Global Positioning System Reference
In-Depth Information
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between them. It can happen that GPS is adversely affected even when daily sunspot
numbers are actually low. Kunches and Klobuchar (2000) point out that GPS opera-
tions are more problematic during certain years of the solar cycle and during certain
months of those years. The years at or just after the solar maximum will be stormy,
and the months near the equinoxes will contain the greatest number of storm days.
Sunspots are good for long-term prediction of ionospheric states.
The Appleton-Hartree formula is usually taken in the literature as the start for
developing the ionospheric index of refraction that is applicable to the range of GPS
frequencies. The formula is valid for a homogeneous plasma that consists of electrons
and heavy positive ions, a uniform magnetic field, and a given electron collision
frequency. Following Davies (1990, p. 72), the Appleton-Hartree formula is
X
n
2
=
1
−
(6.60)
[21
Y
T
Y
T
Y
L
1
−
iZ
−
iZ )
±
iZ )
2
+
2
(
1
−
X
−
4
(
1
−
X
−
Lin
—
3.4
——
Nor
PgE
Si
nce the goal is to find the ionospheric index of refraction that applies to the GPS
fre
quency
f
, several simplifications are permissible. The element
Z
= ν
/f
is the
ra
tio of the electron collision frequency
ν
and the satellite frequency. This term
qu
antifies the absorption. We simply set
Z
=
0. The index
n
now becomes a real
nu
mber; in the notation of (6.3) we have
n
(f )
0. The symbols
Y
T
and
Y
L
relate
to
the magnetic field with reference to the direction of the wave normal, i.e., phase
pr
opagation. The commonly used first-order ionospheric delay expression is obtained
by
setting
Y
T
=
0. An excellent summary of the higher-order ionospheric
te
rms and their effects on the GPS observables is given in Odijk (2002). With these
si
mplifications we obtain
=
Y
L
=
[21
f
N
f
2
n
2
=
1
−
X
=
1
−
(6.61)
The plasma frequency
f
N
is a measure of the electron motion (oscillation) around the
heavy ions. It is a basic constant of plasma. Davies (1990, pp. 21, 73) gives
N
e
e
2
f
N
=
2
ε
0
m
e
=
80
.
6
N
e
[el
/
m
3
]
(6.62)
π
4
10
−
19
coulombs
denotes the electron charge with
In (6.62) the symbol
e
=
1
.
60218
·
10
−
12
faradays/m is the permittivity
of free space. The relevant term is the electron density
N
e
, which is typically given in
units of electrons per cubic meter [el/m
3
]. Substituting (6.62) in (6.61) and developing
a series gives
=
·
10
−
31
kg;
ε
0
=
·
mass
m
e
9
.
10939
8
.
854119
f
N
f
2
f
N
n
=
1
−
=
1
−
2
f
2
+··· =
1
−
N
I
(6.63)
