Transcript Solar-Terrestrial Relations

Acknowledgments
• The lecture notes are based on lecture notes
from Prof. Reinisch who taught the course
in the last few years and lecture notes form
Prof. Ray Walker of UCLA.
• Prof. Reinisch’s lecture notes were
benefited from Prof. Jeff Forbes of
University of Colorado, Boulder.
Solar-Terrestrial Relations
• 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 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 deflected
by the geomagnetic field.
• 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 magnetopause.
Regions in Space, cont.
• Magnetotail: the magnetosphere is stretched by the
solar wind on the nightside.
• Radiation belts: where most energetic particles are
trapped, (major issue for space mission safety).
• Plasmasphere: inner part of magnetosphere 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.
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 magnetostorms: ~ 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
magnetic 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 particleselectric 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
– Magnetic force
– Lorentz force
– Neutral forces
FE = qE
FB = qvxB
F = qE + qvxB
Fg =mg,
Single Particle Motion
Consider the Lorentz force when E  x, t  and B  x, t  are specified.
 Is this normally the case??
dv
m
 q E  v  B
dt
dx
 v  x, t 
dt
To determine the motion of a single charged particle in the fields
we can solve above DEs. 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.
Electric Field Added to a Plasma (B=0)
Eexternal
Uniform magnetic field, and E = 0 :
dv
m
 qv  B
dt
It is customary (and very useful) to set v  v //  v  (natural comp.)
Note that  v //  v    B  v   B. Then
dv //
dv 
m
 0, m
 qv   B or
dt
dt
dv  q
 v   B  v   b with
dt
m
qB
B

, b  ,  is the angular gyrofrequency (Lamor frequency)
m
B
–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 b direction.
Then
x y z
v   v x  v y , v  v z , b  z, and v   b  vx v y vz
0 0 
dv y
dvx
dvz
 v y ,
 vx ,
0
dt
dt
dt
These are coupled DE's that can be "uncoupled" by differentiating:
dv y
dvx
 v y ,
 vx . Differentiate re t:
dt
dt
dv y
d 2v y
d 2 vx
dvx

,
 
. Substitute:
2
2
dt
dt
dt
dt
d 2v y
d 2 vx
   vx  ,
   v y 
2
2
dt
dt
Solve ordinary DE
d 2 vx
2


vx  0. Try
2
dt
vx  v0 exp i   t   
d 2 vx
2
2
2

i

v
exp
i

t




v
exp
i

t




vx .






0
0
2
dt
From x-component of momentum equation :
1 d 2 vx
1
v y   2 dt    2 vx dt 
 dt

  vx dt  ivx
v y  iv0 exp i   t    . The minus sign for the electron.
Take the real parts:
vx  v0 cos   t   
v y  v0 sin   t   
v 2  vx2  v y2  v02 .
vx  v0 cos   t    ,
v y  v0 sin   t    . Integrate:
v0
v0
x  x0 
sin   t    , y  y0 
cos   t   


This is a cicular motion in the x,y plane.
Discuss right/left hand circles.
dvz
We had for the z component
 0. Therefore in the
dt
z direction, the charge moves with constant velocity v // :
dz
d 2z
z  z0  v// t 
 v//  2  0
dt
dt
v
v
x  x0 
sin   t    , y  y0 
cos   t   


Lamor or gyro radius:
rL 
 x  x0    y  y0 
2
2
v


v mv
rL 

 qB
The circumference of the gyro orbit is 2 rL , and the time for 1 orbit:
2 rL 2
m
T

 2
v

qB
• 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

 
  
dv  d ( 12 mv2 )
F v  m
v 
 qv  (v  B)  0
dt
dt
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
I  2rLp nve 
2nm p
Bz
I   jdx
rLp  (vm p ) (eBz )
j  evn
v2
The Magnetotail current sheet: Particle motion
Pitch angle and magnetic moment
The perp velocity v   vx2  v y2  v0 is constant, and so is v // , so the
v v0
ratio is constant: tan  

,
v// v// 0
 is called the pitch angle.
The magnetic moment of a current loop is
m  I A where I=current, A=area.
q
1
For gyrating charge q, the current is I  
q 
T 2
 v
The area is A   r   

2
L



2
2
2
1 q v 1 q mv
m  I A 

,
2 
2 qB
1 2
mv
W
m  2
 
B
B
Single particle theory: guiding center drift
• The electric field can modify the particles motion.

– Assume E  0 but B still uniform and Fg=0.
 
– Frequently in space physics it is ok to set E  B  0


• Only E can accelerate particles along B
• Positive particles go along E and negative particles go along  E
• Eventually charge separation wipes out E
– E has a major effect on motion.

• As particle gyrates it moves along E 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.
 

EB
uE 
B2


• Drift velocity is perpendicular to E and B
• No charge dependence, (electrons and ions move in the direction and
speed) therefore no currents
Drift Motion: General Form
• Any force capable of accelerating and decelerating
charged particles can cause them to drift.
 

