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Acknowledgement
A number of slides are courtesy of
Jeff Forbes
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1
The Solar Wind
The solar wind is ionized gas emitted from the Sun flowing radially
outward through the solar system and into interstellar space.
The solar wind is the extension of the solar corona to very large
heliocentric distances.
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Observed Properties of the Solar Wind at
Electron density
7.1 cm-3
1 AU
Proton density
6.6 cm-3
425 kms-1
He2+ density
0.25 cm-3
Flow speed
Magnetic field
6.0 nT
Proton temperature 1.2 x105 K
Electron temperature
1.4 x105 K
Derived Properties of the Solar Wind
The pressure in an ionized gas with equal proton and electron densities is
Pgas = nkB (Tp + Te)
where kB is the Boltzmann constant and Tp and Te are proton and electron
temperatures. Thus,
Pgas = 25 pico pascals (pPa)
Similarly, a number of other solar wind properties can be derived (see
following table)
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Solar wind properties are often expressed in terms of flux densities at
1 AU; by multiplying by the area of a sphere at 1 AU (2.82 x 1027 cm2),
these can be converted to total fluxes:
Solar wind flux densities and fluxes at 1 AU
Flux through sphere
Flux density at 1 AU
Protons
3.0 x 108 cm-2 s-1
8.4 x 1035 s-1
Mass
5.8 x 10-16 gcm-2s-1
1.6 x 1012 g s-1
Radial
Momentum Flux* 2.6 x 10-9 Pa
7.3 x 1014 N
Kinetic
Energy
1.7 x 1027 erg s-1 (1 erg/s = 10-7 W)
0.6 erg cm-2 s-1
Thermal
Energy
0.02 erg cm-2 s-1
_____________
0.05 x 1027 erg s-1
*sometimes called dynamic pressure, because of its role in confining the
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magnetospheric magnetic field
The Solar Wind is Highly Variable – V[m/s]
fast streams
Historical
Note:
Recent observations
shock
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The solar
wind was
first
sporadically
detected by
the Soviet
space
probes
Lunik 2 and
3.
5
Solar Wind Statistics
“slow”
“fast”
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The Solar Wind
The solar wind exists due to the huge pressure difference between the hot
plasma at the base of the corona and the interstellar medium. This pressure
difference drives the plasma outward despite the restraining influence of
solar gravity.
Historical
Note:
The existence
of a continuous
solar wind was
first suggested
by Lugwig
Biermann
based on his
studies of the
acceleration of
plasma
structures in
comet tails.
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How Does the Solar Wind Escape
the Sun’s Gravity ?
We will now quantify the basic ideas behind coronal heating
and the solar wind through a simplified analytic model. More
sophisticated treatments retain the fundamental ideas below.
First, we will invoke several simplyfying assumptions:
The solar wind is an ideal isothermal gas;
The solar wind flows radially away from the sun;
Magnetic field effects are neglected;
Steady-state solution
Let us now outline the basic equations.
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9
Ch 6 The Solar Wind
Why is the corona flowing outwards, i.e., why is there a solar wind?
Assume "hydrostatic equilibrium. Possible?
GM
p
p
2k
  g    2   , and substitute  
, R B
r
r
RT
mp
GM  dr
dp

p
R r 2T  r 
 GM r dr 

p  p exp  

2



R
r
T
r


R



 refers to the base of the corona
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 6.2 
10
n
R 
If we assume T  r   T    , the integral in (6.2) can be solved
 r 
(problem 6.1). For n = 0 (isothermal corona T = T ) the pressure
profile becomes
 GM   1 1  
p  r   p exp  

