Geoscience Reference
In-Depth Information
7.2
Satellite orbits
The first spectacular result from satellite observations, well advertised by
NASA around 1960, was the “discovery” that “the earth is not an ellipsoid
but rather shaped like a pear”. This pear shape is caused by the spherical
harmonic
J
3
. Its effect, at the North and South Poles, is on the order of
30 m, by three orders of magnitude less than the ellipticity coming from
J
2
,
whose linear effect
a
b
is about 20 km (!).
The first real result, also found around 1960, was a dramatic improvement
in the accuracy of the flattening
f
itself, which lead to a change from 1/297.3,
generally believed before, to 1/298.25, corresponding to a linear improvement
of the earth size of about 70 m!
The earth's flattening causes the largest but not the only deviation of the
earth gravitational field from that of a homogeneous sphere. Generally, the
gravitational potential can be expanded into a series of spherical harmonics
according to Sect. 2.5, Eq. (2-78):
−
1
a
r
n
J
n
P
n
(cos
ϑ
)
∞
V
=
GM
r
−
n
=2
r
n
[
C
nm
cos
mλ
+
S
nm
sin
mλ
]
P
nm
(cos
ϑ
)
.
(7-1)
Here the terms containing
J
n
are the zonal harmonics, and those containing
S
nm
and
C
nm
are the tesseral harmonics.
The former notations
J
nm
=
a
+
∞
n
n
=2
m
=1
S
nm
are not used any
more for the
tesseral
harmonic coecients; for the
zonal
harmonics, the use
of
J
n
has prevailed so far, but also
C
n
0
=
−
C
nm
and
K
nm
=
−
J
n
is being used.
Considering the moon, the only term of appreciable influence is
J
2
,which
represents the flattening. Artificial satellites are, compared to the moon,
much closer to the earth; typical heights above ground of a geodetically
used satellite range from some 300 km up to 20 000 km. Hence, they are
also influenced by harmonics other than
J
2
and can, therefore, be used to
determine harmonics of low degree. For this purpose, we must study the
effect of gravitational disturbances on the orbits of close satellites.
Before we can do this, we must briefly review the theory of an undisturbed
orbit, which means that the gravitational potential has the form
V
=
GM
r
−
,
(7-2)
all
C
sand
S
s being zero. This represents the gravitational field of a point
mass or a homogeneous sphere. Then the motion of a satellite is described
