Geoscience Reference
In-Depth Information
Second-order expansion
We now expand the closed formulas of the two preceding sections into series
in terms of the second numerical eccentricity
e
and the flattening
f
,in
general up to and including
e
4
or
f
2
. Terms of the order of
e
6
or
f
3
and
higher will usually be neglected.
We start from the series
E
u
3
E
u
5
E
u
7
tan
−
1
E
u
=
E
1
3
+
1
5
1
7
u
−
−
+
···
,
q
=2
1
3
·
5
,
E
u
3
E
u
5
E
u
7
2
5
·
7
3
7
·
9
−
+
− ···
(2-181)
q
=6
1
3
.
E
u
3
E
u
5
E
u
7
1
1
−
+
− ···
·
5
5
·
7
7
·
9
The first two series have already been used in the preceding section in (2-
154) and (2-156), respectively; the third is obtained by substituting the
tan
−
1
series into the closed formula (2-133) for
q
.
On the reference ellipsoid
S
0
,wehave
u
=
b
and
E
u
=
E
b
=
e
,
(2-182)
so that
1
3
e
3
+
5
e
5
tan
−
1
e
=
e
−
···
,
15
e
3
1
···
,
2
6
7
e
2
q
0
=
−
q
0
=
5
e
2
1
−
···
,
(2-183)
3
7
e
2
e
q
0
q
0
=3
1+
7
e
2
···
.
We also need the series
=
a
1
···
.
a
√
1+
e
2
1
2
e
2
+
8
e
4
b
=
−
(2-184)
