Geoscience Reference
In-Depth Information
is by definition an equipotential surface
U
(
x, y, z
)=
U
0
.
(2-99)
It is now convenient to introduce the ellipsoidal-harmonic coordinates
u, β, λ
of Sect. 1.15. The ellipsoid
S
0
is taken as the reference ellipsoid
u
=
b
.
Since
V
(
u, β
), the gravitational part of the normal potential
U
, will be
harmonic outside the ellipsoid
S
0
, we use the second equation of the series
(1-174). The field
V
has rotational symmetry and, hence, does not depend
on the longitude
λ
. Therefore, all nonzonal terms, which depend on
λ
,must
be zero, and there remains
Q
n
i
u
E
V
(
u, β
)=
∞
i
A
n
P
n
(sin
β
)
,
(2-100)
b
E
Q
n
n
=0
where
E
=
a
2
− b
2
(2-101)
is the linear eccentricity. The centrifugal potential Φ(
u, β
)isgivenby
Φ(
u, β
)=
2
ω
2
(
u
2
+
E
2
)cos
2
β.
(2-102)
Therefore, the total normal gravity potential may be written
Q
n
i
u
E
U
(
u, β
)=
∞
A
n
P
n
(sin
β
)+
2
ω
2
(
u
2
+
E
2
)cos
2
β.
i
(2-103)
b
E
Q
n
n
=0
On the ellipsoid
S
0
we have
u
=
b
and
U
=
U
0
. Hence,
∞
A
n
P
n
(sin
β
)+
2
ω
2
(
u
2
+
E
2
)cos
2
β
=
U
0
.
(2-104)
n
=0
This equation applies for all points of
S
0
, that is, for all values of
β
.Since
b
2
+
E
2
=
a
2
(2-105)
and
cos
2
β
=
3
1
P
2
(sin
β
)
,
−
(2-106)
we have
∞
A
n
P
n
(sin
β
)+
3
ω
2
a
2
1
3
ω
2
a
2
P
2
(sin
β
)
−
−
U
0
= 0
(2-107)
n
=0
