Geoscience Reference
In-Depth Information
we finally obtain
γ
=
γ
a
(1 +
f
∗
sin
2
ϕ
1
4
f
4
sin
2
2
ϕ
)
,
−
(2-202)
where
f
∗
=
γ
b
−
γ
a
=
f
2
+
f
4
(2-203)
γ
a
is the “gravity flattening”.
Coe
cients of spherical harmonics
Equation (2-167) for the principal moments of inertia yields at once
me
q
0
C
A
ME
2
−
=
1
2
45
3
−
.
(2-204)
Expanding
q
0
by means of (2-183), we find
e
2
3
e
2
7
e
2
m
.
C
A
ME
2
−
1
1
2
=
−
3
m −
(2-205)
Substituting this into (2-170) yields
−C
20
=
J
2
=
C
A
ME
2
−
=
3
e
2
1
1
3
e
4
+
21
e
2
m
1
−
3
m −
(2-206)
=
3
f
1
1
3
f
2
+
2
−
3
m
−
21
fm,
1
5
e
4
+
7
e
2
m
=
4
5
f
2
+
7
fm.
−
C
40
=
J
4
=
−
−
(2-207)
The higher
C
or
J
, respectively, are already of an order of magnitude that
we have neglected.
Gravity above the ellipsoid
Denoting the height above the ellipsoid as ellipsoidal height
h
, then, in case
of a small height, the normal gravity
γ
h
at this height can be expanded into
aseriesintermsof
h
:
∂
2
γ
γ
h
=
γ
+
∂γ
∂h
h
+
1
∂h
2
h
2
+
···
,
(2-208)
2
where
γ
and its derivatives are referred to the ellipsoid, where
h
=0.
The first derivative
∂γ/∂h
may be obtained by applying Bruns' formula
(2-147) together with (2-148) to the ellipsoidal height
h
(instead of
H
):
∂h
=
−γ
1
∂γ
1
N
−
2
ω
2
,
+
(2-209)
M
