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On the consequences of bi-Maxwellian distributions
on parallel electric fields.
Scott, Lewis J.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/26662
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
ON THE CONSEQUENCES OF BI-MAXWELLIAN
DISTRIBUTIONS ON PARALLEL ELECTRIC FIELDS
by
Lewis J. Scott
December, 1991
R.C. Olsen
Thesis Advisor:
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ON THE CONSEQUENCES OF BI MAWELLIAN DISTRIBUTIONS ON PARALLEL ELECTRIC FIELDS
PERSONAL AUTHOR(S)
12
Scott,
Lewis
J.
DATE OF REPORT
December
13a TYPE OF REPORT
13b TIME COVERED
14
Master's Thesis
From
1991,
To
(year,
month, day)
15
PAGE COUNT
78
SUPPLEMENTARY NOTATION
16
The views expressed
Government.
17 COSATI CODES
in this thesis are those of the
18 SUBJECT
GROUP
FIELD
author and do not reflect the
SUBGROUP
ABSTRACT (continue on
Observations made by the
reverse
if
if
necessary and identify by block number)
Electric Fields,
SCATHA,
1
necessary and identify by block number)
SCATHA and DEI
by a bi-Maxwellian distribution function.
models the latitudinal density
(continue on reverse
Department of Defense or the U.S.
Plasma Physics, Space Physics, Bi-Maxwellian Distributions,
DE
19
TERMS
official policy or position of the
profiles
spacecraft reveal the existence
A resultant parallel electric field
and the resultant parallel
of
equatonally trapped plasmas. These plasmas
may
be described
arises as a consequence of this distribution function. This thesis
electric field that occurs
by integrating the particle distributions
density, and assuming quasi neutrality to solve for the electric potential and hence the electric
to obtain the
show that the density profile is a
maximum at the equator and the equatorially trapped plasma is confined closer to the equator for higher anisotropy ratios. The modeled density
profiles are in agreement with some observations. The electric fields that result are on the order of 0.1 uV/m pointing away from the magnetic
equator with greater anisotropy leading to larger electric field strength. Density minimums have also been observed at the magnetic equator.
This
minimum can
be explained by the presence of a
field
22a
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SAMl AS RtPORl
21
J
OTIC UStKS
RESPONSIBLE INDIVIDUAL
R.C.Olsen
DD FORM
1473, 84
MAR
83
The
results
aligned electron distribution.
20 DISTRIBUTION/AVAILABILITY OF ABSTRACT
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ON THE CONSEQUENCES OF BI-MAXWELLIAN DISTRIBUTIONS
ON PARALLEL ELECTRIC FIELDS
by
Lewis J. Scott
Lieutenant, United States
B.S.,
Navy
New Mexico State University,
1985
Submitted in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE IN PHYSICS
from the
NAVAL POSTGRADUATE SCHOOL
^
December, 1991
11
ABSTRACT
Observations
made by
the
distribution.
results
The modeled
that the density profile is
plasma
is
of this
on the order
in
been observed
explained by the presence of a
at the equator
and the
field
agreement with some observations.
of 0.1 ^iV/m pointing
equator with greater anisotropy leading
also
a maximum
electric
confined closer to the equator for higher anisotropy
density profiles are
electric fields that result are
minimums have
and hence the
neutrality to solve for the electric potential
show
equatorially trapped
ratios.
a consequence
resultant parallel electric field arises as
occurs by integrating the particle distributions to obtain the density,
and assuming quasi
The
of
This thesis models the latitudinal density profiles and the resultant parallel
electric field that
field.
A
DE-1 spacecraft reveal the existence
These plasmas may be described by a bi-Maxwellian
equatorially trapped plasmas.
distribution function.
SCATHA and
to
at the
larger
electric
away from
field
magnetic equator.
aligned electron distribution.
in
the magnetic
strength.
This
The
Density
minimum can be
C.I
TABLE OF CONTENTS
I.
INTRODUCTION
1
II.
BACKGROUND AND THEORY
3
A.
B.
III.
IV.
THEORY
3
1.
Plasma
2.
Debye Shielding
3
3.
Plasma Parameter
4
4.
Motion
5.
Magnetic Mirror
6
6.
Geomagnetic
6
7.
Statistical Distribution
3
Definition
in
a Uniform Magnetic
Field
Field
5
7
PREVIOUS OBSERVATIONS
8
1.
P78-2 (SCATHA)
2.
Dynamics Explorer-1 (DE-1)
18
BI-MAXWELLIAN DISTRIBUTIONS
28
A.
SCATHA, DAY 179 OF 1979
B.
DE-1,
DAY
C.
DE-1,
DAY 315 OF
126
9
28
OF 1982
34
37
1983
MODEL
40
A.
BI-MAXWELLIAN
40
B.
SELF CONSISTENT ELECTRIC FIELD
44
1.
Bi-Maxwellian ions and Isotropic electrons
IV
48
2.
C.
V.
Bi-Maxwellian and isotropic ions, Isotropic electrons
COMPARISON OF OBSERVATIONS TO MODEL
CONCLUSIONS
LIST
OF REFERENCES
INITIAL
DISTRIBUTION LIST
#2
53
54
67
68
70
ACKNOWLEDGEMENT
The author
is
The author would
grateful for support of this
work by the Naval Postgraduate School
also like to thank R. C. Olsen for his invaluable assistance
wife Kelly for her support.
Thanks
to
Dr.
Don
Gurnett, University of Iowa, for
plasma wave data, from which densities were determined, and
NASA/MSFC,
for
DE/RIMS
data.
VI
and my
Dr.
DE
Thomas Moore,
I.
Most
INTRODUCTION
a plasma
of the matter in the universe exists in
The study
the region of space outside the earth's atmosphere.
environment around the earth has practical applications
of our understanding of the earth's
electromagnetic
This
true
in
plasma
the understanding of
in
environment comes from
wave measurements taken by
is
of the
propagation of electromagnetic waves
spacecraft charging, and
Much
state.
in
plasma.
particle
and
Such measurements
satellites.
provide basic information on the satellite environment, and clues about basic
plasma processes, such as wave-particle
case
of basic
interactions.
A
particularly interesting
plasma physics comes from the geophysical phenomenom known
as equatorially trapped plasmas.
Equatorially
plasmas are those
trapped
plasmas which are trapped from the magnetic mirror geometry
magnetic
These plasmas are
field.
latitude of the
of this thesis is to
We
will
distribution function,
consequence
to the
show
examine the nature
of equatorially trapped
may be described by a
that they
and explore the consequences
bi-Maxwellian
be an
such a
distribution is that there will
magnetic
field in
order to ensure quasi neutrality of the plasma.
the magnetic
integration
field
of the
observed density
line.
results allow
The density
particle
mapping
profiles
distributions.
profiles for
of the
electric field parallel
In
the
and
field distributions
plasma
distribution along
self-consistent (equatorial) particle
be developed. These
One
of this distribution.
of
work developed below,
will
a few degrees
magnetic equator.
The purpose
distributions.
typically confined to within
of the earth's
can then be obtained from the
These
profiles
will
be compared
to
a comparison between the model and the data
1
obtained along magnetic
field lines.
These
results will
be derived
state
plasma where the
particles are experiencing motion along
line,
and the density
is
collisionless yet high
Section
which
will
trapped.
II
low enough
enough so
that
a
so that
statistical
Section
III,
it
will
be shown
of
a magnetic
treatment
field will
on
be explored
in
plasma physics
that the equatorially trapped
latitudinal density profiles
Section
IV.
be
plasmas which are equatorially
described by a bi-Maxwellian distribution function.
of this distribution
to
field
is valid.
observed by the P78-2 (SCATHA) and Dynamics Explorer-1 (DE-1)
may be
a steady
may be assumed
of this thesis will present the basic theories of
be used and show examples
In
it
for
and
plasmas
satellites
The consequences
resultant parallel electric
II.
BACKGROUND AND THEORY
THEORY
A.
Plasma
1.
Definition
A plasma
is
a collection
of discrete ionized
The
and
neutral particles, which
physical dimensions of the
plasma must
has overall
electrical neutrality.
be large
comparison with a characteristic length X D called the Debye
The
than
in
number
total
of
charges
in
a sphere with radius X D must be much greater
1.
2.
