asen 5335 aerospace environments -- radiation belts1 the radiation belts a radiation belt is a...

ASEN 5335 Aerospace Environments -- Radiation Belts 1

The Radiation Belts

A radiation belt is a population of energetic particles stably-trapped by the magnetic field.


THE RADIATION BELTS

Particles in Magnetic Fields

The motions of electrons and ions in a constant magnetic field (without external forces) are governed by a balance between the so-called Lorenz force (F = qV x B) and the centrifugal force, which results in circular motion:

XX

--+

F = q V x B

B - field intopaper

Plasma particles are diamagnetic in that their orbital gyrations produce a magnetic field that opposes the background magnetic field.


gyroradius q = charge

B = magnetic field strength

• external force independent of charge

• external force dependent on charge

• non-uniform B-field

• curvature in B-field geometry

• converging/diverging field lines

Now, we will consider the influences of an external force and a non-uniform B-field. Five cases will be considered:

qB

mgyrofrequency m = mass

v = velocity perpendicular to B

qB

mvr


1. CHARGE-INDEPENDENT FORCE

XX

--+

F = q V x BGravitationalForce

effective B is larger

rcmvqB larger

gravitational force adds to F here

gravitational force subtracts from F here

effective B is smaller

rcmvqB

smaller


gravitational force gravitational forceBXX

X X X

X

X

X X X

XXXX

+ + --

This represents current flow to the right

In general, vd FBqB2 ;

if F mg

then vd gBqB2

m

vd depends on sign of charge; + drift to right and - to left.


2. CHARGE-DEPENDENT FORCE

Now if we replace g by E, then F mg becomes F qE.

i in , e en

Recall that this assumes only the Lorenz force acting on the particle. In the presence of a neutral gas, this will be true if

Therefore both + and - particles move in the same direction and there is no current.

which is charge independent

In this case, vd FBqB2

EBB2


XX

--+

F = q V x BF = q E F = q E

r larger here (effective B decreased)

r smaller here(effective B increased)

r smaller here(effective B increased)

r larger here (effective B decreased)


BXX

X X X

X

X

X X X

XXXX

E E

Drift of + and - particles due to E-field in presence of B-field

++ --


3. NON-UNIFORM MAGNETIC FIELD

Now consider a non-uniform magnetic field (i.e., |B| varies spatially).

X

BXX

X X

X

X

X

X

X

X

XX X X

X

X

XX XXXXX

<--------- net current flow

+

+

-

-

rc mv

qBThe gyroradius is smaller when |B| is larger:

(reason in-part for “ring current”)


In this case the force on the particle is proportional to the

gradient of B :

where is the particle's velocity around the magnetic field.

v

Note: for “cold particles” (i.e., ionosphere/plasmasphere) is small

gradient drift is small (i.e., they co-rotate)

F 1

2mv

2 BB

vgrad 1

2

mv2

e

BBB3 BB

B3

The so-called gradient drift is

where = the particle's "magnetic dipole moment"


We will be referring to the magnetic moment of a particle later in these lectures, so it is appropriate to take a minute to explain the origin and meaning of this term.

The magnetic moment of a current loop is just defined as

(current in X (area of the loop) the loop)

The equivalent current of a gyrating particle (going around once per ) is

2 /

i dq

dt

e2

Substituting the following,

We obtain

"magnetic moment" of a particle

i A e

2

r2

eB

m

r mv

eB

1

2

mv2

B


4. MAGNETIC FIELD CURVATURE

As a gyrating particle moves along a B - field that is curved, some additional force must act on the particle and make it turn and follow the field line geometry.

v R

F mv 2

R

R

BSince this depends on the sign of q,positive and negative particles “curvature drift” in opposite directions,thus producing electric current. Notethat these are “collisionless currents”,and so do not produce ohmic losses.Complements the non-uniform, B-fieldGradient Drift

R = radius of curvature offield line.

RF ˆ2||

R

mv

vcurv mv||

2RBqR2B2

v = v|| = velocity || to B

F ma m 2R

As a particle follows B, a force is exerted

F


5. CONVERGING/DIVERGING FIELD LINES

Rather, the net force is now in the direction of weaker B-field (diverging field lines).

