Global Positioning System Reference
In-Depth Information
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3.1.1 Kepler Elements
Si
x Kepler elements are often used to describe the position of satellites in space. To
si
mplify attempts to study satellite motions, we study so-called normal orbits. For
no
rmal orbits the satellites move in an orbital plane that is fixed in space; the actual
pa
th of the satellite in the orbital plane is an ellipse in the mathematically strict sense.
O
ne focal point of the orbital ellipse is at the center of the earth. The conditions
le
ading to such a simple orbital motion are as follows:
1. The earth is treated as a point mass, or, equivalently, as a sphere with spherically
symmetric density distribution. The gravitational field of such a body is radially
symmetric; i.e., the plumb lines are all straight lines and point toward the center
of the sphere.
2. The mass of the satellite is negligible compared to the mass of the earth.
3. The motion of the satellite takes place in a vacuum; i.e., there is no atmospheric
drag acting on the satellite and no solar radiation pressure.
4. No sun, moon, or other celestial body exerts a gravitational attraction on the
satellite.
[54
Lin
—
2.0
——
Nor
PgE
The orbital plane of a satellite moving under such conditions is shown in Figure 3.1.
The ellipse denotes the path of the satellite. The shape of the ellipse is determined by
the semimajor axis
a
and the semiminor axis
b
. The symbol
e
denotes the eccentricity
of the ellipse. The ellipse is enclosed by an auxiliary circle with radius
a
. The principal
axes of the ellipse form the coordinate system
(
ξ
,
η
)
.
S
denotes the current position
of the satellite; the line
SS
[54
axis. The
coordinate system
(q
1
,q
2
)
is in the orbital plane with its origin at the focal point
F
of the ellipse that coincides with the center of the earth. The third axis
q
3
, not shown
in Figure 3.1, completes the right-handed coordinate system. The geocentric distance
from the center of the earth to the satellite is denoted by
r
. The orbital locations closest
to and farthest from the focal point are called the perigee and apogee, respectively.
The true anomaly
f
and the eccentric anomaly
E
are measured counterclockwise, as
shown in Figure 3.2.
is in the orbital plane and is parallel to the
η
q
2
S
circle
orbital path
S
satellite
r
b
E
f
e
perigee
q
1
apogee
F
Figure 3.1
Coordinate systems in the orbital plane.