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ESTIMATION OF THE SPECIFIC REAL PHASE AND GROUP REFRACTIVE INDEXES BY THE ALTITUDE IN THEEARTH’S IONIZED REGION USING THE FIRST ORDER APPLETON-HARTREE EQUATIONS

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Khac An Dao, Dong Chung Nguyen
,
Diep Dao
ESTIMATION
OF THE SPECIFIC REAL
PHASE AND GROUP REFRACTIVE
INDEXES BY THE ALTITUDE
IN THE
EARTH’S
IONIZED REGION
USING THE
FIRST ORDER
APPLETON
HARTREE
EQUATIONS
Khac An Dao
1,2
, Dong C
hung
Nguyen
3
, and Diep Dao
4
1
Instituite of
Theoretical and Applied Research (ITAR), Duy Tan University, Ha Noi 100000,
Vietnam
2
Faculty of Electrical and
Electronic
Engineering, Duy Tan University, Da
N
ang 550000, Vietnam
3
Institute of Research and Development, Duy Tan University, Da
N
ang 550000, V
ietnam
4
Department of Geography and Environmental Studies,
University of Colorado,
Colorado
Springs,U.S.
A
1
Abstract
:
The specific phase and group refractive
indexes concerning the
specific
phase and group velocities
of single an
d packet electromagnetic waves
contain all
interactions between the electromagnetic wa
ves and the
propagating medium.
T
he
determination of the sp
ecific
refractive indexes vs. altitude is
also
a
challenging
and
complicated problem.
B
ased on
t
he
first
order Appleton
Hartree equations
and
the
values of
free electron density by
altitude
,
t
his paper
outlined
the numerical estimated results
of the specific real phase, group refractive indexes vs. the
altitude from 100 km up to 1000 km in the ionized
region.
The specific real phase refractive index has
a
value smaller
than 1, corresponding to this value, the spe
cific phase
velocity
is
larger than the light speed (
c
)
meanwhile
the
value of the specific real group refractive index is larger
than 1, the specific group velocity will always be smaller
than light speed (
c
). These estimated results are agreed with
the t
heory and forecasted
model
predicted
.
These
results
could be applied for both the experiment and theoretical
researches, especially for application in finding the
numerical solution of mathematics problems of Wireless
Information and Wireless Power Transmi
ssions.
Keywords
:
Specific real phase and group r
efractive
index
es
by altitude,
The
First order
Appleton
Hartree
equations
,
the Earth’s
ionized region,
Microwave
propagati
on
.
Co
r
responding Author: Khac An D
ao
Email:
Sending to Journal:
9/2020;
Revised
: 11/2020;
Accepted
:
12/2020.
I.
INTRODUCTION
The developments of the theoretical aspects
of
the
refractive indexes
concerning
the electromagnetic waves
(EMW)
propagation in the Earth’s ionized region
always
have been
studying
up today
. The
refractive index of the
EMW
is
an essential concept that reflects
the interactions
between the EMW and a given medium. Depending on the
features of
a
given
propagating
medium
and the forms of
EMWs
, the refractive index
is changed
and it has
be
en
discussed and
formulated
in
different
forms
,
such as by
Sellmeyer formula
and
Lorentz formul
a
[2
5].
During the
time f
rom
1927
to
1932, the essential formula for the
refractive index of
the Earth
‘s
atmosphere’s
ionized
region
i
n a magnetic field
h
as
been
developed and
called
by
the
name
of
the
Appleton
Hartree
equation.
This
equation
describes
generally
the
refractive
index
for
EMW
propagation in a cold magnetized
plasma
region
the ionosphere region
.
Since then there
were
m
any
aspects concerning th
is
refractive index expression
that
have been stud
ied
and published in Literat
ure
, for example
:
the determination of constants being in the Appleton
Hartree equation [4, 5];
the study of effect of electron
collisions on the formulas by magneto
ionic theory; the
development of theory, mathematical formulas concerning
the
complex
refractive indices of an ionized medium
[4, 5
and 7]; the conditions and the validity of some
ESTIMATION OF THE SPECIFIC REAL PHASE AND GROUP REFRACTIVE INDEXES BY
….
approximations related to the refractive index have also
been studied including the high order ionosphere effects on
the global positioning system observables a
nd means of
modeling [6, 7, 8 and 9]; the proposed model and predicted
values of the refractive index in the different layers of the
earth atmosphere medium [10]; the scattering mechanisms
of EMW [11]; the variation of the ionosphere conductivity
with diff
erent solar and geomagnetic conditions [12]; the
ionosphere absorption in vertical propagation [13]; the
atmospheric influences on microwaves propagation[14];
the stochastic perception of refractive index variability of
ionosphere [16]; and a lot of other
aspects have been
studied in references [15
19].
Recently
there are also
many works
continuing to
stud
y
deeply different problems
such as
determin
ation of
the
specific phase and group refractive indexes in different
propagating
environments
,
the calculation of the discrete
refractive indexes based on some conditions
,
the
calculation
of
the refractive index at
F
region altitudes
based on the global network of
Super Dual Auroral Radar
Network (SuperDARN)
[17
21]
.