 RT  R  r  
 GM  
-8
2
p     p exp 
10
N
m

 RT R  
 6.5 
The interstellar medium is at pISM  10 13 N / m 2 , i.e., p     pISM .
This is not possible in equilibrium. One can try other reasonable
temperature profiles with n < 2 7. They all produce a finite pressure > 0.
We need a dynamic solution of the momentum equation!
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11
6.1.2 Heat conduction in the solar corona
Heat conduction equation for steady state conditions:
  Q  0, Assume Q  Qr rˆ
dT
Qr   K
, K  CT 5 2 eV /  m / s / K   with C  7.7  107.
dr
In spherical coordinates:
1  2
  Q  2  r Qr   0  r 2Qr  const.
r r
dT
 dT 
2
2
2 5 2 dT
2 5 2  dT 
r K
 R K  
 RT 
 6.8 
 or r T

dr
dr
 dr 
 dr 
 7  dT  R  R  
 T  T 1  
 1  (Problem 6.2)



 2  dr  T  r
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 6.9 
12
 7  dT  R  R

T  r   T 1  
 1 



 2  dr  T  r
27
27
 6.9 
 7  dT  R 
 7 
T     T 1  
 T 1   


 2 
 2  dr  T 
2
 dT  R
with   
. For    T     0, and

7
 dr  T
 R 
T  r   T 

r


27
27
 6.11
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How does the corona acquire the necessary energy for the
mean thermal energy of the coronal gas to increase outward
from the sun and overcome the sun's gravity ? A source of
coronal heating is required. Four possibilities have been
suggested:
•
•
•
•
Acoustic wave dissipation
Alfven wave dissipation
MHD wave dissipation
Microflares “magnetic carpet”
The currently favored mechanism, evolved from multi-instrument
observations from SOHO, is that “short-circuit” electric currents
flowing in the loops of the “magnetic carpet”, and extending into
the corona, provide the energy necessary to raise the coronal
temperatures to millions of degrees K. Microflares are thought to
accompany these intense currents.
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14
Small magnetic loops permeate the surface of the Sun,
much like a magnetic carpet
Each loop carries as much energy
as a large hydroelectric plant
(i.e., Hoover Dam) generates in
about a million years !
More sensitive instruments are
needed to actually observe the
microflares thought to exist.
Energy flows from the loops when
they interact, producing electrical
and magnetic “short-circuits”. The
very strong currents in these short
circuits are what heats the corona
to high temperatures.
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WHAT ABOUT THE EFFECTS OF THE
SOLAR MAGNETIC FIELD ?
To understand the importance of a magnetic field to the
behavior of a plasma, it is convenient to define a "magnetic pressure”
Pmag =
B2
20

0
where B is the magnetic field strength and
is the magnetic
permeability. Another way of thinking of the magnetic pressure is as the
energy density stored at any point where the magnetic field strength is
B.
A relevant comparison then is between
the plasma gas pressure P and Pmag .
This ratio is defined as "beta”
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 P
Pmag
16
Return to Cravens:
4.5.1 Magnetic pressure
F  I l  B  J  B  N
F  J  B is the force per unit volume on a plasma
Maxwell for static fields:
1
1
J
B  JB 
  B   B
0
0
Vector identity:
  A  B    A   B   B   A  A     B   B     A  , or
   A   B    A  B    A   B   B   A  A     B  . If A  B :
   B   B    B  B    B   B   B   B  B     B 
2    B   B   B 2  2  B   B
B2
1
 J  B  
  B Reinisch_85.511_ch_6
B
20 0
17
Momentum equation  4.80  :
Du
 p  J  B   g    u  u n   Pm

