Radiation of VLF/ELF Waves from a Magnetospheric Tether
by
Maria de Soria-Santacruz Pich
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
ARCHVES
Master of Science in Aeronautics and Astronautics
OF
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUL 07 211
June 2011
© Massachusetts Institute of Technology 2011. All rights reserved.
Author............."
Department of Aeronautics and Astronautics
May 19, 2011
........
C ertified by.......................
Accepted by ........
.f.. ..
. ........................
Manuel Martinez-Sanchez
Professor
Thesis Supervisor
v ..........................................
Eytan H. Modiano
Associate Professor of Aeronautics and Astronautics
Chair, Graduate Program Committee
..
.
71
2
Radiation of VLF/ELF Waves from a Magnetospheric Tether
by
Maria de Soria-Santacruz Pich
Submitted to the Department of Aeronautics and Astronautics
on May 19, 2011, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
The high energy particles of the Van Allen belts coming from cosmic rays, solar storms, high
altitude nuclear explosions (HANEs) and other processes represent a significant danger to
humans and spacecraft operating in those regions, as well as an obstacle to exploration and
development of space technologies. The "Radiation Belt Remediation" (RBR) concept has
been proposed as a way to try to solve this problem through VLF/ELF transmissions in the
ionosphere, which will create a pitch-angle scattering of these energetic particles with some
of them falling into their loss cone, thus reentering the Earth.
The aim of this thesis is to develop an analytical model of propagation and radiation of
Electromagnetic Ion Cyclotron Waves (EMIC) from a high-voltage magnetospheric tether,
which are the waves proposed to scatter protons. The plasma is anisotropic due to the
external Earth's magnetic field and it is assumed to have sufficiently low density, temperature
and degree of ionization so that collisions and thermal velocities can be neglected.
An asymptotic analysis is developed to calculate the fields and power flux radiated by the
tether that reach a specified observation point located in the far-field region. The effect of
the antenna-plasma interaction in the far-field region is studied by adding to the conventional
triangular source current distribution along the antenna a radial current arising from the
sheath region. The near-field case and the radiation impedance are as well studied. Finally,
the results are analyzed and compared with previous models for limiting cases.
Thesis Supervisor: Manuel Martinez-Sanchez
Title: Professor
4
Acknowledgments
I would like to thank my sponsor, Fundacid La Caixa for providing me with a two year
Fellowship. I am extremely grateful to my advisor, professor Manuel Martinez-Sanchez, for
his guidance, support and friendliness. I deeply enjoyed learning from his very wide knowledge
about the physics of propulsion. I also want to thank Prof. Paulo Lozano for his support,
as well as my fellow graduate students at the Space Propulsion Laboratory - Carmen, B.J.,
Chase, Dan, Pablo, Tim, Steve, Taylor, Ryan and Rui - for being so nice to me and helpful
whenever I needed it. I am very thankful to my very good friends Jorge, Dani, Carlos, Maite,
Alessandro, Roberto and Sunny for making these years at MIT so wonderful. Above all, I
want to thank my family and specially my parents, Isabel and Joan Carles, for their support
and advice and for living all the steps of my career as intensely as their own.
I would like to dedicate this thesis to my Grandma, Pilar Pifiol Falc6; without her love
and support any of my achievements would have been possible.
6i
Contents
1
2
Introduction
17
1.1
The Van Allen Belts .........
1.2
Radiation Belt Remediation (RBR) Concept . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3
The Magnetic Mirror Effect and the Loss Cone . . . . . . . . . . . . . . . . . . . . . . . 19
1.4
The Role of EMIC Waves in the Inner Van Allen Belt.......
1.5
Literature Review
1.6
Thesis Outline
......................................
17
. .. . ..
. . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Characterization of EMIC Waves
27
2.1
Dispersion Relation in a Uniform Cold Plasma . . . . . . . . . . . . . . . . . . . . . . . 27
2.2
Evanescence, Resonance and Cutoff Frequencies . . . . . . . . . . . . . . . . . . . . . . . 33
2.3
Polarization
2.4
Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5
Importance of Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7
3
Formulation of the Radiation Problem: The Fourier Transform
3.1
Problem Definition .......
......................................
3.2
Fourier Transform of the Radiation Equation
. . . . . . . . . . . . . . . . . . . . . . .
4 Source Current Distribution
4.1
4.2
5
48
Model of the Source Current Density ......................
. . . . . . . . . . 49
4.1.1
Antenna Source Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2
Radial Source Current Density arising from the Sheath
. . . . . . . . . . . . . . 49
Fourier Transform of the Source Current Density . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1
Fourier Transform of the Antenna Source Current Density . . . . . . . . . . . . . 56
4.2.2
Fourier Transform of the Radial Source Current Density arising from the Sheath
Inverse Fourier Transform
. . . . . . . . . . . . . . . . . . . . .
5.1
The Formulation of the Inverse Fourier Transform
5.2
Integration over w
5.3
K integration: The Cauchy's Residue Theorem
5.4
Tether at an Arbitrary Angle a with respect to Bo: The Stationary Phase Method (SPM)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1
The Stationary Phase Method (SPM)
5.4.2
Integration of the Exponential Term eik
5.4.3
Integration of the Exponential Term for 0. ~ 62
<D (0) around
5.4.4
5.5
. . . . . . . . . . . . . . . . . . . ....
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
, : Third Order Expansion of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tether Aligned with
So
(a = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1
Far-Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2
Near-Field Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
57
95
6 Radiation Pattern Analysis
6.1
Radiated Fields in the Far-Field Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2
Energy Flux in the Far-Field Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3
6.4
6.2.1
General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2
Energy Flux Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Radiation Power and Resistance in the Far-Field Region . . . . . . . . . . . . . . . . . . 110
6.3.1
Integration of the Power Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2
Balmain's Model for the Radiation Impedance
. . . . . . . . . . . . . . . . . . . 110
Very Low Frequency Case with a Dipole Antenna: Comparison of Results . . . . . . . . 114
7 Conclusions and Future Work
119
7.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.119
7.2
Future Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.121
Bibliography
129
10
List of Figures
1.1
Schematic of Van Allen belts ........
1.2
Magnetic mirror configuration...
1.3
Loss cone . . . ... ..
1.4
Mirroring effect and precipitation. ...
2.1
k and Bo configuration... ...........
2.2
Kivs KII
2.3
Wavelength
. . . . . . .
ys0
....
. . .
....
18
.................................
...
...
. ..
...
. ..
. . ... ...
...
....
...
...
.
. . . . . . . .. .......
. ...
.
. . . ..
..
... 22
. . ... .
... .23
......
... ........
20
..
28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .
32
.....
....
. ...
...
....
....
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Resonance 0vsY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5
Polarization coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6
Velocity of the protons for Y=0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7
Group velocity diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Antenna, observation point 9(0,#2) and k(0,#) . . . . . . . . . . . . . . . . . . . . . . 43
4.1
Antenna related axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1
K poles in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
11
5.2
Phase vs 0 for different 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3
Real part of the integrand of E,, vs 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4
Plot of stationary points as a function of 0
5.5
Regions in the plot of stationary points for the proton branch . . . . . . . . . . . . . . . 72
5.6
K plot: Region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.7
K plot: Region II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.8
Normalized Fresnel integrals C(x) and S(x)
5.9
Dimensionless group velocity: Characteristic points . . . . . . . . . . . . . . . . . . . . . 81
. . . . . . . . . . . . . . . . . . . . . . . . . 71
. . . . . . . . . . . . . . . . . . . . . . . . . 77
5.10 Dimensionless group velocity: Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.11 Near antenna analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1
Electric field in the x-direction (radial) for Y = 0.8 and a = 0 . . . . . . . . . . . . . . . 9 6
6.2
Electric field in the y-direction (azimuthal) for Y
0.8 and a = 0 . . . . . . . . . . . . 97
6.3
Electric field in the z-direction for Y = 0.8 and a
0 . . . . . . . . . . . . . . . . . . . 97
6.4
Magnetic field in the x-direction (radial) for Y = 0.8 and a = 0 . . . . . . . . . . . . . . 98
6.5
Magnetic field in the y-direction (azimuthal) for Y = 0.8 and a = 0
6.6
Magnetic field in the z-direction for Y = 0.8 and a = 0
6.7
Electric field in the x-direction (radial) for Y = 0.8 and a = 7r/2 rad . . . . . . . . . . . 9 9
6.8
Electric field in the y-direction (azimuthal) for Y = 0.8 and a = 7r/2 rad . . . . . . . . . 100
6.9
Electric field in the z-direction for Y = 0.8 and a = 7r/2 rad . . . . . . . . . . . . . . . . 100
. . . . . . . . . . . 98
. . . . . . . . . . . . . . . . . . 99
6.10 Magnetic field in the x-direction (radial) for Y = 0.8 and a = 7r/2 rad . . . . . . . . . . 101
6.11 Magnetic field in the y-direction (azimuthal) for Y = 0.8 and a = 7r/2 rad . . . . . . . . 101
6.12 Magnetic field in the z-direction for Y = 0.8 and a = 7r/2 rad . . . . . . . . . . . . . . . 102
6.13 Electric field in the x-direction around 0,,,,t for Y = 0.8 and a = ir/2 rad (continuation
of F ig. 6.7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.14 Magnetic field in the x-direction around
2,Oit
for Y = 0.8 and a = ir/2 rad (continuation
of Fig. 6.10) ... . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 103
6.15 Projection of the Poynting vector over r for Y = 0.8 and a = 0 . . . . . . . . . . . . . . 108
6.16 Projection of the Poynting vector over r for Y = 0.8 and a = 7r/2 . . . . . . . . . . . . . 109
6.17 Projection of the Poynting vector over r for Y = 0.4 and a = 0: fundamental and second
harm onics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.18 Impedance of the antenna versus Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.19 Impedance of the antenna versus a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.20 Radiation resistance vs antenna length for a = 0 and conditions as in Table 6.1 . . . . . 114
6.21 Projection of the Poynting vector over r for Y = 0.01 for an x-oriented dipole . . . . . . 117
14
List of Tables
1.1
Environmental parameters.........
. . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1
Importance of thermal effects.......
. . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1
Input parameters ... ...
3.2
Characteristic frequencies ...........
6.1
Input parameters ..........
....
.. . ..
. . . . . . . . . . . . . . . . . . . . . . . . . . 41
. . . . . . . . . . . . . . . . . . . . . . . . . . 42
....
. . . . . . . . . . . . . . . . . . . . . . . . . . 95
16
Chapter 1
Introduction
1.1
The Van Allen Belts
The Van Allen belts are concentrations of high energy charged particles generated by cosmic rays, solar
storms, and other processes that are trapped by the magnetic mirror formed by the Earth's magnetic
field. High altitude nuclear explosions (HANEs) would inject as well large amount of energetic particles
into the radiation belts. These particles bounce rapidly back and forth between mirror points above
the Earth's atmosphere. The altitude of the mirror point of a particle depends upon the pitch angle of
the particle's velocity with respect to the magnetic field. Only those particles with pitch angles greater
than a certain level are trapped, while particles with lower pitch angles will be lost to the atmosphere.
Prior to the Space Age, the possibility of trapped charged particles had been investigated by Kristian
Birkeland, Carl Stormer and Nicholas Christofilos. The existence of the belt was confirmed by the
Explorer 1 and Explorer 3 missions in 1958, under Dr James Van Allen at the University of Iowa (18].
The higher energy particles become concentrated into three major radiation belts: a broad proton
belt with energies up to 400 MeV (L = 1.2-2, where L is the equatorial altitude of a magnetic line
in Earth radii), an inner electron belt with energies around 1 MeV and an outer electron belt with
energies ranging from 0.1-10 MeV (L = 4-5). The existence of a safe-gap between the inner and outer
.........
..
.....
electron belts indicates that there are certain L-shells that do not trap significant amount of electrons
of any energy for long periods of time or, equivalently, that there are precipitation mechanisms that
are stronger there, probably due to some resonant effects. In addition, the belts contain lesser amounts
of other nuclei, such as alpha particles. There is as well a low-energy background plasma, with much
higher density but lower energy; this plasma is quasi-neutral. The Earth's atmosphere limits the belts'
particles to regions above 700-1000 km, while the belts do not extend past 4 Earth radii. The belts
are confined to an area which extends about 65* from the celestial equator.
Van Allen belts radiation represents a significant danger to humans and spacecraft operating in those
regions, as well as an impediment to exploration and development of space. The high fluxes of energetic
particles in the radiation belts will rapidly damage electronic and biological systems. Shielding to
protect against this radiation would be extremely expensive and, even with hardening measures, the
lifetime and reliability of space systems will be limited by degradation caused by the trapped particles.
The Radiation Belt Remediation (RBR) concept is presented as a way of remediating the Van Allen
belts by the use of orbiting antennas deployed in them that will scatter the very energetic particles,
thus causing some of them to reenter the Earth.
Figure 1.1: Schematic of Van Allen belts
1.2
Radiation Belt Remediation (RBR) Concept
The energetic particles in the Van Allen belts will steadily degrade electronics, optics, solar panels,
and other critical systems and they also pose a significant threat to humans and biological systems
in orbit, thus transit times through the belts must be minimized. The presence of the high radiation
fluxes in the Van Allen belts limits long-duration manned missions to operation below 1200 km of
altitude.
For these reasons, research is being undertaken in order to have human control of the radiation belts.
Abel and Thorne [1] showed that wave-particle interactions caused by VLF transmissions may dominate
losses in the radiation belts. This fact suggested that it could be possible to have practical human
control on the radiation belts to protect the systems orbiting the Earth from natural or HANEs
injections of high energy particles. This idea has been named "Radiation Belt Remediation" (RBR).
This thesis considers a space-based RBR system, which would probably involve a constellation of
several spacecraft operating in the Van Allen belts.
1.3
The Magnetic Mirror Effect and the Loss Cone
The Earth's magnetic field configuration produces a magnetic mirror effect due to the gradient in the
magnetic field intensity. The Earth's magnetic lines constrict near the magnetic poles, thus increasing
their intensity. This results in a tendency of charged particles to bounce back close to the high field
region.
In the presence of a constant magnetic field, charged particles will describe a helical orbit (Larmor
gyrations) around the magnetic field lines. If there is not electric field, both perpendicular and parallel
energy will be conserved. However, if there is a gradient in the magnetic field strength, this will result
in a force parallel to the field lines in the direction of decreasing field. If the particles don't have
enough parallel energy, they will bounce back in the high magnetic field region.
Consider a particle of charge e in a magnetic mirror configuration which is injected with zero initial
. .........
.
.
-
- -
=--
-
---
-
-
--
-
-
--
--
-
-
- -
-
-
-
Bmax
Z
VIl
Figure 1.2: Magnetic mirror configuration
parallel velocity and is gyrating around a magnetic field line at a distance equal to the Larmor radius
rL. The force that it will experience due to the gradient in the magnetic field can be expressed as
follows
S=e 6 x r-'B
Where the magnetic field is B = Brer + Bo 0 + Bzz. The 0-component is parallel to the initial velocity
and will give no force. Thus, the force in the parallel direction will be
F iBr
(1.2)
Where it has been used that
'rL =
-=
wci
-(13
eBo
At the same time, the Gauss law for the magnetic field in cylindrical coordinates states that
1
-
V-B
=
ro9r
(rB) +
1 8
(Bo) +
r,90
8
(1.4)
-Bz = 0
OZ
Considering that OBz/Oz is a very slow varying function inside a particle orbit and taking the average
over one gyration, it can be found that
OBr
Or
/
OB
--
(1.5)
2 Oz
Substituting into Eq. 1.2, the averaged parallel force can be expressed as
(Fli)
mv 2B
_
2Bo Oz
B
(1.6)
az
(1.7)
(F 1)
Where the quantity y is called magnetic moment and is an invariant of the orbital motion
V2
(1.8)
2Bo
Thus, a particle entering a region with converging magnetic lines will experience a net parallel retarding
force given by Eq. 1.7. Depending on the particle's initial parallel energy, it may be reflected back
along the magnetic line at the turning point.
