Is there a coordinate system which is separable?

32. 3D Coordinate Diffusion ( Roederer 1970 ) Basis set 1 bounce-reson 2 longitud. avg. 3 J~0 only Distribution Function RadialAsymm Pitch- angle Symm Pitch- angle Energy Loss ,J,M ƒ (X, ,J,M,t) ,J,M 1 J,M 1 J,M L,K,Mƒ (X,L,K,M,t) LL,K,M 1 K,M 1 M L, B m,Tƒ (X,L,B m,t) L,TL, B m BmBm T L,  T ƒ (L, ,T,t) L,T L,  2  T L,  ƒ (L, , ,t) L L,  3 33  Diffusion

33. 4-D UBK Hamiltonian The Quadrupole trap is located at a specific MLT as well as high-latitude. Thus we need to keep the 3 rd invariant phase  4-D We use two, energy conserving, 4D coordinate systems: – ƒ 1 (M,K,U,B m,n,t): “radial” diffusion + convection. FLUXONS – ƒ 2 (T, ,X,Y,t): Coulomb collisions + pitchangle diffusion –Where “n” separates quadrupole & dipole trapping Operator splitting enables us to carry out radial transport on ƒ 1, and pitchangle+energy loss on ƒ 2.

34. Summarize 4D Transport The UBK transform has the unique property of solving several requirements at once: –Separates convection from diffusion (MLT-dependence) –Permits Hamilton-Jacobi solution to equations of motion –Treats both Quadrupole & Dipole Regions (hi-latitudes) –Permits rapid calculation of diffusion coefficients –Can use operator splitting to diagonalize Diffusion tensor Cannot handle time- or spatial- scales that violate 1 st invariant. E.g. rare shock acceleration events (1991). It should indicate whether a hi-latitude source exists.

35. Conclusions Cusp source term may explain the mysterious origin of outer radiation belt particles. We need to develop the theory of both the acceleration and transport of hi-latitude sources. The generalization of Fermi acceleration applied to cusp, followed by a diffusive transport to radiation belts. Such a theory compares favorably with radiation belts in other magnetospheres. It may even solve Fermi’s problem of the origin of the galactic cosmic rays. SDG