i i u  un 
Dt

B2  1
   p  
   B   B   g    u  u n   Pm
i i u  un 
20  0

B2
pB  
20
p ne k B Te  Ti 

Plasma  
B 2 20
pB
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A "high beta plasma" ( >> 1) is one
which is controlled principally by the
plasma gas dynamics .
A "low beta plasma" ( << 1) is one
which is dominated by the intrinsic
magnetic field.
 P
Pmag
IN THE CORONA
If the magnetic field is small, we would expect the expanding
corona to drag the magnetic field with it --- this is called a "frozenin" magnetic field, characteristic of a high-beta plasma.
If the magnetic field is large, we expect the magnetic field to
"contain" the plasma, or at least to inhibit its expansion.
For the real corona, where the magnetic pressure is a few times
the gas pressure, a mixture of these extreme behaviors is expected.
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Magnetic-field lines deduced from the isothermal MHD coronal
expansion model of Pneuman and Kopp (1971) for a dipole field at
the base of the corona. The dashed lines are field lines for the
pure dipole field.
MHD modeling shows that the
inner magnetic field lines (R <
2) near the equator are closed,
and that at higher latitudes the
field lines are drawn outward
and do not close.
These field lines that do not
close nearly meet at low
latitudes, but do not reconnect;
this abrupt change in the
magnetic field polarity is
maintained by a thin region
of high current density called
the interplanetary current sheet. This current sheet separates the plasma
flows and fields that originate from opposite ends of the dipole-like field.
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20
Coronal Holes and Solar
Wind Speed and Density
The interplay between the inward
pointing gravity and outward pointing
pressure gradient force results in a
rapid outward expansion of the
coronal plasma along the open
magnetic field lines.
At low latitudes the direction of the
coronal magnetic field is far from
radial. Therefore the plasma cannot
leave the vicinity of the Sun along
magnetic field lines. At the base of
low-latitude coronal holes, however,
the magnetic field direction is not far
from radial, and the expansion of the
hot plasma can take place along open
magnetic field lines without much
resistance  fast solar wind.
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THE INTERPLANETARY MEDIUM AND IMF
Intermixed with the streaming solar wind is a weak magnetic field,
the IMF.
The solar wind is a
“high-” plasma, so
the IMF is "frozen in”;
the IMF goes where
the plasma goes.
Consequently, the "spiral"
pattern formed by
particles spewing from
a rotating sun is also
manifested in the IMF.
The field winds up because
of the rotation of the sun. Fields in a low speed wind will be
more wound up than those in high speed wind.
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22
Loci of a
succession
of fluid parcels
(eight
of them in this
sketch)
emitted at a
constant
speed from a
source
fixed on the
rotating
Sun.
Loci of a succession of fluid
particles emitted at constant
speed from a source fixed
on the rotating Sun.
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23
Plasma leaves the sun predominantly at high latitudes and flows
out and towards the the equator where a current sheet is formed
corresponding to the change in magnetic field polarity.
The Sun’s magnetic field is dragged out by the high-beta solar
wind. The current sheet prevents the oppositely-directed fields from
reconnecting.
The current sheet is tilted with respect to the ecliptic (about 7°),
ensuring that earth will intersect the current sheet at least twice during
each solar rotation. This gives the appearance of "magnetic sectors".
1
j=
2

 B
0
3
1
j
B
 j =  B =  1 iˆ
o3
x 3
2
1976 (max 1979) 1986
1998 (max 2001) 2008
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At Earth, the
IMF can be
directed
either inward
or outward
with
respect to the
Sun, forming
a pattern of
“magnetic
sectors” that
appear to
rotate with
the Sun.
-
-
-
+ ++
+
+
+
--
-
SUN
-
-
+ + +++
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+
+
25
Wavy Structure of the Interplanetary Current
Sheet
Where Earth’s orbit intersects this current sheet determines whether
Earth “sees” a positive or negative magnetic sector.
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Acceleration of High-Energy Particles:
Near the Sun & in Interplanetary Shocks
The measured spectra of
energetic particles near Earth
indicate 2 spectral regimes.
The time history indicates
the high-energy component
was accelerated near the Sun,
and the low-energy
component in interplanetary
space, probably in association
with shocks.
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Two Classes of Solar Particle Events
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28
WHAT IS MEANT BY SUPERSONIC ?
Let us now examine more quantitatively the behavior of a
charged particle in a constant, uniform B-field. This will allow us to
comment on the notion previously introduced of a "supersonic" solar
wind.
Particle Motion in a Magnetic Field
The force on a charged particle in a magnetic field is called the
Lorenz force
F qV B
q = charge intensity
V = charged particle velocity
B = magnetic field strength
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29
In a plane  to B , the force on a particle is always  to
V , similar to a ball on a string; the particle executes a
circular motion where the centrifugal force is balanced by
:
F
F  ma  m dV  qVB
dt
V
For circular motion:
r V
2
V
2
 r r
VxB
B
Therefore