Debye Shielding
The
electrostatic potential of
V=
where
(eo
If
y
= 8.85x1
the
is
0" 12
the charge
is
electrostatic
Farads/m) and
immersed
in
a point charge q
—5—
potential,
r is
electrons.
The
potential then
11/2
ne 2
a vacuum
is
given by
eo
is
the
permittivity
constant
the distance from the point charge [Ref.
a plasma, a positive charge
becomes
47ceor
where
in
Volts
while repelling ions, and similarly, a negative charge
Xn =
length.
will attract
will attract
1].
electrons
ions and repel
k
is
Boltzmann constant
the
temperature, which
density
equilibrium
(e
= 1.6xl0 -19
potentials
is
in
=
{k
a measure
of
the
Coulombs).
1
.38x1
0" 23
average
of the
plasma,
and
has
the
This
Tg
Joules/K),
e
effect
a plasma. The electron temperature
charge
used
in
electron
n
out
a surplus or a
deficit of
the
electric
the definition of X D
because the electrons are more mobile than the ions and do most
shielding by creating
is
electrons
of
screening
of
is
the
energy,
kinetic
the
is
is
negative charge.
of the
[Ref. 2]
Plasma Parameter
3.
order for a collection
In
of
collisionless plasma, three conditions
be much less than the dimensions
ionized
must be
of the
particles
The Debye
satisfied.
plasma.
be considered a
to
The number
length must
of particles in
a
enough number
of
Debye sphere defined as
must be much greater than
particles for
Debye
for
a
must be
collisionless
Typical
considered
in
10~9
plasma.
.
Thus the
low.
the
for
thesis,
is,
there must be a large
of
particles
Finally the
The plasma paramater g
is
frequency of
defined as:
2,3]
plasmasphere,
are kTg =
dimensions are on the order
g ~
that
plasma g*A. [Ref
values
this
,
shielding to be statistically valid.
collisions of particles
and
1
1
eV,
which
n = 1x10 6
is
m-3
1000 km. This gives X D =
considered
may be
the
,
7.4
region
and the
m,
ND
to
be
typical
~ 10 9 and
,
treated as a collisionless
4.
Motion
A
The
in
a Uniform Magnetic Field
collisionless
individual
charged
plasma
behave as a
will
move
particles will
applied electric and magnetic fields.
from the particle
collection of individual particles.
For space applications the fields resulting
may be
motion are often small and
particularly true for the
magnetic
field,
moving
acting on a charged particle
determined by the
trajectories
in
in
neglected.
The
less so for the electric field.
a combined
P and
£? field is
This
is
force
given by the
Lorentz equation:
P=q(P = Vxg)
where P
V
the Lorentz force, and
is
For the case where
particle will
P
=
0,
the particle velocity.
is
and the magnetic
field is
uniform, a charged
execute simple cyclotron gyration with a frequency
m
and radius
„
_ vperp
where mand v are the mass and
angle a
is
magnetic
field.
The
line.
[Ref. 3]
\q\B
IQIB
velocity of the
velocity parallel to the
a guiding center which
The
mvsina
charged
particle
the angle between the velocity vector of the
not affected by the magnetic
Vpar.
mvperp
is
field.
magnetic
particle
field line,
v^
pitch
and the
= vcosa,
is
This motion describes a circular orbit about
travelling along the
trajectory of this particle
and the
is
a
magnetic
helix with
it's
field line with velocity
axis parallel to the field
Magnetic Mirror
5.
The magnetic moment
of
a gyrating
particle is defined to
be
2B
If
we now
consider a particle gyrating
the magnetic force
of the orbit
the direction of lower B.
in
remains
invariant.
v^^ must
the parallel
becomes
Thus as the
strong
enough
particle
a stronger magnetic
v^
must then decrease.
the plane
field line
to
line,
a strong B
\i
field
If
the magnetic
the particle
will
field
be reflected
a magnetic mirror where there
in
region on either side of a
weak
is
field region.
Field
The geomagnetic
at the
weak
field,
energy must also remain constant,
total
become zero and
will
along the the
is
from a
particle travels
of velocity
is in
the particle travels along the field
may become trapped
field
Geomagnetic
6.
As
Since the
increase.
component
Thus a
back.
One component
have two components.
will
about the guiding center, and the other
pointing
region
a slowly converging magnetic
into
field
may be represented as a magnetic
center of the earth. The strength of the magnetic
i^ (1+3sin
e =
2
3
X)
field is
dipole fixed
given by
i/2
Anr
where u
earth, X
earth.
is
is
the permeability constant,
the geomagnetic latitude and
The equation
RE
distance,
is
in
the magnetic dipole
r is
=
r
2
cos X =
the radius of the earth, and L
earth
is
moment
of the
the distance from the center of the
of the field lines is
r
where
M
radii,
at
which that
LRE cos 2 X
is
a
label for
field line
6
a
field line,
equal to the
crosses the magnetic equator.
Figure
now
is
1
a schematic representation
magnetic
of the
field lines [Ref. 2].
If
we
define the strength of the magnetic field at the equator to be
B
r
4k(LR e Y
the strength of the magnetic
then given by
field is
8
B =
(1
2
+
3sin X)
1
*
6
cos X
The magnetic
of the earth,
field
which
is
The distance s along the magnetic
equator forms a natural magnetic mirror.
field line,
defined to be
s =
Figure
7.
.
sinX(1
2
-i-
3sin X)
1/2
magnetic
latitude
by
1
sinrr (V3sin?i)
t
2V3
2
Magnetic Field
lines for
a dipole
[Ref. 2].
Statistical Distribution
When
number
at the equator, is related to the
LRE
1
stronger at the poles than at the
dealing with
of individual particles
a system which
it
becomes
is
composed
of
a very large
impractical to solve the equations of
motion for the system.
may be
Instead the particles
the use of the distribution function.
treated statistically through
Classically the distribution function gives the
and momenta
probability distribution of the values of the coordinates
particles.
The density
of particles in coordinate
momentum
the distribution function over
n
where
and
n
is
=J
the density, and f(fy)
is
space
is
obtained by integrating
or velocity space.
f(f,V)d
3
of the
That
is:
V
the distribution as a function of position
velocity integrated over three dimensional velocity space.
The two
distribution
functions of interest
this
in
thesis are the
bi-
Maxwellian, and the Maxwellian or isotropic distribution functions, given by
fbi
m
2nkT,perp
2nkT,par
= n
fiso
where
and
B.
T^
and
T^
= n
m
1/2
m
" 'poip
1 3/2
m
2nkT
e
" 'par )
mv2
~2kT
are the characterise temperatures
parallel directions with respect to the
magnetic
in
the perpencicular
field line.
PREVIOUS OBSERVATIONS
Observations
(DE-1) spacecraft
made by
at
the
the P78-2
earth's
equatorially trapped plasmas.
A
(SCATHA) and
the Dynamics Explorer-1
magnetic equator reveal the existence of
description of the spacecraft instrumentation
and environment along with several examples
presented here.
8
of
observed distributions are
1.
P78-2 (SCATHA)
SCATHA
The
spacecraft
synchronous, near earth
satellite is
and
in
UCSD
approximately
the orbital plane.
SC-9 Auroral
spacecraft
in
The body
orbit.
and diameter
with a length
particles
1
The
.75 meters.
experiment shown
in
a near
shape
spin rate of the
to the earth-sun line
in
relationship to the
the
SCATHA
Figure 2.
in
5 Electrostatic Analyzers (ESA's),
of
two Rotating Detector Assemblies (RDA's). Each
contains a pair of ion and electron ESA's, and can be rotated through 220
degrees providing measurements
for detecting ions, is
a high energy
The other
RDA
of particle flux at various angles.
mounted
in
SC-9 experiment
configuration of the
is
1979
of
The observations discussed here were taken from
four of which are contained
ESA,
January
an axis perpendicular
with
The SC-9 experiment consists
RDA
in
of the satellite is cylindrical in
of approximately
rpm
1
was launched
(HI) detector
the Fixed Detector
is
shown
(LO detector) and the
Figure 3 [Ref.
1
The energy
at
resolution at
1
4].
eV
One RDA
to 81 keV.
(FIX detector) are low energy
e V to 2 keV.
64 exponentially spaced energy
in
seconds, with an option to dwell
seconds.
