A High Latitude Source for Radiation Belt Particles? Robert Sheldon, NSSTC October 19, 2004...
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A High Latitude Source for Radiation Belt Particles?Robert Sheldon, NSSTCOctober 19, 2004Huntsville Modelling Workshop
CorrelationsHighest 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 (2nd moment) or density starved (0th moment).
MotivationThe 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 problemRecent 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.
OutlineData support for high-latitude sourcePOLAR dataSimulations and invariantsComparison of accelerator efficienciesDefinition of efficiencyDipole / Fermi / QuadrupoleTransport from the cuspa) 4D Fokker-Planck
Sheldon et al., GRL 1998POLAR/ CAMMICE data1 MeV electron PSD in outer cusp
Maxwell 1880 Chapman 1930
QuadrupolarT87 MagnetosphereSince Chapman & Ferraro 1937, weve known the magnetosphere is a quadrupole. All trapping models have assumed a dipole.If Quadrupole is source, how does it change the transport?
The Simulated T96 Quadrupole TrapLousy TrapGreat AcceleratorCan be made to trap better though.
e- Trap Regions of T96 Cusp
- Cusp Provisional InvariantsMinimum energy is defined by separatrix energy (ExB = B) ~ 30 keVThe max energy defined by rigidity.~ 4 MeV e-Mapping C-shell limits to the dipole give 5
Accelerator EfficiencyWhy would the cusp accelerate at all? Why not just use standard well-known accelerators?
Plasma ThermodynamicsA neglected fieldso this discussion is very qualitative. (See Krall & Trivelpiece)In ideal gases the Maxwell equilibrium is extremely well maintained by collisionsIn 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
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.
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 Q1, 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 = PowerA 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.
The Dipole Trap AcceleratorThe 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 motion1)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 3rd invariant, L-shell
The Fermi-Trap AcceleratorWaves convecting withthe solar wind, compresstrapped ions between the local |B| enhancementand the planetary bow shock, resulting in 1-Dcompression, or E// enhancement. Pitchangle diffusion keeps it in.
The Quadrupole TrapA quadrupole is simply the sum of two dipoles.Dipole moving through a magnetized plasma, heliosphere, magnetosphere, galactosphere. Maxwell showed how conducting plane (plasma) reflects imageA binary system of magnetized objects binary starsA distributed current system-Earths core, supernovaeQuadrupoles 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 1st & 3rd invariants
Fermi I,II vs Alfven I,II 1-D compression, E//Upstream Alfven waves impinging on barrierScattering inside trap due to wavesMax Energy due to scale size of barrier (curvature) becoming ~ gyroradiusAcceleration time exponentially increasingCritically depends on angle of SW B-field with shock2-D compression, E Compressional waves impinging on cuspScattering inside trap due to quadrupole null pointMax energy due to gyroradius larger than cusp radius (rigidity).Acceleration time exponentially increasingCritically depends on cusp angle with SW
Cusp FeedbackBut the cusp is turbulent! How can the REAL cusp trap anything? It doesnt. Usually.
Quadrupole Trap in the Laboratory(Two 1-T magnets, -400V, 50mTorr)
Cusp Diamagnetic Cavities
Diamagnetic Effect of Cold Plasma
Time-Energy Dispersed Events
Cusp ORBE Scaling LawsMaximum energy from rigidity cutoffs, scaled by distance of planetary cusp to surface of planet. Assuming:Brad ~ Bsurface= B0Bcusp ~ B0/Rstag3Erad= 5 MeV for EarthEcusp ~ v2perp~ (Bcuspr)2 ~ [(B0/Rstag3)Rstag] m = E/B is constantErad-planet~(Rstag-Earth/Rstag-planet)(B0-planet/B0-Earth)2Erad-Earth
Scaled Planetary ORBEPlanetMercuryEarthMarsJupiterSaturnUranusNeptuneERAD4 keV5 MeV< 1.5 eV150 MeV1.2 MeV1.4 MeV0.42 MeVR STAG1.410.41.2565202025B0 (nT) 330 31,000 < 6430,000 21,000 23,000 14,000
High Latitude TransportIf energetic particles are trapped in the cusp, how do they get to the dipole radiation belts?
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?
Fokker-Planck Equationdfk/dt= fk/t+ Cf/y+ Dz2fk/z2+ /xi(Dij fk/xj)i,j= index for (M,K) or (E,a); k = quad or dipole trap1st term is the source + loss terms2nd term is convective velocity in (U,Bm)-space3rd term is diffusion in (U,Bm)-space4th term is remaining diffusion in (E,a)-spaceMixed parabolic & elliptic PDE solved with standard finite element techniques. Operator splitting, mapping between 1 2 (Fast, efficient, not numerically diffusive).
Diffusive-Convective TransportIf coordinates are not separable, ==>new physics:Vortices, convectionAnomalous, or enhanced diffusion, migrationConvection + diffusion = confusionIf coordinates are separable, then D = D1 * D2 ==> diagonally dominant, easily generalized from 1-D.DLL, Daa, Dmm ==> radial diffusion models (Schulz 74)Is there a coordinate system which is separable?
3D Coordinate Diffusion (Roederer 1970)Diffusion
4-D UBK Hamiltonian The Quadrupole trap is located at a specific MLT as well as high-latitude. Thus we need to keep the 3rd invariant phase4-DWe use two, energy conserving, 4D coordinate systems: 1(M,K,U,Bm,n,t): radial diffusion + convection. FLUXONS 2(T,a,n,X,Y,t): Coulomb collisions + pitchangle diffusionWhere n separates quadrupole & dipole trappingOperator splitting enables us to carry out radial transport on 1, and pitchangle+energy loss on 2.
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 motionTreats both Quadrupole & Dipole Regions (hi-latitudes)Permits rapid calculation of diffusion coefficientsCan use operator splitting to diagonalize Diffusion tensorCannot handle time- or spatial- scales that violate 1st invariant. E.g. rare shock acceleration events (1991). It should indicate whether a hi-latitude source exists.
ConclusionsCusp 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 Fermis problem of the origin of the galactic cosmic rays.SDG
Recent hypotheses putting the ORBE source inside the belts have neglected, what I think are still viable, alternative explanations. I hope this talk will broaden the proposed ORBITALS mission goals to consider the types of data that might support or eliminate the hypothesis of high latitude sources for ORBE (and, as we even more tentatively hypothesize, the ring current ions themselves.)Theres another whole talk justifying the need for another hypothesis, but I assume we are here because we believe the need exists. This data shows MeV electrons in a region of the dipole that is NOT dipole trapped. What is it then? The pink PAD distributions show it is peaked aroun