Geology Reference
In-Depth Information
The reference level surface is an equipotential
U
0
of the total geopotential, or
gravity potential,
U
=
V
+
W
,
(5.1)
where
V
is the gravitational potential and
W
is the centrifugal potential. With
Ω
as the Earth's mean rate of rotation,
r
the radius from the geocentre and θ the
geocentric co-latitude, the centrifugal force per unit mass (3.76) becomes
2
r
sinθ.
In turn, this can be expressed as the negative gradient of the centrifugal potential
Ω
2
x
1
+
x
2
1
2
Ω
W
=−
1
2
Ω
2
r
2
sin
2
=−
θ
1
3
Ω
1
3
Ω
2
r
2
2
r
2
P
2
(cosθ),
=−
+
(5.2)
1
/2 being the second-degree Legendre polynomial. The gravita-
tional potential is
3cos
2
P
2
=
θ
−
=−
G
ρ
(
r
)
|
d
V
,
V
(
r
)
(5.3)
r
|
r
−
where the potential at the field point at
r
is found by integrating over all source
points at
r
. Further,
G
is the universal constant of gravitation and ρ is the mass
density. Just as expression (1.168) is the solution of the Poisson equation (1.163),
the gravitational potential obeys the Poisson equation
2
V
∇
=
4π
G
ρ.
(5.4)
Thus, the equation governing the geopotential is easily shown to be
2
U
2
∇
=
4π
G
ρ
−
2
Ω
.
(5.5)
Identification of the particular equipotential surface,
U
0
, being used for reference
can be made through a number of geometrical parameters, but the
mean equivolu-
metric radius
,
d
, appears most convenient to use. The mean equivolumetric radius,
or more simply the
mean radius
, is the radius of a sphere containing the same
volume as the reference surface. In considering the global figure problem, it is
illuminating to make (5.5) non-dimensional by expressing the variables in terms of
d
and the gravitational attraction
GM
/
d
2
on the surface of a spherically symmetric
mass distribution of radius
d
enclosing the same mass
M
and volume
as the
Earth's surface. The geopotential
U
is then expressed in units of the gravitational
V
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