Global Positioning System Reference
In-Depth Information
The third law can be stated mathematically as
T
2
a
s
=
4
π
2
µ
4
π
2
GM
≡
(
3
.
18
)
=
3
.
986005
×
10
14
meters
3
/
sec
2
(ref. 12) is the gravitational
constant of the earth. Thus, the right-hand side of this equation is a constant. In
this equation the semi-major axis
a
s
is used rather than the mean distance from
the satellite to the center of the earth. In reference 11 it is stated that
a
s
can be
used to replace the mean distance because the ratio of
a
s
to the mean distance
r
s
is a constant. This relationship can be shown as follows. If one considers the
area of the ellipse orbit equal to the area of a circular orbit with radius
r
s
,then
where
µ
=
GM
a
s
r
s
=
r
s
b
s
πr
s
πa
s
b
s
=
or
(
3
.
19
)
Since
a
s
,
b
s
,
r
s
are constants,
a
s
and
r
s
is related by a constant.
3.11 KEPLER'S EQUATION
(
11,13
)
In the following paragraphs Kepler's equation will be derived and the mean
anomaly will be defined. The reason for this discussion is that the information
given by the GPS system is the mean anomaly rather than the actual anomaly
that is used to calculate the position of a satellite.
In order to perform this derivation, a few equations from the previous chapter
will be repeated here. The eccentricity is defined as
a
s
−
b
s
c
s
a
s
e
s
=
≡
(
3
.
20
)
a
s
where
c
s
is the distance from the center of the ellipse to a focus. For an ellipse,
the
e
s
value is 0
<e
s
<
1. When
a
s
0, which represents a circle.
The eccentricity
e
s
can be obtained from data transmitted by the satellite.
In Figure 3.7 an elliptical satellite orbit and a fictitious circular orbit are shown.
The center of the earth is at
F
and the satellite is at
S
.Thearea
A
1
is swept by
the satellite from the perigee point to the position
S
. This area can be written as
=
b
s
,then
e
s
=
A
1
=
area
PSV
−
A
2
(
3
.
21
)
In the previous chapter Equation (2.24) shows that the heights of the ellipse
and the circle can be related as
QP
SP
a
s
b
s
=
(
3
.
22
)
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