FB
uF 
qB 2
– If the force is charge independent the drift motion will
depend on the sign of the charge and can form
perpendicular currents.
Homework
• 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, Hint: radius of curvature: calculus.
Lecture II
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 
0
• 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  0
• B is the magnetic field
• Magnetic field line has neither beginning nor end.
Maxwell’s Equations (II)
– Faraday’s Law
E  
– Ampere’s Law
B
t
1 E
B  2
  0J
c t
• c is the speed of light.
• 0 is the permeability of free space,
• J is the current density
• 00 = 1/c2
 0  4 10 7H/m
Integral Form of Maxwell’s Equations
• Maxwell’s equations in integral form
1
 E  ndA    dV
Gauss’ integral theorem
0
A
 T dA   TdV
 dl T  dA   T
A
l
V
A
– 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


A
C
B  nd A  0
E  ds   
B '

 n dF  
t
t
– 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.
1 E '
'
B

d
s


n
dF


J

n
dF
0
2 
C

c t
Nonuniform B Field:
Gradient B drift
Assume B along z has a gradient  B,
B  x  B  x z
dB
B 
y
dx
The lamor radius is smaller where B is larger, since rL  mv eB ,
etc. This leads to the grad-B drift velocity
1
B  B
uB   v rL
, B   B
2
2
B
sign for ions, -sign for electrons  current!
In a dipole field: ring current
Centrifugal Force: Curvature drift
Assume a charged particle moving along a curved field line.
Centrifugal force:
mv//2
Fc  2 R c
Rc
For radius of curvature R c ,
Fc  B mv//2 R c  B v//2 rc  b
u cB 
 2

2
2
qB
Rc qB
Rc qB m
v//2
u cB  
rc  b
Rc 
" " sign for ions, "-" sign for electrons  current!
In a dipole field: ring current
Total drift velocity in non-uniform B field:
u B  u B  u cB
1

v//2
B  B
u B  uB  u cB    v rL

rc  b 
2
B
Rc 
2

Formation of ring current
Adiabatic Invariants
Hamiltonian mechanics, working with a generalized coordinate
q and its conjugate momentum p, shows for periodic motions that
the action remains invariant for slow changes (adiabatic)
in the system!!!! The action is defined as the integral over one or
several periods of the motion: J 
 pdq
Every symmetry has a constant of integral.
For our gyromotion, a good coordinate is the azimuthal angle  ,
and the conjugate momentum is the angular momentum l  mv rL . Then
J   pdq 
2
 mv r d  2 mv r
 L
 L
0
1 2
mv
2 mv v
4 m W 4 m
2 mv rL 
 4 2


m


q B
q
 m  const First adiabatic invariant
Magnetic mirrors
Let's look at a B field that converges in space.
Within a neighborhood r >> rL , the field can be considered
cylindrical around the central axis in direction z. Then
B  Br  z  r  Bz  z  z with Br  Bz .
From Maxwell's equation   B  0, and in cylindrical coordinates
1 d
dBz
B 
0
 rBr  
r dr
dz
dBz
rBr    r
dr for Br  const
dz
1 dBz
The two components are related as required
Br   r
by the divergence-free of the magnetic field
2 dz
Assume a particle moves with velocity v // in the z direction, i.e.
parallel to the magnetic field. The magnetic moment m remains
constant when the particles moves into larger B fields, from
B0 to B:
W0 W
v2 0 v2
B 2
2

, or

 v 
v0
B0
B
B0 B
B0
v increases proportional to B.
Can v increase indefinately?? No. The total energy of the particle
1
1
m  v2 0  v//2 0   m  v2  v//2  .
2
2
increases, v//2 decreses until v//2  0  mirror reflection!
is conserved: W 
When v2
The reflected particle will go back to the point with B=B0 and
onward. If the field becomes stronger again, v //
decreases again until it reflects again: magnetic bottle.
v
The pitch angle  is defined as tan  
or
v//
2
v
W
2

sin 
 sin   2

where W = const.
2
2
2
v  v// W
v  v//
v
B 2
sin 2
v2
B
From v 
v0 , we have


B0
sin 2 0 v2 0 B0
2

Here  0 is the initial pitch angle at z  z0 . At reflection sin 2  1, or
B0
1
v2
B
2
 2 
 sin  0 
2
sin  0 v0 B0
B
If the max field strength is Bmax , then all pitch angles  0 for which
sin 0 
B0
are reflected (confined in the bottle).
Bmax
Loss cone: formation.
• The force is along B and away from the direction of
increasing B.
• If E||  0 and kinetic energy must be conserved
1
2
mv2  12 m(v||2  v2 )
a decrease in v|| must yield an increase in v
• Particles will turn around when B  12 mv 2  m
Magnetic bottle bounce period
A charged particle in a magnetic bottle bounces back between
the mirror points. The time to move from the minimum at z 0 to the
reflection point z max is T 
zmax

z0
Tb  4
zmax

z0
Tb  4
Bz 2
dz
2
and v//  v cos   v 1  sin   v 1 
sin  0
v//
B0
zmax

z0
dz
. The total bounce period is then:
v//
dz
v 1
Bz 2
sin  0
B0
Second adiabatic invariant:
JL 
 mv// dz  4mv
zmax

z0
1
Bz
B0
sin 2  0 dz
• 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).
s2
J   2mv||ds  const .
s1
– Using conservation of energy and the first adiabatic
invariant
J 
s2
s1
B 12
2mv(1 
) ds  const .
Bm
– If the field is a dipole their trajectories will take them
around the planet and close on themselves.
• 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    B  ndA inside the
orbit must be constant.
• Limitations on the invariants
–  is constant when there is little change in the field’s strength over
a cyclotron path.
B
1

B
c
– All invariants require that the magnetic field not change much in the time
required to one cycle of motion
1 B
1

B t

where  is the orbit period.
  ~ 10  10 s
 J ~ 1s
 ~ m
6
3
• 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 ( vc)
v  v  v  v  u D  vc  u gc  vc
– 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.