Earth’s Quadrupole Cusp: Implications for ORBE R. B. Sheldon, NASA/MSFC/NSSTC/USRA T. Fritz & J.-S. Chen, CSP/BU GEM 2005, Santa Fe July1, 2005

The Oldest Physics Problem How does point A influence point B? –Aristotle: mind, “spooky action at a distance” –Democritus to Descartes: particles –Newton: gravity (e.g. tides) –Huygens: waves –Faraday: fields How does the Sun transfer energy to Earth? 1.Photons & protons (DC equilibrium): pressure, heat 2.Waves & impulsive events (AC mechanical): Alfvenic, compressional, shocks 3.Electric & Magnetic fields (AC/DC): currents

CEP (Ions)

Sun-Earth Transducers Proton pressure  Bow shock, hot plasma (100eV electron, 1 kev/nuc ion), thermalized ram energy “Frictional” or “viscous” (  V 5/2 ) Impulsive  SSC, shock acceleration, Fermi, radial diffusion, Kp, “mechanical” (  V 2 ) Fields  Polar cap potential, convection, ring current, Dst, AE, “electrical” (V*Bz) [ICME] What transducer is CIR  ORBE? Poor correlation of ORBE with all of the above! Best with Vsw.

Springs & Shock Absorbers Why does a car have BOTH springs & shocks? –Springs are “reversible”, adiabatic, they “bounce back” (and ruin the tire tread). –Shock absorbers are “irreversible”, non-adiabatic, they convert the energy to heat. –Ex: manual dynamo with lightbulb or with 1F capacitor. Vsw energy transducer must be irreversible. –Cannot be too “stiff”, ideally it is “critically damped” Magnetic fields are “springy”, what are “shocks”? –Something responding to Vsw, yet not stiff…

I. The Quadrupole Cusp -- Static Equilibrium

The Dipole Trap Great Trap Poor accelerator ENA of E >1 keV particles outside trap.

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

Maxwell 1880 Chapman 1930

Quadrupolar T87 Magnetosphere All modern B-models have high latitude cusps. Since Chapman & Ferraro 1937, we’ve known the magnetosphere is a quadrupole. Why is this important?

The 2 nd Cusp Invariant |B| N.Ionosphere Equator S.Ionosphere 3 wells 2 wells Bouncing on a field line without crossing the equator s-distance

T96 Cusp Topology Solstice 16UT Solstice 4UT Equinox 16UT Equinox 16UT,-Bz

Ionospheric Footpoint of the HiLatitude Minima: Tilt vs Press -3.67deg +1.75deg+7.3deg 5dyn 3.3dyn 1.7dyn

Ionospheric Footpoint of HiLatitude Minima: Tilt vs Dst -3.67deg +1.75deg+7.3deg -50nT -30nT -10nT

Ionospheric Footprint of HiLatitude Minima: Press v Dst -50nT -30nT -10nT 1.7dyn 3.3dyn5dyn

Minima “size” Dependencies Well depth in nT |Dst|, nanoTesla Tilt, degrees Pressure, dynes >10 1.5%0.5%14% >1 1%0.5%12% From a linear fit to the previous simulations, we found the percentage change in area (for well depths above the threshold) projected on the ionosphere) for each nT, degree, or dyne increase, to illustrate the dependencies.

Optimal Quadrupole Geometry Both sunward (positive) tilt and/or high solar wind pressure are needed to produce the poleward cusp minima. Without the poleward minima, the 3 rd drift invariant is not well defined (as we show next.) |Dst| alone doesn’t develop the poleward side of the cusp, but it amplifies or magnifies what is already there. (Significant for statistical correlations.) Bz northward (not shown) is also positively correlated to poleward cusp minima.

II. The 3 rd Cusp Invariant

Cusp Equator B-field lines |B| C=1 C=2 C=1.5

|B| B-field lines

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

Chaotic, nearly trapped

H + Trapping in T96 Cusp Hi E cutoff Numerical Roundoff Loss-cone cutoff Red= None Green=Quasi- Blue= Yes

Hi E cutoff Numerical Roundoff Loss-cone cutoff Red= None Green=Quasi- Blue= Yes e - Trapping in T96 Cusp

Cusp Provisional Invariant Limits Energy Limits (1 st invariant at 100nT) –Minimum energy, Emin, is defined by cusp “separatrix” energy (ExB =  B) ~ 30 keV in the dipole? –Max energy, Emax, defined by rigidity.~ 4 MeV e - (20keV H+) –Consequently, no protons are expected to be trapped. Pitchangles locally o, (2 nd invariant) Low C-shells are empty below 1 Re for all energy, with a high-Cshell cutoff ~6 inversely dependent on Energy. 1 < C <~6 Re

Mapping Cusp to Dipole Conserving the 1 st invariant, and pitchangle scatter the particles into the cusp-loss cone (<40 o ), then the particles can appear in the dipole trap, or radiation belts. What would their distribution look like? –Energy limits to the rad belts, give ~ keV for protons, and 1-15 MeV for electrons. –C-shell limits to the dipole give ~5<L<∞?  very close to the PSD “bump”. –Mapping pitchangles  50 o <  < 90 o at dipole eq? Cusp particles look like ORBE injections.