2
F  ma  m  2r  m Vr 





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







30
And we may now define the "gyroradius" or "cyclotron radius":
r  mV
qB

Since V  r , the "gyrofrequency" or "cyclotron frequency"
may be defined:
  qB
m
The above results assume that the B-field is uniform, and that
there are no external forces applied (i.e., an electric field).
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The Notion of "Supersonic"
The transmission of sound as we know it depends on
collisions, the average distance between collisions and the
average frequency of collisions being important parameters.
For an ideal (thermalized) gas, the speed of sound is
given by:
average kinetic speed
of particles
Cs  kT
m V l
collision
frequency
mean
free path
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32
Now, the mean free path calculated for the solar wind
near the earth is nearly 1 AU (!), implying an essentially
collisionless fluid. How can the idea of a "supersonic" solar
wind be at all credible under these circumstances ?
The mean free path is the "interaction distance" for individual
particles, i.e., the average distance a particle moves before
changing direction.
For particles in a magnetic field, the "effective interaction
distance" is the gyroradius, and the "effective collision
frequency" is the gyrofrequency.
 
2
In a plasma, a charged particle can make its presence felt with
out colliding with another charged particle.
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33
By analogy with the ideal thermalized gas, we can
calculate an ”interaction speed" as follows:
mV

eB
eB
m
V
~ 
2
 r
Cs  kT


l~
m
2










Therefore the effective mach number is
V
solar wind
1
V 2

So, in the above crude sense the solar wind can be described as
"supersonic".
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34
An alternative analogy is found by asking what is the physical
mechanism in plasmas that "transmits information" in a manner
similar to sound waves in a gas with collisions?
We all know that waves can travel on a string, and that a standing
wave pattern is set up when you pluck the string. Think of beads on a
string. If some mechanism makes the string shake, then all the beads
are affected even though they do not actually collide with each other.
From classical physics, the velocity with which waves travel along the
string is
V  T
T
Where T = tension and  is the mass per unit length.
Charged particles similarly gyrate around a magnetic field line,
like “beads on a string.”
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Alfven waves are solutions to the hydromagnetic equations, and are
analagous to the classical physics waves traveling along a string in the
sense that waves replace collisions as a means of transmitting information
Alven waves have a velocity
B2
2
B
V  
A
o
o
= magnetic field tension
 = mass density of particles
Alfven waves are "magnetic" waves traveling along the field lines, and
represent the limiting speed at which information can be carried in a
collisionless plasma. Near earth,
Vsolar wind
~10
V
A
so the solar wind is "superalfvenic".
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The Heliosphere and its Interaction with the Interstellar Medium
Heliopause
Interstellar Medium
Termination Shock
Heliosphere
Heliospheric Bow Shock ?
The heliosphere and heliopause represent the region of space influenced
by the Sun and its expanding corona, and in some respects encompass
the true extent of the solar system.
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37
The radially-expanding supersonic solar wind must
be somehow diverted to the downstream direction to
merge with the flow of the interstellar medium. This
diversion can only take place in subsonic flow, and
therefore the supersonic expansion of the solar wind
must be terminated by an “inner shock” or
“termination shock”.
Flow lines of the
interstellar plasma
do not penetrate
into the region
dominated by the
solar wind flow but
flow around a
“contact surface”
called the
heliopause,
which is
considered to be
the outer boundary
of the heliosphere.
The interstellar
medium (ISM) will
form a heliospheric
bow shock if it is
supersonic with
respect to the
heliopause
26 km/sec
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