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
=
G ρ.
(5.4)
Thus, the equation governing the geopotential is easily shown to be
2 U
2
=
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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