Geology Reference
In-Depth Information
The second-order terms in (5.83) are not required to obtain (5.84), and substitution
of
d
for
a
is su
cient to obtain (5.85). Geodetic formulae have traditionally been
expressed in terms of the ratio of centrifugal force to total gravity at the equator,
2
g
e
,
a
Ω
m
1
=
(5.86)
the parameter used by Newton, rather than
m
. Their connection can easily be
derived from the dimensional form of expression (5.56) for equatorial gravity and
the expansion (5.83). These give
2
d
1
+···
Ω
+
(1/3)
f
m
1
=
[
GM
/
d
2
]
1
+···
−
(3/2)
m
+
(1/3)
f
=
m
1
3
2
m
+···
+
,
(5.87)
first-order approximation being adequate for subsequent applications. Again to first
order,
m
1
1
m
1
1
3
2
m
3
2
m
1
m
=
−
+···
=
−
+···
.
(5.88)
With
m
1
as a replacement for
m
, relations equivalent to (5.84) and (5.85) are
1
3
m
1
2
3
f
1
2
21
m
1
f
1
8
21
κ
+···
,
2
m
1
+
3
f
2
J
2
=−
+
+
−
+
(5.89)
4
7
m
1
f
4
32
35
κ
+···
.
5
f
2
J
4
=
−
−
(5.90)
Equatorial gravity is neither calculable nor easily observed, and it is desirable to
introduce
2
a
3
2
d
3
GM
(1
m
1
=
Ω
GM
=
Ω
+
f
+···
)
=
m
(1
+
f
+···
),
(5.91)
or
m
=
m
1
(
1
−
f
+···
)
.
(5.92)
The first-order form (5.92) is su
cient to convert (5.84) and (5.85) to
1
2
3
f
3
1
8
21
κ
+···
,
3
m
1
+
7
m
1
f
3
f
2
J
2
=−
+
−
+
(5.93)
4
4
32
35
κ
+···
.
7
m
1
f
5
f
2
J
4
=
−
−
(5.94)
J
2
,
J
4
,
a
and
GM
are well determined by tracking measurements on satellites and
spacecraft, while
is very well known from centuries of astronomical observa-
tions. Thus, equations (5.93) and (5.94) can be solved for the remaining unknowns
Ω
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