FDA
fifth
Assembly (FDA). The
covering an energy range from
detectors covering an energy range from
the energy range
in
The
These detectors scan
levels over
a period
of 16
energy levels from 2 to 128
specific
each step
is
approximately 20 percent.
[Ref. 4,5]
On day 179
axis of the satellite,
direction.
The
HI
of
1979 the
LO
detector
detector, which
line
was used
is
parallel to the spin
and the HI and FIX detectors were looking
and FIX detectors provide
LO
was parked
in
the radial
pitch angle distributions, while the
looking approximately perpendicular to the magnetic field
to provide
energy
distribution information.
Figures 4 and 5
show
Figure
2.
SC-9 Auroral
P78-2
particles
SCATHA
10
experiment on
spacecraft.
[Ret. 4]
-O
E
a>
«/>
to
<s
o
2-+ -
cr
ai
rO
UJ
^
o
oo
c
QJ
O
5
•
o
_J
o
o
cc
J3
6
a>
</5
i/>
<r
>^
o C7>
o a>
c cnj
•
a>
UJ CD
-= OO
c
31
**
O
UJ
-o
o
^
o
<->
a>
CT>
o
o 5" en
CJ
a O)
c oo
O
O
cr
5
Figure
3.
SC-9 Auroral
Particles experiment.
11
[Ret. 4]
EQUATORIAL ION PITCH ANGLE DISTRIBUTION
DAY
179 OF 1979
21:37
TO 21:41 UT
_
£
ios
§
*
103
TIME
(SUNj
10
45
90
135
PITCH ANGLE
Figure
4.
Ion Pitch
Angle
225
180
Distribution,
12
270
315
(°)
SCATHA Day
179
of 1979.
[Ref. 5]
36(
EQUATORIAL ELECTRON PITCH ANCLE DISTRIBUTION
DAY
OF 1979
HI DETECTOR
179
10?
TIME
523 eV
:38:05
TO
21:39:0
CRX100
103
160
180
225
315
270
PITCH
Figure
5.
ANGLE
Electon Pitch Angle Distribution,
13
45
360/0
(
90
°)
SCATHA Day
179
of 1979.
[Ret. 5]
the equatorial ion and electron pitch angle distributions as observed by the
SCATHA
These data are taken
spacecraft on day 179 of 1979.
time, L = 5.5, at the magnetic equator.
The count
rate
is
plotted as
degrees corresponding
The count
to looking earthward.
rate
electron distribution
near
This indicates that there
Figure
6.
The
90°.
The
The
inset
in
character of the plasma.
and
will
of
in
greater detail
41
plasma seen by the
in
Section
SCATHA
pitch
angle,
A mass
equatorially trapped.
these
show evidence
data
show a
of
again
a field-aligned
characteristic
for pitch
portion
angles
the
of
a non-thermal
will
III.
an
of
These data were
H+
the
particles
The lowest energy
.
distribution,
in
the
as well
width
(Fig. 7).
of
the
This characteristic of the data
may be
be pursued below.
14
r*
Both Figures 7 and 8
that
indicating
decrease
Figures 4,5,7,and 8 suggests that the data
This
energy
analysis of these data with the Light Ion
distribution with increasing energy.
distribution function.
distributions are
Figures 7 and 8 are plots of the log 10
Spectrometer shows that the ions are primarily
(7.25 eV)
to the
spacecraft.
versus pitch angle for ions and electrons respectively.
90°
component
1979 provide a second example
of
taken at 2100 local time, and L =5.3.
at
eV
41
This data has been previously examined by Olsen
The data from day
show peaks
The
peaks
flattening out of this portion indicates
be examined
equatorially trapped
a function
at pitch
addition to
in
shows the low energy
figure
this
and 180°,
and electron energy
plotted as
is
r"
at 0°
also a field aligned
is
equatorial ion
log 10
The
distribution for ions.
[Ref. 5]
shows peaks
Figure 5
in
electron distribution.
in
a function
peaked
is
angles of 90° and 270° indicating equatorially trapped distributions.
shown
local
through 360 degrees, with pitch angles greater than 180
of the pitch angle
at 90°/270°.
1000
at
fitted
are
Mass
ions
All of
trapped
shown
in
with a bi-Maxwellian
I
I
I
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EQUATORIAL DISTRIBUTION FUNCTIONS
DAY 179 OF 1979
10
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21 :41
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ENERGY
Figure 6.
•
1
400
1
1
800
(eV)
and Electron Energy Distribution,
179 of 1979. [Ret. 4]
SCATHA Day
15
o
X
•
4
«31
1
1000
!
10-
o,o
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DET.
FIX
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SCATHA/UCSO
DAY 41 OF 1979
06:59 - 07:08 UT
aO
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X
CD°
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a*
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X
x
.
a#
a**
XX
eV
FIX DET.
22.8
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xx 3
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3£3PT-A.
s
xx
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x/x
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\<
3
3 a aa a a
° 3
-.
8.0loa
a
o
102 eV
HI
7.5j-
DET
r
«»
t
7.0I-
•
»
6.5
343 eV
HI
DET
i»
6.0
i
5.3-
20
Figure 7.
40
60
80
PITCH
ANGLE (DEGREES)
100
Ion Pitch Angle Distribution,
16
120
140
SCATHA Day
160
41 of 1979.
4.5
21eV
SCATHA/UCSD
DAY 41 OF 1979
06:59 - 07:08
ELECTRONS
HI DETECTOR
IS
<§
n
102eV
T
2.01l
°
1
5
342 eV
1.0
-*..-.•*
•«V
0.5
1050 eV
20
40
60
80
100
120
140
160
180
PITCH ANGLE (DEGREES)
Figure
8.
Electon Pitch Angle Distribution,
17
SCATHA Day
41 of 1979.
Dynamics Explorer-1 (DE-1)
2.
The Dynamics
1981
3,
Re
in
an
elliptical
geocentric.
(DE-1) spacecraft
Explorer-1
polar orbit with an
The general shape
with a diameter of approximately
1
DE-1 spacecraft and the location
The RIMS instrument
sensor)
is
m.
.4
The
mounted
RE and
a
spin rate
perigee of
1.1
1
meter long polygon
is
6 seconds, with an
The data analyzed here were taken using
Mass Spectrometer (RIMS)
the Retarding Ion
of 4.9
of the spacecraft is
axis perpendicular to the orbit plane.
(radial
apogee
was launched on August
of the
RIMS
Figure 9
instrument.
detector on
it.
shows the
[Ref. 6.7]
consists of three sensor heads.
One head
to look radially out of the spacecraft perpendicular to
the spin axis while the other 2 (±Z) sensor heads look along the spin axis.
Each
of the
sensor heads consists of a Retarding Potential Analyzer (RPA)
followed by an Ion
to
50
in
the
Mass Spectrometer
Volts, providing
to
1
(IMS).
The RPA voltage sweeps from
energy analysis, while the IMS provides
32 amu range.
for
mass
analysis
Pitch angle distributions are obtained from the radial
50 eV range.
This sensor has a radial aperture of
sensor
for ions in the
±10°
the plane perpendicular to the spin axis and ±55° perpendicular to the
in
The
spin plane.
in
to
spin axis detectors have a resolution of ±55°,
the direction perpendicular to the magnetic
Lower
altitude
(e.g.,
trapped ions have been
day
1
26
of
satellite is
1
RE
[Ref 6,7]
below SCATHA) observations
made by
RIMS
the
982 provides one example
near 4.6
field.
of
detector on DE-1.
H+
1100 UT.
and He + from 0930 to 1230 UT.
taken by the radial detector with a
,
Volt retarding potential.
18
of
equatorially
The data from
an equatorially trapped plasma. The
with an equator crossing at
spin-time spectrogram for
and are looking
Figure 10
is
a
The data was
The spin-phase
is
1>
pvvi
WIRE
A
*
±L
55°
55°
HALF-ANGLE
CONE
HALF-ANGLE
CONE
±55<
K
Figure
9.
RIMS
PWI
WIRE
Detector on the Dynamics Explorer-1 spacecraft.
19
[Ret. 7]
SINDOO
NOI 001
<
CD
O
o
o
o
o
6
CN
<
en
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q tn
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ro 6
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CN
to
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00
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to or
a:
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LiJ
bJ
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oo
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en
oo
oo
en
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en od
I
0*3
Figure 10.
x
«-
+H
+9H
DE-1 Spin-time Spectrogram, Day 126 of 1982.
