Geoscience Reference
In-Depth Information
Expressing
P
2
as
P
2
(sin
β
)=
2
1
2
sin
2
β
−
(2-125)
and, finally, adding the centrifugal potential Φ =
ω
2
(
u
2
+
E
2
)cos
2
β/
2from
(2-102), the normal gravity potential
U
results as
3
+
2
ω
2
(
u
2
+
E
2
)cos
2
β.
(2-126)
The only constants that occur in this formula are
a, b, GM
,and
ω
.Thisis
in complete agreement with Dirichlet's theorem.
q
0
sin
2
β
U
(
u, β
)=
GM
E
tan
−
1
E
u
+
2
ω
2
a
2
q
1
−
2.8
Normal gravity
Referring to the line element in ellipsoidal-harmonic coordinates according
to (1-155), replacing
ϑ
by its complement 90
◦
−
β
,weget
ds
2
=
w
2
du
2
+
w
2
(
u
2
+
E
2
)
dβ
2
+(
u
2
+
E
2
)cos
2
βdλ
2
,
(2-127)
where
u
2
+
E
2
sin
2
β
u
2
+
E
2
w
=
(2-128)
has been introduced. Thus, along the coordinate lines we have
u
=variable,
β
= constant,
λ
= constant,
ds
u
=
wdu,
ds
β
=
w
√
u
2
+
E
2
dβ ,
β
=variable,
u
= constant,
λ
= constant,
ds
λ
=
√
u
2
+
E
2
cos
βdλ.
(2-129)
λ
=variable,
u
= constant,
β
= constant,
The components of the normal gravity vector
γ
=grad
U
(2-130)
along these coordinate lines are accordingly given by
∂U
∂s
u
1
w
∂U
∂u
,
γ
u
=
=
∂U
∂s
β
1
w
√
u
2
+
E
2
∂U
∂β
,
γ
β
=
=
(2-131)
∂U
∂s
λ
1
√
u
2
+
E
2
cos
β
∂U
∂λ
γ
λ
=
=
=0
.
