Space Radiation & It's Effects On Space Systems & Astronauts Course Sampler

SPACE RADIATION AND ITS EFFECTS ON SPACE SYSTEMS
AND ASTRONAUTS

Instructor:
V.L. Pisacane

ATI Course Schedule: http://www.ATIcourses.com/schedule.htm

Single
Particle
Motion

SPACE RADIATION AND ITS EFFECTS ON SPACE
SYSTEMS AND ASTRONAUTS

Single Particle Motion

by
V. L. Pisacane

Space Radiation and its Effects on Space Systems and Astronauts 5─ 1 ©VLPisacane,2012

Single
TOPICS Particle
Motion

 Introduction
 Equation of Motion
 Gyration Motion
 Guiding Center Motion
 Mirror Points
 Summary


INTRODUCTION Single
Particle
Background Motion

 Trapping of particles by magnetic fields was first
studied by Kristian Birkeland in Norway in ~1895
– He aimed beams of electrons at a magnet inside a
vacuum chamber and noted that they seemed to
be channeled towards its near magnetic pole
 Birkeland interested Poincaré who analyzed the
motion of charged particles in a magnetic field and
showed that they spiraled around field lines and
were repelled from regions of strong field
 Birkeland also interested Carl Stormer who carried
out more detailed analyzes
 Existence of trapped radiation was confirmed by the
Explorer 1 and Explorer 3 missions in early 1958, http://www.centennialofflight.gov/essay/Dictionary/R
under Dr. James Van Allen at the University of Iowa ADIATION_BELTS/DI160.htm

 As a consequence, the trapped radiation are often
called the Van Allen radiation belts


INTRODUCTION Single
Particle
Summary Motion Motion

 Motion of the trapped radiation as illustrated in Figure 6.4 consists of three primary
components:
– Gyration ~ milliseconds
• Particles rotating around field lines
– Mirroring ~0.1 ─ 1.0 s
• Particles traveling from one hemisphere to the other and back
– Longitudinal drift ~ 1 ─ 10 min
• Particles driting east or west

Figure 6.4 Motion
of charged
particles trapped
5– 4 in the Earth’s
magnetic field,


EQUATION OF MOTION Single
Particle
Introduction Motion

 Force on a particle in a magnetic and av electric field with charge q is
given by the Lorentz force
FL  qE  v  B 
where
B = magnetic flux density or magnetic field vector, T
E = electric field, V m-1
FL = Lorentz force, N
q = charge, C
v = velocity, m s-1
F = applied force
 Equation of motion with the addition of an applied force F is then
dv
m  F  q(E  v  B)
dt


GYRATION MOTION Single
Particle
Introduction Motion

 For a uniform magnetic flux density and no electric field, the force will
be constant and perpendicular to the motion resulting in circular motion
 Speed will remain constant
 Positive particles will rotate clockwise around an out of plane field line
 Negative particles will rotate counterclockwise around an out of plane
field line

g

g

Positive charge Negative charge


Particle
Gyro-Frequency and Larmor Radius Motion
 Gyro-frequency or relativistic cyclotron frequency wc and Larmor radius rc follow
from the gyration equation of motion

dv g rlwc  v g
m  qv g  B
dt
vg
qv gB   mrl w 2
rl 
c
wg
qrlwg  B   mrl w2
c v gm0 g  mc  m0 gc
rl   
qB  qB  qB  v2 qB  qB  qB 
wg    1 2
m m0 g m0 c v gm0 g p pc 1 R
rl    
qB  qB  q cB cB
qB  qB  vc q q B  vc q B  c2
wg    B  vc  
m mvc pc q R q R
where
F = force, N
m = gm0, relativistic mass, kg rl = Larmor radius, m
m0 = rest mass, kg R = magnetic rigidity, V
a = acceleration, m s-2 g ≡ (1-v2/c2)─1/2
q = charge, C wg = gyrofrequency, s-1
v = magnitude of particle velocity, m s-1 fc = wc/ 2p, gyrofrequency, Hz
vg = magnitude of gyration velocity, m s-1 B = magnetic flux density perpendicular to plane of
motion, T


Particle
Gyro-Frequency for Electrons and Protons Motion

 For electrons
qB  qB   1.60  10 19 B  B
wg     1.76  1011  rad s -1
m m0 g 9.11  10 31 g g
w B
fg  c  2.80  1010  Hz
2p g
 For protons
qB  qB  1.60  10 19 B  B
wg     9.58  107  rad s -1
m m0 g 1.67  10 27 g g
w B
fg  c  1.52  107  Hz
2p g

 Ratio of gyro-frequencies for the electron and proton is
fg ,electron m0 ,proton 1.67  10 27
   1833
fg ,proton m0 ,electron 9.11  10 31
where
me= 9.11 x 10-31 kg
qe = ─ e = ─1.60 x 10-19 C
mp= 1.67 x 10-27 kg
qp = e = 1.60 x 10-19 C


Particle
Larmor Radii for Electrons and Protons Motion

 For electrons
m0 gc 9.11  10 31 gc gc
rl ,electron    5.68  10 12
qB   1.60  10 19 B B

