Geoscience Reference
In-Depth Information
Then the abbreviated series
a
2
b
2
e
2
1+
e
2
−
=
e
2
e
4
,
=
−
a
2
(2-193)
bγ
b
−
aγ
a
aγ
a
=
−e
2
+
2
m
2
+
e
4
13
7
e
2
m
+
1
4
m
2
−
are introduced and we obtain, upon substitution,
γ
=
γ
a
1+
−
m
2
sin
2
ϕ
1
2
e
2
+
2
m
+
2
e
4
13
7
e
2
m
+
1
4
−
(2-194)
8
e
4
+
4
e
2
m
sin
4
ϕ
.
+
−
1
We may also express these quantities in terms of the flattening
f
by substi-
tuting the equation
1
e
2
=
1=2
f
+3
f
2
+
f
)
2
−
···
.
(2-195)
(1
−
The flattening
f
is most commonly used; it offers a slight advantage over
the second eccentricity
e
in that it is of the same order of magnitude as
m
:
it is not immediately apparent that
m
2
,e
2
m
,and
e
4
are quantities of the
same order of magnitude. We obtain
GM
=
abγ
a
1+
2
m
+
7
fm
++
4
m
2
,
(2-196)
U
0
=
aγ
a
1
7
fm
+
1
4
m
2
,
2
3
f
+
1
6
m
1
5
f
2
4
−
−
−
(2-197)
γ
=
γ
a
1+
− f
+
2
m
+
2
f
2
7
fm
+
1
4
m
2
sin
2
ϕ
26
−
(2-198)
2
f
2
+
2
fm
sin
4
ϕ
.
+
−
1
The last formula is usually abbreviated as
γ
=
γ
a
(1 +
f
2
sin
2
ϕ
+
f
4
sin
4
ϕ
)
,
(2-199)
so that we have
f
2
=
−f
+
2
m
+
2
f
2
26
7
fm
+
1
4
m
2
,
−
(2-200)
1
2
f
2
+
2
fm.
f
4
=
−
By substituting
sin
4
ϕ
=sin
2
ϕ
1
4
sin
2
2
ϕ,
−
(2-201)