20
shown on
the
The
direction.
from -180° to 180° with 0° being
left
The center
along the spectrogram.
and the top and bottom
angle,
log 10
counts)
(ion
sensor
pitch angle of the
line
shown by
is
shows the
line
show
the satellite velocity
in
phase
spin
the white lines running
for 0°
pitch
for 180° pitch angle.
The
spin
phase
scaled according to the "color" bar to the
is
right
spectrogram with white being minimum and black a maximum.
we can see
spectrogram
peaked
peaked
1
90° and 270°.
1035 UT [Ref
to 2 orders of
of the
plot
H+
there
,
becomes an
which
at 90°,270°) at
rate being
line
0°,180°)
at
that for
magnitude lower than those
log 10 (count
These data were taken
at
1055
approximately 0545 LT and L=4.6.
0700 LT. High count
to
13
an RPA-time spectrogram
for the
RPA
voltage from
Figure 14
peaks
These
is
at 90°
a
figures are
line plot of
and 270°.
for
H+
50 V.
seen
is
will
may be
fitted with
be examined
in
.
Figure 11
is
a
and L =
4.6.
The data
comes
crosses the equator
at
a spin-time spectrogram from
for ot=90°
near the equator.
Figure
and He + taken by the +Z (a=90°) sensor,
These spectrograms show low energies
fluxes
and energies near the
both indicative of equatorially trapped plasmas.
the log 10 (count rate) versus pitch angle for ions with
This data
was taken
at
14 indicate equatorially trapped plasmas, and
data
H+
[Ref. 5].
satellite
and counts away from the equator and high
equator.
(a
similar with count
trapped ions at L = 4.6
The
Figure 12
rates are
to
for
local time,
of equatorially
from the data taken on day 315 of 1983.
0430
this
versus pitch angle for ions with peaks at
rate)
The second example
From
trapped distribution
equatorially
The He + data are
8].
of the
aligned distribution (a
field
day has been previously examined by Olsen
for this
is
a
is
(RAM)
0555
their
shapes also suggest
a bi-Maxwellian. The data
further detail
in
Section
21
III.
local time. Figures
for both
1 1
and
that the
days 126 and 315
o
to
——————— ————— ——— ——— —————
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Figure 11.
_l
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(8)DJ }unoD)
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u
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Ion Pitch Angle Distribution, DE-1
22
I
Day 126
of 1982.
[Ret. 8]
SlNflOO NOI 001
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Figure 12.
x
+H
+3H
DE-1 Spin-time Spectrogram, Day 315
23
of 1982.
0*0
Figure 13.
1
x
+H
+s>H
DE-1 RPA-time Spectrogram, Day 315
24
of 1982.
———————
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Figure 14.
Ion Pitch Angle Distribution, DE-1
Day 315
of 1983.
[Ret. 8]
Olsen
observed
at
[Ref. 9]
the
measurements.
has shown that there
magnetic
Figure 15
shows such a density
DE-1 spacecraft on day 296
of density
In
often a relative density
means
by
equator,
is
of
of
total
profile
electron
minimum
density
as observed by the
1983. The bottom portion of the figure
is
a
plot
versus magnetic latitude as determined from the plasma wave data.
the top portion of the figure the density has been normalized by the factor
(L/4.5)
4
to eliminate the variations in density
the satellite
observed.
magnetic
orbit.
This density profile
The trapped
latitude
is
induced by the
typical
when a
ions typically observed with
during this
orbit,
and are
mimina are observed.
26
RIMS
typically
radial
component
of
minimum
is
density
are confined to ±10°
found whenever such
Dynamics Explorer
1
100
j
in
*
(7)
c
23 October 1983
CD
Q
0139 - 0456 UT
83/296
100
10
=
L
-
4.7
5.7
E
u
>^
'en
c
3.3 -
4.7
20.7 — 21.3
j
30
i
-20
<D
R E Geocentric
Magnetic Local Time
i
-10
10
10
i
i
20
30
Magnetic Latitude (degrees)
Figure 15.
Density
Minimum
Profile,
DE-1 Day 296
of 1983.
[Ref. 9]
BI-MAXWELLIAN DISTRIBUTIONS
III.
The purpose
of
Dynamics Explorer-1 (DE-1) spacecraft
observations
(
Tperp
section
be
T^
and
with
a
that
observed distributions of
of the
the
characteristic
In fitting
temperatures
plasma may be determined.
This
present three examples of equatorially trapped plasmas which
will
fitted with
a bi-Maxwellian
may
distribution.
SCATHA, DAY 179 OF 1979
A.
Figures
previously
LO
and
16
shown
in
show
17
the
(47c3/7/dft)
eV range
of 7.2
cm
-3
gives a temperature
The
.
temperature of 55 eV and density of 2.8 cm
2 to 100
to the
cm -3
.
eV range gives a temperature
100
to
the 100 to
in
fit
of
-3
.
The
fitting
in
these
of
A
fit
to the ion
25 eV and a
350 eV range gives a
to the electron
data
of 1.7
cm -3
in
.
the
The
eV and a
density of
process assumes the distributions are
isotropic.
Hence, the density determined
"density" obtained
fit
(7^)
27 eV and density
1000 eV data gives a temperature
Here, the
distributions,
These data were measured by the
Figure 6, up to 500 eV.
the 2 to 100
in
density
2.4
and electron energy
ion
detector at the magnetic equator with a pitch angle near 90°.
data
fit
show
to
bi-Maxwellian,
and the density
)
is
plasmas may be described as a bi-Maxwellian.
equatorially trapped
the
P78-2 (SCATHA) and the
examining data from the
in
fits is
these
fits
of 251
does not give the
true density.
therefore designated {AndnidO).
are consistent with those reported by Olsen.
28
[Ref. 5]
These
The
results
-
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Figure 16.
Ion
Energy
1
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co
Distribution Function,
29
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CO
SCATHA Day
in
179
of 1979.
T
I
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Figure 17.
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Electron Energy Distribution Function,
30
SCATHA Day
179
of 1979.
The
pitch angle distributions
and hence a more accurate
can be used
density.
The
to obtain the parallel temperature,
eV
Figure 4, were examined for the 11.2, 41 and 103
shown
pitch angle distributions,
The
pitch angle distributions,
and electrons are shown
cos
2
a
in
electron
Figure 5, were examined at 41 .3 and 523 eV.
in
Figures 18 and 19.
The
for ions
log 10 f is linearly related to
is
given by
1/21
m
m
£R>Ki perp
2nkTpar
Elog 10 (e)
Elog 10 (e)
kT
« perp
K'perp
K'perp
"'pa.
»
the plasma can be described as a bi-Maxwellian, the data
straight line,
each
and
in
with a slope proportional to the ratio of the perpendicular to the parallel
logio' =logio n
of
The
energies.
as measured by the HI and FIX detectors,
temperature. This relationship
If
shown
ion pitch angle distributions,
19.
when
the angular distribution
pitch angle distribution follow
The angular
distributions
2
plotted vs cos a.
is
a reasonably
show a
greater than approximately .05, particularly
will
The
fall
2
cos a.
along a
portion
initial
straight line in Figures
18
2
deviation from linearity for cos a
the ion data.
in
This
due
is
to the
presence of a warm isotropic background.
The
18 and 19 are the results of a bi-Maxwellian
solid lines in Figures
using the perpendicular temperature and the density obtained from the
the energy distributions
is
in
The
Figures 16 and 17.
the parallel temperature.
In
addition a
warm
only free parameter
isotropic
background
with the isotropic density being five percent of the total density.
temperatures of the ions and electrons were estimated from the
as summarized
in
appears
to
the
fit
Table
1.
be too
function flattens below
quasi-linear diffusion.
In
Figure 18, for the 11.2
high.
fitted
The
left
to
then
included
parallel
to the data,
ions the amplitude of
This stems from the fact that the distribution
20 eV, as shown
The
eV
fits
is
fits
fit
in
Figure 6. This
is
a symptom
density obtained from the energy distributions
31
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z .00D
/
_J
_l
Ld
-
-
+/
^
X
o
<
~
/+
>u >
1
?
