1. ESS 154/200C Lecture 15 The Inner Magnetosphere II

2. Date Day Topic Instructor Due
ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869)
Date Day Topic Instructor Due
1/4 M A Brief History of Solar Terrestrial Physics CTR
1/6 W Upper Atmosphere / Ionosphere CTR
1/8 F The Sun: Core to Chromosphere CTR
1/11 M The Corona, Solar Cycle, Solar Activity
Coronal Mass Ejections, and Flares CTR PS1
1/13 W The Solar Wind and Heliosphere, Part 1 CTR
1/15 F The Solar Wind and Heliosphere, Part 2 CTR
1/20 W Physics of Plasmas RJS PS2
1/22 F MHD including Waves RJS
1/25 M Solar Wind Interactions: Magnetized Planets YM PS3
1/27 W Solar Wind Interactions: Unmagnetized Planets YM
1/29 F Collisionless Shocks CTR
2/1 M Mid-Term PS4
2/3 W Solar Wind Magnetosphere Coupling I CTR
2/5 F Solar Wind Magnetosphere Coupling II;
The Inner Magnetosphere I CTR
2/8 M The Inner Magnetosphere II CTR PS5
2/10 W Planetary Magnetospheres CTR
2/12 F The Auroral Ionosphere RJS
2/17 W Waves in Plasmas 1 RJS PS6
2/19 F Waves in Plasmas 2 RJS
2/26 F Review CTR/RJS PS7
2/29 M Final

3. Introduction
In previous lectures, we showed how the solar wind interacted with the magnetosphere, causing the plasma in the magnetosphere to circulate.
In steady state, a slab of magnetized solar wind reconnects with a sector of the magnetopause, transferring magnetic flux into the tail. That flux then reconnects in the tail and returns to the dayside.
It may appear that an electric field has been applied to the magnetosphere by the solar wind as if by capacitor plates along the flanks of the magnetosphere, but this is not correct.
At all times the magnetosphere is being pulled by the connected solar wind against the drag of the ionosphere and the atmosphere at low altitude.

4. Field-Aligned Currents
The correct approach to understanding both the corotation and solar wind-driven convection of the magnetospheric plasma considers the magnetic and plasma stresses in both the ionosphere and magnetosphere.
Plasma momentum equation – force balance – leads to a fundamental driver of field-aligned currents that couple magnetosphere to ionosphere.
Pressure gradient term
Inertial term
Vorticity dependent terms (w =U)
Assumptions: •j = 0, E + UxB = 0.

5. Region-2 Currents – Shielding
Integrating over a flux-tube, slow flow:
where V is the flux tube volume.
If gradients are not parallel then field-aligned currents flow.
Currents close in ionosphere - imply electric field in ionosphere. If mapped to magnetosphere opposes convection.
The idea of simply mapping the electric field down on the ionosphere is popular but incorrect.

6. Inner Edge of Ring Current
Data from Polar and ISEE to determine current intensity in Inner magnetosphere
The inner edge (L ~ 3) has eastward current
Also evidence or partial ring current (nightside currents stronger than dayside)

7. Particle Drift Paths
or LT-R
Treating particles as test particles drifting in prescribed electric and magnetic fields, the particles conserve their total energy.
For 90˚ pitch angle particles µ is conserved
Assuming corotation and uniform convection electric field:
Potential:  = 91.5/L - EconvL sin 
Energy: µB + q = constant
specifies the local time, 0˚ at noon, 90˚ at dusk
Electrons: positive slope in  - B space
Ions: negative slope in  - B space

8. Electron and Low Ion Magnetic Moment Drift Paths
In Ф-B space, our trajectories are straight lines.
Electron paths are either horizontal or slope upward.
Open trajectories are on the left. The Alfven layer is tangent to dusk.
Trajectories to the right are closed corotating.
Low μ ions are similar.

9. Ion Drift Paths: Medium μ and High μ
Trajectories all tilt down from left to right.
Proton nose ions refer to an early local time arrival of medium energy protons in data obtained by Explorer 45.
Closed banana orbits circulate within the Alfven layer.
Lines joining Dawn to Dusk describe closed drift paths, mainly gradient drift paths.

10. Near-Geosynchronous Data
Near geosynchronous orbit many energetic particles are drifting through the magnetosphere rather than trapped in periodic orbits.

11. Radiation Belt
Ion Energy Distribution
Electron Slot
Energetic Protons
The flux of low energy particles is quite variable at all distances.
The very energetic particle fluxes in the inner magnetosphere are rather stable and change only under unusual circumstances.
At high latitudes or L-values, the fluxes of very energetic particles are quite variable.
Particles can move radially by diffusion (scattering) across magnetic field lines. If they do this slowly they conserve their magnetic moment and gain energy as they reach stronger magnetic fields.