Consider now that the particle has initial parallel and perpendicular velocities given by vilo and vo,
respectively. At the turning point (TP), the parallel velocity of the particle will be zero VIirP
=
0.
Assuming weak electric fields, the total energy and magnetic moment must be conserved
1
±Q-=e
= const
=>
vO
+
V 1O= v=
TP1.9
(1.9)
I..........
2
= const
-mvI
2Bo
2
2
mvIT
0
2BTP
mo
y
-
(1.10)
2BTP
Loss
cone
Vi
Figure 1.3: Loss cone
The initial perpendicular velocity can be expressed as vIo = vo sin~o, where 0 is the initial pitch angle
of the particle as shown in Figure 1.3. Substituting this expression into Eq. 1.10, it can be derived
that
sin~o =B
=B
(1.11)
0
B
TP
According to this expression, a particle injected with an angle
6 > Oo will keep trapped into the mirror
configuration bouncing back at the turning point. On the other hand, if it is injected with
a
BTP
that is more than B 0 , or if
BTP
6 < 6o for
occurs below some altitude (around 120 km) where collisions
are important, it will fall into its loss cone and it will be lost.
All this theory can be applied to the Earth's magnetic field lines, which converge towards the magnetic
poles and expand near the equator. If the high energy particles are injected with a pitch angle larger
than the one given by Eq. 1.11, (or equivalently, their initial parallel energy is not high enough), they
will not be able to overcome the mirror forces close to the magnetic poles, thus they will bounce back
at the turning point. On the other hand, if their pitch angle is sufficiently small, they will be lost into
the atmosphere. The Van Allen Belts correspond to the magnetic field lines along which the mirror
trapping is particularly prevalent.
The idea of the Radiation Belt Remediation concept is to originate pitch angle diffusion by waveparticle interaction in order to randomly reduce the pitch angles of the particles in the belts so that
they precipitate into the atmosphere along the magnetic field lines.
Figure 1.4: Mirroring effect and precipitation
1.4
The Role of EMIC Waves in the Inner Van Allen Belt
The inner Van Allen belt extends from approximately L = 1.2 to L = 2 and naturally traps high
energy protons. Although the analysis presented in the following chapters is more general, numerical
examples will assume that the tether operates at L = 1.5 at equatorial magnetic latitude, or equivalently
at an altitude of 3000 km. The characteristics of the natural plasma environment at this location are
summarized in Table 1.4 [7].
According to Table 1.4 the environment of the inner Van Allen belt corresponds to a plasma with
very low collisionality and a strong external magnetic field imposed by the Earth, with protons as the
predominant ionic species. For this reason, collisions are neglected in the development presented in
Table 1.1: Environmental parameters
10Earth magnetic field [T]
0.4
Daytime plasma temperature [eV]
0.2
[eV]
temperature
Nightime plasma
1010
Plasma density [m-3]
3
Ion composition
[m-
Neutral density
[m-3]
]
H+ = 5.109
He+ 109
10
0+
1010
H = 1010
Neutral composition [m- 3 ]
Larmor radius [m]
Debye length [m]
Mean free path [km]
He
1010
o =109
re =0.15
r= 6.57
0.047
Ae= 135
A, = 192
the following chapters and the plasma is considered to be formed by protons and electrons with the
same density (so plasma is quasineutral).
Electromagnetic Ion Cyclotron Waves (EMIC) are the waves selected to scatter the high energy protons
trapped in the inner belt. The frequency of these waves is below the lower hybrid frequency and below
any characteristic electron frequencies, but it is of the order of the ion cyclotron frequency. This
fact will favor the transmission of energy and interaction with these energetic particles. Pitch-angle
scattering of the protons will lead to diffusion in pitch, with some of them falling into their loss cone,
thus removing them from the radiation belts. The ability of EMIC waves to scatter high energy protons
has been proved by several authors, which are detailed in the next section.
1.5
Literature Review
The spontaneous generation of electromagnetic ion cyclotron waves (EMIC) in the magnetosphere
has been investigated by different authors. These waves have been studied using observations on the
ground and in space. Erlandson and Ukhorskiy [8] reported the occurrence of EMIC waves during
magnetic storms using observations from the Dynamic Explorer 1 (DE1) and compared their diffusion
model with the observed proton flux in the loss cone. Recently, the propagation of the EMIC energy
was studied by Loto'aniu et al. [121 based on data from CRRES satellite.
Many publications can be found as well about the interaction between EMIC and the high energy
particles trapped in the belts. The pitch angle scattering of protons and electrons due to ion cyclotron
waves was derived by Lyons and Thorne [131 using a quasi-linear diffusion model.
The effect of
heavy ions on the diffusion coefficients was introduced by Jordanova et al. [9]. Albert and Bortnik
[2] considered the nonlinear response of the resonant electrons due to the interaction with EMIC
waves in a multi-ion plasma. Kennel and Petschek [101 and Cornwall et al. [6] modelled the loss
of high energy protons in the inner belt due to ion cyclotron waves and more recently, Yahnin and
Yahnina [19] summarized recent findings related to particle precipitation associated with EMIC waves.
However, the literature concerning the scattering of protons is much more limited than the one related
to electron-wave interaction.
All these papers are related to the unintentional generation of EMIC waves and to particle interaction
studies. However, no research has been undertaken concerning the propagation of electromagnetic
ion cyclotron waves radiated from a magnetospheric antenna, which is the objective of this thesis.
Special attention will be paid not only to the wave propagation, but also on the radiation problem,
which requires to deal with the antenna and its coupling to the plasma. The mathematical procedure
presented in the following chapters is based on the study of radiation of whistler waves from an
magnetospheric tether developed by Takiguchi [17].
Concerning the antenna, Bell, Inan and Chevalier [5] studied the current distribution on a VLF dipole in
a magnetized plasma and showed that, under certain conditions, it can be approximated as triangular.
The impedance of a short-dipole antenna in a magnetized plasma was modelled by Balmain [3] and
the antenna-plasma interaction was recently studied by Song et al. [15]. The formulation of these
publications has been used in this thesis to model the source current density responsible for radiation.
Kuehl calculated the analytical solution of the fields radiated from an electric dipole in a cold plasma
for very low frequencies. His results are compared with the formulation of this thesis in Chapter 6.
1.6
Thesis Outline
This chapter presented an overview of the idea and the physics related to the Radiation Belt Remediation concept, and the role of EMIC waves in the scattering of the high energy protons in the belts.
Chapter 2 develops a model of propagation of these waves, deriving the dispersion relation and analyzing its behavior. Chapter 3 sets up the problem of radiation of EMIC waves from a magnetospheric
tether, deriving the formulation in the Fourier domain. The antenna is analyzed in Chapter 4 with
the main purpose of getting analytical expressions for the source current distribution. A mathematical
methodology to find the analytical solution for the radiated fields is presented in Chapter 5. In Chapter
6 the resultant radiation pattern and antenna impedance are discussed and the model developed in
the previous chapters is compared with the literature. Finally, Chapter 7 summarizes the conclusions
and points out the path to future work.
Chapter 2
Characterization of EMIC Waves
2.1
Dispersion Relation in a Uniform Cold Plasma
The linearized system of equations formed by the momentum and Maxwell's equations is presented
next. Convective accelerations and the displacement current have been neglected as well as the contribution of ni~ (i - V-e) in the Ampere's law, where nei represents the perturbed density of the charged
particles due to the presence of the wave. Protons are the only positive charged species considered,
since they are the most abundant population in the inner belt. In order to get the dispersion relation
of EMIC waves no current sources have been considered at first. The resulting equations are as follows
= -e(E 1 +
me
t
= e(E1 + 6ili
V
x
E1 -
e,
x Bo)
x Bo)
(2.1)
(2.2)
(2.3)
........
......
.....
V x B1
= poeno(6 1 -
Vei)
(2.4)
The subscript 1 refers to the perturbed fields and it is omitted in the following derivations. The external
magnetic field Bo is taken along the z-axis and the wavenumber vector k is located at an angle 0 with
respect to Bo in the X-Z plane, as shown in Figure 2.1. The magnitude of the wavenumber vector is
defined as
27r
A
(2.5)
where A is the wavelength.
tB
0
Figure 2.1: k and BO configuration
It is assumed that the plasma responds linearly to the electromagnetic disturbances excited by the
tether source. In order to solve the equations it is convenient to express them in the Fourier domain.
Fourier analysis implicitly assumes that the field is a superposition of components of different frequencies which are stationary in both time and orientation. Taking B = Boe-z + B, k = ki- + ki1lez
k sin6 eL
+ k cosO ez and assuming a dependence of the type Re [ei(k-Wt)
=
, the linearized equations
can be expressed as follows
imewve = e(E +
-i-i
= e(k
(2.6)
x Bo)
Ve
+
x Bo)
(2.7)
k x E =wB
(2.8)
k x B = -ip.oeno(O Where E, B, dzh, d6e
(2.9)
I)
f (k, o) are the wave fields and velocities in the Fourier domain. From the
momentum equations it is possible to get the following expressions for the velocity of the charged
particles
1i
e =
-0
"
Ex + Ey
2
Bo1-
-i
BO
Ve
i-Ex + i wEy
1 -~~WC"C
= B1
eyBo
Ex + Ey1
W2
-Ex
-i
1-
1--ABo
wE.
e
Vez
-i2
e
e Ez
wine
e
Wm2
i
(2.10)
(2.11)
Where we and wc are the ion and electron cyclotron frequencies, respectively. These frequencies are
defined as follows
wJci
Wce
-
-
eBo
mi
(2.12)
eBo
me
(2.13)
The lower hybrid frequency
WLH
will appear later on in the formulation and it is defined as the square
root of the product of the cyclotron frequencies
WLH
eBo
ciD-wce
=(-
(2.14)
Combining Faraday and Ampere equations
k x k x E = -iwpoeno(i - 6e)
(2.15)
Where the velocities are given by Eq. 2.10 and Eq. 2.11.
Introducing the following quantities
Y
Wci
(2.16)
L2 =(2.17)
2
e ,pono
K = kL
(2.18)
Where Y is the dimensionless frequency, L has units of longitude and is called "skin depth" and K
is the non-dimensional wavenumber. It is possible to eliminate the velocity components and get the
following system of equations for the electric field components, which is homogeneous due to the
absence of current sources. These equations imply some simplifications of Eqs. 2.10, valid as long as
W << WLH. More specifically, they neglect the terms with w/we in the transverse motion of electrons
of Eqs. 2.10, leaving only the E x B drift.
F
/
2
g)
2- K 2 cos
Y3
JY2
EX
0
Y
Y2
_y2
1-Y21
iK 2 sin~cosO
_12
iK 2sinOcosO
0
-K2
Ey
=
(2.19)
0
-i (K2sin20 +
0
The determinant of the matrix above must be zero for the system to be consistent, which gives the
dispersion relation for EMIC waves
2
1+cos
2
K
0
2
Y2sin o
sjn40 (
\
4
2M
iY
Mi
(2.20)
2 + y 2 cog 2 g
Y2
2
2
1+cos g
2
Where M = m. The assumption w <<
2
y2sin o
2M
+
InL
y4 4
( + y2
_
(2.21)
Y 2 cog 2g
has been used. Physically, this approximation assumes
WLH
that electrons only do E/B drift in the transverse direction but it keeps their complete longitudinal
dynamics. This derivation is also presented in Appendix I using Stix's formulation [16], which leads
to the same result.
2
These expressions show that there are two possible solutions for K , named Ki and Ke
,
which
correspond to the proton and electron branches, respectively. The branch of electrons is closed and
right-handed, thus unguided. Conversely, the left-handed mode corresponding to the proton branch
has resonances for a given value of 0 close to 7r/2, or equivalently, for a given frequency at a fixed 0 (see
Section 2.2). Thus, the left-handed mode is guided along the field lines, so most of the propagation
occurs for a group velocity (Section 2.4) approximately parallel to the external magnetic field.
Considering Y 2/M << 1 (thus
W << WL),
the dispersion relation can be written as follows
K2
1+cos
2
2
±
4
sn O
4
+ Y 2cos 2O
(2.22)
However, this approximation would not be valid close to the resonance, where the movement of protons
becomes important. For this reason, Eq. 2.22 will not be used in the rest of the analysis, except when
specified.
The dispersion relation for different values of the dimensionless frequency Y is presented in Figure
2.20, where K 11 = Kcos6 and K 1 = Ksin9 have been used, which are the wavenumber vector parallel
and perpendicular to the magnetic field of the Earth, respectively. In this figure, the electron branch
is represented with a dashed line and the proton branch with a continuous line. It can be observed
that the electron branch propagates for any range of frequencies (below
WLH),
while the proton branch
only propagates for Y < 1.
From Eq. 2.5 it is possible to obtain the relationship between the wavelength and 0. This plot is
presented in Figure 2.3 for Y = 0.8 and it will be used in Chapter 3 to size the length of the antenna.
K plot
3
Proton branch
- -- - Eectron branch
2.5
Y =0.9
2
S1. 5
0.5
-0
-
Y =1.2
Z
It-Y
0
0.5
Y = 0.4
= 0.2
\
1
1.5
2
IK I
2.5
Figure 2.2: Kivs Kii
3
3.5
4
Wavelength vs 0for Y = 0.8
Proton branch
Electron branch
20 -
0
0
1
0.5
1.5
0 [rad]
Figure 2.3: Wavelength vs 0
2.2
Evanescence, Resonance and Cutoff Frequencies
Evanescence
The wave will be evanescent when Im(K) > 0 =- K 2 < 0, which corresponds to Y > 1 for the proton
branch. For the electron branch, the wave cannot be evanescent.
Evanescent waves can be important near the antenna, where they must be included when calculating
fields and imposing boundary conditions.
Resonance
The wave will be resonant for K -+ oo, where absorption may occur by the background and low
energy plasma. For the electron branch, the wave cannot be resonant. Physically, this is because
electron inertia has been neglected in their transverse dynamics, and so electrons only do E/B drift,
but not gyrations. Only the dispersion relation corresponding to the proton branch shows a resonance
condition.
If the proton mass is considered infinite like in Eq. 2.22, the resonance always occurs at
Ores
7r/2. However, this condition cannot correspond to a mechanical resonance, because the asymptote
is perpendicular to the direction of the external magnetic field. For whistler waves, the resonance
appeared due to the projection of Blo in the direction of K [17]. However, if the resonance of EMIC
waves occurs at 7r/2, there is no projection of the external magnetic field that can resonate with the
protons, thus no mechanical resonance. The mistake here is that the proton mass has been considered
infinite. This assumption cannot be made close to the resonance condition where the proton movement
matters.
For this reason, the proton mass close to the resonance must be considered as in Eqs. 2.20 and 2.21.
The Ores becomes smaller than 7r/2 rad due to the effect of the finite proton mass, which occurs at
different 0 angles depending on the value of Y, and are close to 7r/2 rad for most of the frequency
range. As Y increases, the resonant 0 decreases. The resonance condition is given by
tan 2 0Oes = M 1
(2.23)
Figure 2.4 shows the resonance condition presented above.
It is interesting to analyze the behavior of the dispersion relation around the asymptote. After a little
algebra, it is possible to observe that K goes like (cosO - cosOres)-1/ 2 around the asymptote
K
Y
~
y22
2 (1 - Y )
ysty=-1
_
C
oC
(cosO
-
cosres)
(2.24)
VI(cosO
-
coSOs~r)
The expression above shows that EMIC and whistler waves behave similarly around the asymptote,
since the dispersion relation for whisters goes like (cosO
-
W/Wce) -12.
This result could be expected
because the physics of the resonance of both waves is the same, with the the difference that whistlers
resonate with electrons and EMIC with protons.