CD
L
CO
o
o
cm
27.00
69
•
7
•
-
CO
cm
-
251.00
64
CO
°
>u >
O
q
+/
CN
O
••
°
f
II
II
I"
II
a
i
1—
c
1
1
l
l
1
1
1
1
.*
1
1
1
-*
1
1
1
1
1
41
1
1
t
1
1
1
ii'l
fO
in
dS-
1
I
I
i
i
I
i
JX
^
I
ii
i
i
mm
CM
(,w>l/t s)
Figure 19.
C
+t
0l
i
&°l
Electron Pitch Angle Distribution.
33
-
1
1
o d
(Figures
16,17)
must be corrected
account
to
anisotropy and the
for the
spacecraft potential by the factor
9<t>
n =
kT,peip
Aiidn
dft
The data suggest
that the
"V
'
perp' 'par
spacecraft potential
is
temperatures and the corrected density are summarized
TABLE
1.
Ions (11.2
k^
-3
DE-1,
DAY
Figure 20
41 eV)
is
a
Table
1.
1979
Electrons (41 .3 eV)
Electrons (523 eV)
0.7
1.6
55
0.7
126
of
Ions (103 eV)
25
(eV)
kTpar (eV)
B.
&
5.3
)
in
DENSITY AND TEMPERATURES, SCATHA
Day 179
Density (cm
The
approximately +8V.
0.6
27
251
8
19
1.4
OF 1982
plot of log 10 (count rate)
versus cos 2 a for the data taken from
10:55 to 10:56 at a magnetic latitude of -0.9°. Figure 21
is
a
taken from 10:40 to 10:41, at a magnetic latitude of -4.4°.
these figures are the angular distribution which
equatorial bi-Maxwellian with
kT^^ =
1 1
eV,
34
kT^
similar plot for data
The
solid lines in
RIMS would show
= 0.4eV, and a density
for
of
an
TT
i
i
i
i
i
i
i
I
i
i
i
i
i
i
i
i
i
i
i
i
j
n t\
i
i
1
i
i
n
I
I
M
I
I
I
I
I
I
I
I
I
I
[
oo
d
CD
CO
CD _J
d
O^
UJ
en
o
l
CO
o
U
Si!
CO
2?
00
II
1-
CN
c-
csi
-
1
d
d
CN
U
d
ii.(75
cc
x> o
«2
CD
ir <t
II
<
en
<
Id Id 0_
QKa:
'
CO
'
1
*-
_s (8}DJ
o o
cn
*n
}unoD) 0l
6o|
k
Figure 20.
DE-1 Pitch Angle
Distribution,
35
Day 126
of
1982, 10:55 to 10:56
oo
d
CO
CO
d
° S
9
7 o §
in
o
o
o
a
£
° 5 *~
2
co
o
u
I
-
CM
r
CM
°°
IO
<°
"
II
+
{
^
d
>^
O
d
> >
o o
o
r- O
1^
CM
CN
or
II
_,
UJ
to
I > UJ
o
<
or
e t <
UJ UJ 0_
Q
OT
I—
CN
d
h"
o
o d
or
_s (s}dj
}unoo) ol 6o|
(
Figure 21.
DE-1 Pitch Angle
Distribution,
36
Day 126
of
1982, 10:40 to 10:41
approximately 5 cm
latitude
where the
-3
.
The
equatorial distribution
has been mapped
measurements were taken, assuming there
particle
to
is
the
no
parallel electric field.
C.
DE-1,
DAY 315 OF 1983
Figures 22 and 23
5:25 to 5:26 for X =
8.9°.
show
An
the data from 5:55 to 5:56 at X = -0.6°
equatorial bi-Maxwellian
giving temperatures of kTpg rp =
defined,
due
10-30 cm"3
to questions
can be
20 eV, kTpar = 2.5 eV.
fitted to
.
37
these data
The density
about the detector efficiency, but
is
and from
is
poorly
approximately
i
i
i
i
i
i
|
««
i
i
i
i
i
i
i
i
i
i
i
i
i
i
I
i
i
i
i
i
i
i
i
i
i
i
i
i
|
00
d
to
-^
u
mQ
I
O
UD
CO
d
lu
°.
O
°w
d
>
>
OO
n
CD
CO
CN
tf>
c
d
>
CN CM
II
II
o
8
i
i— t—
-* .*
<
Ld Ld 0_
Q
ce a:
'
I
I
'
'
'
I
I
I
I
I
ro
I
I
I
I
I
I
I
I
I
I
I
CN
I
I
I
I
I
I
I
I
I
I
o
d
^s (apj ;unoD) 0l 6o|
Figure 22.
DE-1 Pitch Angle
Distribution,
38
Day 315
of
1982, 5:55 to 5:56
1
1
1
i
i
it
i
t
T
i
i
i
m
\
1
1
i
i
i
i
i
i
i
i
i
i
1
ii
i
i
i
i
i
i
i
1
ii
i
i
i
i
1
1
1
|
*
*
1
-
1
*
-
i *
—
1
-
i $
^
.
-
1
m
i *
0*
•
•
-
-
1
ii
CO _l
CN
1
s
(jl
in <j
r
/
/
#"
o
o
o
,—
O
U
-
/
f
*• *
—
/
-
/
**
7
-
/
>^
>
/
is
o
CO
•
Som
1
1
CO
T
CN
o^
CN
d
ZL:
S
•
*
LJ
X >o
CO oo z
~5 * <
cr
5T<
11
*%•
II
a.
/
-
i i
1
-*?
>- 1—
-¥ -¥
r**
_
/
ii
a
-
CD
/
C" <N
l_J
d
/
^
II
in
/
<r-
5
"
7
m~^ c1—
lO
m co >
^oi U
o
°°7 d
i—
±rv
1
d
•"
i
1
ii
CO
1
,
>
^2 LJo
Qd
O)
O CO
SG
CN
00
d
LJ Q_
en cr
1
1
!
i
/"*
»#i
i
i
1
i
i
i
i
i
i
i
i
i
1
i
i
i
rO
cO
.s
i
1
1
1
1
1
1
1
CN
1
1
1
1
1
1
1
1
1
1
1
1
1
o o
(apj }unoo) 0L 6o|
(
Figure 23.
DE-1 Pitch Angle
Distribution,
39
Day 315
of 1982, 5:25 to 5:26
MODEL
IV.
A.
BI-MAXWELLIAN
has been seen from the data taken by the UCSD/SC-9 and RIMS
It
SCATHA and
instruments on the
DE-1
satellites that there is
a thermal plasma
population trapped within a few degrees of latitude of the magnetic equator.
These populations may be characterized as bi-Maxwellian
want
to
examine how bi-Maxwellian
and the consequences
a bi-Maxwellian distribution
color bar to the right of the figure
Figure 25
a contour
is
10° latitude.
distribution
latitude
is
10
now added,
,
nV/m
same
away from
Figure 26
electric field.
to
to
of
increase
the
these figures that the
in
perpendicular
the
relatively
at 10° latitude.
the equator, the
a constant
If
10° latitude.
The black
initial
distribution function that
in
the middle
is
maps
mapped
away from
a
24 and
to
off of
X=
will
10° for
a
the equator,
the equator.
This
mapped
to
the low energy portion of the
has been excluded. Outside
40
to
equatorial distribution
distribution
distribution function is
portion
weak
electric field is
Figure 26 where the distribution function has been
in
The
has been mapped to
equatorial density for both Figures
the electric field accelerates the low energy ions
can be seen
plot of
for the distribution function.
distribution that
shows how the
As the
a contour
the perpendicular velocity direction as the
The
cm -3
is
latitude
magnetic equator.
as the particles travel from a
field region.
pointing
in
due
is
dropping to 4.6
not be changed.
0.1
This
of the velocity
cm -3
at the
shows the scale
plot of the
function decreases
stronger magnetic
25
Figure 24
can be seen from the comparison
It
increases.
component
space
velocity
in
magnetic
distributions evolve with
for parallel electric fields.
We
distributions.
this region the distribution
Bi
— Maxwellian
L
=
4.5
xm
=
Electric Field
n
=
200
10.00 cm'
I
I
=
ons
Distribution
0.00°
0.00^V/m
5
kTp. rp
I
I
=
Potential
kT^
5.00 eV
I
I
=
I
0.00 V
=
0.50 eV
1
1
160
-
120
-
12
1
10
-9
8
80
40
Hra|l*H"^
7
6
5
''&$&*^$mHM
E
$>!^%^f»|^H^^H
E
4
I
l1"
-40
—
3
2
^^^^^
^^t^gj
-80
-
120
-
•160
-
-200
1
-1
-2
I
I
1
1
1
-200-160-120 -80 -40
V perpendiculor
Figure 24.
