Space Physics and Space Weather

• Space: “empty” volume between bodies (solid bodies are excluded)

• Space physics: space within solar system (astrophysics is not space physics)

• Solar-terrestrial relations: space physics focused on solar wind and terrestrial space

• Space plasma physics: application of plasma physics to space• Space physics: Coriolis force and gravity not important (unless

noted)• Space weather: space physics applications. Space phenomena that

endanger space assets and applications and human in space• Space physics: electromagnetic field + charged particles• Require significant math:

– Working on but not solving partial differential equations in this class– Vector operations

• Require: electromagnetics (additional reading may help)

Regions in Space• Solar wind (sun’s atmosphere, but not bonded by gravity): plasma (ions and

electrons in equal number but not attached to each other) stream flows out continuously, but with variations, from the sun with extremely high speeds into the interplanetary space. Note: in space, all ions are positively charged.

• Formation of the magnetosphere: the solar wind disturbs geomagnetic field; geomagnetic field deflects solar wind.

If there is no SW SW effect on thegeomagnetic field

Regions in Space• Magnetopause: the boundary separates the magnetosphere from

the solar wind (crucial for any solar wind entry).• Bow shock: standing upstream of the magnetopause, because the

solar wind is highly supersonic.• Magnetosheath: the region between the bow shock and the magneto-pause.• Magnetotail: the mag-netosphere is stretched by the solar wind on the nightside.

• Radiation belts: where most energetic particles are trapped, (major issue for space mission safety).

Regions in Space

• Plasmasphere: inner part of magnetosphere (1000km – 4 Earth radii) with higher plasma density of ionospheric origin.

• Ionosphere: (80 ~ 1000 km) regions of high density of charged particles of earth origin.

• Thermosphere: (> 90 km) neutral component of the same region as the ionosphere. The temperature can be greater than 1000 K.

Regions in Space

Space Weather Phenomena•Magnetic storms (hurricanes in space)

•Global-scale long-lasting geomagnetic disturbances

•Magnetic substorms (tornadoes in space)

•Impulsive geomagnetic disturbances

•Auroras (rains from space)

•Enhanced energetic particle precipitations associated with storms/substorms

•Ionospheric plasma density disturbances (fog?)

•Destruction of the layered structure of the ionosphere.

Evidence for Space Processes

• Aurora: emissions caused by high energy charged particle precipitation into the upper

atmosphere from space.• Geomagnetic field: caused by electric currents below the earth’s surface.• Geomagnetic storm/substorm: period of large geomagnetic disturbances.• Periodicity of magnetic storms: ~ 27 days.• Rotation of the Sun: 26 ~ 27 days.

• Space physics started with observations of the aurora.– Old Testament references to auroras.– Greek literature speaks of “moving

accumulations of burning clouds”– Chinese literature has references to auroras

prior to 2000BC

– Galileo theorized that aurora is caused by air rising out of the Earth’s shadow to where it could be illuminated by sunlight. (Note he also coined the name aurora borealis meaning “northern dawn”.)

– Descartes thought they are reflections from ice crystals.– Halley suggested that auroral phenomena are ordered by the

Earth’s magnetic field. – In 1731 the French philosopher de Mairan suggested they are

connected to the solar atmosphere.

• By the 11th century the Chinese had learned that a magnetic needle points north-south.

• By the 12th century the European records mention the compass.

• That there was a difference between true north and the direction of the compass needle (declination) was known by the 16th century.

• William Gilbert (1600) realized that the field was dipolar.

• In 1698 Edmund Halley organized the first scientific expedition to map the field in the Atlantic Ocean.

Plasma• A plasma is an electrically neutral ionized gas.

– The Sun is a plasma– Interplanetary medium: the space between the Sun and the Earth is “filled” with a

plasma.– The Earth is surrounded by plasmas: magnetosphere, ionosphere. – Planetary magnetospheres, ionospheres– A stroke of lightning forms plasma– Over 99% of the Universe is plasma.

• Although neutral a plasma is composed of charged particles- electric and magnetic forces are critical to understand plasmas.

• Plasma physics: three descriptions– Single particle theory– Fluid theory– Kinetic theory

Forces on charged particles(single particle theory)

– Electric force FE = qE

– Magnetic force FB = qvxB

– Lorentz force F = qE + qvxB

– Neutral forces Fg =mg,

Negative Charge

Positive Charge

Use right hand rule to find the direction of F.