12. Gyroradii of Radiation Belt Particles
Gyroradius in km
The gyroradius of a charged particle is
where V is the perpendicular velocity, A is the mass in amu, E is the perpendicular energy and B is the magnetic field strength.
When the particle gyroradii are small compared to the curvature of the field lines, the particles remain trapped.

13. Time Scales of Radiation Belt Motions
Gyro Period
Bounce Period
Drift Perpendicular to B
Drift Period

14. Time Scales Continued
Three motions can be studied separately through the use of adiabatic invariants:
pdq is conserved under slow changes of the system where q is the coordinate and p is the momentum
The first adiabatic invariant is
The energy of a particle is conserved
constant
Particle must mirror when particle reaches a critical value of
Second adiabatic
is conserved on periods long compared to the bounce period.
Plots have relativistic corrections.

15. Radiation Belt Electrons
Annual Variation of energetic electrons versus L-value at low altitudes and high
Diffusion can occur along the magnetic field when waves resonate with the gyromotion.
Diffusion can occur radially across the magnetic field when the magnetic field changes on the time scale of the drift of the particle around the magnetosphere.
SAMPEX was a polar orbiting, low-altitude spacecraft.
POLAR was an excentric orbiter in polar orbit with apogee at 9 RE.

16. Proton and Electron Radiation Belts
The electron radiation belt contains a slot of low flux caused by enhanced loss rates due to whistler-mode waves (both lightning-generated and instability (self) generated waves).
During large disturbed periods, the radial diffusion rate can be much greater and the particles can fill the slot and move into an inner stable radiation belt.

17. Loss of Particles from Radiation Belts: Electrons
Particles that drift into the magnetopause or tail can be lost from the magnetosphere.
If the drift paths do not intersect a boundary, the particles can be trapped for a long time.
If the gyroradius is much smaller than the scale size of the field changes and no time variations occur near the gyrofrequency, the first adiabatic invariant is preserved
If Bm > Batm, then the particle will be lost. Sin (αe) has to be < (Be/Batm)½
If waves do exist near Ωe, then the pitch angle can change. If the pitch angle moves toward 0°, it will hit the atmosphere and be lost.
Sources of such waves include lightning and natural instabilities in the plasma.

18. Loss of Particles from the Radiation Belts: Protons
Height
H Density
3 RE
800 cm-3
4 RE
300 cm-3
5 RE
150 cm-3
3 keV
2.2
5.7
11.5
30 keV
3.2
8.5
16.8
100 keV 40
110
215
Lifetimes (Hours) for charge exchange for H+ with α = 45°
Protons can decay like electrons by being pitch angle scattered into the loss cone. This can be due to externally caused signals or by instability.
They also can be lost via charge exchange which creates a fast neutral not tied to any magnetic field line.

19. Loss of Particles: Cyclotron Resonance and Scattering into the Loss Cone
When circularly polarized waves resonate with the cyclotron motion of a charged particle, the interaction exchanges energy between the wave and the particle.
The resonance also changes the pitch angle of the particle. If the change in pitch angle forces the particle into the loss cone, it will hit the atmosphere and be lost. This maintains the pitch angle anisotropy despite the scattering that otherwise would lead to isotropy.
The maintenance of the anisotropy can lead to sustained wave growth by the “loss cone instability” mechanism.
Energy Change in Resonance with Right-Hand Cyclotron Waves
Particle Type
Resonance
Particle Energy
Wave Energy
Parallel Energy
Perpendicular Energy
Pitch Angle
Electron
Head-on
Decrease
Increase
Positive Ion
Overtaking
An analogous relationship exists for resonance with left-handed waves.

20. Loss of Particles from the Radiation Belt: Radial Diffusion
Particles drift around the magnetosphere on closed drift paths that contain a fixed amount of magnetic flux.
If the magnetosphere changes its magnetic configuration as the particle drifts around, it can move in or outward. Net diffusion will occur if there are gradients.
The first and second invariants remain the same unless there is noise at the gyro or bounce frequency.
If the particle moves to higher fields then it must gain energy if is constant.
The parallel momentum integrated along the bounce path is the second adiabatic invariant.
or where
This is independent of the energy of the particle.
If the field line becomes shorter the parallel momentum increases.

21. The Radiation Belts
Energetic particles can be trapped in the dipole mirror geometry of the Earth’s magnetic field.
These intensities build up and can be dangerous to spacecraft.

22. Summary
Field-aligned currents transfer solar wind momentum to the ionosphere and the atmosphere.
The inner magnetosphere can store much energy from the solar wind in the ring current.
At low energy, particle motion is affected by both the electric field and the gradient drift.
At high energies, the curvature and gradient drift are dominant. These are the radiation belts.
Particles can be lost from the radiation belts by radial drifts and by pitch angle scattering into the loss cone.
Charge exchange can turn ions into fast neutrals.