Geology Reference
In-Depth Information
If our assumption that the reference figure of
the Earth is a level spheroid is correct, then this
equation will be satisfied for any colatitude ™.By
hypothesis, the flattening
f
is a small quantity.
Therefore, the following approximations are ap-
plicable:
Alternatively, assigning the flattening
f
,the
parameter
J
2
must be calculated as follows:
1
3
.2f
m/
J
2
D
(14.32)
The Stokes' parameter
J
2
plays a fundamental
role in geodesy and is termed
dynamic form
factor
or
ellipticity coefficient
. It can be deter-
mined by observation of satellite orbits. In fact,
by (
14.23
), to the second order the normal gravity
potential at distance
r
from the Earth's centre is
given by:
a
1
C
f cos
2
™
I
1
R.™/
Š
1
1
R
2
.™/
a
2
1
C
2f cos
2
™
I
R
2
.™/
Š
a
2
1
2f cos
2
™
1
Š
(14.27)
GMa
2
r
3
GM
r
Substituting into (
14.26
) and dividing both
sides by
a
/
GM
gives:
U.r;™/
Š
J
2
P
2
.cos™/
1
3
2
r
2
ŒP
2
.cos ™/
1
GM
D
1
C
f cos
2
™
aU
0
(14.33)
1
2
1
C
2f cos
2
™
3cos
2
™
1
J
2
While the first term in (
14.33
) maintains a
satellite on an elliptical orbit, the second one
introduces a precession in the satellite orbit, at
a rate that depends precisely on
J
2
. Therefore,
measuring the precession rate gives a measure of
the dynamic form factor. A recent estimate is:
2
1
2f cos
2
™
3cos
2
™
1
(14.28)
m
where the quantity:
J
2
D
1:081874
10
3
(14.34)
2
a
3
GM
m
(14.29)
The
World Geodetic System
1984 (WGS84)
represents the most recent consistent set of con-
stants and model parameters for the definition of
the normal gravity and the reference ellipsoid.
In this reference system, the basic parameters
assume the values listed in Table
14.1
.
The WGS84 reference can be used to calculate
the parameters of the theoretical normal gravity
associated with the ellipsoid. It can be shown
(e.g., Heiskanen and Moritz
1993
) that the gravity
field on the ellipsoid is given by:
represents the ratio between centrifugal potential
and gravitational potential for a sphere and is
termed
rotation parameter
. Neglecting all the
quadratic terms in
f
2
,
mf
,and
fJ
2
in (
14.28
)we
obtain:
aU
0
GM
D
1
3
2
J
2
cos
2
™
C
fcos
2
™
C
m
2
1
2
mcos
2
™
(14.30)
To obtain an equation that is independent from
™, it is necessary that the sum of all terms in cos
2
™
be zero. Therefore, the following relation must be
satisfied:
a”
a
cos
2
§
C
b”
b
sin
2
§
p
a
2
cos
2
§
C
b
2
sin
2
§
”.§/
D
(14.35)
1
2
.3J
2
C
m/
In this formula, which is known as
formula of
Somigliana
, § is the
geodetic latitude
,thatis,the
f
D
(14.31)