F=q v x B

Single Particle Motion

Consider the Lorentz force when , and , are specified.

Is this normally the case??

,

To determine the motion of a single charged particle in the fields

we can solve above DEs.

t t

dm q

dtd

tdt

E x B x

vE v B

xv x

Consider different situations:•SI Units

–mass (m) - kg–length (l) - m–time (t) - s–electric field (E) - V/m–magnetic field (B) - T–velocity (v) - m/s–Fg stands for non-electromagnetic forces (e.g. gravity) - usually ignorable.

//

//

//

, :

It is customary (and very useful) to set (natural comp.)

Note that . Then

0, or

with

, , is

dm q

dt

d dm m q

dt dtd q

dt mqB

m B

Uniform magnetic field and E = 0

vv B

v v v

v v B v B

v vv B

vv B v b

Bb the angular gyrofrequency (Lamor frequency)

–If q is positive particle gyrates in left handed sense–If q is negative particle gyrates in a right handed sense

Orient the z axis of the cartesian coordinate system in the direction.

Then

, , , and

0 0

, , 0

These are coupled DE's that can be "uncoupled" by diffe

x y z x y z

yx zy x

v v v

dvdv dvv v

dt dt dt

b

x y z

v v v v v b z v b

rentiating:

22

2 2

22

2 2

22

2

0

22 2

0 02

, . Differentiate re t:

, . Substitute:

,

Solve ordinary DE

0. Try

exp

exp exp

yxy x

y yx x

yxx y

xx

x

x

dvdvv v

dt dt

dv d vd v dv

dt dt dt dt

d vd vv v

dt dt

d vv

dt

v v i t

d vi v i t v i t

dt

2 .xv

22

2

0

0

0

2 2 2 20

From x-component of momentum equation :

1 1

exp . The minus sign for the electron.

Take the real parts:

cos

sin

.

xy x x x

y

x

y

x y

d vv dt v dt v dt iv

dt

v iv i t

v v t

v v t

v v v v

0 0

0 00 0

cos , sin . Integrate:

sin , cos

This is a in the x,y plane.

Discuss right/left hand circles.

We had for the z component 0. Therefore in the

z dir

x y

z

v v t v v t

v vx x t y y t

dv

dt

cicular motion

//

2

0 // // 2

ection, the charge moves with constant velocity v :

0dz d z

z z v t vdt dt

0 //z z v t

00

00

sin

cos

vx x t

vy y t

0

// //0

tan , is called the pitch anglev v

v v

0 0

2 2

0 0

sin , cos

Lamor or gyro radius:

The circumference of the gyro orbit is 2 , and the time for 1 orbit:

2 22

L

L

L

L

v vx x t y y t

vr x x y y

v mvr

q B

r

r mT

v q B

• Gyro motion– The gyro radius is a function of energy.

– Energy of charged particles is usually given in electron volts (eV)

– Energy that a particle with the charge of an electron gets in falling through a potential drop of 1 Volt- 1 eV = 1.6X10-19 Joules (J).

• Energies in space plasmas go from electron Volts to kiloelectron Volts (1 keV = 103 eV) to millions of electron Volts (1 meV = 106 eV)

• Cosmic energies go to gigaelectron Volts ( 1 geV = 109 eV).

• The circular motion does no work on a particle

0)()( 2

21

Bvvqdt

mvdv

dt

vdmvF

Only the electric field can energize particles!

Current Produced by Particle Motions A Particle View of the Magnetopause

• When an electron or ion penetrates the boundary they sense a v x B force. After half an orbit they exit the boundary.

• The electrons and ions move in opposite directions and create a current. The ions move farther and carry most of the current. The number of protons per unit length in the z-direction that enter the boundary and cross y=y0 per unit of time is 2rLpnu . (Protons in a band 2rLp in y cross the surface at y=y0.) Since each proton carries a charge e the current per unit length in the z-direction crossing y=y0 is

where 22

2 pLp

z

nmI r nve v

B

( ) ( )Lp p zr vm eB

j evn

jdxI

Electric Field Added to a Plasma (B=0)

Eexternal

Homework• 2.13, 2.15 (no (d) for under), 2.16, 2.18, 2.4*

• Errors in the book.– 2.4, gamma => 1/gamma– 2.13, page 32, line 2 above the figure, delB=-3B/r– 2.15, alpha is a constant, not pitch angle.– 2.18, 10^6 km, not used. – 2.18: assume parallel for curvature drift and

perpendicular for gradient drift– 2.18, Hint: radius of curvature: calculus.

