Geoscience Reference
In-Depth Information
Potential and gravity
By substituting these expressions into the closed formulas (2-123), (2-141),
(2-142), and (2-143), we obtain, up to and including the order
e
4
,thefol-
lowing relations.
Potential:
1
−
3
e
2
+
5
e
4
+
3
ω
2
a
2
.
U
0
=
GM
b
1
(2-185)
Gravity at the equator and the pole:
ab
1
14
e
2
m
,
γ
a
=
GM
3
3
−
2
m
−
(2-186)
a
2
1+
m
+
7
e
2
m
.
γ
b
=
GM
Clairaut's theorem:
1+
9
35
e
2
.
ω
2
b
γ
a
f
+
f
∗
=
5
2
(2-187)
The ratio
ω
2
a/γ
a
may be expressed as
ω
2
a
γ
a
=
m
+
2
m
2
,
(2-188)
which is a more accurate version of (2-180).
From the first equation of (2-186), we find
GM
=
abγ
a
1+
2
m
+
14
e
2
m
+
4
m
2
,
3
(2-189)
which gives the mass in terms of equatorial gravity. Using this equation, we
can express
GM
in Eq. (2-185) in terms of
γ
a
, obtaining
U
0
=
aγ
a
1
m
2
.
1
3
e
2
+
1
6
m
+
5
e
4
2
7
e
2
m
+
1
4
−
−
(2-190)
Here we have eliminated
ω
2
a
by replacing it with
GM m/b
.
Now we can turn to Eq. (2-146) for normal gravity. A simple manipula-
tion yields
1+
bγ
b
−aγ
a
aγ
b
sin
2
ϕ
γ
=
γ
a
1
.
(2-191)
a
2
−b
2
sin
2
ϕ
−
a
2
The denominator is expanded into a binomial series:
1
x
=1+
2
x
+
8
x
2
+
√
1
···
.
(2-192)
−
