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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”

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earth s quadrupole cusp implications for orbe

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
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?
    • Photons & protons (DC equilibrium): pressure, heat
    • Waves & impulsive events (AC mechanical): Alfvenic, compressional, shocks
    • Electric & Magnetic fields (AC/DC): currents
sun earth transducers
Sun-Earth Transducers
  • Proton pressure  Bow shock, hot plasma (100eV electron, 1 kev/nuc ion), thermalized ram energy “Frictional” or “viscous” (rV5/2)
  • Impulsive  SSC, shock acceleration, Fermi, radial diffusion, Kp, “mechanical” (rV2)
  • 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
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…
the dipole trap
The Dipole Trap
  • Great Trap
  • Poor accelerator
  • ENA of E >1 keV

particles outside trap.

quadrupolar t87 magnetosphere
QuadrupolarT87 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
The 2nd Cusp Invariant

Bouncing on a field line without crossing the equator

|B|

N.Ionosphere Equator S.Ionosphere

3 wells

2 wells

s-distance

t96 cusp topology
T96 Cusp Topology

Solstice

4UT

Solstice

16UT

Equinox

16UT

Equinox

16UT,-Bz

ionospheric footpoint of the hilatitude minima tilt vs press
Ionospheric Footpoint of the HiLatitude Minima: Tilt vs Press

+1.75deg

+7.3deg

-3.67deg

5dyn

3.3dyn

1.7dyn

ionospheric footpoint of hilatitude minima tilt vs dst
Ionospheric Footpoint of HiLatitude Minima: Tilt vs Dst

+1.75deg

+7.3deg

-3.67deg

-50nT

-30nT

-10nT

minima size dependencies
Minima “size” Dependencies

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
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 3rd 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.
cusp equator

C=1.5

C=2

B-field lines

Cusp Equator

|B|

C=1

the simulated t96 quadrupole trap
The Simulated T96 Quadrupole Trap
  • Lousy Trap
  • Great Accelerator
  • Can be made to trap better though.
h trapping in t96 cusp
H+ Trapping in T96 Cusp

Hi E cutoff Numerical Roundoff Loss-cone cutoff

Red= None

Green=Quasi-

Blue= Yes

e trapping in t96 cusp
e- Trapping in T96 Cusp

Hi E cutoff Numerical Roundoff Loss-cone cutoff

Red= None

Green=Quasi-

Blue= Yes

cusp provisional invariant limits
Cusp Provisional Invariant Limits
  • Energy Limits (1st 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 40-90o, (2nd 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
Mapping Cusp to Dipole
  • Conserving the 1st invariant, and pitchangle scatter the particles into the cusp-loss cone (<40o), 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 ~ 0-100 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  50o < a < 90o at dipole eq?
  • Cusp particles look like ORBE injections.
slide29

Sheldon et al., GRL 1998

POLAR/ CAMMICE data

1 MeV electron

PSD in outer cusp

iv accelerator efficiency

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 “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 3rd invariant, L-shell

the 1 d fermi trap accelerator
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
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 2nd invariant.
  • 2D betatron acceleration increases E┴ , violating the 1st & 3rd invariants
  • Efficiency Product: hT = h1 h2 h3 h4 h5 h6…
model
Model
  • Fast solar wind is trapped in the cusp
    • 27 day recurrence, non-linear with Vsw
  • High Alfvenic turbulence of fast SW heats the trap
    • Low Q-value, compressional, BEN
  • 2nd Order “Fermi” accelerates electrons
    • Low energy appear first, then high w/rigidity cutoff.
  • 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
1. Non-Linear Vsw Dependence

30keV

100eV

10keV

1keV

Vsw

Flux

Flux

seed

trap

seed

trap

E

E

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

V. Cusp Feedback, CDC, and Ion Trapping

But the cusp is turbulent! How ca n the REAL cusp trap anything for 2 days!?

It doesn’t.

Usually.

real life
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.
cluster observations
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.
cusp scaling laws
Cusp Scaling Laws
  • Maximum energy from rigidity cutoffs, scaled by distance of planetary cusp to surface of planet.
  • Assuming:
    • Brad ~ Bsurface= B0
    • Bcusp ~ B0/Rstag3
    • Erad= 5 MeV for Earth
    • Ecusp ~ v2perp~ (Bcuspr)2 ~ [(B0/Rstag3)Rstag]
  • m = E/B is constant

EPlanet~ EEarth(RPBPlanet/REBEarth)2 (RE-Stag/RP-Stag)4

slide48

Scaled Planetary ORBE

Planet

Mercury

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

R STAG

1.4

10.4

1.25

65

20

20

25

B0 (nT)

330

31,000

< 6

430,000

21,000

23,000

14,000

ERAD

0.66 MeV

5 MeV

< .5 eV

7.1 MeV

1.6 MeV

0.81 MeV

0.12 MeV

conclusions
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
Kolmogorov, Arnol’d, Moser (applied to Jupiter perturbation of Earth)………

Earth orbit as

Perturbed by

Jupiter.

Poincaré slice

x vs. vX taken

along the E-J

line.

Earth orbit if

Jupiter were 50k Earth masses.

empirical prediction
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
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.