In addition, presently
many attempts are devoted
to
researches of
The
Wireless Power Transmission (WPT) problems
using
high
power
microwaves and Laser
power b
eams
. During
propagation of
high power
beams,
the Earth atmosphere
region will be ionized
,
this fact
has
generated
some
research
problems
concerning
the propagating theory
development of EMWs power beams with
Gaussian
energy
distributions, the real
interactions
of High power beams
and
the Earth atmosphere this fact brought about
the
modified concepts of
the relative permittivity,
EMWs
velocit
ies
,
and
refractive indexes
[25
32, 39, 40
].
So far, it has a few
systematic
data of the specific phase
and group refractive indexes
vs. altitude of the ionized
region published
in the
L
iterature
[10,
27,
28,
37,
42,
43]
.
In our
previous
published work
[28
, 39
]
,
we have
studied
and
outlined
the relative permittivity and the
numerical
data of the
complex phase refractive index
by altitude
based on the
free electron density (
N
e
) distribution
[38].
In
t
his paper
using
t
he
first
order Appleton
Hartree
equations
by
pass
ing
the imaginary
parts due
to their values
are
very
small, we
estimated and outlined
the systematic numerical
results
of both
kinds of
the real phase and group refractive
indexes (
n
ph
and
n
gr
) vs. the altitude
concerning the single
and packet EMWs forms propagating
in the ionized regions
from 100 km to 1000 km
depending on the frequency range
of from 8 MHz to 5.8 GHz
.
II.
THE
EXPRESSIONS
OF
R
ELATIVE
PERMITTIVITY
AND
REAL
REFRACTIVE
INDEXES EXPRESS
IONS
FOR THE EARTH’S
IONIZED REGION
II.1.
Briefly on electromagnetic waves propagation in the
ionized region
The features of the ionosphere region
strongly influence
microwaves
propagation.
The
m
echanism of refraction
mainly occurs
in
the following ways
: when
t
he EMW
comes to the ionosphere
region
, the electric field
of EMW
forces the free electrons being in the ionosphere into
oscillation with the same frequency as that of the EMW.
Some of the
radio
frequency energy is transferred to this
resonant oscillation, and the oscillating electrons will then
either be lost due to recombination or will re
radiate the
original wave energy. The total refraction can occur when
the collision frequency of the
ionosphere is less than the
EMW frequency
,
and the electron density in the ionosphere
is high
enough [9,
14,
15,
25].
When the EMW frequency increases
to
higher
values
,
the
number
of reflection decreases and
then
not the
refraction. So there will be
a
defined limiting frequency
(
so
called,
critical frequency or
plasma frequency
) where
the signals could pass throu
gh the ionosphere layer
[9,14,
33]
.
If the propagating EMW’s frequency is higher than the
plasma frequency
of the ionosphere, then the free ele
ctrons
cannot respond fast enough, and they are not able to re
radiate the signal. The
expression determin
ing
the critical
frequency
has the form
:
f
critical
=9.
N
e
.
Herein,
N
e
[m
3
]
is
a
free
electron density
being in
the
ionosphere region
.
If we
do not take into account the number collision of ionized
particles (O, N, H…), then
the
effective permittivity (
ε
eff
)
as a function of
critical frequency or
plasma frequency
(
p
)
and EMW’s frequency (
)
that can be written as the
following form [32,
38,
39]:
ε
eff
=
ε
0
(
1
ω
p
2
ω
2
)
(a)
;
ω
p
=
N
e
e
2
m
ε
0
(b)
(
1
)
Based on this formula, the plasma frequency (
p
) of the
ionized region has been calculated, and its value is about
8 MHz [32
34].
II.2.
The expressions of the complex relative
permittivity
in the ionized region
In the ionized region,
the dielectric permittivity has been
accepted as a complex number.
Various processes are
labeled on the imaginary part: ionic and dipolar
relaxation, atomic and electronic resonances at higher
energies. As the response of the ionized region to external
fields
that
strongly depends on the
EMW
frequency
, the
response must always arise gradually after the applied
field
,
which can be represented by a phase difference
leading to the formation of the imaginary part.
The
complex relative permittivity in the ionized region can be
expre
ssed in the following form [
36,
38
,
39
]:
r
(
) =
1
4
π
N
e
.
e
2
ε
o
m
e
1
(
ω
2
+
S
2
)
i
4
πσ
ω
=
ε
r
(
)
+ i
ε
r
(
)
(2)
σ =
N
e
.
e
2
m
e
S
(
ω
2
+
S
2
)
(3)
Herein,
N
e
is the
free
electron density,
ω
is
the
angular
frequency,
m
e
is electron mass,
o
is the
vacuum dielectric
constant,
σ
is the conductivity, and
S
is the
collision
angular frequency of ionized particles
in the ionized
region. The
(
)
and
′′
(
)
are
denoted as
the real part
and imaginary part of the relative permittivity,
respectively.
Based on the graphic curves of free electron
density by altitude in the ionized region
,
the different kind
s
of conductivities and relative permittivity vs. the altitude

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