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Radiation belt particle dynamics
Prepared by Kevin Graf
Stanford University, Stanford, CA
IHY Workshop on
Advancing VLF through the Global AWESOME
Network
Basic Motion
 Motion of charged particle q in presence of electric and magnetic
fields governed by Lorentz Force Equation with external force:
F m
dv
 qE  v  B   Fext
dt
 For static, uniform magnetic field, no electric field, and no external
force, particle gyrates around magnetic field line:
v x  v  cos c t   
dv| |
B  B0 zˆ
dv q
v y   v  sin c t   
 0,
 v  B 
dt
dt
m
v z  v||
Note the pitch
angle of the motion.
 v 
  tan  
 v|| 
1
Gyrofrequency and Gyroradius
derived from force balance.
mv 2
 qv  B
rc
rc 
mv 
qB
c rc  v 
c 
 qB
m
Particle Drifts
 Imposing external force causes particle drift across B,
but not simply in the direction of the external force:
m
dv| |
 Fext| |
dt
dv
m   qv  B  Fext 
dt
dv
m
 qv  B  Fext
dt
v||t  
Fext  B
 Drift Velocity
2
qB
Force affects radius of
gyromotion, resulting in drift
orthogonal to both B and Fext.
m
t  v||0
v t   vm t   v D
v m t   Gyromotion (same as before)
vD 
Fext||
Specific Particle Drifts
 Gravity:
vg 
mg  B
qB 2
 Electric Field:
vE 
EB
B2
 B-Gradient:
 B-Curvature:
mv 2 B  B 
v 
2qB 3
vR 
mv ||2 RC  B 
qRC2 B 2
Magnetic Mirror
A charged particle traveling along a magnetic field line
can be reflected by converging magnetic field.
Force Picture
As Derived From Adiabatic
Invariance of Magnetic Moment μ
mv 2

 constant
2B
sin 2   sin 2  0 

B
B0
Increasing B as field lines
converge leads to increase
in pitch angle α until particle
reflects.
• Particles can be confined in a magnetic
mirror configuration.
• Note that if a particle is traveling very
parallel to the magnetic field line (small
α) it can escape through the ends of
the mirror rather than reflecting.
Geomagnetically Trapped Radiation
 Energetic, charged particles (occasionally referred to
as “radiation”) trapped in the Earth’s Magnetosphere:
 Gyrate around and travel along the geomagnetic field lines.
 Are trapped in a magnetic mirror, bouncing from North to
South and back.
 Experience gradient and curvature drifts to the West for
protons and to the East for electrons (drift due to gravitational
force is present, but it is of significantly smaller magnitude).
Mechanism of electron precipitation
by whistler waves
South Atlantic Anomaly
 Earth’s dipole is
not centered
 South Atlantic
Anomaly – weak
spot along Earth’s
surface
 Smaller B  larger
drift loss cone
 Particles
precipitate due to
larger loss cone
Loss Cones
 Bounce loss cone
 Particles with sufficiently small pitch
angles will be precipitated
 Drift loss cone
 Charge dependent drift
 |B| lowest over South
Atlantic Anomaly (SAA)
 Particle can drift into
region of low |B| and
precipitate
Precipitation of Energetic Electrons

Particles escaping the geomagnetic mirror, colliding with the denser
atmosphere of the lower ionosphere, are said to “precipitate” and can
create such phenomena as the aurora borealis.
Application to VLF Research
(This material is discussed more thoroughly in the tutorials on LEP.)



VLF electromagnetic waves, created by
lightning, transmitter, or otherwise,
can induce precipitation of energetic
electrons by altering the pitch angle of
their motion.
The precipitation results in an
ionospheric enhancement which perturbs
subionospherically propagating VLF
signals beneath the disturbed region.
The perturbation on this subionospheric
VLF signal can be detected in data
acquired by AWESOME VLF receivers.
Example Detection of Transmitter
Induced Precipitation
Periodic precipitation induced by
periodic keying of NPM transmitter
is detected on NLK-MI signal using
superposed epoch averaging and
Fourier analysis.
Texts to Reference
Single Particle Dynamics & Plasma Physics



Fundamentals of Plasma Physics by J.A. Bittencourt
Introduction to Plasma Physics and Controlled Fusion by F. F. Chen
Introduction to Plasma Physics: With Space and Laboratory Applications by
D. A. Gurnett and A. Bhattacharjee
Geomagnetically Trapped Radiation

Introduction to Geomagnetically Trapped Radiation by Martin Walt
Transmitter-Induced Precipitation


Abel, B., and R. M. Thorne (1998), Electron scattering loss in Earth’s inner
magnetosphere - 1. Dominant physical processes, J. Geophys. Res., 103, 2385-2396.
Inan, U. S., M. Golkowski, M. K. Casey, R. C. Moore, W. Peter, P. Kulkarni, P. Kossey,
E. Kennedy, S. Meth, and P. Smit (2007), Subionospheric VLF observations of
transmitter-induced precipitation of inner radiation belt electrons, Geophys. Res. Lett.,
34, L02106, doi:10.1029/2006GL028494.