1
40
1
80
i
120
i
160
200
( km /s)
Bi-Maxwellian Distribution Function.
41
en
O
Bi
— Maxwellian
L
=
4.5
=
Electric Field
n
=
=
Xm
10.00 cm"
5
Distribution
—
Ions
10.00°
0.00/zV/m
kT^
=
Potential
5.00 eV
=
k^
0.00 V
=
0.50 eV
f 12
11
10
9
8
7
6
5
E
E
U)
4
>"
3
2
-40 -
1
-1
-2
-200-160-120 -80 -40
Vp«rpendicukjr
Figure 25.
40
80
120
160
200
(kfTl/s)
c
Bi-Maxwellian Distribution Function at X = 10
42
en
O
Bi
— Maxwellian
L
=
4.5
Xm
=
Electric Field
n
=
10.00 cm"
3
=
—
Distribution
Ions
10.00°
0.10/zV/m
kT
=
Potential
5.00 eV
=
0.51
kT,™ =
V
0.50 eV
12
"
I"
11
1
1
8
7
6
E
5
en
4
3
2
en
o
1
-1
-2
-200-160-120 -80 -40
^perpendicular
Figure 26.
40
80
(km/Sj
120
Bi-Maxwellian Distribution Function at
43
X=
160
200
10° (E
=
0.1
^V/m
).
does not decrease as
function
causes an increase
X=10° when
there
consequence
of
a
an
is
away from the
the density
in
as when there
rapidly
electric field of 0.1 jiV/m
then
parallel electric field
away from
of the distribution for ions
energy
the circular sector with the white circle with
is
= 0.5 V. The
Figure 27.
work
B.
to
effect of
a constant
The density
becomes
12.8
cm -3
.
at
The
energy portion
region excluded by conservation of
in
off
equator.
to exclude the low
increase
density
electric field. This
the magnetic equator which causes an
The
the equator.
is
no
is
mv 2 /2 =
parallel electric field
g<j>.
Here
on density
is
= JEds
<|>
shown
in
This result, coupled with previous observations (Fig. 15), motivates
develop a model
for self consistent electric fields.
SELF CONSISTENT ELECTRIC FIELD
The
distributions
kl^
temperatures
shown thus
= 10-50 eV and
and kTpar = 8-19 eV
distributions,
isotropic
for
plasma population
cold isotropic background
magnetic
[Ref.
field
distributions
electric fields
may be described as
kl^
electrons.
that
for ions,
addition
the
In
11,12]
ions
[Ref.
to
that
an
ensure quasi
and electrons are
on observed
of effects of parallel electric fields
on
Whipple
neutrality
the
field
parallel
when
the pitch angle
The signature
of
to
parallel
have been previously examined
[Ref. 15]
has extended the theory
particle distributions.
44
This
of the total density.
electric
different.
particle distributions
by Croley and others [Ref. 13,14].
observed plasma
"hidden" by the spacecraft potential.
has shown
to
kTperp = 27-250,
10], that there is often cold
may comprise from 10-90%
must be present
for
is
bi-Maxwellian with
= 0.5-1.5 eV
has been shown by Olsen
it
Persson
far
He obtained an
T
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
^
o
V)
0)
0)
k_
en
<u
0>
ol
CN **
1)
c
o
it
O
6 > >
^ ^ d
n
"*
J
I
I
L
m
J
I
I
I
_l
UJ
I
I
in
CN
°
*>
II
q
o
CD
||
X
2
X &
h- i.* JC
I
I
L
o
d
'u/u
Figure 27.
Density versus Magnetic Latitude (E=0.1 n-V/m
).
expression for the electric potential as a function of latitude for arbitrary
distribution functions.
This theoretical development
the basis for the work
is
developed below.
The purpose
field
of this section is to
a magnetic
parallel to
field
line,
determine an expression for the electric
and the
given that the equatorial distribution for ions
superimposed upon a cold
is
isotropic.
The choice
isotropic
of
bi-Maxwellian, or a bi-Maxwellian
background, while the electron distribution
plasma
maintain reasonably analytic forms.
is
latidunal profile of the density,
distributions
Results from
there are regions where this approximation
motivated by a desire to
is
AMPTE
is valid.
The
for
a steady state plasma where the density
to
experience collisionless motion along a magnetic
motion
will
be analyzed
terms
in
of the
is
[Ref. 16] indicate that
solution
so low that
it
field
will
be derived
may be considered
line.
The
particle
conserved quantities along the magnetic
the total Energy and the magnetic moment.
field line, that is,
Whipple [Ref
15]
has obtained the following expression
density for an arbitrary distribution function
3/2
B
n.Mj m
OO
f
for the
species
:
«
•M E=q^B VE-O 6-Lie
where
E=
Km
<t>(s)
y W+»W+9<i>(s) =
(
rnv£^
u
mvWa
2B(s)
28(s)
is
=
Total Energy
a
moment
the electrical potential as a function of the distance s along the magnetic
field line,
and
f
+
positive (ds/dt>0)
and t~ are the distribution functions
and negative
{ds/dt<0) directions.
46
for particles
going
in
the
theorem states
Liouvilles'
plasma
constant along the particle trajectories.
is
expressed
In this
performing
required
the
must
integrations
0,
to
the
obtain
E and
and n
density,
the
of velocities to
Defining the electric potential,
these invariants of the motion.
equator to be
distribution
can be
combinations of variables which remain constant.
of
be converted from a function
first
The
constants are the invariants of the motion
case, these
function
terms
entirely in
a collisionless
that the distribution function for
p..
Before
distribution
a function
and density
of
at the
respectively, the distribution functions at the equator
,
are:
fbi
= n
r
m
m
liB Q
-
1/2
i
kT,perp
e
i
E-\iB
+
kT,par
J
ZnkTpar
J
t
and
Since
expected that there
directions,
trapped
equatorially
is,
f
kT
2%kT
particles
being
are
be symmetry between
will
+
that
3/2
m
ficn—TlQ
ISO
=
f~
=
f
,
and
this
considered
it
is
particles travelling in opposite
be assumed
will
here
for
this
model.
Therefore
3/2
If
oo
oo
f
=kB
n
dEd\i.
m
<7<HrB
the bi-Maxwellian distribution function
density
is
is
VE-(7<^-|iB
now
substituted into this equation the
given by
\iB
>
*
n
= kB 2
m
3/2
r
m
C.TIK
1
perp
-\
*
*
m
2KkTpar
1/2
OO
OO
kTp&p
"
+
'
E-)iB
kT'pot
dEd\i
Jt=0 <E= 9<HiB
-iE-qfy-^B
and a straightforward
integration yields
q<t>
n
n
kT.par
e
0)
1-7
where
Y=
1—Bo
B
.
kTperp
*•
kT.par
J
For an isotropic distribution
JL =
e~
kT
(2)
.
n
1.
Bi-Maxwellian ions and Isotropic electrons
If
may be
the condition of quasi neutrality
is
now invoked
the electric potential
solved for as a function of the distance s along the magnetic
Using subscripts
/
and e
for ions
the only species of ions present,
and electrons, and assuming
that
field line.
Hydrogen
is
we now have
00
60
kT,par
j
= e
kTa
1-Y/
Solving for $
V ln(1-Y/
1
-
+
-
kTe
k'par,
Figure 28
shows
several ratios of
this potential
kT^^
to
plotted
kT^. The
(3)
)
1
as a function
potential
is
magnetic
a maximum
with larger anisotropy ratios leading to a steeper slope.
48
of
latitude for
at the equator,
o
T
If
1
1
1
'
'
'
CO
/
'
1
o
«~
/
/
1
1
1
J
.
/
CM/
/
.
-
-
II
f
"
.
-
*
"
t—
-*
\£
'
1
-
-*
"
1
I
-
-
"*
/
-
Ld
Q
o
-
Z>
(—
(—
m
V)
S
Ul
>
.