For a proton in a diverging B-field as shown in the following figure, the force acting at right angles to the B-vector does not lie in the plane of circular motion of the charged particle.

The same holds true for an electron; that is, the net force is away from the region of stronger (converging) field strength.

When the magnitude and duration of the force are sufficient to actually cause the charged particle to reverse direction of motion along the line of magnetic force, the effect is known as mirroring, and the location of the particle's path reversal is known as the mirror point for that particle.

Therefore, as a charged particle moves into a region of converging B, a force acts to slow the particle down.


The three types of charged particle motion in Earth’s magnetic field

Some typical periods of particle motion for 1Mev* particles at 2000 km altitude at the magnetic equator

Tg Tb Td gyroperiod bounce period drift period

electrons 7s (r = .3km) .1s 50 min.

protons 4ms (r = 10km) 2s 30 min.

Note that Td >> Tb >> Tg , so that these processes can essentially be assumed decoupled from each other

* 1 eV = energy acquired by a particle of unit charge when accelerated througha potential of 1 volt = 1.6 x 10-12 ergs.


According to Faraday's Law of Magnetic Induction, a time rate of change of magnetic flux will induce an electric field ( and hence a force on the particle ):

dl

B

t Edl B BdA

Bt 0

Therefore, the requirement that no forces act in the direction of motion of the rotating particle demands that

Let us examine the mirroring problem more quantitatively. Assuming that the Lorenz force F = qV x B is the only force acting on the particle (i.e., no external forces), the kinetic energy (K.E.) of the particle does not change (Lorenz force is perpendicular to V and therefore does no net work); only the direction of motion changes.

Constants of Particle Motion


In other words, that the magnetic flux enclosed by the cyclotron path of the charged particle is constant:

1

2

mv2

BWe previously defined the magnetic moment , implying

B BdA constAssuming B does not vary spatially within the gyropath,

where r = gyroradius =

mvqB

B B dA Br2


B m2v

2

q2B2

B

2m

q2 const

Note: same assaying that K.E.does not changeif there are noforces parallelto v

First Adiabatic Invariant

Or, = constant . This is called the first adiabatic invariant of particle motion in a magnetic field.

We should note that the above has assumed that constant

within at least one orbital period of the particle.

This is only approximately true, and the term "invariant" is also an approximation, but one that reflects the first-order constraints on the particle motion.

V

B

A charged particle in a magnetic ‘bottle’

Conservation of the first adiabatic invariant can cause the spiraling particle to be reflected where the magnetic field is stronger.

This causes the particle to be trapped by the magnetic field.

v v sin and 1

2mv 2

sin2 B

const

Consequences of Conservation of the First Adiabatic Invariant


If increases to 90° before the particle collides vigorously with the neutral atmosphere, the direction of v|| will change sign (at the "mirror point") and the particle will follow the direction of decreasing B.

Since , (i.e, the K.E. of the particle

remains constant since the only forces act to V), then must increase as B increases, and correspondingly the distribution of K.E.

between and changes:

1

2mv

2 const

v

v||

v v sin

v|| v cos


For a given particle the position of the mirror point is determined by the pitch angle as the particle crosses the equator (i.e., where the field is weakest) since

sin2 B

const

sin2 eqBeq

1

BM

BM Beq

sin2eq

Therefore, the smaller eq the larger BM , and the lower down in altitude is the altitude of BM.


Loss Cone and Pitch Angle Distribution

Obviously this will happen if eq is too small, because that requires a relatively large BM (|B| at the mirror point).

Particles will be lost if they encounter the atmosphere before the mirror point.

B

losscone

The equatorial pitch angles that will be lost to the atmosphere at the next bounce define the loss cone, which will be seen as a depletion within the pitch angle distribution.


Magnetic Mirroring in a Dipolar magnetic Field

Trajectory of particleoutside the atmosphericbounce loss cone. Thisparticle will bouncebetween mirror points

Trajectory of particleinside the loss cone.this particle will encounter the denser parts of the atmosphere (i.e., below 100 km) and precipitate from the radiation belts.

asen 5335 aerospace environments -- radiation belts1 the radiation belts a radiation belt is a...

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