Geoscience Reference
In-Depth Information
Fig. 1
Centrifugal force
z
ω
P
p
Fz
x
y
p
y
x
The vector
F
z
is given by
⎛
⎞
δΦ
δ
⎛
⎞
⎝
x
δΦ
δ
⎠
2
x
ω
2
p
⎝
⎠
=
2
y
F
z
=
ω
=
ω
grad
Φ
=
.
(12)
y
δΦ
δ
0
x
The corresponding centrifugal potential function is
1
2
ω
2
x
2
y
2
Φ
=
(
+
).
(13)
As mentioned above the gravity vector
g
is the resultant of the gravitational force
F
and the centrifugal force
F
z
. Accordingly, the potential of gravity
W
is the sum of
the potentials of the gravitational potential
V
and the centrifugal potential
Φ
:
G
v
r
d
v
1
2
ω
2
x
2
y
2
W
=
W
(
x
,
y
,
z
)
=
V
+
Φ
=
+
(
+
).
(14)
Combining the Laplace expression of
with Poisson's equation for
V
(Eq.
8
), leads
to the generalized Poisson equation for the gravity potential
W
:
Φ
2
Δ
W
=−
4
π
G
ρ
+
2
ω
.
(15)
The gradient of
W
is called gravity vector and describes the total force acting on a
unit mass.
δ
W
δ
x
δ
W
δ
y
δ
W
δ
g
=
grad W
=
.
(16)
z
The vector magnitude is called
gravity
and has the unit of an acceleration, and the
direction of the vector is the direction of the plumb line.
