Single Particle Motion

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

SPACE RADIATION AND ITS EFFECTS ON SPACE SYSTEMS AND ASTRONAUTS

Single Particle Motion

by V. L. Pisacane

Single Particle Motion

©VLPisacane,2012 Space Radiation and its Effects on Space Systems and Astronauts 2 5 ─

TOPICS

Introduction Equation of Motion

Gyration Motion

Guiding Center Motion

Mirror Points

Summary

Single Particle Motion

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INTRODUCTION Background

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, under Dr. James Van Allen at the University of Iowa

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

http://www.centennialofflight.gov/essay/Dictionary/RADIATION_BELTS/DI160.htm

Single Particle Motion

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5– 4

INTRODUCTION Summary 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 in the Earth’s

magnetic field,

Single Particle Motion

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EQUATION OF MOTION Introduction

Force on a particle in a magnetic and av electric field with charge q is given by the Lorentz force

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

BvEF qL

)(qdt

dm BvEF

v

Single Particle Motion

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GYRATION MOTION Introduction

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

Single Particle Motion

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GYRATION MOTION Gyro-Frequency and Larmor Radius

Gyro-frequency or relativistic cyclotron frequency wc and Larmor radius rc follow from the gyration equation of motion

where

F = force, N m = gm0, relativistic mass, kg m0 = rest mass, kg a = acceleration, m s-2

q = charge, C v = magnitude of particle velocity, m s-1

vg = magnitude of gyration velocity, m s-1

R

cB

q

q

R

vcB

q

qvcB

pc

q

mvc

vcqB

m

qB

c

v1

m

qB

m

qB

m

qB

mrBrq

mrBqv

qdt

dm

2

g

2

2

00

g

2

clgl

2

clg

g

g

w

g

w

ww

w

Bvv

g

g

g

w

w

cB

R

cB

1

q

pc

qB

p

qB

mvr

qB

cm

qB

mc

qB

mvr

vr

vr

0g

l

00g

l

g

g

l

gcl

rl = Larmor radius, m R = magnetic rigidity, V g ≡ (1-v2/c2)─1/2

wg = gyrofrequency, s-1

fc = wc/ 2p, gyrofrequency, Hz B = magnetic flux density perpendicular to plane of

motion, T

Single Particle Motion

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GYRATION MOTION Gyro-Frequency for Electrons and Protons

For electrons

For protons

Ratio of gyro-frequencies for the electron and proton is

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

Hz B

1080.22

f

s rad B

1076.11011.9

B1060.1

m

qB

m

qB

10cg

1-11

31

19

0

g

g

p

w

g

g

gw

Hz B

1052.12

f

s rad B

1058.91067.1

B1060.1

m

qB

m

qB

7cg

1-7

27

19

0

g

g

p

w

g

g

gw

18331011.9

1067.1

m

m

f

f31

27

electron,0

proton,0

proton,g

electron,g

Single Particle Motion

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GYRATION MOTION Larmor Radii for Electrons and Protons

For electrons

For protons

Ratio Larmor radii for the electron and proton is

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

18331011.9

1067.1

m

m

r

r31

27

electron,0

proton,0

electron,l

proton,l

g

g

g

B

c1068.5

B

c

1060.1

1011.9

qB

cmr 12

19

31

0electron,l

g

g

g

B

c1004.1

B

c

1060.1

1067.1

qB

cmr 8

19

27

0protons,l

Single Particle Motion

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Magnetic Rigidity

Kinetic Energy

Speed [% of c]

Gyro-frequency

Corona (10 mT)

Interplanetary 1 AU (5 nT)

Earth 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

GYRATION MOTION Larmor Radii and Gyro-frequency in Terms of Magnetic rigidity

Magnetic Rigidity

Kinetic Energy

Speed [% of c]

Larmor Radius, Re = Earth radii

Corona (10 mT)

Interplanetary 1 AU (5 nT)

Earth 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

Examples of Larmor radii and gyro-frequency for any charged particle with a given rigidity at

Sun near Earth surface of the Earth

Single Particle Motion

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GYRATION MOTION Summary Motion in Uniform Magnetic Field

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

Single Particle Motion

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GUIDING CENTER MOTION Introduction

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

so that the equation of motion

can be approximated by

Bv

v g

g qdt

dmBvEF

v gc

gc qqdt

dm

gcg vvv

BvEFv

qqdt

dm

Single Particle Motion

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GUIDING CENTER MOTION Guiding Center Equations

Equation of motion of the guiding center is

Partitioning the velocity and force into components parallel and normal to the magnetic field

gives

If for E|| and F|| are constant, guiding center motion parallel to the magnetic field is