 For protons
m0 gc 1.67  10 27 gc gc
rl ,protons    1.04  10 8
qB  1.60  10 19 B B

 Ratio Larmor radii for the electron and proton is
rl ,proton m0 ,proton 1.67  10 27
   1833
rl ,electron m0 ,electron 9.11  10 31

where
me= 9.11 x 10-31 kg
qe = ─ e = ─1.60 x 10-19 C
mp= 1.67 x 10-27 kg
qp = e = 1.60 x 10-19 C

Particle
Larmor Radii and Gyro-frequency in Terms of Magnetic rigidity Motion

 Examples of Larmor radii and gyro-frequency for any charged particle with a given
rigidity at
 Sun
 near Earth
 surface of the Earth
Larmor Radius, Re = Earth radii
Magnetic Kinetic Speed
Rigidity Energy [% of c] Corona Interplanetary Earth
(10 mT) 1 AU (5 nT) Surface (30 μT)
1 GV 0.43 GeV 73% 330 m 6.6x108 m ≈ 100 Re 1.1x105 m ≈ 0.017 Re
5 GV 4.1 GeV 98% 1.65 km 3.3x109 m ≈ 520 Re 5.5x105 m ≈ 0.086 Re
20 GV 19.1 GeV 99.8% 6.60 km 1.3x1010 m ≈ 2100 Re 2.2x106 m ≈ 0.340 Re

Gyro-frequency
Magnetic Kinetic Speed
Rigidity Energy [% of c] Corona Interplanetary Earth
(10 mT) 1 AU (5 nT) Surface (30 μT)
1 GV 0.43 GeV 73% 105.6 kHz 0.053 Hz 316.9 Hz
5 GV 4.1 GeV 98% 28.4 kHz 0.142 Hz 85.1 Hz
20 GV 19.1 GeV 99.8% 7.2 kHz 0.004 Hz 21.7 Hz

Space Radiation and its Effects on Space Systems and Astronauts 5 ─ 10 ©VLPisacane,2012

Particle
Summary Motion in Uniform Magnetic Field Motion

 Orbit of a charged particle in a uniform, static magnetic field is a
spiral trajectory
 Particle gyrates in plane perpendicular to magnetic field in a
circular orbit
 Radius of orbit is the Larmor radius, rl
 Sense of gyration depends on sign of the particle’s charge
 Positive and negative charged particles will rotate in opposite
directions
 Gyro-frequency is given by wg or fg
 Gyro-period given by 1/fg or 2p/ wg
 Tangential velocity is constant
 Particle’s energy is conserved


GUIDING CENTER MOTION Single
Particle
Introduction Motion

 When geomagnetic field in not constant and uniform and electric field is not zero,
motion is more complicated
 However, when the variation in the forces over distances comparable to the Larmor
radius is small, these effects can be treated as perturbations
 Partitioning the motion into gyro motion and motion of the guiding center where
– Gyration motion – circular motion of the particle around the magnetic field in the
plane perpendicular to the magnetic field, vg
– Guiding center motion – motion of the center of the gyrating particle (all motion
but gyration motion), vgc
 Let
v  v g  v gc

so that the equation of motion
dv
m  F  qE  qv  B
dt
can be approximated by
dv gc dv g
m  F  qE  qv gc  B m  qv g  B
dt dt


Particle
Guiding Center Equations Motion
 Equation of motion of the guiding center is
dv gc
m  F  qE  qv gc  B
dt
 Partitioning the velocity and force into components parallel and normal to the
magnetic field
v  v gc||ε||  v gc  F  F ε||  F
|| E  E||ε||  E
gives
dv gc|| dv gc
m  F||  qE|| m  F  qE  qv gc  B
dt dt
 If for E|| and F|| are constant, guiding center motion parallel to the magnetic field is
F||  qE||
v gc||  t  v gc||(0)
m
 Forces parallel to magnetic field accelerates particle along the field line with direction
depending on sign of the force or charge of the particle
 Forces perpendicular to the magnetic field will accelerate the particle orthogonal to
the magnetic field line


Particle
Motion in Uniform Magnetic Field Motion

 General motion of proton and electron in a constant magnetic field with an initial
velocity along the field line results in the direction of the magnetic field results in
helical motion

g

g


Particle
Orthogonal Drift Velocities of the Guiding Center 1/2 Motion

 Gyration without perturbations (see A)
 General Force Drift (see C)
– Drift due to presence of general force F
FB
v gc ,F 
qB 2

 Electric Field Drift (see B)
– Drift due to presence of electric field E
EB F  qE
v gc ,E 
B2
 Gradient Drift (see D)
– Drift due to gradient in t magnetic field ∇B
– Dominates for the Earth B

mv 2
v gcB  
3
B  B
2qB

http://en.wikipedia.org/wiki/File:Charged-particle-drifts.svg

Particle
Orthogonal Drift Velocities of the Guiding Center 2/2 Motion

 Gravitational drift
– Drifts due to presence of gravity vector g

mg  B
v gcg 
qB 2

 Magnetic Curvature Drift B ||

– Drift due to Rc radius of curvature of magnetic field

2 2
mv|| mv||
v gcR  2 2 Rc  B  4 B  B   B
qB Rc qB

Drift out for positive charge
Drift into for negative charge
 Polarization Drift
– Drifts due time derivative of the electric field dE/dt