Resonance 0 vs Y
0.8k
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Y = O)<,
Figure 2.4: Resonance 0 vs Y
Cutoff frequencies
The frequencies at which K -+ 0 are called cutoff frequencies. From the dispersion relation it is possible
to observe that there are no cutoff frequencies for these waves.
2.3
Polarization
The polarization of the wave describes the orientation of its oscillations. In order to get as much
physical meaning as possible, a new coordinate system will be used. This coordinate system is formed
by the vectors (ki,
t,
I
which are shown in Figure 2.5 and represent the direction of the wavenumber
vector k, the direction perpendicular to k (transverse) contained in the plane X-Z and the y-direction,
respectively.
In this new coordinate system, the components of the electric field will be
z
Be
jy
y
Figure 2.5: Polarization coordinates
Ek = E.sin + Ezcos6
(2.25)
Et = EscosO - EzsinO
(2.26)
From Eq. 2.19 it can be seen that
Ex
=
-i
Ez
=
-i
Y2 _-K
-
2
(1
y3
_y
2
)
Ey
K 2sin cos [y2 - K 2 (1 _ y 2 )]
y 3 (K 2 sin2o + M)
Substituting into Eqs. 2.25 and 2.26, the fields in the new coordinates can be expressed as
(2.27)
(2.28)
Ek
Et
=
=
-i (K 2 + M) 3[Y 2 2- K 2 (I _ y 2 )] Sing £
Y (K sin2o + M)
2
2
2
(1 - y )] cosO
- E3
-i [Y - K
(K2sin2 + M)
(
(2.30)
Different conclusions can be drawn from the expressions above. Eq. 2.30 shows that the projection of
the electric field on the plane perpendicular to the wavenumber vector is elliptically polarized around
k. It can be shown that it corresponds to a left-hand elliptical polarization, which is the direction of
rotation of protons under the influence of the projection of Bo on k.
At the protons resonance, K
-
oc according to Eq. 2.20. In this limit, and making use of the
resonance condition in Eq. 2.23, the expressions above simplify as follows
Ek
-+
(2.31)
oo
K-*oo
Et
vy2
-iEY
-
y
2
M + M
(2.32)
K-+oo
Thus, the electric field is mainly along the direction of the wavenumber vector or in other words, the
wave becomes electrostatic. At the resonance, E is along k and this means that B
=
0 (so k x E = 0).
For any other direction of propagation, the wave is formed by an electromagnetic component given by
Eq. 2.30 and the associated magnetic field, plus a longitudinal oscillation along k given by Eq. 2.29.
In reality, Ek will be finite so thermal effects would become important around the assymptote.
From Eqs. 2.11 it is possible to get the corresponding relations for the velocity of the protons. Figure
2.6 shows the velocity of the protons in the ik and it directions in semilogarithmic scale. This plot
clearly shows the divergence of the velocity at the resonance direction (which is
Y = 0.8).
0
re,
= 1.5397rad for
............
Vkand v vs 0 for Y = 0.8
10
10
10
-
10
103
0
0.2
0.4
0.6
0.8
0 [rad]
1
1.2
1.4
1.6
Figure 2.6: Velocity of the protons for Y=0.8
2.4
Group Velocity
The group velocity of a wave specified by
F (J',t) = f00 -00
-00
f
e
is the velocity of the constructive interference maximum of the wave. When
(2.33)
f k is a slowly varying
function, this maximum at each instant of time occurs at the point of stationary phase. The locus of
these points is characterized by the group velocity, which is defined as the gradient of the frequency
in the k space
-
_=
(2.34)
Fe k
For an homogeneous loss-free plasma, the group velocity equals the ratio between the total energy flux
and the total energy density, or in other words, the direction of energy flow in loss-free media is along
the group velocity vector. This direction is called the ray direction. In an anisotropic medium, the ray
direction is generally different from the direction of the k vector.
The group velocity can be found as follows
g =
V
Where the components
(ik,
Ow
1 Ow--
-
k ik +
8 |k|
|k| 0
i
(2.35)
iO) are indicated in Figure 2.7.
The angular difference a between V- and k can be calculated as follows
1
tana
Bw
1 &Ikl
Jk| a0
= 71-5k
aw
a|k| 0
The ray direction lies along the perpendicular to the k , 6 surface on which w is constant.
Plots for the group velocity of EMIC waves are presented and analyzed in Section 5.4.4.
Vo
zI
2B0
0x- 0= a
Figure 2.7: Group velocity diagram
(2.36)
2.5
Importance of Thermal Effects
The collisionless cold-plasma model developed so far will not be valid when thermal or collisional effects
become important. This is the case of the propagation around the asymptote, where the mechanical
resonance of ions will translate into thermal dissipation.
One way to estimate the importance of thermal effects is to calculate the range of 0 around the
asymptote where the phase velocity vph becomes three times the thermal velocity
Vth
or less. The
phase velocity is in general greater than the thermal velocity, so the wavenumber vector k is usually
small compared to
' . However, k -4 oo close to the resonance, thus thermal effects will become
important. These velocities are defined as follows
Vth =
Vph
=
2,Tj
(2.37)
Ti
m
|ki (0)}|
(2.38)
Where K is the Boltzmann constant and Ti ~ 3500 K is the temperature of the low energy protons in
the inner Van Allen belt, whose collective effect will scatter the high energy protons.
Table 2.1 shows the range of 0 where the thermal velocity becomes one third of the phase velocity.
This table shows that thermal effects will be noticeable in a very narrow band around the asymptote,
which expands with increasing frequency.
Table 2.1: Importance of thermal effects
Y
0as ym [rad]
Oth range [ra]
0.2
1.5660
1.5609 - 1.5660
0.4
1.5606
1.5570 - 1.5606
0.6
0.8
0.9
1.5533
1.5397
1.5226
1.5503 - 1.5533
1.5366 - 1.5397
1.5188 - 1.5226
0.99
1.0901
1.0477 - 1.0901
Chapter 3
Formulation of the Radiation
Problem: The Fourier Transform
3.1
Problem Definition
The following chapters present the analysis of a tether radiating EMIC waves in the plasma of the
magnetosphere. For numerical applications, the antenna is assumed to be operating at L = 1.5 at equatorial magnetic latitude. Typical input parameters for this surrounding medium and a representative
antenna are summarized in Table 3.1.
no [m-3]
1010
Bo [T]
10-5
Table 3.1: Input parameters
Io [A]
-ci [Sm-1] r [m]
L [m]
10
1.602.10-4
5-10r 2.277-103
La [m]
4-103
a [m]
0.1
Where no is the density of the plasma, Bo is the magnetic field of the Earth, r is the distance from
the antenna to the observation point, L is the skin depth given by Eq. 2.17, La is the length of the
antenna, a is the radius of the antenna, oj is the electrical Hall conductivity (defined later) and lo is
the intensity in the antenna.
The total antenna length is taken to be 4 km according to Figure 2.3, which is approximately A/2 of
the wave propagating parallel to the magnetic direction for Y = 0.8.
The characteristic frequencies of the plasma for the case being studied are presented in Table 3.1.
Table 3.2: Characteristic frequencies
wei [radis]
wce [rad/s]
wpi [rad/s]
wpe [rad/s]
957.88
1.76.106
1.32-105
5.64-106
[rad/s]
4.11 - 104
WLH
are the ion and electron cyclotron frequencies given by Eqs. 2.12 and 2.13, re-
Where wei and
We
spectively;
is the lower hybrid frequency given by 2.14, and opi and wpe are the ion and electron
WLH
plasma frequencies, respectively. The plasma frequencies are given by the following expressions
e 2n
(3.1)
eom
(3.2)
60mi
Wpe
=
0Me
The aim of the analysis presented in the following sections is to get the radiation pattern from a
long tether antenna, operating in the EMIC range of frequencies, as seen from an observation point
located in the far-field region. The region closer to the tether but outside the sheath will be studied as
well; however, it will be assumed that the observation point is far enough for the linear analysis to be
applicable. Special attention will be paid to the current distribution in the antenna and to the effect
of the sheath surrounding the tether.
Figure 3.1 presents a schematic of the situation, where the observation point is located at the coordinates if(O2 , #.) and the wave is characterized by the wavenumber vector k(O,
#).
.........
......
--
Z'BfrBo
YIB
X'B
Figure 3.1: Antenna, observation point
3.2
f*(O, #,) and k (0, 4)
Fourier Transform of the Radiation Equation
Adding the source current density
j. (which includes both, particle and displacement currents) as a
given source term to Eq. 2.9, Ampere's law can be written as
Vx $
=
e) ± po
(3.3)
[oeno(6i - ie)+ poi]
(3.4)
poeno(iU -
Combining Ampere's and Faraday's law
V xV
x
E=
As was done in Chapter 2, the linearized equations can be found by introducing a dependene of the
type Re
[ei(k.Wt)I.
The linearized equation presented above is as follows
k x k x E
=
+ poJs]e)
-iw [poeno(vi -
(3.5)
Where J, is the Fourier transform of the source current density. Substituting the velocities from Eqs.
2.10 and 2.11 and introducing the parameters defined in Eqs. 2.16 to 2.18, it is possible to find the
following system of equations for the electric field
Y
1 -y
K 2 cos2
2
t
E
-
y2
y
Y
Y
1-y22
Y
Y
Z
=
K2
+ iK 1 -y2
iK2sin0cos0
K 2sin0cosO
y
2 $+i
(K28in20 +
Jsx
i
2
Y
1
M)
. s
o-Uci
z=1$z
(3.6)
(3.7)
(3.8)
o-ci
This system can be identified with the linearized expression for Ohm's law, which can be written as
follows
71
1
[A] -E =
;
(3.9)
Js
-ci
where [A] is the same tensor that appeared in Eq. 2.19 but divided by Y. This tensor is given by
K2cos2O)
y
-
1-Y2
Y2
1-Y
2
2
Y2
-K sinOcosO
2
y
0_y2
-
K 2
Y
J-1y2
0
y
2
K sin0cos0
Y
0
S(K
2
sin2 0 +M) j
and o-ci is the Hall conductivity of the plasma
S2 0
cini
=
en0
Bo
(3.10)
Which appears in the formulation due to the magnetization of protons, represented by the Hall parameter vector
3
i given by
#-.p~
we
vi Bo
Bo
-
(3.11)
Where pi is the mobility of protons
Pi =
(3.12)
miVi
And vi is their collision frequency. The conductivity along the external magnetic field Bo is defined as
follows
0.1
e2 n0
(3.13)
-i
Ignoring the motion of protons and neutrals, the inertia of protons as well as the gradients of pressure,
it is possible to get the following expression of the Ohm's Law
.Ii=
$8
2
+
ji,=op il
$
(3.14)
(3.15)
From the expression above, two conductivities perpendicular to B 0 can be defined, the Pedersen
conductivity
orp and the Hall conductivity oH, which govern the current components parallel and
perpendicular to the electric field E, respectively. These parameters can be expressed as follows
o)\
minovi
JP=1 + #f2
A3
+I|| '
1+#
UH =
(3.16)
BO
eno
-2
Bo
(3.17)
From these expressions it can be easily seen that up << OH << o-||, the ratio being
#3i
in each
case. Thus, the Pedersen conductivity is very small in the ionosphere and the relevant term in the
perpendicular motion is the Hall conductivity.
Inverting matrix [A], the electric field in the transformed domain can be expressed as
E(k) =
[Z] Js
aci
(3.18)
where [Z] is the impedance matrix given by (in the k-related axes of Figure 2.1)
i (K 2 sin2 0 + M) (Y 2
[Z]
=
SiK
-Y
3
-
(K 2 sin2 0 + M)
2sin0cos0
2
(Y - K
Y3
K 2 + K2Y 2 )
2
+K
i (Y 2 K 2sin20 + Y 2 M - K 2 McC s20
2
Y
2
)
2
Mcos
2
K yysin0cos0
2y 3
-K
2
- K2
+ K2Y 2)
(3.19)
sin0cos0
+ K 2 y 2 + K 2 Y 2 COS 2G + K'
Y 2 cos
2
0 - K 4 cos 2O)
Y 3M
Whee = -
+ K 2y
2
iK 2sin0cos0 (Y
i (-y 4
(K 2 sin2 0 + M)
(K2
-r
Kp{ K2i-ely.
Where Kj2 and K,, are given by Eqs. 2.20 and 2.21, respectively.
(3.20)
)
Finally, the Fourier transform of the electric field needs to be expressed in the antenna related axes
(Figure 3.1) rather than in the k-related axes of Figure 2.1. By doing so, the coordinates will not
change during the computation of the inverse Fourier transform. In these axes, the antenna is on the
XB - ZB
plane at an angle a with respect to Bo, which is along the zB axis. k is oriented according
to the polar angles (6,
#).
In the development presented so far, k was expressed as a function of 0 in
the X-Z plane and Bo was already along the
ZB
axis. Thus, it is required to rotate the reference frame
an angle -# around the zB axis. The final expression for the Fourier transform of the electric field in
the antenna related axes is the following
EB(k)
o-ci
(3.21)
[ZB] JB
where
cos#
T
[ZBI
[R] [Z]
[R]
,
[R]=
sinp 0
-sin4 cos$ 0
0
cosa
JB
[Aa is
0
-sina
0
(3.22)
1
0 sina
1
0
is
(3.23)
0 cosa
and the transformed magnetic field is given by
7,
B=
kx E
(3.24)
In order to continue with the inverse Fourier transform of the fields, it is required to find an expression
for the transformed source current density Js. This study is presented in Chapter 4.
Chapter 4
Source Current Distribution
The source current density
3j
introduced in Chapter 3 can be defined as the sum of the antenna current
and the displacement current densities
Js =Ja
Where
3a
+
o
-Esh
at
is the current density in the antenna and
Esh
=
ja + jsh
(4.1)
is the electric field inside the sheath, since the
displacement current is neglegible outside this region.
The antenna driving current couples to the plasma through particle currents and a reactive radial
current through the sheath capacitance, which is identified with the displacement current above.
In the following sections both contributions to the source current density are modelled and transformed
to the Fourier domain.
4.1
4.1.1
Model of the Source Current Density
Antenna Source Current Density
The antenna source current density can be expressed as follows
(za) cos (Wot)
ja (Za, t)
Where a is the antenna radius and
Iz and
Za
Iz.
for 0 < R < a
(4.2)
are the unit vector and distance along the antenna,
respectively.
The antenna current distribution in a magnetized plasma can be approximated initially by a triangular
current distribution if the antenna is short compared to c/wo, where wo is the local angular plasma
frequency [3, 14, 5]. This ratio equals the proton skin depth L if wo = wpi. Defining an antenna lenght
La, the triangular current distribution is
Ia = I0 (1
-(
a4.3)
With this assumption, the source current density inside the wire will be
j;a(Za, t)
4.1.2
=
-
7ra2
1 - 2|za)
La)
cos (wot) z.
for 0 < R < a
(4.4)
Radial Source Current Density arising from the Sheath
Song et al. [15] modelled the antenna-sheath-plasma system in the inner magnetosphere for the whistler
range. The model is time-dependent and one-dimensional. They considered the ion density inside the
sheath as frozen at the ambient density and a negligible effect of the magnetic field of the wave (or of
Bo). The cylindrical approximation does not consider the effect of the center and tips of the antenna,
which could be important in the close-field region. The modelling of the center and tips is out of the
scope of this thesis and is left for future work.
With these assumptions, the Poisson's equation inside the sheath can be written as follows
10
08<
R
R&R OR
---
eno
(4.5)
Eo
Where 4) is the electric potential inside the sheath. Integrating the expression above, the electric field
in the sheath will be
$4)
-2r
OR
=
eR [Rs2 (t) - R 2 ] ir
2,E
0R
(4.6)
Where ir is the unit vector in the radial direction. It has been imposed that
Esh,
(R
Rs (t)) = 0,
where R. (t) represents the cylindrical radius of the boundary between the sheath and the plasma,
which is a function of time. This boundary condition was previously used by Song [15] to define the
sheath radius, and it is equivalent to saying that the net charge on the antenna and in the sheath
equals zero.