Lecture II

Pitch angle and magnetic moment2 2

0 //

0

// // 0

The perp velocity v is constant, and so is v , so the

ratio is : tan , is called the pitch angle.

The magnetic moment of a current loop is

where I=current

x y

m

v v v

vv

v v

I A

constant

2

2

22 2

, A=area.

1For gyrating charge q, the current is

2

The area is

11 1 2,2 2

L

m m

qI q

T

vA r

mvq v q mv WI A

q B B B

Single particle theory: guiding center drift• The electric field can modify the particles motion.

– Assume but still uniform and Fg=0.– Frequently in space physics it is ok to set

• Only can accelerate particles along• Positive particles go along and negative particles go along • Eventually charge separation wipes out

– has a major effect on motion. • As particle gyrates it moves along and gains energy • Later in the circle it losses energy.• This causes different parts of the “circle” to have different radii - it doesn’t close on itself.

• Drift velocity is perpendicular to and• No charge dependence, (electrons and ions move in the direction and speed) therefore no

currents

0E

0BE

E

B

E E

E

EE

2B

BEuE

E

B

B

Drift Motion: General Form

• Any force capable of accelerating and decelerating charged particles can cause them to drift.

– If the force is charge independent the drift motion will depend on the sign of the charge and can form perpendicular currents.

2qB

BFuF

Electric and Magnetic Fields: Simple situations

• Single electric charge (monopole):– Positive charge– Negative charge– Net charge– E field (intensity): + => -

• Electric dipole• No magnetic monopole.• Magnetic field (magnetic dipole)

– Magnet: N and S (pointing to), geomagnetic poles: located oppositely, – B (mag flux density, including magnetization): N=>S– (H: mag field intensity)– current loop

• E and B are chosen in plasma physics because of the Lorentz force.

Maxwell’s Equations

• Maxwell’s equations– Poisson’s Equation (originally from Coulomb's law)

• E is the electric field is the electric charge density 0 is the electric permittivity (8.85 X 10-12 Farad/m)• Positive charge starts electric field line• Negative charge ends the line.

– Gauss Law (absence of magnetic monopoles)

• B is the magnetic field• Magnetic field line has neither beginning nor end.

0

E

0 B

Maxwell’s Equations (II)– Faraday’s Law

– Ampere’s Law

• c is the speed of light. 0 is the permeability of free space, H/m

• J is the current density 00 = 1/c2

t

B

E

02

1

c t

E

B J

70 104

Integral Form of Maxwell’s Equations• Maxwell’s equations in integral form

– A is the area, dA is the differential element of area– n is a unit normal vector to dA pointing outward.– V is the volume, dV is the differential volume element

– n’ is a unit normal vector to the surface element dF in the direction given by the

right hand rule for integration around C, and is magnetic flux through the

surface. – ds is the differential element around C.

0

1A

dA dV

E n

0

'

A

C

d A

d dFt t

B n

BE s n

02

1' '

Cd dF dF

c t

E

B s n J n

A V

l A

d dV

d d

T A T

l T A T

Gauss’ integral theorem

Nonuniform B Field:Gradient B drift

2

Assume B along has a gradient ,

The lamor radius is smaller where B is larger, sin

grad-B dri

ce ,

etc. This leads to the

1,

2

sign for ions, -

ft v

s

elocityL

B L

x B x

dBB

dxr mv eB

Bv r B

B

z B

B z

x

Bu B

ign for electrons !

In a dipole field: ring

cur

cur

ren

t

t

ren

Centrifugal Force: Curvature drift

2//

2

c

2 2// //

2 2 2

2//

Assume a charged particle moving along a curved field line.

Centrifugal force:

For radius of curvature R ,

" " sign for ions, "-" s

c cc

c c ccB

c c

cB cc

mv

R

mv v

qB R qB R qB m

v

R

F R

F B R B r bu

u r b

ign for electrons !

In a dipole field: ring

cur

cur

ren

t

t

ren

2//

2

Total drift velocity in field:

1

2

Formation of ri

non-uniform

ng current

B B cB

B B cB L cc

vBv r

B R

u u u

Bu u u r b

B

Adiabatic Invariants, working with a

, shows for periodic motions that

the remains invariant for slow changes (adiabatic)

in the system!!!!