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

||||

||gc qEFdt

dvm

gc||||gcv vεv

BvEFv

gc

gc qqdt

dm

FεF ||||F

)0(vtm

qEFv ||gc

||||

||gc

EεE ||||E

BvEFv

gc

gc qqdt

dm

Single Particle Motion

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GUIDING CENTER MOTION Motion in Uniform Magnetic Field

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

Single Particle Motion

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GUIDING CENTER MOTION Orthogonal Drift Velocities of the Guiding Center 1/2

Gyration without perturbations (see A)

General Force Drift (see C) – Drift due to presence of general force F

Electric Field Drift (see B) – Drift due to presence of electric field E

Gradient Drift (see D) – Drift due to gradient in t magnetic field ∇B – Dominates for the Earth

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

B

2E,gcB

BEv

2F,gcqB

BFv

EF q

BBv

3

2

BgcqB2

mv

Single Particle Motion

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GUIDING CENTER MOTION Orthogonal Drift Velocities of the Guiding Center 2/2

Gravitational drift – Drifts due to presence of gravity vector g

Magnetic Curvature Drift – Drift due to Rc radius of curvature of magnetic field

Polarization Drift – Drifts due time derivative of the electric field dE/dt

2gcgqB

m Bgv

BBBBRv2

4

2

||

c

c

2

2

||

gcRqB

vm

RqB

vm

dt

d

qB

m

dt

d

B

12

g

gc

EEv

p

w

|| B

Drift out for positive charge Drift into for negative charge

dt

dE

Single Particle Motion

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MIRRORS POINTS Introduction

If the electromagnetic fields do not vary in time, the energy of the particle is constant so that

The energy can be expressed as

in regions where charged particles can drift in either direction along magnetic field-lines

However, particles are excluded from regions where v|| is imaginary

Charged particles must reverse direction at those points on magnetic field-lines, these points are termed bounce points or mirror points

E = Energy

t = time m = particle mass

v|| = velocity parallel to field line vd = drift velocity

0dt

d

E

qBvvm2

1 2

D

2

||E

2/1

2

D||v)qB(

m

2v

E

qBmv2

1 2

DE

qBmv2

1 2

DE

= electric field potential, E = ─

Vm = magnetic field potential q = charge = orbital dipole moment of particle

=IA = 2riwc/2, I = current, A = area

Single Particle Motion

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MIRRORS POINTS Mirror Points in Dipole Field

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

where lm = magnetic latitude of mirror point aeq = pitch angle at the equator

6 – 18

From: http://www.altfuels.org/sampex/losscone/index.html

Pitch Angle 60o

Pitch Angle 40o

0 10 20 30 40 50 60 70 80 90

-80

-60

-40

-20

0

20

40

60

80

Equatorial Pitch Angle, degs

Mirro

r Latitu

e,

degs

Mirror magnetic latitude as function of magnetic equatorial

pitch angle

2/1

m

2

m

6

eq

2

sin31

cossin

l

la

Single Particle Motion

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MIRRORS POINTS Mirror Points in Dipole Field

Mirror points occur at same field strength on each bounce, i.e., at same magnetic latitude and drift equatorially

If pitch angle close to 0 or 180 degrees (nearly aligned with magnetic field), mirror point will fall below planet surface

Particle with equatorial pitch angle 90 degrees, will mirror at equator and remain in magnetic equatorial plane

How far the particle travels from magnetic equator before "mirroring" is determined by pitch angle at the equator

Single Particle Motion

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MIRRORS POINTS Equatorial Loss Cone

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

lca

Loss -one geometry

http://www-spof.gsfc.nasa.gov/Education/wtrap2.html

4/156

lcL3L4sin

a

Single Particle Motion

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MIRRORS POINTS Mirror Points of Trapped Radiation 1/2

Spenvis simulation Initial Conditions

– L-shell: 2 Re – Equatorial pitch

angle: 30o

6 – 21

3D view of L shell

Altitude of mirror points

Cylindrical projection of the shell Footprints in Northern and Southern hemisphere

Single Particle Motion

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MIRRORS POINTS Mirror Points of Trapped Radiation 2/2

3D view of L shell Altitude of mirror points

Cylindrical projection of the shell Footprints in Northern and Southern hemisphere

Spinvis simulation Initial Conditions

– L-shell: 2 Re – Equatorial pitch

angle: 15o

Mirror Points – Some mirror

points subsurface

Single Particle Motion

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6 – 23

SUMMARY Typical Particle Characteristics

Table 6.5 Characteristics of typical radiation belt charged particles

Single Particle Motion

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SUMMARY Summary Single Particle 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 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

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

6 – 24

2

FqBBFv

Single Particle Motion

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DISCUSSION