Integrating again and imposing the boundary condition 4 (R = a) =
<P (R, t) = <Da (t) - -
4)a
R2 - a 2 - 2R2 (t) Ln
(t), the potential is given by
(-)]
(4.7)
and
op
OR
which satisfies Esh, (R = R) = 0.
Imposing boundary conditions
ena
-=-- R - Rs2 (t)
2eO
R
(4.8)
ESh
(R
= a+) _ -
0R
R
IR=a+ =
(4.9)
o
Substituting Eq. 4.8, the surface charge can be expressed as follows
eno
o-r= --2a
2a
-
R (t)
(4.10)
where
U
1 laa
8t
27ra 49 Za
(4.11)
Using
(4.12)
Cos (wot)
(o1 - 2 za)l
Ia(Za, t) =
and differentiating Eq. 4.10 with respect to time, it is possible to get the following relationship
eno 2R, dR,
2
a
1 (F) 2Io
2wra
dt
(4.13)
La
or
R2 (t)
=
(0)
sin (wot)
2I
R(enowoLaa
(4.14)
where the upper sign corresponds to the upper branch and the lower sign to the lower branch.
Notice that R, (0) is simply the value of R, at t = 0, not necessarily the "steady state" sheath radius,
which has no relevance in this dynamic situation. Rather than taking R. (0) from this steady state
interpretation, the problem should be closed by imposing
#
(R,) = 0 (in addition to E (R,) = 0). Song
et al. [151 avoid this, and by using the steady state Rso instead, they find a non-zero
# (Rs), which they
interpret as "the radiation field". This is not warranted, because the model is purely electrostatic, and
should not yield radiation, only capacitance, or in general reactance information. Using
# (R,)
= 0,
from Eq. 4.7 it can be obtained that
[R2 (t)
<>a (t) - e
-
a2 - 2R 2 (t) Ln
RS(t)
(4.15)
where R, (t) is explicitly given by Eq. 4.14. From Eq. 4.15, it can be observed that
=- -
49R2
SO
#a (t)
reaches its maximum when Rs (t)
4-
Ln
a2
/)
(4.16)
a, and still from Eq. 4.15, this maximum is
In reality, some ions are captured during most of the cycle, so
allow some electron capture, but for me/mi
From Eq. 4.14, the minimum Rs
< 1, #_=
#ama_
#a_
=
0.
should be slightly positive to
0 is a good approximation.
for the upper branch happens when sin (Wot) = 1 (wot =r/2 +
27rn), and imposing this minimum is a in Eq. 4.14, then
a2 = R 2(0)-
2o(4.17)
renowoLa
which identifies R, (0). Substituting back in Eq. 4.14 then (for the upper branch),
R(
=
[1 - sin (wot)] + a2
21o
7renowoL0
(4.18)
and finally, from Eq. 4.15, in the form
,())
2E-
enoa 2
For compactness, define
a
1 - Ln
a
-2 1
(4.19)
A=
2k
enowoLa
p =(t)
W'
(4.20)
(t2
(4.21)
enoa2
For the upper branch, it can be obtained that
p (t) = [1 + A - Asin (wot)] [1 - Ln (1 + A - Asin (wot))] - 1
(4.22)
For the lower branch, the sign of sin (wot) would be reversed (180Q shift).
The potential given by Eq. 4.22 is slightly non-sinusoidal, although its fundamental harmonic is
sin (wot), in quadrature with the current (capacitive).
The sheath radius given by Eq. 4.18 varies sinusoidally, with an average value (R2)
=
3
a2 +
210
renowoL.
a 2 (1 + A), and a minimum R, = a. No "static" R, appears.
The current in the sheath is purely a displacement current (consistent with
4>m_
=
0, which ignored
charging/discharging). Since the total current is divergence-free, this means that (for the upper branch)
27rRjsh (R, t)
jsh (R, t) =
Which is consistent with
Jsh
-=
-
Io Cos (WO t)-
(
rRLa
ti,
Jocos (wot)
ir
I
for a <R < R, (t)
(4.23)
(4.24)
= C ah. The sign is reversed for the lower antenna
branch.
Finally, the source current density considering both, the driving current along the antenna and the
radial contribution arising from the sheath, can be expressed as follows
S(1
t(a
for 0
-zcosOo)z
ro cos(Wot)
cos t)=
)
cosOiZ + sinqpIy)
< R <a
(4.25)
for a < R < R, (t)
The first term is the antenna's current along the wire carrying a triangular source current distribution
and the second term corresponds to the radial displacement current in the sheath.
4.2
Fourier Transform of the Source Current Density
In order to introduce the source current density in the linearized equations it is required to compute
its Fourier transform. The Fourier transform of
J(,W) = (2
(, t) can be expressed as follows
4
a
t) e'('''-El: d 3 dt
(4.26)
Let k1 and kii be the projections of k perpendicular and parallel to the antenna, respectively. k 1
makes an angle x to the
xa
axis and X'has cylindrical components (R,
,
za) in (xa, ya, za). Because
X and '&are measured from the same arbitrary origin, only the term (v - X) matters, then
k = ki + k1 = kcosX I- + kisinx iY + k|Ii,
RcosV
k.-
E
+ Rsino E + zatz
k 1 R cos (0 - x) + k 1|za
(4.27)
(4.28)
(4.29)
(
Zat I BO
Ja
Xa
Figure 4.1: Antenna related axis
Substituting the integration volume element by d3 X = RdRdVdza, the Fourier transform is
J (k, t)
(2i
f
acos(-X)+klz.] eiwtR
(X, t)e-i[kR
dRd4dzadt
(4.30)
The Fourier transform can be separated into the two contributions due to the antenna and the sheath
J,(k, w) = Ja(k, w) + Jsh(k,W)
Where
(4.31)
00
Jak,w)
Se- iWOteiWtdt
=2iz
(2 7r)4
00
2
za
,i[kiRcos(b-X)+kiiza]
RdRdpdza
fantenna 72(
(4.32)
00
Jh(k,w)
(2)
(27r
e- WOte iWtdt
4
-
-oo
ffat
I
sh e ath
Iosig
aa)
7rRLa
ScoSV)ti
- sim/i) e-i[kRcos(b-x)+kjza|RdRd~dza
(4.33)
The following sections detail the calculation of the Fourier transform of both contributions.
4.2.1
Fourier Transform of the Antenna Source Current Density
Due to the harmonic nature of the driving current source in the antenna, it is enough to calculate the
spatial part of the integral of the Fourier Transform. The remaining terms are
Ji(Ic) = (2r)3
3 e
d
(4.34)
The integration over 0 is easily done using Bessel functions
e- ikRcos(b-X0dp
= 2,rJo (k1 R)
(4.35)
And the integral on the radius follows immediately
a
S27rJ
(kiR) R dR
0
=
J1 (k a)
0
The integration over za can be done using integration by parts it gives
(4.36)
L./2
za e-ikllzadza
(1-
k La 1
-Cos
kl
L
(4.37)
-L./2
Finally, the current distribution in the Fourier domain can be expressed as follows
a
k
(27r) 3 aLa kk 2
Where k1i = k (cosO cosa + sinO cos sina) and k
-
cos
(kji
zz.
perpendicular to the antenna, respectively. In general, aki << 1, so J1 (ak±) t
a
-k)
1 -
4Io
(kii
cos
k2La
(27r) 3
(4.38)
are the k-components parallel and
2-
=
)
,2
"z
aki, and then
(4.39)
Adding the Dirac delta term from the integration over time, the expression in the (k, w) space will be
-+ -
For long waves
4Io [1 - cos (kii
3 ) =
(2Jr)k
(27r)
k1L
L)]
6 ( - WO) Zz.
(4.40)
k112La <<1
1 )oLa
ja~,W =(27r) 3
2 .
5
(w - wo) z
(4.41)
This last expression corresponds to an elementary dipole of strength 1IoLa located at the origin.
4.2.2
Fourier Transform of the Radial Source Current Density arising from
the Sheath
The integration of the contribution of the radial current arising from the sheath can be expressed as
follows
R,(t)
0
Jsd,
=0
w)
ei(w-wo)tdt
"
(27r)4,r,
dR
J
-o
a
2r
J
La/2
cos i
.+-sin* e-ik*Rcos(O-X)d
0
e-ikllzasign(za) dza
(4.42)
-La/2
The integration over Za gives
L./2
e- *kizsign(za) dZa =
(4.43)
[1 - cos (kiiLa/2)]
-L./2
The integration over 0 gives
27r
cosiix + sin@I2,
e-ikRcos( b-x)d4 = -27ri J1 (kR) c
+
sinxy )
(4.44)
0
The integral over R is obtained by using the Bessel function properties as follows
R.,(t)
I
(t))
= Jo (k_ a) - Jo (k-.R,
J1
Jk((kiR)
RIRdR
1
(4.45)
It is just left to compute the integration over time
Jsh(k,W)
Io
[1 - cos (k11 La/2)]
47r4 La
k
.
cosx
+ sinXty
f [Jo (k-La) - Jo (kiRs (t))]
ci(u-w")tdt
-00
Assuming k 1 R <«1
and using Eq. 4.18
(4.46)
Jo (k-La)
-
JO (kiLR, (t))
ki
k-L
2
2
kLa2
[R (t) - a 2
R
k
A [1 - sin (wot)]
(4.47)
[6 (w - 2wo) - 6 (w)]
(4.48)
Where A was defined in 4.20. The integration in time will be
[1-
sin (wot)] e(---'00)dt =
27r 6 (w - wo) +
-00
The term 6 (w) is only different from zero when w
0, which corresponds to a constant in time. For
this reason, this term will not contribute to the radiated power.
Finally, the source current density arising from the sheath can be expressed as follows
Jsh(kW )
=
-
47r 4 L~ween 0 [1 - cos (k 1 La/2)] (cosX2x + sinXSY)
6 (W- Wo) +
[6 (w - 2wo) -
(w)]
(4.49)
It is possible to observe that the non-linear effects of the sheath contribution are the 2wo-harmonics
in the current spectrum. The ratio between the fundamental and second harmonics is of the order of
1/2. However, it must be kept in mind that the second harmonic of the waves most likely to interact
with the protons of the belts would not propagate because 2wo would exceed we ; however, this effect
needs to be taken into account at lower frequency ranges.
Finally, the total source current density in the Fourier domain is given by
ZI
4- Io4k/k
I-
(k, w) = Io [1 - cos (kil)]
cosX {6 (w - wo) + j [(w - 2wo) - 6 (w)]}
n sinX {6 (w - wo) +
[r
7
(-wo)
[6 (w - 2wo) - 6 (w)]}
(4.50)
Chapter
Inverse Fourier Transform
5.1
The Formulation of the Inverse Fourier Transform
The inverse Fourier transform of the fields can be calculated as
E(r,t)
=
fE(k,
(t)=
ffB(,w)e(
w)e-(k''t)dskdo
(5.1)
--wt)dskdw
(5.2)
Where from Eq. 2.8
BB(k, w) =
Introducing d3 k = k 2sinO dk dop dO and substituting the terms from Chapter 3
$(ft)
=fif
I7O~L
L3
3 [ZB] Bei(k-: wt)K2sinO dKdpd/xdw
(5.3)
The process to perform this quadruple integral is as follows
1. Integration over w
2. Fix 0,
#
and integrate on K using Cauchy's Residue Theorem
3. Integrate over the spherical angles 0, #
(a) Using the Stationary Phase Method (SPM) for a tether located at an arbitrary angle a with
respect to Bo
(b) Exact integration over
#
and SPM for 0 integration for a tether aligned with Bo (a = 0) in
the far field region
(c) Exact integration over # and numerical integration over 0 for a tether aligned with Bo
(a = 0) in the near-field region
5.2
Integration over w
The only term that depends on the frequency is the source current distribution JB. By observing Eq.
4.50, the integration over w can be expressed as the contribution of the two first harmonics of the
driving frequency. Thus, the electric field can be expressed as follows
$(i,t) =
1
(Y)
ei''+
Y=Yo
$ 2 (r)
e
(5.4)
Y=2Y
Where
$
$2(i
1
Y
=
Y=YO
Y=2Yo
=
1aL
f
ci L3
[ZB] Blei'K2 sin0dKd~d6
[ZB]I BeL
YO
(5.5)
[ZBI B2e& gK2
(5.6)
sinO dKd~d6
2 Yo
And the source current density integrated over w can be separated into the part responsible for the
first harmonic and the part responsible for the second harmonic
JB(k)
jBl1(k)
jBi(k) + JB
=
[Aa] Io 11 -
2
(5.7)
(k)
Cos k
Iok- L/k
9]
eno cs
4.Lwo
1 1
.Iok
/kI
8r 4Lawoeno coSX
JB
2
(k)
[Aa]Io
-
Cos k
(5.8)
1o-/jIsinX
L)]
wo I sin
0L
J
(5.9)
0
Where [Aa] is given by Eq. 3.23. However, for the range of frequencies of interest of EMIC waves, the
second harmonic of the proton branch will not propagate, because it corresponds to 2wo > wej. The
contribution of the electron branch to the second harmonic is much smaller than the field due to the
first harmonic, and it will be considered negligible. Thus, the approximation can be made that the
result is only due to the first harmonic
E Y=Yo e-iwot
In the following derivation, and in order to simplify the notation, JB(k) will be used instead of
(5.10)
JB1(k)
to refer to the contribution of the first harmonic.
5.3
K integration: The Cauchy's Residue Theorem
For this step of the integration process, K is allowed to be a complex variable and the integration
is performed in the complex plane using the Cauchy's Residue Theorem, although the K integration
is in principle along the real axis. With the Cauchy's Residue Theorem it is possible to calculate
the integral around a closed curve of a function that is analytic except at a finite number of isolated
singularities. This theorem is stated as follows
Let f(z) be analytic in and on a simple, closed, positively-oriented curve C except for possibly finitely
many isolated singularities inside C. Then,
N
f(z)dz = 27ri
C
Res [f;z= zn]
(5.11)
n=1
where z 1 , z 2 ... z are the locations of the isolated singularities inside C.
The first step must be to extend the range of K integration from (0, oo) to (-oo, oo). To continue
covering the full range of K, the integration over 0 should be simultaneously restricted to (0, 7r/2)
rather than (0, 7r). Both integration regions are mathematically identical, as described in Section
Appendix II.
From the dispersion relation presented in Eqs. 2.20 and 2.21, it is possible to see that there are four
possible poles for each direction of K, which are K -and K
corresponding to the two signs of the
square root. The analysis presented so far has been repeated taking into account the drag term due
to collisions, which is responsible for the imaginary part of K. The poles obtained from this complex
expression of the wavenumber vector have been represented in the complex plane shown in Figure 5.1.
The closing of the path of integration must be done in the half plane in which the term eik' does not
diverge, or in other words, the imaginary part of k- must be positive. The exponent can be written
as follows
,r
r
ik-x = i-K [sin0sin0,cos(# - $x ) + cos0cos0x] = i-Kcos-y
L
L
Where -y is the angle between k and Y, (0,
#)
(5.12)
are the coordinates of the wavenumber vector and (Ox, #,)
are the coordinates of the observation point. Two cases are possible
Im
Integration path
when cosy > 0
Re
K1
K-e
Figure 5.1: K poles in the complex plane
" cosy > 0 for 0 < 0 <
Ores.
The exponential term will be convergent for Im (K) > 0, which
implies that the integration path must enclose the poles K/ and K,.
* cosy < 0 for
7r
- 0res < 0 <wr. The exponential term will be convergent for Im (K) < 0, which
implies that the integration path must enclose the poles K,- and Ke-.
As stated above, the integration region is K : -oc to oc, 0 : 0 to 7r/2 and
4
: 0 to 27r. As it can be
easily observed from the K-plots, cosy is always positive in this domain, thus the valid poles in the
calculation of the residue are K/ and K,. Detailed explanation of the symmetry of the problem and
the limits of integration is provided in Appendix II.