Hamiltonian mechanics generalized coordinate

q and its conjugate momentum p

action

The action is defined as the integral over one or

several periods of the motion:

Every symmetry has a constant of integral.

For our gyromotion, a good coordinate is the azimuthal angle ,

and t

J pdq

2

0

2

he conjugate momentum is the angular momentum . Then

2

First adiabatic invariant

12 4 422 4

L

L L

L m

m

l mv r

J pdq mv r d mv r

mvmv v m W mmv r

q B q

const

Magnetic mirrors

L

Let's look at a field that converges in space.

Within a neighborhood r >> r , the field can be considered

cylindrical around the central axis in direction . Then

with .

From Maxwellr z r zB z B z B B

B

z

B r z

's equation 0, and in cylindrical coordinates

10

for

1

2

zr

zr r

zr

d dBrB

r dr dzdB

rB r dr B constdz

dBB r

dz

B

B

The two components are related as required by the divergence-free of the magnetic field

//

m

0

Assume a particle moves with velocity v in the direction, i.e.

parallel to the magnetic field. The magnetic moment remains

constant when the particles moves into larger B fields, from

B to B:

W

z

2 22 20 0

00 0 0

2 2 2 20 // 0 //

2

W, or

B B B B

increases proportional to B.

Can increase indefinately?? No. The total energy of the particle

1 1is conserved: .

2 2

When increases,

v v Bv v

B

v

v

W m v v m v v

v v

2 2// // decreses until 0 mirror reflecti

on!v

0

//

The reflected particle will go back to the point with B=B and

onward. If the field becomes stronger again, v

decreases again until it reflects again:

The pitch angle

magnetic bottle.

is defined as //

22

2 22 2////

tan or

sin sin where W = const.

v

v

v v W

v v Wv v

2 22 2

0 2 20 0 0 0

20 0

22 0

02 20 0 0

max

sinFrom , we have

sin

Here is the initial pitch angle at . At reflection sin 1, or

1sin

sin

If the max field strength is B , then all pitch an

B v Bv v

B v B

z z

Bv B

v B B

0

00

max

gles for which

sin are reflected (confined in the bottle).

: formation.

B

B

Loss cone

• The force is along B and away from the direction of increasing B.

• If and kinetic energy must be conserved

a decrease in must yield an increase in

• Particles will turn around when

0|| E

||v v21

2 mB mv

)( 22||2

1221

vvmmv

Magnetic bottle bounce period

max

0

0

max//

A charged particle in a magnetic bottle bounces back between

the mirror points. The time to move from the minimum at z to the

reflection point z is . The total bounce period is then:z

z

b

dzT

v

T

max

0

max

0

max

0

2 2// 0

// 0

20

0

2L // 0

0

4 and cos 1 sin 1 sin

4

1 sin

:

J 4 1 sin

z

z

z

b

z

z

z

B zdzv v v v

v B

dzT

B zv

B

B zmv dz mv dz

B

Second adiabatic invariant

• In general, the second adiabatic invariant– The integral of the parallel momentum over one complete

bounce between mirrors is constant (as long as B doesn’t change much in a bounce).

– Using conservation of energy and the first adiabatic invariant

– If the field is a dipole their trajectories will take them around the planet and close on themselves.

.22

1|| constdsmvJ

s

s

.)1(2 212

1

constdsB

BmvJ

s

sm

• The third adiabatic invariant– As particles bounce they will drift because of

gradient and curvature drift motion.– As long as the magnetic field doesn’t change

much in the time required to drift around a planet the magnetic flux inside the orbit must be constant.

dA B n

• Limitations on the invariants is constant when there is little change in the field’s strength over a

cyclotron path.

– All invariants require that the magnetic field not change much in the time required to one cycle of motion

where is the orbit period.11

t

B

B

m

s

s

J

~

1~

1010~ 36

cB

B

1

• The Concept of the Guiding Center

– Separates the motion (v) of a particle into motion perpendicular (v) and parallel ( v||) to the magnetic field.

– To a good approximation the perpendicular motion can consist of a drift (uD ) and the gyromotion ( vc)

– Over long times the gyromotion is averaged out and the particle motion can be described by the guiding center motion consisting of the parallel motion and drift.

c cD gc v v v v u v u v