Global Positioning System Reference
In-Depth Information
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acting on the satellite, normal orbits are indeed usable for orbital predictions over
short periods of time when low accuracy is sufficient. Thus, one of the popular uses
of normal orbits is for the construction of satellite visibility charts.
The normal motion of satellites is completely determined by Newton's law of
gravitation:
k
2
mM
r
2
=
F
(3.1)
In
(3.1),
M
and
m
denote the mass of the earth and the satellite, respectively,
k
2
is
th
e universal constant of gravitation,
r
is the geocentric distance to the satellite, and
F
is the gravitational force between the two bodies. This force can also be written as
F
=
ma
(3.2)
[56
w
here
a
in this instance denotes the acceleration experienced by the satellite. Com-
bi
ning (3.1) and (3.2) gives
Lin
—
1.8
——
Nor
*PgE
k
2
M
r
2
a
=
(3.3)
This equation can be written in vector form as
r
r
3
r
r
3
k
2
M
r
=−
=−µ
(3.4)
[56
where
k
2
M
µ =
(3.5)
is
the earth's gravitational constant. Including the earth's atmosphere, it has the value
µ =
10
8
m
3
s
−
2
. The vector
r
is directed from the central body (earth) to
th
e satellite. The sign has been chosen such that the acceleration is directed toward
th
e earth. The colinearity of the acceleration and the position vector as in (3.4) is a
ch
aracteristic of central gravity fields. A particle released from rest would fall along
a s
traight line toward the earth (straight plumb line).
Equation (3.4) is valid for the motion with respect to an inertial origin. In general,
on
e is interested in determining the motion of the satellite with respect to the earth.
Th
e modified equation of motion for accomplishing this is given by Escobal (1965,
p.
37) as
3,986,005
×
m)
r
r
3
k
2
(M
r
=−
+
(3.6)
Because
m
M
, the second term is often neglected and (3.6) becomes (3.4).
Figure 3.1 gives the position of the satellite in the
(q)
orbital plane coordinate
system
q
q
3
]
T
=
[
q
1
q
2
as