1 dE m dE
v gcp   2
wgB dt qB dt
dE
dt


MIRRORS POINTS Single
Particle
Introduction Motion
 If the electromagnetic fields do not vary in time, the energy of the particle is
constant so that
dE
0
dt
 The energy can be expressed as
1/2

v||   (E  B  q)  vD 
2
E  mv||  vD   B  q
1 2 2
2
m
 

2
 in regions where
1 2
E  mv D  B  q
2
charged particles can drift in either direction along magnetic field-lines
 However, particles are excluded from regions where v|| is imaginary
1 2
E  mv D  B  q
2
 Charged particles must reverse direction at those points on magnetic field-lines,
these points are termed bounce points or mirror points
E = Energy  = electric field potential, E = ─
t = time Vm = magnetic field potential
m = particle mass q = charge
v|| = velocity parallel to field line = orbital dipole moment of particle
vd = drift velocity =IA = 2riwc/2, I = current, A = area

Particle
Mirror Points in Dipole Field Motion
 Distance a particle travels from the equator before
mirroring is determined by its pitch angle at the
equator
 Equatorial pitch angle for a particle that mirrors at a
given magnetic latitude in a dipole field is
cos6 lm Pitch Angle
sin a eq 
2

1  3sin2 lm 1 / 2 60o
where
lm = magnetic latitude of mirror point
aeq = pitch angle at the equator
80

60

40
Mirror Latitue, degs

20
Pitch Angle
0

-20
40o
-40

-60 6 – 18
-80

0 10 20 30 40 50 60 70 80 90
Equatorial Pitch Angle, degs

Mirror magnetic latitude as function of magnetic equatorial
pitch angle From: http://www.altfuels.org/sampex/losscone/index.html

Particle
Mirror Points in Dipole Field Motion

How far the particle travels from magnetic equator before Particle with equatorial pitch angle 90 degrees, will mirror at
"mirroring" is determined by pitch angle at the equator equator and remain in magnetic equatorial plane

If pitch angle close to 0 or 180 degrees (nearly aligned with Mirror points occur at same field strength on each bounce,
magnetic field), mirror point will fall below planet surface i.e., at same magnetic latitude and drift equatorially

Particle
Equatorial Loss Cone Motion
 Equatorial Loss Cone is a cone of velocities of
charged particle whose apex is on the equator
and axis along a magnetic field line that
represents the charged particles that will be lost
due to interaction with the atmosphere or the
surface in a dipole field
 Loss-cone angle for intersection with the Earth
depends solely on L-shell value and not on
particle mass, charge or energy where http://www-spof.gsfc.nasa.gov/Education/wtrap2.html

sin alc  4L6  3L5 
1 / 4

a lc

Loss -one geometry


Particle
Mirror Points of Trapped Radiation 1/2 Motion

 Spenvis simulation
 Initial Conditions
– L-shell: 2 Re
– Equatorial pitch
angle: 30o

3D view of L shell

Altitude of mirror points

6 – 21

Footprints in Northern and Southern hemisphere Cylindrical projection of the shell

Particle
Mirror Points of Trapped Radiation 2/2 Motion

 Spinvis simulation
 Initial Conditions
– L-shell: 2 Re
– Equatorial pitch
angle: 15o
 Mirror Points
– Some mirror
points subsurface

3D view of L shell Altitude of mirror points

Footprints in Northern and Southern hemisphere Cylindrical projection of the shell

SUMMARY Single
Particle
Typical Particle Characteristics Motion

6 – 23

Table 6.5 Characteristics of typical radiation belt charged particles

SUMMARY Single
Particle
Summary Single Particle Motion Motion

 Motion along uniform magnetic field
– Uniform B, no E: constant speed along B
– Uniform B, E parallel to B: charge dependent acceleration along B
 Gyration motion
– Circular orbit in plane perpendicular to B
– Sense of gyration depends on charge sign
– Larmor radius rl and gyration frequency fg
 Drift motion orthogonal to magnetic flux density B from force F
– Drift perpendicular to B and F according to vF  F  B qB2 may depend on charge sign
– If F proportional to q then drift motion independent of charge
– If F not proportional to q then drift motion dependent on charge sign
 Non-uniform and time-varying magnetic field
– Gradient drift perpendicular to B and ∇B depends on charge sign
– Converging/diverging B: deceleration/acceleration along B with mirroring
6 – 24
 In a dipole field
– Charged particles will mirror at a latitude depending on the equatorial pitch angle
– Particles will intersect Earth if equatorial pitch angle is within equatiorial loss cone


Single
Particle
Motion

DISCUSSION


Space Radiation & It's Effects On Space Systems & Astronauts Course Sampler

More Related Content

What's hot

Viewers also liked

Similar to Space Radiation & It's Effects On Space Systems & Astronauts Course Sampler

More from Jim Jenkins

Recently uploaded

Space Radiation & It's Effects On Space Systems & Astronauts Course Sampler