III. The Discovery of Cusp MeV Electrons

POLAR: Oct 12-16, 1996

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

POLAR 4/1/97 Cusp Traversal

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

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

The 1-D 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.

The 2-D Quadrupole Trap A quadrupole is simply the sum of two dipoles. Quadrupoles have “null-points” which stably trap charged particles (eg. Paul trap) Motion of the dipoles results in a 2D constriction of the volume. This is just a generalization of 1D Fermi- acceleration to 2D. 1D Fermi acceleration increases E //, violating the 2 nd invariant. 2D betatron acceleration increases E ┴, violating the 1 st & 3 rd invariants Efficiency Product:  T =  1  2  3  4  5  6 …

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 e - Min Energy 45 keV2.5 keV30 keV Trap Volume m m m 3 Trap Lifetime > s10 4 s10 9 :10 5 s Accel. Time > 300,000s8,000s25,000s Trap Power < 5x10 8 W10 6 W5x10 7 W

Model 1.Fast solar wind is trapped in the cusp –27 day recurrence, non-linear with Vsw 2.High Alfvenic turbulence of fast SW heats the trap –Low Q-value,  compressional, BEN 3.2 nd Order “Fermi” accelerates electrons –Low energy appear first, then high w/rigidity cutoff. 4.Trap empties into rad belts simultaneous L=4-10 –“gentle” evaporation, or “rapid” topology change –Initially “butterfly” around 70-deg equatorial

1. Non-Linear Vsw Dependence E Flux E 30keV 100eV 1keV 10keV Vsw seedtrap seedtrap The Reason that Vsw interacts non-linearly is that it does several things at once. It heats the seed population, while also making the trap deeper.

V. Cusp Feedback, CDC, and Ion Trapping But the cusp is turbulent! How can the REAL cusp trap anything for 2 days!? It doesn’t. Usually.

Real Life Up to this point, we have developed the theory of cusp trapping and acceleration in an ideal, vacuum quadrupole. However, real life is far more interesting. POLAR data, which triggered this investigation, shows trapped ion flux and a highly modified magnetic field, which we argue is a Cusp Diamagnetic Cavity. The positive feedback between the quadrupole and trapped ions, suggests that CDC are ubiquitous and important.

Cusp Diamagnetic Cavities a.k.a Magnetic Bubbles

Turbulence, Power, Spectra…

Schematic CDC

Stability of Infinitesimal Dipole

Stability of Finite Ring

Cluster Observations POLAR sees thick (1-6 Re) CDC, whereas Cluster sees thin (< 1Re). We interpret this as a radial dependence on the thickness of the CDC. When we tried to model this with current loop stably superposed on T96, we did not reconstruct the observations. We plan to use hybrid code to find a new plasma equilibrium with cusp B-fields.

VI. An Interplanetary Test by Scaling

Cusp 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 Planet ~ E Earth (R P B Planet /R E B Earth ) 2 (R E-Stag /R P-Stag ) 4

Scaled Planetary ORBE Planet Mercury Earth Mars Jupiter Saturn Uranus Neptune E RAD 0.66 MeV 5 MeV <.5 eV 7.1 MeV 1.6 MeV 0.81 MeV 0.12 MeV R STAG B 0 (nT) ,000 < 6 430,000 21,000 23,000 14,000

Conclusions The quadrupole is a nearly universal trap and cosmic accelerator more efficient than Fermi (and shocks). The quadrupole cusp has ideal properties to couple AC mechanical energy from SW into the magnetosphere. The peculiar correlations of ORBE with SW can be explained by requiring an intermediate stage of the non-linear cusp. A test of the mechanism using comparative magnetospheres shows the correct energy scaling. Soli Deo Gloria

Kolmogorov, Arnol’d, Moser (applied to Jupiter perturbation of Earth)……… Earth orbit as Perturbed by Jupiter. Earth orbit if Jupiter were 50k Earth masses. Poincar é slice x vs. v X taken along the E-J line.

II. The Outer Radiation Belt Electrons (ORBE)

McIlwain, 1966

ORBE (McIlwain 1966)

McIlwain 1966

Empirical Prediction McIlwain 1966: Geo MeV e increases Paulikas & Blake 1979: Vsw best external Nagai 1988: Kp best internal predictor Baker 90 LPF, Koons&Gorney 90 NN Dmitriev&Chao03 Log-Linear Ukhorskiy et al., 04 NonLinear

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. There is a Dst signature with ORBE, but magnitudes are uncorrelated, only occurrence.

1996

Hi-Latitude Minima: Dst vsPress