Geoscience Reference
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then the
C
2
n
are given by
1
− n
+5
n
C
ME
2
.
3
e
2
n
(2
n
+ 1)(2
n
+3)
−
A
C
2
n
=
−J
2
n
=(
−
1)
n
(2-170)
Here we have introduced the first eccentricity
e
=
E/a
.For
n
=1thisgives
the important formula
C
A
Ma
2
−
C
20
=
−
(2-171)
or, equivalently,
J
2
=
C
A
Ma
2
−
,
(2-172)
which is in agreement with the respective relation in (2-95) when taking into
account the rotational symmetry causing
A
=
B
.
Finally, we note that on eliminating
q
0
=(1
/i
)
Q
2
(
i
(
b/E
)) by using
Eq. (2-167), and
U
0
by using Eq. (2-122), we may write the expansion of
V
in ellipsoidal harmonics, Eq. (2-110), in the form
E
GM Q
0
i
u
E
V
(
u, β
)=
i
3
ME
2
Q
2
i
u
P
2
(sin
β
)
.
2
E
3
G
C − A −
15
i
1
+
E
(2-173)
This shows that the coecients of the ellipsoidal harmonics of degrees zero
and two are functions of the mass and of the difference between the two
principal moments of inertia. The analogy to the corresponding spherical-
harmonic coecients (2-91) is obvious.
Thisisaclosedformula,notatrun-
cated series!
2.10
Series expansions for the normal gravity field
Since the earth ellipsoid is very nearly a sphere, the quantities
E
=
√
a
2
b
2
,
−
linear eccentricity,
e
=
E
a
,
first (numerical) eccentricity,
(2-174)
e
=
E
b
,
second (numerical) eccentricity,
f
=
a
−
b
,
flattening,
a