-
en
-
T3
-
_l
<
Z
—
UJ
1—
-
O
-
Q_
-
-
1—
CN
O
-
OH
\—
—
C
D
-
o
UJ
_
_l
UJ
> >
O
q o
q
0)
cu
*
-
-
II
II
a.
1—
•
'
1
l
O
d
I
I
1
1
1
1
.
iT>
d
(s>|0A)
Figure 28.
1
1
in
Electric Potential
<f>
as a function
49
of
Magnetic Latitude.
The
Of
may now be
parallel electric field
obtained by taking the gradient
(J).
dS
from which
we
obtain
1—
/c7,pe^c,
4
cos ^sinX 3+5sin 2 X
/(T,par,
£=-
1
LRE e
t
The
29.
2
as a function
of the electric field
strength of the electric field
on the order
<j>
(3) into (1),
fields.
degree
The
shown
Figure
in
of anisotropy with
strength of the electric
of 0.1 n-V/m.
The density
for
of latitude is
related to the
is
greater anisotropy leading to stronger electric
field is
1+3sin X
1-Y/
*Te
^'par
The magnitude
1
-+-
profile
may
also be obtained by substituting the expression
which after rearranging gives
—
kT,par
j
1+-
n
kT„
'-
This density profile
a maximum
mirror.
shown
at the equator.
in
(1-Y,)
Figure 30, from which
This
is
we see
that the density
the effect of particles trapped
in
a
self consistent electric field differ
and gives the opposite trend
contrary to our goal.
consequences
in
Partly for this
of adding
an
isotropic
from that
for
the density profile.
reason,
we
background.
50
a constant
In
is
a magnetic
Greater anisotropy confines the particles closer to the equator.
result for
field,
is
=
a sense,
This
electric
this is
continue by looking at the
>'
o
—
i
1
1
1
1
I
,
,
|
1
,
I
|
i
i
ITI
1
>
o
9 o o
-'
q
a>
ij
0)
'
-
Q
1
1
//
1
/
•
j*
"
l
«/
II
UJ
1
f
n/
I*-'
i
I
//
1/
ll
hj*
j
II
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(w/At/)
Figure 29.
i
1
in
o
ppii,
Parallel Electric Field
i
i
i
o
o
oupsG
as a function of Magnetic Latitude.
51
—
i
r
i
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1
T
1
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to
CN
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1
in
d
CN
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d
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Figure 30.
Density as a function of Magnetic Latitude.
52
2.
Bi-Maxwellian and isotropic ions, Isotropic electrons
we add an
If
ion temperature
is
background and assume that the
isotropic ion
equal to the parallel temperature
of the ions (=1 eV),
isotropic
which
is
reasonable, the ion and electron density profiles are then given by
e6
kT,pat
Hi = e
1-fl
R+
l
(4)
1-Y,
e<t>
Hi = e
kT.
n<
where
R
the ratio of the isotropic to the total ion density.
is
Invoking the
condition of quasi neutrality yields
kT,par.
R+
1-R
kTe
= e
1-Y,
Solving for
<{>,
the result
is
-,
f
kipar.
r
R+
e
which upon
1
i
+
1 -"
-
kT.
.
(5)
1-Y,
differentiating to obtain the electric field is
3kTpar, \-R
£=-
kT,perp,
kT,parj
kT,par,
eLRc
1+-
kT
4
2
cos XsinX 3+5sin X
1-Y,
R+
1-fl
1-Y,
53
2
1+3sin X
The density
profile is
again obtained by substituting for $
equation
in
(5) into (4)
to obtain
1-r-l
k'par.
1+-
kTm
R+'- R
n
Figures 31
,
32 and 33 are
electric field
and density
1-7/
magnitude of the
plots of the electric potential, the
profiles
as a function of magnetic
are plotted for ratios of the isotropic to total ion density from
The
results are similar to those for
isotropic
component has the
The density and
distribution.
increases,
and the strength
however any
qualitative
The
latitude.
1
a purely bi-Maxwellian ion
to
results
90 percent.
The
distribution.
effect of lessening the effect of the anisotropic
potential
do not decrease as
of the electric field is
changes
in
as
rapidly
decreased.
the shapes of the electric
latitude
There are
not,
or
field, potential,
density distributions.
C.
COMPARISON OF OBSERVATIONS TO MODEL
Figure 34
eliminate the
shown
in
plot of the density profile
radial
density profile.
for
a
as observed by the DE-1
The density has been normalized by the
on day 315.
model
is
#2
dependence, introduced by the
The dashed
line is
satellite
the density profile which
temperatures and density determined from the
Figure 22.
density of 40
cm -3 and
These are
WT^
= 20 eV,
kT^
the density of the bi-Maxwellian
54
factor
fit
is
satellite
(Z./4.5)"
orbit,
4
to
from the
predicted by the
to the distribution
= 2.5 eV, and a
component
is
30 cm
total
-3
.
—
r
i
i
|
O
d
i
—V
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T
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"4"
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ii
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1—
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,
1
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,L_.._.
oo
d
d
l
(S)|0A)
Figure 31.
Electric Potential
as a function
55
of
Magnetic Latitude.
r
-
i
i
—r
> >
o o >
o o o
d
-'
q o
CM
V
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II
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a:
i
i
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1
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i
i
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1
in
o
d
d
(w//\t/) p|8JJ
Figure 32.
Parallel Electric Field
i
#
i
IS
o
o
DUp9|S
as a function of Magnetic Latitude.
56
r
-i
"i
—
r
i
d
1
—
i
1
r
d
ll
o
to
Ld
Q
3
X)
3
i
o
CO
>
c
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00
Ld
Q
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o o o
d
o
CM
0)
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a.
t—
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J
q
a
(—
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I
o
HJ*
I
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m
Figure 33.
J
I
I
L
o
d
Density as a function of Magnetic Latitude.
57
(c
Figure 34.
_UJO) X)|SU8Q pSZl|DLUJON
Density as a function of Magnetic Latitude, DE-1
58
Day 315
of 1983.
The observed
profile is consistent with the predicted results.
This agreement
is
obtained without any free parameters, aside from the assumption that the
electron distribution
is
isotropic.
Other measurements
typically
a density minimum
with the
predicted results.
where the density
latitudes
it
is
in
magnetic equator,
at the
A
characteristic density
increases
abruptly
[Ref.
degree
1
apparent contradiction
minimum
profile
that
is
In
12].
of
such
cases,
the
Such a density
latitude.
density
profile
is
Figure 35. Instead of invoking the condition of quasi neutrality to
determine the electric potential and
parallel electric field, the potential
solved for by substituting the given ion density
resultant electric potential
is
electrons are
still
(3).
and
known the
the potential
equation
in
more
is
approximately constant between X = -10° to 10°, at which
approximately doubles over
shown
have shown that there
of the density profile
assumed
Figure 38
is
density profile that results.
electron densities
field is
shown
profile into
Figures 36 and 37.
in
electron density profile
to
a
be
equation
may be
solved
(4).
The
Now
that
for.
the
If
isotropic the electron density profile is given
There
is
a
and
relatively large difference in the ion
The
the equator.
difference
increases with greater anisotropy ratios for the ions. This
in
by
and the electron
plot of the given ion density profile
away from
may be
the density profiles
is in
clear violation of
the principle of quasi neutrality of plasmas.
A
the
possible explanation of this
electrons are
not
is
that
when a
aligned distribution
is,
is
minimum
is
observed,
described by an isotropic distribution, but by a
Maxwellian with a parallel temperature that
temperature, that
density
the electrons are
seen by the
is
greater than the perpendicular
field aligned.
SCATHA
59
bi-
satellite
One example
on Day 179
of
of 1979.
a
field
Till
Ld
_l
Lu
o
CN
1
1
1
1
1
|
1
1
1
1
|
1
1
1
1
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I
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1
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vn
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en
Q_
2
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00
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Q
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0)
c
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o
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Q_
V
1—
.
1
1
1
l
1
in
q
csl
CN
l
l
l
l
1
l
l
i
i
i
in
i
i
i
1
in
d
i
i
i
i
o
d
>u/ u
!
Figure 35.
Typical Ion Density
60
Minimum
Profile.
m
(S}|OA)
Figure 36.
Electric Potential
as a function
of
Magnetic Latitude.
