Geoscience Reference
In-Depth Information
direction of the vector
p
=[
x, y,
0]
(2-3)
and is, therefore, given by
f
=
ω
2
p
=[
ω
2
x, ω
2
y,
0]
.
(2-4)
The centrifugal force can also be derived from a potential
Φ=
1
2
ω
2
(
x
2
+
y
2
)
,
(2-5)
so that
∂
Φ
∂x
,
.
∂
Φ
∂y
,
∂
Φ
∂z
f
=gradΦ
≡
(2-6)
Substituting (2-5) into (2-6) yields (2-4).
In the introductory remark above, we mentioned that gravity is the resul-
tant of gravitational force and centrifugal force. Accordingly, the potential of
gravity,
W
, is the sum of the potentials of gravitational force,
V
, cf. (1-12),
and centrifugal force, Φ:
W
=
W
(
x, y, z
)=
V
+Φ=
G
v
l
dv
+
1
2
ω
2
(
x
2
+
y
2
)
,
(2-7)
where the integration is extended over the earth.
Differentiating (2-5), we find
∂
2
Φ
∂x
2
+
∂
2
Φ
∂y
2
+
∂
2
Φ
∂z
2
=2
ω
2
.
∆Φ
≡
(2-8)
If we combine this with Poisson's equation (1-17) for
V
,wegetthe
general-
ized Poisson equation
for the gravity potential
W
:
4
πG
+2
ω
2
.
∆
W
=
−
(2-9)
Since Φ is an analytic function, the discontinuities of
W
are those of
V
:some
second derivatives have jumps at discontinuities of density.
The gradient vector of
W
,
∂W
∂x
,
∂W
∂y
,
∂W
∂z
g
=grad
W ≡
(2-10)