Once the poles are selected, the integration over K can be performed using the Residue Theorem as
follows
2(ri
LV
]]LI:Resr
2
ZB B
[ZB] JekYL
'2 sn
dd
sinO
dkdO
(5.13)
where
Res ([ZB] fBerk 2
L6 [ZB]JBeikg-i)2
K K
(K 2 -K)
(5.14)
(K + Ke)
Ke
KK
)
Y3M~ )(K
[ZB]JBeik-Z (
2
2
(5.15)
- K2) (K + K)
Ki
The term A is given by Eq. 3.20 and it is introduced to cancel the denominator of the [ZB] matrix.
5.4
Tether at an Arbitrary Angle a with respect to B 0 : The
Stationary Phase Method (SPM)
The Stationary Phase Method (SPM) is used to integrate over both spherical angles 0 and
# when
the
antenna is at an arbitrary angle a with respect to the external magnetic field Bo, and the observation
point - is in the far field region. The method and application to the radiation problem are presented
in the following sections.
5.4.1
The Stationary Phase Method (SPM)
The Stationary Phase Method is stated as follows
Define a function f(x, t)
f (x, t) =
e*
F (() d(
(5.16)
Say that F
(()
is a slowly varying function of ( (except in the immediate vicinity of the singular
points), whereas the phase is large and rapidly varying. The integrand averages to almost zero due to
the rapid oscillations of b
occur only when <b
(()
over most of the range of integration. Exceptions to this cancellation
(() is stationary; i.e., when the phase has an extreme value. The integral can
therefore be estimated by finding places where <b (() has a vanishing derivative. Then, the integral
can be approximated by evaluating it in the neighbourhood of each of these points, and summing the
contributions of all of them. This procedure is called Stationary Phase Method (SPM).
Suppose that <b (() has a vanishing first derivative at
=
4s. In the neighbourhood of this point, <b ()
can be expanded as a Taylor series as follows
1 d2<b
Since F (() is slowly varying, the contribution to the integral from this stationary phase point is approximately
f (x, t) = F (Cs) e
I CC0) e i
d2< 4'i s(
s
_2
d(
(5.18)
In the far-field case being studied, the parameter r/L is large, and the exponential term eik'Z oscillates
much more rapidly than the other terms. However, at the stationary points the exponential term does
not vary rapidly. Near the stationary points (0s, Os) it is possible to take the other terms out of
the integral and only integrate the exponential term. Since only the vicinity of the stationary points
contributes, the exponent can be expanded in Taylor series about these points.
The exponent of the exponential term eikZ can be written as
ik
.r
r
= i K [sinOsin~xcos($ - $&)+ cosOcosOx] = ijYb
(5.19)
The phase <Dis given by
<b (0, #, OX,
sin~sin,,cos($ -
#ox) =
1+C0S20
2
2
y sin o ±
4x) +
i"| (1
cos~cosOx
+
)
2
(5.20)
2
2
+ Y cos g
Where the positive sign corresponds to the electron branch and the negative sign to the proton branch.
The stationary points of the phase correspond to extreme values of this function. These extreme values
can be observed in Figure 5.2, where the phase is represented for different values of Ox. From this plot,
it is possible to see that there exist a critical value of Ox = Ox,
This situation corresponds to
2
= 0 and
where the stationary points merge.
9 = 0 at the same time.
<bvs0forY=0.9
0
0.2
0.4
0.6
0.8
1.2
1.4
0 [rad]
Figure 5.2: Phase vs 0 for different Ox
Provided Ox is not near its critical value, the phase can then be expanded in Taylor series about these
stationary points (Os, 0s)j
_
i
9)2.1
(6-6sj
2
042
1 02q)
<by (0, 4,' 0", 4)
Where the subscript
j
= <D(Osj , Osj ) + 2 862
(0_Oj2
S,
4-4j)
(.1
refers to a particular stationary point. This second order approximation corre-
sponds to a fitting with as many parabolas as stationary points of the phase <b.
The integration over the spherical angles reduces to a summation over the stationary points of the
integration of the exponential term
$(-
=
sin6, [ZB (6,)] JB (OS)
Res
o,
lf
oo
e
(+$s
2+
KL4
eg L y
j~e
Z
Stationary Points
=0 and
The stationary points can be found by imposing that
(5.22)
d~d4
4 =0, which correspond
to the
following spherical angles
= 0
O<D
=0
=
---
Os3
Os
(5.23)
(5.24)
The calculation of the expression of Osj is tedious but pretty straightforward, and for this reason it
has not been detailed here.
These stationary points can be observed in Figure 5.3, where the ordinate axis represents the xcomponent of the real part of the integrand of the electric field for Y = 0.8, O2 = 0.1 rad and a = 0.
6
6
The stationary points for the proton branch for this case are s1 = 0.208 rad and S2 = 0.983 rad.
..........................
....
..
......
Eq. 5.24 is represented in Figure 5.4 for Y = 0.8. This equation gives a relationship between the
O2-direction of the observation point and the 0-direction of the wavenumber vector of the waves that
will propagate their information to that point. Points 1, 2, 3 and 4 can be traced in Figures 5.5, 5.7
and 5.9. Another more physical intuitive explanation of this fact was provided in Section 2.4 through
the concept of group velocity.
As it is possible to observe in Figure 5.4, two branches appear, which correspond to proton and electron
waves. For the proton branch, there is a maximum
62_
such as if 0. >
_ there would be no
proton waves reaching the observation point. The maximum 0 = res is set by the resonance condition
given in Eq. 2.23.
Integrand of Re(E,) for O,= 0.1 rad, 4 = $, and Y = 0.8
x 10,
0.2
0.3
0.4
0.5
0.6
0 [rad]
0.7
0.8
0.9
1
1.1
Figure 5.3: Real part of the integrand of E. vs 0
For 02 < 62_
but bigger than a certain value depending on the frequency, there would be a total of
three stationary points, two from the proton branch and one from the electron branch. On the other
hand, for small values of 02, there would be a total of four stationary points, three from the proton
branch and one from the electron branch. This extra proton wave that propagates to the observation
direction is due to the presence of an asymptote at 0 = Ores (Eq. 2.23).
Plot of stationary points for Y = 0.8
=0 0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
1.2
0 [rad]
Figure 5.4: Plot of stationary points as a function of 02
These different regions are highlighted in Figure 5.5, which shows the plot of the stationary points for
the proton branch. In Region I, three proton waves will reach the observation point, while in Region
II just two waves will have the correct direction of propagation. From this plot, it is possible to check
that the stationary points for 0. = 0.1 rad marked in red correspond to the case showed in Figure 5.3.
Plot of stationary points for the proton branch for Y = 0.8
0.2
2
0.180.160.14-
0.120 1---
- - -- -- -- -- -- -- - --- -- -- -- -- --
-- -- -- -- -- -- -- -- -- -+-- --
0.080.060.04Region
: Two waves4
Region I: Three waves
0.02
11
0
3
0.4
0.2
0.6
0.8
1
1.2
1.6
1.4
0 [rad]
Figure 5.5: Regions in the plot of stationary points for the proton branch
Another way to clearly visualize these different regions is through the K-plots. In these plots, the
direction of propagation of the information of the waves is perpendicular to the curve at each point.
This direction is the same as the direction of the group velocity of the waves fi , and must match the
observation direction (02,
)
Figure 5.6 shows the K-plot of the proton branch where the selected 02 falls in Region I. It is possible
to see that three proton waves will propagate at the desired angle, which are K (Osi), K (0s2) and
K ( 9s3). In this region, the two first stationary Os from Eq. 5.23 are Osi,2 =
d..
The third of the
stationary points is due to the presence of the asymptote, and it corresponds to OS3 =
4,. +
7r.
Figure 5.7 shows the K-plot of the proton branch where the selected 0. falls in Region II. In this case,
just two proton waves will propagate to the observation point, which are K (Osi), and K (0s2). In this
region, the stationary Os from Eq. 5.23 is Os1,2 = Ox.
Kplot for Y=0.8
1=
01
20
1
15
1
10
5
I ________________
0
IK I
5
II
10
15
10
15
_
-j
20
Figure 5.6: K plot: Region I
15
10
5
0
5
IKJl
Figure 5.7: K plot: Region II
20
The next step is to perform the integration of the exponential term presented in Eq. 5.22.
The
procedure is presented in the following sections.
5.4.2
Integration of the Exponential Term ezk-
Using the second order expansion of the phase <I (0) around the stationary points, the integral of the
exponential term can be expressed as follows
II e-dOdO
N
-
fieEy
3
=
[L
~
a
(O-Osj )2+1 aq2d Sj(~~
&
e
i
dO
f)
j=10
)2
(5.25)
Where N is the total number of stationary points to be considered, which can be three or four depending
on the value of O0, as detailed in the previous section.
The integrals appearing in Eq. 5.25 have been calculated in two ways, which are described next.
Asymptotic expansion
The radiated field can be described using analytic formulae from an asymptotic solution based on the
SPM. There are two asymptotic ideas here:
" Expanding near (Os, Os).
* Extending the limits of integration near(Os, Os) to ±oc.
Because of the asymptotic treatment of the problem, the solutions derived will be only valid in the far
field region, away from the antenna.
In the far field region, several wavelengths away from the antenna, it is possible to apply the SPM to
asymptotically evaluate the inverse Fourier integrals presented in Eq. 5.25. The integration region can
be set from -oo to oo, since only the vicinity of the stationary point will contribute to the integral.
These integrals can be calculated as follows
00j1
2
ry
a
~j
((_('Sj)2
______
(5.26)
d( =
r y 82
Finally, the solution for the electric field provided by the Residue Theorem for K-integration and the
asymptotic approach for the integration of the spherical angles is as follows
E
)
=
(
i E Res
sinos [ZB(Os)] B(Os)
K(S))
e
t
s,4)
StationaryPoints
2x
(5.27)
Where the expression of the residue is given by Eq. 5.14.
The magnetic field can be easily found from the linearized Faraday's relationship in Eq. 2.8, which
gives the following expression for $(r)
S
1
2ri YwZi
Lo-ci
[sinOsk(Os) X [ZB(Os)I fB(OS) (K(OS))
E
yIos,
StationaryPoints
(5.28)
1 r y~a(
2 g2D
The expressions above clearly show that with the second order fitting the fields decay like oc 1/r and
the Poynting vector goes like oc 1/r 2
Fresnel Integrals
Instead of extending the ranges to too in the far field region, the integral of the series-expanded
exponential term can be calculated exactly using normalized Fresnel Integrals (i.e., with a finite range
of integration, but still accepting a local quadratic approximation). This fact will allow to compare
the asymptotic approximation with the exact solution of the integrals, which is analyzed in Chapter
6. The normalized Fresnel Integrals are defined as follows
C(x)
Cos (2 t)
=
dt
(5.29)
sin (2t ) dt
(5.30)
0
S(x)
=
0
These integrals are presented in Figure 5.8.
After some algebra, the inverse Fourier integrals in Eq. 5.25 can be exactly calculated as follows
Ie
2
1r
y
2
I
(A~~
3
')2
d
=
2
4=0
C
[4-C
-
2A- (27r - #s
r
)
s
ps7rS
is S
A
(27r - #sj)
(5.31)
Normalized Fresnel Integrals
0.5
0
1
1.5
2
3
2.5
x
3.5
4.5
4
5
Figure 5.8: Normalized Fresnel integrals C(x) and S(x)
I
7r
ry
(0-O~j
.92,p
e
S
"
dO
-
OSj,
is - 2 s +
C2A
1 2Ar~ (7r - OS))
Where AC
~
(- v
0=0
= !.Y
2L
to a2,<0.
<
0
iiS(
. The expressions with the plus sign correspond to
(7r-
9
OS))]
(5.32)
> 0 and the minus sign
5.4.3
Integration of the Exponential Term for O ~ O2,
Expansion of <D (0) around 9
2
=
: Third Order
0
Going back to Figure 5.4, it was possible to observe that for a given value of Y there is a maximum
Ox at which the proton wave can propagate. At this critical value of 0., =
6,t
both stationary points
from the proton branch merge (Figure 5.2), thus the second order fitting does not seem to correctly
fit the phase anymore. At 62,
the stationary points merge at a point where a
In order to better approximate the integral around
6
series to the third order around the point where a
*
= 0.
~xr
62 , the phase <b can be expanded in Taylor
= 0. This point is denoted with the superscript
The remaining integrals of the expansion are presented below
ffeiY*d0d4
=
j=10
e
- +
-- (0-6**(-42)2dd4
(5.33)
0
Using the asymptotic expansion, these integrals can be calculated as follows
if
J
>
e iY
[
(-)+
1*
(0-0*)1] do = 2 [(a)
(b)]
(5.34)
dO = 2 [(a) + (b)]
(5.35)
-
0=-oo
if
-
<
0
e
-+
0=-oo
Where
Y
6
0-
(a)
=
6A
(F A kB
3A)+
B Lk
3A [2B/
[Jl13
6
(b)
FB
(
B)
L
+ J-13'
2B
B)
(B
\
[/A3A
-
K''
113
(/32B
3A
B, )A
I
-
3
B
,>
BAB>0
(2B
7A~r1/
AB>O0
3A
,/
Where J, (z) is the Bessel function of the first kind, K, (z) is the Bessel function of imaginary argument,
A =
-Y
* and B = 'Y
And the double integral can be calculated as follows
eif2o
[
*+ a
a07d*a
(0-
)](44
)
O
2dd#
(5.36)
=
2
2
It must be noted that the cross derivative disappears because it vanishes when evaluating it at 0* after
integrating over
#.
In contrast to the second order fitting, with the third order approximation the fields decay c
1/r 5 / 6
and the Poynting vector goes like oc 1/r5 / 3 . This implies an intensification of the wave in the vicinity
of 0,,,
although the effect turns out to be weak.
5.4.4
Group Velocity Analysis
The group velocity of the wave was introduced in Eq. 2.35. The dimensionless group velocity f=
in the coordinate axis
(XB, YB, ZB)
can be expressed as follows
1
ogx
Vo
-=-
wei L
V9z
-
wctL
og
-
1
weiL
O
.1
wei L (VgksinO+
vgocosO) =
1
1sin+
wei L (0 |k|
|k| 00
1
(vgkcos0 - vgosinO) =
-co
wciL
0Uo
COuk
1
-
cos6
(5.37)
sinO
(5.38)
ow
Ik| O0
The orientation of the waves that will propagate their information to the observation point must equal
the orientation of that point, or in other words
09
=
ox
(5.39)
#g
=
Ox
(5.40)
Where the orientation of V is given by (0g, 4g).
Figures 5.9 and 5.10 show the plot of (V.9,
Vz) with 0 as a running parameter. These figures can be
related to the properties of the waves described in the previous sections (see Figures 5.4, 5.5 and 5.7
for points 1, 2, 3 and 4).
At 0 = 0, the group velocity is only in the z-direction. As 0 increases, ,
point where 09 =
x,,..
increases until it reaches the
At this point only one wave propagates. This case corresponds to
a
= 0
seen in Section 5.4.3. After that, 09 decreases with increasing 0 to the point that it becomes zero,
which corresponds to the beginning of the resonance branch. At the resonance branch, 09 increases
again until the group velocity goes to zero at 0 = O,,e, where to the resonance condition given by Eq.
2.23 is satisfied. The wave does not propagate for 0 > Ores.
The regions presented in Figure 5.5 can as well be observed in the group velocity plot (see Figure 5.10).
091
corresponds to the orientation of the three waves that will propagate to Ox =
Region I. On the other hand, if Ox =
it falls into Region II.