—
-1
i
1
—
r
o
n
0)
o
_
O
m
_
Lu
Q
3
o
o
c
ii
ill
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V)
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QL
CN *-
o
c
or
I—
en
o
o
Ld
_l
LU
J
O
L
J
L
(uj/Ar/)
Figure 37.
J
I
I
L
lO
Electric Field
pisij
i
i
i
oupaG
as a function
62
i
m
of
Magnetic Latitude.
o
1
1
I
1
I
1
1
1
|
I
1
1
1
1
1
|
1
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Ion
1
1
1
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in
6
}
Figure 38.
1
in
6
u/ u
!
and Electron Density as a function
63
of
Magnetic Latitude.
The
pitch
angle
distribution
for
distribution is
shown
pitch angles.
The bi-Maxwellian
temperatures
kTp^^.6 and
in
dashed
line in this figure is
profile.
There
X=10°.
Beyond
fit
field
The
Figure 39.
log 10
aligned
f
is
eV.
profile,
When
the result
that the density
good agreement between the
the
electron
2
plotted against cos a, for low
these values are used
is
shown
ion
in
in
Figure 40.
the
The
and electron densities up
because the
no longer described by a bi-Maxwellian.
minimum
of
the electron density profile for the given ion density
this latitude the analysis is invalid
distributions are
portion
to this portion of the distribution gives for the
/(Tpar =21
determination of the density
is
the
that
is
observed
electron distributions.
64
is
ion
These
associated with
to
and electron
results imply
field
aligned
i
i
—
o
_,._._
-p
o
i
4»
-•
*
*
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i
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1
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c
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1
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to
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r
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CD
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1
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0_
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c
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-i
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1
1
i
1
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CD
CN
(,uJ>|/t s )
Figure 39.
i
0l5
l
d
°l
Electron Field Aligned Distribution,
65
SCATHA Day
179
of 1979.
1
1
i
i
i
i
i
i
1
.
i
i
i
i
1
1
'
1
'
1
'
1
o
CN
1
'
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-
-
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LU
_l
L_
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in
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or
Q.
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CI)
£
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Ld
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o
(0
L.
-
C/)
.
CO
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or
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en
o
LjJ
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q
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0)
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m"
CN
0.50
>
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q
a>
CN
CO
o
II
a
i
CO
z
g
i
i
i
i
o
CN
CN
i
II
o
Ld
a.
i
1
i
i
i
i
in
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1
i
_l
LJ
C
i
i
in
-
II
1—
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t—
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z
o
II
-
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l
1
1—
1
in
ci
,
,
1
"
o
d
>u/>u
Figure 40.
Ion
and Electron Density
66
Profile (electrons field aligned).
CONCLUSIONS
V.
It
has been shown that equatorially trapped plasmas may be described by
a bi-Maxwellian
that
an
electric field
preserve quasi
at the
The
profile
the magnetic
field
must
result
order to
in
strength of the electric field that results
is
on the
density profile that results from this distribution
effect of the electric field is to
the density decreases.
density
The
to
of this distribution is
is
a
equator with greater anisotropy confining the particles closer to
The
the equator.
parallel
neutrality.
order of 0.1^V7m.
maximum
The consequence
distribution function.
A comparison
observed density
of the
by the model, shown
predicted
reduce the amount by which
Figure
in
profile to the
shows good
34,
agreement.
Data have also shown that a density minimum sometimes exists
magnetic equator with an abrupt increase
latitude
in
density at approximately ±10°
where the density approximately doubles.
the electric potential, and hence the electric
results for the
solution
when
gradient
in
potential
quasi
potential
and strength
neutrality
the density profile
If
field,
may be
with
is
calculated.
of the electric field are
was assumed,
and strength
at the
in
given
The
similar to the
exception that the
the
of the electric field are very
large at the
boundary where the density abruptly increases. The calculated electron density
profile
shows
assumed
to
neutrality
is
that the condition of quasi neutrality
be
isotropic.
If
is
violated
is
preserved up to the boundary
observed
the electrons are
the electrons are instead field aligned, quasi
in
the ion density profile.
implies the presence of a field aligned electron distribution
minimum
if
at the equator.
67
This
whenever a density
LIST
OF REFERENCES
1.
Halliday, D., and Resnick, R., Fundamentals of Physics, 3rd
John Wiley & Sons, Inc., 1988.
2.
Parks, G. K., Physics of Space Plasmas, pp. 21-24,55, Addison-Wesley
Publishing Company, 1991.
3.
Chen,
F.
ed., v. 1,
4.
5.
F., Introduction to Plasma Physics and Controlled Fusion, 2nd
Plenum Press, 1983.
DeForest, S. E., and others, Handbook for UCSD SC9 SCATHA Auroral
Particles Experiment, University of California at San Diego, Revised edition,
1980.
Olsen,
R.
C,
"Equatorially
Geophysical Research,
6.
Hoffman
v.
Trapped Plasma Populations", Journal of
86, pp. 11235-11245,
and others, Dynamics
Company, 1981.
R. A.,
Publishing
7.
ed., p. 598,
1
December 1981.
Explorer, pp. 477-499, D.
Reidel
National Aeronautics and Space Administration Report NASA TM-82484,
Instrument Manual for the Retarding Ion Mass Spectrometer on Dynamics
Explorer-1, by S. A. Fields, and others, pp. 1-5, 33-41, May 1982.
8.
Olsen, R. C, and others, "Plasma Observations at the Earth's Magnetic
Equator", Journal of Geophysical Research, v. 92, pp. 2385-2407, 1 March
1987.
9.
Olsen, R. C, "The Density Minimum at the Earth's Magnetic Equator",
press, Journal of Geophysical Research, 1 991
10.
Olsen, R. C, and others, "The Hidden Ion Population: Revisited", Journal
of Geophysical Research, v. 92, pp. 2385-2407, 1 December 1985.
11.
Persson, H., "Electric Field along a Magnetic Line of Force in a LowDensity Plasmaa", The Physics of Fluids, v. 6, pp. 1756-1759, December
1963.
12.
Persson, H., "Electric Field Parallel to the Magnetic Field in a Low-Density
Plasma", The Physics of Fluids, v. 9, pp. 1090-1098, 1966.
68
in
13.
14.
Croley, D. R., and others, "Signature of a Parallel Electric Field in Ion and
Electron Distributions in Velocity Space", Journal of Geophysical Research,
v. 83, pp. 2701-2705, 1 June 1978.
F., and Fennell, J. F., "Signatures of Electric Fields From High
Altitude Particles Distributions", Geophysical Research Letters, v.
Mizera, P.
and Low
4, pp.
311-314, August 1977.
15.
Whipple, E. C, "The Signature of Parallel Electric Fields in a Collisionless
Plasma", Journal of Geophysical Research, v. 82, pp. 1525-1530, 1 April
1977.
16.
Braccio, P. G. Survey of Trapped Plasmas at the Earth's Magnetic Equator,
Naval Postgraduate School, Monterey, California,
Master's Thesis,
December 1991.
69
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Station
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Naval Postgraduate School
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2.
Library,
3.
Professor K. E. Woehler, Code PH/Wh
Chairman, Department of Physics
Naval Postgraduate School
Monterey, California 93943-5000
4.
Dr. R. C. Olsen, Code
Department of Physics
PH/0S
Naval Postgraduate School
Monterey, California 93943-5000
5.
LT
P. G. Braccio
10 Brookside Dr.
Greenwich, Connecticut 06830
6.
Dr. E. C.
Whipple
NASA/HQ/SS
Washington
7.
Dr. T. E.
DC 20546
Moore
NASA/MSFC/ES53
Huntsville,
8.
Alabama 35812
Dr. J. L. Horwitz
CSPAR
University of Alabama
Huntsville, Alabama 35899
9.
Singh
Department of
Dr. N.
Electrical Engineering
University of Alabama
Huntsville, Alabama 35899
10.
MSP
Mr. G. Joiner, Code 11
Office of Naval Research
800 N. Quincy St.
Arlington, Virginia 22217
70
11.
LTL
J.
Scott
3805 Eunice Hwy.
Hobbs,
New
Mexico 88240
71
Thesis
S3832
c.l
Thesis
S3832
c.l
Scott
On the consequence
:es
of bi-Maxwellian
distributions on parallel
electric fields.
Scott
On the consequences
of bi-Maxwellian distributions on parallel
electric fields.