Geology Reference
In-Depth Information
"
1
R
r
2n
J
2n
P
2n
.cos ™/
#
X
GM
r
U.r;™/
D
n
D
1
1
3
2
r
2
ŒP
2
.cos ™/
1
(14.23)
where the zonal coefficients
J
2
n
are analogous to
the Stokes' coefficients of the spherical harmonic
expansion (
14.5
). If
a
and
b
are the equatorial and
polar radii, respectively, then the
flattening f
will
be given by:
a
b
a
f
D
(14.24)
To apply this model to the Earth, we must
assume that the reference figure of the Earth is a
level spheroid
, that is, an ellipsoid of revolution
which is an equipotential surface of a
normal
gravity field
. In this case, the geoid representing
the actual figure of the Earth will be an equipo-
tential surface that deviates from the reference
spheroid because of lateral heterogeneity in the
mass distribution within the Earth. To express
the flattening of the level spheroid in terms of
Stokes' coefficients, let us consider the approx-
imate equation of a spheroid, which allows to
write the Earth's radius,
R
, as a function of the
colatitude:
Fig. 14.2
Mass redistribution (
arrows
) in a homoge-
neous deformable body once that it is put in motion about
a spin axis. The grey zone indicates the area for which
U
(™)
<
U
>
GM
R
I
U. =2/
U.0/
D
U. /
D
GM
R
C
1
2
2
R
2
D
(14.22)
The excess potential along the Equator at dis-
tance
r
D
R
from the centre implies that the body
must redistribute its mass, pushing material away
from the centre along the equatorial belt and
flattening at the poles, as illustrated in Fig.
14.2
.
Consequently, it will acquire an oblate form.
However, such an adjustment will change the
gravity potential
V
D
V
(
r
), because the new shape
does not have anymore a spherical symmetry. It is
quite intuitive that a process would start such that
the new potential first adds a zonal term
P
2
(cos™),
which determines in turn an additional
P
4
(cos™)
term in
U
(™), that controls now a new change of
shape, and so on.
The final shape of the body will be that of
an
oblate spheroid
whose potential is an infinite
series of zonal harmonics of even degree:
R.™/
Š
a
1
f cos
2
™
(14.25)
Substituting
r
in the spherical harmonic ex-
pansion (
14.23
) by this expression, gives the
constant potential
U
0
of the reference ellipsoid.
To this purpose, we will consider the second-
order approximation of the geopotential (
14.23
)
with
R
D
a
,whichis:
1
R
2
.™/
J
2
P
2
.cos ™/
a
2
GM
R.™/
U
0
Š
1
3
2
R
2
.™/ŒP
2
.cos™/
1
(14.26)