0g2,
091,
which falls into
only two waves will propagate to that observation point, so
Dimensionless group velocity Vg for Y = 0.8
0.25
0.2
0.15
0.1 -
0.05-
0-0.005
0.015
V, [-]
Figure 5.9: Dimensionless group velocity: Characteristic points
Dimensionless group velocity Vg for Y = 0.8
0.25
0.2
0.15
0.1
0.05
01
-0.005
0.015
V9 [-]
Figure 5.10: Dimensionless group velocity: Regions
5.5
Tether Aligned with B0 (a
=
0)
If the tether is aligned with the external magnetic field, the integration over the azimuth angle
#
can
be done analytically. The integration over the inclination angle 0 is more complicated and needs to be
done numerically for the near-field case, or using the stationary phase method for the far-field case.
These integrations are described in the following sections.
In the far-field, the radial source current arising from the sheath does not have a noticeable effect
for the frequency range of interest, thus the only term considered is the assumed triangular source
current distribution of the antenna. On the other hand, in the near-field, the current arising from
the sheath needs to be taken into account; the cylindrical approximation used to model this term
neglects the effect of the tips and center of the tether, which is out of the scope of this thesis. A more
detailed analysis of the near-field region should consider the 3D effects of including those regions; for
this reason, the formulation of the near-field derived next will be analyzed keeping in mind all these
approximations.
5.5.1
Far-Field Analysis
In the far-field case with the antenna aligned with
So,
the integration over
4 can
be done analytically
and the stationary phase method can be used to solve the integration over 0 in the same way than in
Section 5.4. The source current in the antenna is taken to be a triangular distribution, which is a good
assumption according to previous studies [3, 5, 14]. The contribution to the source of the radial term
arising from the sheath is neglected in the formulation below because it does not have a noticeable
effect in the far-field region.
Analytical integration over
4
using Bessel Functions
If the tether is aligned with the external magnetic field (a = 0), the source current density derived in
Chapter 4 does not depend on the azimuthal angle, which appeared in k1l in the denominator of the
Fourier Transform of this expression (see Eq. 4.39). For this reason, the integration over # can be
done directly using Bessel functions. The integral presented in Eq. 5.13 with c = 0 can be written as
follows
Res [ZB]fBei (
if(r~'
2)} sinOdbd6
(5.41)
$>0
27ri Io
1
L L1:Res
LPoc 2,r3 L
1 - cos (kijj
)
k
[ZBI
eikxK 2 sin0dpdO (5.42)
And the magnetic field
27ri 10
1
L oci 27 3 LaYwci
0
JfI
(Res
K x
-cos (kij
k2
Z)
[ZB]
0
eik''K
2
sinOdpdxO
(5.43)
Introducing the impedance matrix in the antenna related axes [ZB] given by Eq. 3.22, the components
of the electric field can be expressed as follows
Ex
=
Res{
c:
J
Ey
=
J
Ez
=
-
Cos (-'.Kcos0)
C0820
cifK()cosecose
(5.44)
5.d44
[cos$Z13 (0) - sinpZ2 3 (0)] ei LK()sinosinecos(i)
s
'I
1
Res
1 cos
L
Kcos)cifK(6)coscoAsin
cos20
[sin4Z 13 (0) + cos&Z 23 (0)] ei K()sindsin6ecos(-
d4d
(5.45)
Cos (
Kcos0) Z33 (0) eiiK(0)cos~cosO.sin
2
()esn
COS 0
0Res{
I
K(9)sin~sin6,cos(p-p,)
dd0
(5.46)
Where [Z] is given by Eq. 3.19.
The integrals in
#
can be expressed as Bessel Functions. After some algebra, it can be shown that
27r
I1(0,
06, #$)
27risinO. J1 ( K(6)sin0sinx)
(5.47)
= 21ri coso, J1 (jK(O)sin0sinx)
(5.48)
sin#eiK(O)sinsin6.cos(p-p.)dd=
=
0
12(0, O x ) =
13(0,
cosp
Ox, $X)
ei±K)
sno-)dO
ei - K(0)sinsin6.cos(p-px)d4 = 21rJo( rK()sin0sinx)
=
(5.49)
0
Once the integration over
of 0 look as follows
# is done, the expressions for the
components of the electric field as a function
E
=
o
C
Res
{
1
Q-4tLKcos0)-: csos,
(-Co
ei Kcoscoso [Z13 12 - Z 23 I1]
sinO dO
(5.50)
0
E
= C
1-o(-1 i-KcosO)
sL
Res
0
Ez = C
(Res
Where C =
and
functions of 0, o, and
#..
11,
Kcos
{
es1-cs(I aKcos6)
12
cosc"
1,-cscs,(5.51)
[Z13 1 + Z 23 -215 sinO dO
eiKcos3cosoZs3 13
'1cs~oO
sinO dO
(5.52)
and 13 are the integrals calculated in Eqs. 5.47 to 5.49, which are
If the antenna is aligned with Bo the case is totally symmetric and the fields cannot depend on the
azimuthal angle
#.. For this reason,
and in order to simplify the algebra, the x and y components will
be combined into radial and azimuthal components
=
Ecosd, + Esino,,
=
-Exsin4
The field expressed in these new components is
+ E cos#$
(5.53)
(5.54)
Er
=
2CriC
I
1
Res
cos
(!
- KcosO)
KcosOcosG
Z 13 J1( LKsinsinO) }sin dO
2riC]
Z 23 J1(
Ez=
27r C
Z 33
1
Res
ccos
2
(5.55)
Kcos
Ksinsin6b) sin dO
Res
{
K
s
Jo( K(O)sinsin6 ) sinO dO
(5.56)
iKcoscosO.
(5.57)
Which do not depend on #., as expected. For a more general a, one should still calculate Er, E4,
because they may have some simple #, dependence.
Integration over 0 using the SPM
The stationary phase method can be used to perform the integration over 0. In this case, the product
of the Bessel and exponential functions oscillates much more rapidly than the other terms, except at
the stationary points. Near the stationary points (Os,
the integral and only integrate this product.
The fields can be expressed as follows
#s)
it is possible to take the other terms out of
(1L-K
2 (Os) coss)
1 - cos
2siCZRes
cos
s
Z1 (Os) sin~s
StationaryPoints .
f
EO
=
KsinsinO.)eii Kcoscos& dl
os (I.K
(0s) cos(s)
2 L 22S
Z23
Cos Os
27ri C Z Res
StationaryPoints
J Ksinsin6.) e LKcoso
E,
=
2lrrCZRes
/
Stationary Points
d
(IL.K (Os) cosOs)
2
(0s ) sin~s
(5.59)
O(.9
Ji(Ii~o~oO
7 j~sin~sinO~)
cos
-
o1
:
{
(5.58)
(0s) sin~s
os
.
Jo( Ksin0sinO,) ei KcoscosO= do
(5.60)
Where x = MKsinsinO, is a large number because r/L is large. For this reason, the Bessel functions
can be simplified taking their asymptotic form
JO (x)
J1 (x)
-
cos
~ -cos
irx
(x
-
4
x -
-
1
wx
2 2
(5.61)
2
4
+ e(X
e
)
(5.62)
Substituting these approximations, the remaining integrals that need to be calculated are
00
Jo(jKsinsinOx)eiiKcosLcos6d
1\-i
-
[ei
j itK(cosOcosx +sin~sinOi)
-00
ij
i :-K(cos~cos9c--sinOsinO.)
dO
(5.63)
J1 (
KsinOsin0k)
eif KcosOcosz
1 C2
dO
foo
[
2
-
ie
.
K(cos6cosO,+sin~sinO,)
eLirK(cosOcosO.-sinsinO)]dO
From the expression of the phase 4 (Ow,
#5x,
(5.64)
0, #) in Eq. 5.20, it is possible to see that
ieMK(cos6cosO,+sin6sinO,)
(5.65)
_
iiK(coscosOz -sinsin,)
i MY
(p=p+,r)
(5.66)
And the integral of the Bessel function of zero-th order can be written as
00
1 V 2f
im
-00
JO (L KsinOsinO ) C LKcos Ocoso" dO
L
2
72
=O
1
1
YQ p p'D
(O)
(p=4.+r) 1Y
d6
(5.67)
Where the following relations have been used
e-i4
e
71
1
f -+
1f-1
1
1
(5.68)
(5.69)
The phase can be expanded around the stationary points like in section 5.4. It can be easily seen
that
=
x =
&~
=
TKsin~sin6x
Substituting all these expressions, the
.
resulting integration is as follows
Jo(Ksin6sinOx)eitK
eitY((s, Os=0.)
Ocosc6 dO
(5.70)
StationaryPoints
+
v
(5.71)
iY(s~~'I
r
42 4 42.p]
E TO-T~aq5
OS,OSO+
Similarly
oo
J
Ji( Ksinsin.)ei'KcoscosGgdO
(5.72)
-
St ationaryPoints
I
eY4(0s,+s=0,+r)
82D
+s=e.
Finally, the electric field can be expressed as follows
(5.73)
Er
(r) =
1 - cos (IkK (0s) cosOs)
2 L 2S
Z 13 (Os) sin~s
27ri CSRes
Stationar y Point s.
efiz'YCes, Os=42)
r
a2. a21
V r
ei iYI(Os,+Os=+.+70()74
924
-a-7--a71S,$ks=&.
Ezp(r)
=
=-
L 2S
27r C
{
0s~P~tk.
So2
IStationary Points
a2 12,
r y
L
1 - Cos
[-c
ei
Y (0
D
s, S
2+ a2,
r. 0
=0 +
sinOs
0
(5 .7 5 )
,=~~i
(I-K (s) cosOs)
L
(OS )
Z 33 (0s) sins
.Cosg
3!-YDO,OS=*=)Nds
e
L
e i : -Y1 S ,' S O)
a2 ( a2 (
r
Res
Z23
Cos
I StationaryPoints
N7r
Ez (r)
(I-K (0s) cos~s)
1 -Cos
1:2
27ri C ERes
7 -a-T1s,#s=4.+7r
|6
.
}2
eifYT(0s,+ s=+0.+ir)
(5.76)
V
The magnetic field can be easily obtained from the Faraday's law (Eq. 2.8).
It is interesting to note that this approach gives exactly the same results as using the SPM for both,
0 and #, and so there is no real advantage in going through the Bessel functions. As it will be shown
in the next section, the advantage of integrating exactly over
#
is for the near field, and perhaps for
analysis of the intermediate field (transition).
5.5.2
Near-Field Analysis
Near the antenna, but far enough for the linear approximation to be valid, it is not guaranteed that
the parameter r/L will be large. As a consequence of that, the exponential term will not vary fast
enough for the Stationary Phase Method to be applicable, and the integration over 0 must be done
numerically.
In the near-field, the radial source current density arising from the sheath would be taken into account.
However, the cylindrical approximation presented in Section 4.1.2 fails near the tips and center of the
tether, thus a more detailed analysis would be required to accurately solve the problem in the near-field,
which is outside the scope of this thesis.
BOI
OX small
r
La/4
Figure 5.11: Near antenna analysis
Taking into account these limitations, for a tether aligned with the external magnetic field (a = 0),
the integration over
#
can be done directly using Bessel functions as was done for the far-field case.
The integral to calculate with a = 0 can be written as follows
E(r
2ri
Res [ZB] fBe'
(K)2
sinOd dO
(5.77)
Introducing the impedance matrix in the antenna related axes [ZB] given by Eq. 3.22, the components
of the electric field can be expressed as follows
2rij
Ex
R
{
eiK()cosOcosO.
[C (0) (cos#Zu1(0) - sin#Z2 1(0))
Ey
27i
Res
\\LLs'in6
I
ei-nK(O)sin~sinO~cos(4>-O,)
+ Jz (cos#Zi3 (0) - sin#Z 2 3 (0))] d~d6
2sK
eiK()cosOcosO.
(5.78)
eiK()sinOsin.cos(4-42
4
{e
[C (0) (sin#ZI (0) + cos#Z 2 1(0)) + Jz (sin#Zi3 (0)
Ez
=
R
2ri f
2
LK(O)cosOcos6
(K
2 s)
+ cos#Z
23
(0))] d~d6
(5.79)
eiLK()sinsinOxcos(44)
[C (0)Z3 1 (0)+ Jz Z3 3 (0)] d/d0
(5.80)
Where [Z] is given by Eq. 3.19 and
C (0) =
The integration over
#
[1-cos( 2 !Q-Kcos)] (-r2 K 2cosOsin0
) CL
27r0
4
2 COS20 L,,
-Io
(5.81)
is easily done using Bessel functions. The components of the electric field as a
function of 0 look as follows
E,
Res
=
s
eiMKcoscosO6
}
(5.82)
[C (0)(Zuli + Z2112 ) + J (Z1311 + Z2312 )] dO
2
s iiKcos~cosO. (K) ili
~
27ri
=
oc(
Res ei c s
sinI3
}
(5.83)
d6
(5.84)
[C (0)(Z1112 - Z2111 ) + Jz (Z1312 - Z23 1)] do
ZRes{
2E
Ez
eimKcoscosO.
1
[C (0) Z31 + J"2Z33]
Where I1,
12
(K
)2
and 13 are the integrals calculated in Eqs. 5.47 to 5.49, which are functions of 0, Ox and
Like in the far-field case, the symmetry of the problem allows us to eliminate the dependency on
#
by expressing the fields in cylindrical components. The field as a function of 0 expressed in these new
components is
E
=
{
(2Rri)e2
eiKcosOcosO.
(K~ 2 sinO J1 ( L Ksin0sinx)
[C (0) Zn +±JzZ1
3 ]}d6
E,
=
2 2I
Res
ei LKcoscose
[C (0)
Z21 + JzZ23]
E
(5.85)
0
(27r 'JRes
sinG J1 ( Ksinsinx)
do
eiKcos6coss 2 ,
[C () Z31+ JZ
z 33]
(K )2
dO
(5.86)
KOsnsnx
(K~T2sinG Jo(-KGsiLin~
(5.87)
Close to the antenna, cosO. ~ 1 and the term r/L sinO6 will be small. However, the term K 1 = K sinO
can become very large close to 0 = 0 res, and for this reason the Bessel functions cannot be approximated
using series expansion. Thus, the complete Eqs. 5.85 to 5.87 need to be integrated numerically over 0.
This integration diverges at the resonance condition given by Eq. 2.23, which would be eliminated if
thermal effects had been considered. However, dealing with thermal effects requires to formulate the
problem from kinetic theory, which is outside the scope of this thesis.
One would have expected Ez to tend towards zero the fastest as the cylindrical radius R -+ 0. Instead,
it tends to some nonzero limit, while Er, E, tend to zero. This strange behavior comes from the
Ez dependency on Jo(yK(O)sinOsin ) in Eq. 5.87, which tends to 1 as R -+ 0 and dominates the
integral; something similar happens to Er, E0, which present a dependency on J1 (y KsinO) that tends
to zero as R
-+
0. From the expressions derived above it can be observed that this would happen for
any antenna model, thus it cannot be blamed on the assumed triangular current distribution. This
behavior is possibly due to the cylindrical model used for the radial source current term that does not
reproduce the tips and center of the tether, where the radiation is probably concentrated. The thermal
effects and detailed analysis of the tips and center of the antenna are left as future work.
Chapter 6
Radiation Pattern Analysis
6.1
Radiated Fields in the Far-Field Region
The method presented in the previous chapters allows us to calculate the fields E and B radiated from
the magnetospheric tether. Figures 6.1 to 6.6 show the plots of these fields versus 0, (polar angle of
the observation point) for a = 0, while Figures 6.7 to 6.12 show the case of a = 7r/2 rad. These plots
represent the radiation pattern in the far-field region for a driving frequency of Y =
an observation point located at
#,
w/weg = 0.8 and
= 0.
