Geoscience Reference
In-Depth Information
where the notations of Sect. 2.1 are used. The rectangular coordinate system
x, y, z
will be applied in this chapter in the usual sense: it is geocentric, the
x
-and
y
-axes lying in the equatorial plane with Greenwich longitudes 0
◦
and
90
◦
East, respectively, and the
z
-axis being the rotation axis of the earth.
The sign of the components of
g
,
,
δ
g
, etc., will always be chosen so
that they are positive in the direction of increasing coordinates.
γ
6.2
Normal gravity vector
The gravity field of an equipotential ellipsoid is best expressed in terms of
ellipsoidal-harmonic coordinates
u, β, λ
, introduced in Sects. 1.15 and 2.7.
They are related to rectangular coordinates
x, y, z
by
x
=
√
u
2
+
E
2
cos
β
cos
λ,
y
=
√
u
2
+
E
2
cos
β
sin
λ,
(6-6)
z
=
u
sin
β.
If
x, y, z
are given, then
u, β, λ
can be computed by closed formulas. First
we find
x
2
+
y
2
=(
u
2
+
E
2
)cos
2
β,
z
2
=
u
2
sin
2
β.
(6-7)
Eliminating
β
between these two equations, we obtain a quadratic equation
for
u
2
, whose solution is
− E
2
)
1
.
4
E
2
z
2
(
x
2
+
y
2
+
z
2
2
+
1
u
2
=(
x
2
+
y
2
+
z
2
1+
(6-8)
E
2
)
2
2
−
Then
β
is given by
tan
β
=
z
√
u
2
+
E
2
u
x
2
+
y
2
,
(6-9)
and for
λ
we simply have
y
x
.
tan
λ
=
(6-10)
With known ellipsoidal-harmonic coordinates, the normal potential
U
is
given by (2-126):
q
0
sin
2
β
3
+
2
ω
2
(
u
2
+
E
2
)cos
2
β.
(6-11)
U
(
u, β
)=
GM
E
tan
−
1
E
u
+
2
ω
2
a
2
q
1
−
