Global Positioning System Reference
In-Depth Information
Kepler's three laws are listed below (see Chapter 1 in ref. 11):
First Law:
The orbit of each planet is an ellipse with the sun at a focus.
Second Law:
The line joining the planet to the sun sweeps out equal areas in
equal times.
Third Law:
The square of the period of a planet is proportional to the cube
of its mean distance from the sun.
These laws also apply to the motion of the GPS satellites. The satellite orbit
is elliptical with the earth at one of the foci. Figure 3.6 shows the orbit of a GPS
satellite. The center of the earth is at
F
and the position of the satellite is at
S
.
The angle
v
is called the actual anomaly. In order to illustrate the basic concept,
the shape of the ellipse is overemphasized. The actual orbit of the satellite is
very close to a circle. The point nearest to the prime focus is called the perigee
and the farthest point is called the apogee.
Kepler's second law can be expressed mathematically as (Figure 3.6)
t
−
t
p
A
1
T
πa
s
b
s
=
(
3
.
17
)
where
t
presents the satellite position at time
t
,
t
p
is the time when the satellite
passes the perigee,
A
1
is the area enclosed by the lines
t
t
p
,andthe
ellipse,
T
is the period of the satellite,
a
s
and
b
s
are the semi-major and semi-
minor axes of the orbit, and
πa
s
b
s
is the total area of the ellipse. This equation
states that the time to sweep the area
A
1
is proportional to the time
T
to sweep
the entire area of the ellipse.
=
t,t
=
FIGURE 3.6
Elliptical orbit of a satellite.
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