The parameters used to obtain these results are presented in the following table.
no
[m-3]
1010
Bo [T]
10-5
Table 6.1: Input parameters
L [m]
Io [A] oi[Sm-1] r [m]
5.106 2.277-103
10
1.602-10-4
La [m]
4-103
a[m]
0.1
The figures for the antenna aligned with the external magnetic field (a = 0) show that the fields
increase as they approach the resonant
2,,,,
which equals
62_
= 0.0311 rad for Y = 0.8. It is at this
point where the fields become maximum, dropping drastically after it. The feature responsible for this
behavior is the third stationary point laying on the resonance branch detailed in Section 5.4.1. These
results show that the wave propagates in a very small cone around the external magnetic field. What
is more, for the observation point defined above, the electric field is mostly radial and the magnetic
field azimuthal. From the polarization analyzed in Section 2.3 it can be seen that at resonance the
magnetic field tends to a finite number.
The figures corresponding to the antenna perpendicular to the external magnetic field (a = 7r/2 rad)
show that the fields oscillate strongly and the resonance disappears. In this configuration, the electric
field is as well mostly radial-cylindrical and the magnetic field azimuthal, and they are weaker than
for the aligned antenna case.
x 10-
Real part of the E,-field vs O, for Y = 0.8 anda = 0
2
Q
0
-2
Ox[red]
Figure 6.1: Electric field in the x-direction (radial) for Y = 0.8 and a = 0
Real part of the E -field vs Oxfor Y = 0.8 and a = 0
x 10-8
2
-2
-6
-81
0
0.01
0.02
0.03
ex [rad]
0.04
0. 05
Figure 6.2: Electric field in the y-direction (azimuthal) for Y
x
0.06
0.8 and a = 0
Real part of the Ez-field vs O, for Y = 0.8 and a = 0
10_,
0.01
0.02
0.03
ex [rad]
0.04
0.05
0.06
Figure 6.3: Electric field in the z-direction for Y = 0.8 and a = 0
Real part of the Bx-field vs O, for Y = 0.8 and a = 0
x 10 14
E;
IiII
-2
-4
-6
-8
-10
0
0.01
0.02
0.03
0.04
0.06
e [rad]
Figure 6.4: Magnetic field in the x-direction (radial) for Y
0.8 and a = 0
Real part of the B -field vs 0, for Y = 0.8 anda = 0
x 10-"
2
1.5-
1 -
0.5-
0
-0.5 -
-1-
-2-2.
0
0.01
0.02
0.03
0.04
0.05
0.06
Ox[rad]
Figure 6.5: Magnetic field in the y-direction (azimuthal) for Y : 0.8 and
a a =0
x 10 13
Real part of the Bz-field vs O,for Y = 0.8 and a = 0
0.06
0,[rad]
Figure 6.6: Magnetic field in the z-direction for Y = 0.8 and a
2
x 10_
Real part of the E -field vs
6 for Y = 0.8 and a= x/2
0
rad
1,5
1
0.5
wr
0
0.06
ex [red]
Figure 6.7: Electric field in the x-direction (radial) for Y
0.8 and a = 7r/2 rad
x
Real part of the Ey-field vs Oxfor Y = 0.8 and a = n/2 rad
10.7
2
0.01
0.02
0.03
Ox[rad]
0.04
0.05
0.06
Figure 6.8: Electric field in the y-direction (azimuthal) for Y = 0.8 and a = gr/2 rad
Real part of the Ez-field vs 6, for Y = 0.8 and a = n/2 rad
10.7
/wvvvvvwvvwwAtw~
0
0.01
0.02
0.03
Ox[rad]
0.04
Figure 6.9: Electric field in the z-direction for Y
100
0.05
0.06
0.8 and a = 7r/2 rad
Real part of the B -field vs
x 10~13
0
for Y = 0.8 and a = a/2 rad
0
m-1
0,01
0
0.02
0.03
Ox[rad]
0.04
0.05
0.06
Figure 6.10: Magnetic field in the x-direction (radial) for Y = 0.8 and a = ir/2 rad
x
10~
Real part of the By-field vs fx for Y = 0.8 and a = 7/2 rad
0.06
Ox[rad]
Figure 6.11: Magnetic field in the y-direction (azimuthal) for Y
101
0.8 and a = 7r/2 rad
Real part of the B.-field vs Oxfor Y = 0.8 anda = i/2 red
x 10~12
-1.5 k
0
0.01
0.02
0.03
0.04
0.05
0.06
Ox [rad]
Figure 6.12: Magnetic field in the z-direction for Y
0.8 and a = 7r/2 rad
The two methods of integration over 0 of the exponential term using the asymptotic expansion or the
Fresnel integrals (Section 5.4.2) result in practically identical results, thus the asymptotic approximation is perfectly valid in the far-field region.
The case around the critical value of 6 where,9 = 0 has been studied as well. Figures 6.13 and 6.14
show the results for the radiated fields along the x-direction calculated using the second and the third
order approximations presented in Section 5.4.3. From these plots it is possible to observe that the
third order approximation is accurate very close to
, but is not good far from the critical case, as
expected. The plot shows that in the immediate surroundings of
2,t
the second order approximation
fails and it is the third order expansion the one that gives a reasonable and continuous solution. It
can be observed that there is no major increase of the fields around this region.
102
Real part of the E -field vs O0for Y=0.8
x1
1r
-0.4 -0.6-0.8-1 -
0.1
0.11
0.12
0.13
0.14
0.15 0.16
Ox[rad]
Figure 6.13: Electric field in the x-direction around 0,,
of Fig. 6.7)
x 10-15
0.17
0.18
0.19
for Y = 0.8 and a = 7r/2 rad (continuation
Real part of the B -field vs 0, for Y=0.8
0.15
O, [rad]
Figure 6.14: Magnetic field in the x-direction around 6.,
of Fig. 6.10)
103
for Y = 0.8 and a = 7r/2 rad (continuation
The introduction of the current arising from the sheath in the source term does not modify significantly
the fields in the far field region except for the second harmonic described in Section 5.2. However, for
the range of frequencies of interest of EMIC waves, the second harmonic of the proton branch does not
propagate, since it corresponds to 2wo > wci. Nevertheless, this new term in the source current would
give a propagating second harmonic in other frequency regimes, like in the whistler mode, thus the
effect of the sheath needs to be included to analyze this kind of higher frequency waves. In addition,
this evanescent 2w0 contribution must be kept near the antenna, since its exponential decay occurs
over distances of order 1/k ~ L.
6.2
6.2.1
Energy Flux in the Far-Field Region
General Formulation
The Poynting vector S represents the instantaneous energy flux [W/m 2] of the wave at a given point
S (X, t)
(6.1)
x B
=E
And the time-averaged Poynting vector is
(
B
=(6.2)
Consider a wave propagating with k and w local quantities. The wave can be viewed as a superposition
of plane waves which are spatially limited and 1 gives the point of maximum constructive interference
at a time t when the wave is centered on k and w. The wave electric field can be represented as
S=Re (Ee-
), where the phase
4
varies with i and t. Taking the time-averaged quantities in the
Poynting theorem [161, the Poynting vector will be
= Re E x B* =
2pto
104
Apo
(E x B* + E* x B
(6.3)
However, this expression is not general in the case of more than one mode of propagation. The wave
normal and Poynting vector directions are not necessarily parallel, so the wave group velocity vg differs
from the wave phase velocity
Vph
V,~h.
The angle between
V', and Bo is 0., while 0 is the angle between
and Bo. O6 and 0 may differ. However, independently of the medium, the angle 0. coincides with
the direction of the Poynting vector, and hence with that of the group velocity.
The electric field consisting of a superposition of modes (say four modes, since there can be up to four
stationary points) can be expressed as follows
E (, t)
Re [1e
1E
+±3e
-"w)
+
t
+
-t)
+
2
g-wt) + k 3 ei(J-'t)
eI
+ E 4 ei
+ $2e
*e(A:
-
e±
ke-
+
-
+wE)e
4 ei(I-'t)
+ *e-(I:wt)]
(6.4)
Similarly, the magnetic field is
(z,t)
Re [Biei(k-wt) + A 2 eiQ
2
+B 3 e(
Where Ej, BJ3
f
+
i
-wt)
~e-
+
$
3
e
)+ B 4 e
+ B- 2 ei(Ewt) + Be-
B4e(
+ 3;e(+
-
±
-
-
*e((6.5)
(k3 ) represent the complex electric and magnetic fields in the Fourier domain
evaluated at the points
j, which
correspond to the stationary points. That is
Sy (k
) =
()
-Z(k)
Z
Uci
JB(k)
( x $(E))
105
kj
kj(6.6)
(6.7)
Taking into account the following relations for the time-averaged complex exponentials
"'-wt)
(e ("'*~*)ei(k,
(ei(k-
.- wt)
e-i(kC"--
0 for m, n = 1, 2,3,4
~2iot i(kl+kC)-9
-
_
ot)
a
Ot ei(k -k,)-Z
for m
i (kM '-k',)-
(6.8)
(6.9)
7n
1 form= n
(6.10)
Substituting into Eq. 6.12, the time-averaged Poynting vector simplifies to
-[E
B
1
+1
x B4
±
E 2*xB
+
$*x B2 e(2
x
2*ei(
2
e4
x
2
4
x
Jie
x b$ek
+X*3 e(X
2)-+
3
3
4
xB*+E*xB 1 +Ei
±
x
+
x
±+
3
±
* x
B4 e
±
x
e
x
e
-)
x
xB
ExB
2
3
1x
$ 4 e('
'+* +
e(
+ E* x B3e-2(
2.+E
4
$1x
-'i
x
-
+E 2 x
+E4
E 3 xB+E
+2
x
2 x
1
x
E3 x
±
-)x
'
e(-2)
3
e+
x
1'
x
2
e2--)
(6.11)
= 1 Re
[$1 x
+E1 x
*e
+E2 x
+E3
x
+4
x Be
*+
2po1234
E2
x B* +
E3
x B* +
$4
S*
x
+ E1 x B*e
+
1 x B'e
*e
+ $2
E
*e
+
2
x
'*e
1*e
+E
3 x Bie
+
£3 x
4*e
+
$4
-
x
+E4 x
*e(
-
x
*
-(6.12)
The terms in the second to fifth line represent spatial interferences between the various propagating
waves (one per stationary point). The first line is the simple generalization of the one-wave case,
if interferences are ignored. The interference terms should produce rapid oscillations of
K5)
vs
OX.
However, see below for numerical results.
6.2.2
Energy Flux Results
Figure 6.15 presents the projection of the time-averaged Poynting vector over the unit radial vector
for a = 0.
s=K-
(6.13)
Consistent with the definition of the group velocity, the Poynting vector must be along the radial
direction to the observation point, and this is the only component that results form the calculation.
Form this plot it is possible to see that the radiation concentrates at the resonant angle and it is
practically zero anywhere else. A shoulder (broad maximum) appears near the resonance which is
due to the finite length of the antenna and the angle subtended by it at the observation point. It is
interesting to note that the high frequency oscillations that appeared in the the fields presented in
Section 6.1 disappear when time-averaging the Poynting vector taking into account the interference of
the different modes.
107
......
.....
Figure 6.16 show the projection of the time-averaged Poynting vector over the radial vector for a =
7r/2 rad. In this antenna configuration it is possible to observe that the radiated power flux is about two
orders of magnitude smaller compared to the parallel antenna case. The strong oscillations observed
are probably due to the interference terms present in the formulation of the time-averaged pointing
vector, which are stronger around the parallel direction (OX = 0).
As we pointed out in Section 6.1, the introduction of the radial current arising from the sheath in the
source term generates a resonance at the second harmonic. Figure 6.17 shows this result for Y = 0.4,
where two resonances appear due to the fundamental and second harmonics. However, for the range of
frequencies of interest of EMIC waves, the second harmonic of the proton branch does not propagate,
2
since it corresponds to wo > wcz.
x
Projection of the Poynting vector vsO, for Y = 0.8 and a = 0
10l
4. 5t
F
3.5
2.b
E
an 2
0
I
0.01
-A,
0.02
0.03
6 [rad]
0.04
0.05
0.06
Figure 6.15: Projection of the Poynting vector over r for Y = 0.8 and a = 0
108
. .....
..
...
..........
.....
x
Projection of the Poynting vector vs O,for Y = 0.8 and a = 7/2 rad
10.11
0
0.01
0.02
0.03
0.04
0.05
0.06
O,[rad]
Figure 6.16: Projection of the Poynting vector over
x 10
F for Y = 0.8 and a = 7r/2
9 Projection of the Poynting vector vsOx for W = 0.4 and a = 0
52nd Harmonic
Fundamental
4-
E
3-
2-
10.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
ex [rad]
Figure 6.17: Projection of the Poynting vector over
harmonics
j'
109
for Y
0.4 and a = 0: fundamental and second
6.3
6.3.1
Radiation Power and Resistance in the Far-Field Region
Integration of the Power Flux
The radiation power can be found by integrating the time-averaged Poynting vector in Eq. 6.12 on a
spherical shell around the antenna
P
=
f7, f
.r
=2
r 2sinOxd6xdpx
2
2r
-$
r2 sinOxdxdx
(6.14)
This integration can be done using the trapezoidal rule.
Once the radiated power has been calculated with the expression above, the radiation resistance R can
be found as follows
2P
(6.15)
R=
02
6.3.2
Balmain's Model for the Radiation Impedance
The input impedance of a conducting antenna can be derived from
Zin =
f-
.J
dS
(6.16)
S
Where S is the surface of the antenna,
Ja is the current density on the tether and E is the electric
field on its surface parallel to the antenna when the conducting material is removed.
Balmain [3] derives an expression for the impedance of a short cylindrical dipole in a magnetoplasma
using the quasi-static electromagnetic theory that can be used for any range of frequencies and any
dipole orientation. This approximation is valid when the antenna is short compared to the wavelength
110
of the wave, which is a rational approximation to obtain an estimation of the impedance of the tether
for the purposes of this study. It is to be noted that the electrostatic approximation should be good
in a neighborhood of the resonance cone in k-space (as noted in Chapter 2), and since most of the
radiation concentrates in this neighborhood, Balmain's approach may be appropriate even for not very
short antennas, or not very near the antenna. His model neglects the current arising from the sheath
and assumes that the charge along the antenna is uniformly distributed on its volume, instead of being
concentrated at the surface. Taking into account these approximations, Bahnain derives the following
expression for the radiation impedance
Zin =
2p
Ln
iw7rEoeLaF1/2 I
2a
-1 - Ln
2F
(6.17)
Where a and La are the radius and length of the antenna, respectively. For EMIC waves, the parameters
F, p and Ej are given by
F
=
sin 2 a
+p 2 cos 2a
(6.18)
(6.19)
p2
'El
w2
eCL
1+XL=XJ
-
|-
2
2
X2 w e
(6.20)
(6.21)
Where XL and X1 are derived in Appendix I. It can be observed that the approximations introduced
for EMIC waves imply p = |pl e i/2, because E- > 0 and E|| < 0. Thus, the input impedance has a
positive real part indicating power flow into the plasma [3], and a negative imaginary part that reveals
the capacitive effect of the system. The input impedance can be expressed as follows
Zin = R + iX =IZIe3 = v/R2 + X 2ei arctg(X/R)
(6.22)
Where R > 0 is the resistance of the antenna and X < 0 represents the capacitive reactance.
If the tether is aligned with B 0 , the impedance can be written as follows
Z1| =
Ri1
2
izw,reeLa
.
Ln
(-) -1+
Ln (p)
R11 + iX||
11
=
(6.23)
(6.24)
2
X11 =
2a
La-L
2
ixrieeLa
Ln (2a)-a -1+ LnIpI
(6.25)
On the other hand, if the tether is perpendicular to B 0 , the impedance will be
Z
=
2
iwreoeLa
Ln
2a
- 1 - Ln
2
)
= R + iX1
(6.26)
According to Balmain [3], the positive real part of the input impedance derived above indicates power
flow into the plasma, thus it is the responsible of electromagnetic radiation.
