Geoscience Reference
In-Depth Information
The most significant geodetic result is the reliable determination of
J
2
and, therefore, of the flattening
f
, around 1/298.25. Already in 1964, the In-
ternational Astronomical Union (IAU) adopted the value 298
.
25 correspond-
ing to
J
2
= 1082
.
7
·
10
−
6
(see Sect. 2.11), followed by the IAG International
Geodetic Reference Systems 1967 and then 1980, which in the slightly dif-
ferent form of the World Geodetic System 1984 (WGS 84) is standard even
today (2005).
7.4
Rectangular coordinates of the satellite and
perturbations
We now describe how the rectangular coordinates of the satellite are com-
puted from the orbital elements. Then we will outline how they are affected
by the irregularities of the gravity field. These considerations are necessary
for the determination of tesseral harmonics from satellite observations.
We introduce an equatorial coordinate system
X
0
Y
0
Z
0
that is at rest
with respect to the stars. The origin is at the earth's center of mass. The
Z
0
-axis coincides with its axis of rotation; the
X
0
Y
0
-plane is the equatorial
plane. The
X
0
-axis is the line of intersection of the equatorial plane and
the ecliptic (the plane of the earth's orbit around the sun); according to
astronomical terminology, it points to the
vernal equinox
.Thiscoordinate
system
X
0
Y
0
Z
0
is fundamental in spherical astronomy. Note that the di-
rections of the coordinate axes so defined are not completely constant in
time. This fact requires certain refinements for which the reader is referred
to Moritz and Mueller (1987: Chap. 7). In the present context, we consider
the
X
0
Y
0
Z
0
-system as constant in time.
The relation between the rectangular coordinates of a satellite and the
elements of its osculating ellipse (Sect. 7.2) at a certain time is found as fol-
lows. Consider Fig. 7.3 and the coordinate system
e
1
,
e
2
defining the orbital
plane. Assuming
e
3
orthogonal to this plane,
⎡
⎤
cos
v
sin
v
0
⎣
⎦
r
(7-31)
is the representation of the satellite in this system. This result may be trans-
formed into the equatorial system
X
0
Y
0
Z
0
by a rotation matrix
R
and re-
sults in a vector denoted as
X
0
=[
X
0
,Y
0
,Z
0
]. The transformation is