The following figures show the value of the resistance and the reactance versus the dimensionless
frequency, Y, and the inclination of the tether with respect to Bo, respectively. The parameters of the
selected antenna were detailed in Table 6.1. Figure 6.18 shows that the impedance (both resistance
and reactance) decreases with the antenna length and the inclination of the antenna; from Eq. 6.17,
it can be seen that a resonance occurs at tani = pI.
....
...
.......
.
Antenna impedance versus Yfor a= 0 rad
-30 1
0.1
0.2
0.3
0.5
0.4
Y
0.6
0.7
= o/ci
Figure 6.18: Impedance of the antenna versus Y
Antenna impedance versus a for Y = 0.8
0.8
a [rad]
Figure 6.19: Impedance of the antenna versus a
113
0.8
0.9
Figure 6.20 compares the resistance computed integrating the radiated power and the radiation resistance derived by Balmain [3] detailed in Eq. 6.17, for an orientation of a = 0 and assuming a triangluar
source current distribution. Prom this figure it is possible to see that the differences between the results
of both models are larger for large antenna lengths compared to the wavelength, which corresponds
to the region where Balmain's model is not valid. The discrepancies could also be due to Balmain's
approximations mentioned above, that is neglecting the sheath and assuming the charge in the antenna
is distributed over its volume instead of being concentrated at the surface.
Comparison of the Balmain and integrated resistance vs La
0.
C
.Cu
0
CU
1
2
3
4
7
5
6
Antenna Length [km]
8
9
10
Figure 6.20: Radiation resistance vs antenna length for a = 0 and conditions as in Table 6.1
6.4
Very Low Frequency Case with a Dipole Antenna: Comparison of Results
Kuehl [11] calculated the analytical solution of the far-fields radiated from an electric dipole in a cold
plasma for the very low frequency range that satisfy
114
(6.27)
<<
Ci
>
1
>>
1
WP>e
(6.28)
2
W2
(6.29)
ce Me
Under these conditions, he derived the expression for the radiated electric field in the far field region.
For an x-oriented dipole, the electric field from Kuehl can be written as
E (r-)
=
,e-
-sin#
(6.30)
e
''
2
±
(6.31)
WComi c
2
w me
2
2
sin
cos Ox -
0.
And for a z-oriented dipole
2
E(r-)
--
=
4
me ,rr
2
cos O
2
eo
(6.32)
2
) sin o,
2
Where p. and Pz are the dipole moments in the x and z direction, respectively. e'4 and eO are the
azimuthal and spherical-angle unit vectors and V)1 and
1
02
can be expressed as follows
(6.33)
=
VA
2
=
VA
Cos2
C
Ox
(
(2M
115
in2ox
(6.34)
Where
VA
is the Alfv6n velocity given by
VA =
The wave corresponding to
#b2
Bo
(6.35)
Vy/onomi
contributes only in the region where it is a real number, which is given
by
tan2o < w
(6.36)
wim
However, for the x-directed dipole, the amplitude of this second wave in this region is much greater
than that of the first wave. For this reason, most of the radiation is concentrated in a very narrow
cone around the external magnetic field axis with a half angle given by
tan 2
=w
(6.37)
For the z-oriented dipole, only the second wave propagates in the same small region given by Eq. 6.37.
The code and formulation developed in this thesis is also valid for the very low frequency case analyzed
by Kuehl. What is more, they can be validated by considering the limiting condition given by Eq.
6.27 and comparing the results with the expressions obtained in his paper.
With this purpose, the case of Y =
'
= 0.01 will be considered. The source current is modelled
as an electric dipole given by Eq. 4.41 and the contribution of the sheath is neglected. With these
considerations, the source current can be expressed as
2
J,I(k) =
, 1 IoL,
2Zz.
(27r) 3 2
(6.38)
Introducing these inputs to the model for an x-directed dipole (a = 7r/2), it is possible to obtain the
power flux shown in Figure 6.21. From this figure, it can be observed that most of the radiation occurs
116
in a narrow cone around the axis of Bo at a half angle named 6e
The exact angle of this cone can
be found from Eq. 5.24 introducing 0 = 0res (Eq. 2.23) and finding the corresponding 6
This
calculation gives 0,,,s = 2.24-10~ 4rad. On the other hand, the expression derived by Kuehl in Eq.
6.37 gives 0,
= 2.33-10-4rad. Thus, the difference is less than a 5%.
Projection of the Poynting vector vs O for Y=0.01
0.45
0.40.350.30.250.20.150.1
I
0.05 F
0
0.1
0.2
0.3
0.4
0.5
Ox[rad]
0.7
0.8
0.9
1
a3
X 10
Figure 6.21: Projection of the Poynting vector over r for Y = 0.01 for an x-oriented dipole
In Kuehl formulation the first wave is characterized by 01, which in this thesis corresponds to the wave
defined by the first stationary point k (Osi). Equating both definitions
k K(Osi) [sinOsisinO6 + cosOsicos6.] =
#i4
1rw
Ioco
1
K(Osi)
cosOsicosO.]
J9060 L ) [sinOsisinO. +
w 1i(o
(6.39)
(6.40)
For 0, = 2-10-4 rad, this corresponds to Osi = 0.6707rad, which gives 4'1 = 137.48. On the other
hand, V)1 can be calculating using Eq. 6.33, which gives V)1 = 137.44. This corresponds to an error
117
smaller than the 0.03%.
For a x-directed dipole, the results obtained with the model developed in this thesis show that the
contribution to the field of the first wave inside the radiation cone is about six orders of magnitude
smaller than the one of the second wave, or in other words, most of the radiation occurs inside this
cone and is due to the second wave. This result agrees with the conclusions drawn by Kuehl, where he
showed that for an x-directed dipole the amplitude of the second wave inside the cone is much greater
than that of the first wave.
For a z-directed dipole, similar results are obtained, but in this case the field due to the first wave
is about nine orders of magnitude smaller than that due to the second wave, thus negligible. This
conclusion agrees as well with Kuehl's results, which show that only the second wave contributes to
the field radiated from a z-directed dipole.
118
Chapter 7
Conclusions and Future Work
7.1
Conclusions
The previous chapters presented an analysis of the propagation and radiation of electromagnetic ion
cyclotron waves (EMIC) from a magnetospheric tether under the following assumptions
" The plasma is anisotropic due to the magnetic field of the Earth and it is assumed to have
sufficiently low density, temperature and degree of ionization so that collisions and thermal
velocities can be neglected.
" Protons are the only positive charged species considered; their motion is taken into account in
the formulation together with electrons motion.
" The plasma is assumed to respond linearly to the electromagnetic disturbances excited by the
source.
" The source current has been assumed linearly distributed along the antenna. The distribution
has its maximum at the center and goes to zero at both ends of the antenna.
119
* The effect of the sheath has been modeled by introducing a radial current in the source term.
The cylindrical approximation used to model this term does not consider the effect of the center
and tips of the tether, which may lead to significant errors in the near-field region.
Under these assumptions, a theory of propagation of electromagnetic ion cyclotron waves (EMIC) in
the magnetosphere has been developed, as well as a mathematical formulation to find the analytical
solution of the radiation pattern in the far-field region from a tether at an arbitrary orientation with
respect to the magnetic field of the Earth.
The dispersion relation of EMIC waves shows that there are two branches possible, corresponding
to electron and proton waves.
The proton branch presents a resonance for an orientation of the
wavenumber vector close to r/2 rad, at which the wave becomes electrostatic. This branch disappears
for w > wej. The wave is shown to be elliptically polarized.
The methodology developed to find the radiation pattern in the far-field region consists of calculating
the Inverse Fourier Transform of the fields generated by the specified current source. An analytical
solution of the Inverse Fourier Transform has been found using the Residue Theorem and the Stationary
Phase Method. The last one is only valid in the far-field region, where cancellations do occur except
around the points of stationary phase.
The results in the far-field region are presented for tether orientations parallel and perpendicular to
the external magnetic field. From these results it can be observed that propagation mainly occurs in
a small angle around Bo. For an aligned antenna, the maximum power flux occurs at the observation
point whose orientation corresponds to the resonance condition, while for the perpendicular case the
maximum power flux is localized closer to the magnetic field lines. The introduction of the current
arising from the sheath does not modify significantly the far-field results except for the appearance of
a second harmonic at 2w. However, for the EMIC range of interest, this second harmonic does not
propagate because it corresponds to 2w > wej.
The model developed in this thesis is also valid in the very low frequency limit (w << we1 ), which is
in good agreement with previous results obtained by Kuehl for the Alfven regime [111.
The radiated power obtained from the model has been used to compute the radiation resistance. This
result and the model proposed by Balnain [3] show the same trends. The differences between both
models are larger for large antenna lengths compared to the wavelength, which corresponds to the
region where Balmain's model is not accurate. Balmain neglects the sheath and assumes that the
charge in the antenna is distributed over its volume instead of being concentrated on the surface; these
approximations may also contribute to the discrepancies we observe between both models.
In the near-field region, quasi-analytical solutions have been derived assuming a triangular source
current distribution and a cylindrical model of the radial current arising from the sheath. As the
cylindrical radius tends to zero, the results show that Ez tends to some nonzero limit, while E7 , E4
tend to zero. This strange behavior comes from the fields dependence on the Bessel functions and
it cannot be blamed on the assumed triangular current distribution, but it could be related to the
cylindrical approximation assumed to model the effect of the sheath. This approximation does not
reproduce the center and tips of the antenna, which may have a noticeable effect in the near-field
region. The correct modeling of the near-field requires a more detailed analysis of those parts of the
tether, which is left as future work.
7.2
Future Work
Future research should address the following topics
e Model of the near-field region. The cylindrical approximation used in this thesis for the source
current arising from the sheath may lead to significant errors in the near-field region. For this
reason, a more detailed model needs to be developed to study the surroundings of the tether
outside the sheath paying special attention to the behavior of the tips and center of the tether.
" Study of end effects in the tips and center of the antenna. These effects may change the distribution and the electromagnetic fields in the near-field region.
" Model of the sheath around the antenna. Due to the non-linear nature of the problem, numerical
effort would be required to deal with the analysis of the sheath. The final goal would be to couple
the numerical model of this region with the analytical formulation of the far-field developed in
this thesis.
" Self-consistent antenna source current density. The antenna source current density has been
assumed linearly distributed along the tether. However, this model is not self-consistent with
the developed formulation. Iteration between the analytical model of the far-field region and the
numerical model of the sheath will allow to find the correct source current density.
* Detailed analysis of the radiation impedance. The radiation resistance has been calculated and
compared to previous models [4]. However, a formulation for the reactive part of the radiation
impedance based on far-field results needs to be determined and quantified.
* Tether design work. Structural, electrical and thermal design of the antenna would be required
when thinking about a future tethered satellite. The deployment system, dynamics of the tether
and constraints that the antenna imposes to the design of the satellite are very important points to
be addressed in the near future. The Tethered Environmental Reconditioning Satellite (TERSat)
is being designed at MIT to test the effectiveness of radiated EMIC waves in releasing trapped
high energetic protons from the Van Allen Belts. TERSat has been selected to compete in the
UNP-7 together with other 10 universities for an opportunity to launch into space. The objectives
of this satellite are the following
- Goal Criteria: To scatter through the loss-cone high energy protons and electrons and be
able to measure this scattering.
- Success Criteria: To radiate EMIC waves from the tether and measure this radiation.
- Minimum Success Criteria: To launch TERSat and deploy the tether successfully.
Appendix I: Characterization of EMIC
Waves using Stix Formulation
The system of equations formed by momentum and Maxwell's equations is presented next
e
=
-e($1 +
=e(E 1
V xE
1
Ve x
Bo)
+ i x Bo)
(7.1)
(7.2)
=
t
(7.3)
o+
p os
(7.4)
V x B1 =
The subscript 1 refers to the perturbed fields and it is omitted in the following derivation. j =
.m
qmnom represents the currents due to the velocity of the particles (protons and electrons) in the
plasma and
JS is the distribution of source currents from the antenna. The external magnetic field Bo
is taken along the z-axis and the wavenumber vector k is located at an angle 0 with respect to Bo as
shown in Figure 2.1. Taking B = Boe-z + B 1, k = k1 + klez = k sinO e1
123
+ k cos e&zand assuming a
dependency of the type ei(k-g-wt) , the linearized equations can be expressed as follows
e(
imewve
-imwvi
+ e X Bo)
=e(E +
j x Bo)
k x E= wB
(7.5)
(7.6)
(7.7)
k x B =poj+po
j
(7.8)
The momentum equations can be written as
83e
-
IT
-
+ Wceje x
bo = w joE
(7.9)
BjP
- Wiji x bo =
Where bo =
-
w icoE
(7.10)
and wei, wce, wpi and Wpe are given by Eqs. 2.12, 2.13, 3.1 and 3.2, respectively.
From the momentum equations, the Ohm's law can be expressed as follows
j
Where
-iwco [XI E
(7.11)
01
[x]=
W
x1=~
ix.
Xi
0
0
0
X1I1
2
2
2w
pe ce
2
ci2
Wce
2
-
22
W2
W
-oge
2 +
Wei
<<<
~
2
2
W C2i
L2
ci
(7.13)
E~
i W2
Wce < Wpe
eno
Wpe
X II =
Where the approximations W
-
2
2
Wi
Xx
W2 -W
2
o-
Wci
(7.12)
EoBo
w
w2
-
2
(7.14)
W2
(7.15)
Pe
have been used.
Combining Faraday and Ampere's equations
k x k x E = -iwpoj - iwpoj,
(7.16)
Substituting the expression for j from the Ohm's law
k x k x E+
-
[xE
-
C
k 2 [I] + =
[D]
[x] E
E =-moa
where
125
-impoj
(7.17)
iwo
(7.18)
(7.19)
-i
_ - k
[D]
i
k||k_
CTXx
2
w2
JTX~ 0-k
2
=cXX
0
Multiplying the expression in Eq. 7.19 times L 2
wpi
x-X -
KI
2
$x||
- ki
2
2
-i-
KI
p.
=xx
K-
w2
Kx1
i
(7.20)
0
xP _- K2
Kl; K1
0
0
$xI
,"
E
-i
e 223
nos
- K_
Where K = k L. Finally substituting the expressions from Eq. 7.13 to 7.15 and multiplying by
.K
.y
1
1
g
K±i
y2
SY
1-Y2
1-Y2
Y2
0
1-Y2
SKI; K-L
Y
E = -js
O'ci
(7.22)
Mi
-K K2
Y
1 + me)
0
weim
-c2 =O
Where Y = '.
(7.21)
eno
Bo
This expression is equivalent to the one obtained in Chapter 2.
(7.23)
Appendix II: Symmetry of the Inverse
Fourier Transform Integration
As detailed in Section 5.3, the dispersion relation gives two pair of poles, which have been named
K and K
. It was shown that the integration path must enclose the poles K/
cosy > 0, or K,-- and K-
and K+ when
when cosy < 0, where -yis the angle between k and X-.
One of the first steps in the K-integration was to extend the domain from (0, o) to (-oo, 00), which
required that the integration over 0 should be simultaneously restricted to (0, r/2) rather than (0, r).
The final integration region was K: -oc to oc, 6 : 0 to
7r/2 and #: 0 to 27r, where cosy > 0 in all the
domain.
This region of integration is mathematically identical to the original domain. In the original domain,
the integration should distinguish between cosy > 0 or cosy < 0, corresponding to the poles Kte and
K,-e, respectively. This is equivalent of extending the integration of K from -oo to 00 and restricting
0 from 0 to 7r/2 , where only the poles Kte are considered.
In the K-plot presented in Figure A2 it can be easily seen that the two regions are equivalent, so the
integration is symmetric with respect to 0 = 7r/2. For this reason, it would be enough to integrate 0
between 0 and 7r/2 and multiply the result by two. However, the integral was already counted twice
during the integration in the complex plane using Cauchy's Residue Theorem, thus it is not necessary
to multiply the result by two again.
127
Kperp
Figure A2: Wavenumber plot showing the symmetry with respecto to 7r/2
128
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