Geoscience Reference
In-Depth Information
and similar parameters that characterize the deviation from a sphere are
small. Therefore, series expansions in terms of these or similar parameters
will be convenient for numerical calculations.
Linear approximation
In order that the readers may find their way through the subsequent practical
formulas, we first consider an approximation that is linear in the flattening
f
.
Here we get particularly simple and symmetrical formulas which also exhibit
plainly the structure of the higher-order expansions.
It is well known that the radius vector
r
of an ellipsoid is approximately
given by
f
sin
2
ϕ
)
.
(2-175)
As we will see subsequently, normal gravity may, to the same approximation,
be written
r
=
a
(1
−
γ
=
γ
a
(1 +
f
∗
sin
2
ϕ
)
.
(2-176)
90
◦
, at the poles, we have
r
=
b
and
γ
=
γ
b
. Hence, we may write
b
=
a
(1
For
ϕ
=
±
b
=
γ
a
(1 +
f
∗
)
,
−
f
)
,
(2-177)
and solving for
f
and
f
∗
,weobtain
f
=
a
−
b
,
a
(2-178)
f
∗
=
γ
b
−
γ
a
,
γ
a
so that
f
is the flattening defined by (2-174), and
f
∗
is an analogous quantity
which may be called
gravity flattening
.
To the same approximation, (2-143) becomes
f
+
f
∗
=
2
m,
(2-179)
where
=
ω
2
a
γ
a
=
centrifugal force at equator
gravity at equator
m
.
(2-180)
This is
Clairaut's theorem
in its original form. It is one of the most striking
formulas of physical geodesy: the (geometrical) flattening
f
in (2-178) can
be derived from
f
∗
and
m
, which are purely dynamical quantities obtained
by gravity measurements; that is,
the flattening of the earth can be obtained
from gravity measurements
.
Clairaut's formula is only a first approximation and must be improved,
first by the inclusion of higher-order ellipsoidal terms in
f
, and secondly by
taking into account the deviation of the earth's gravity field from the normal
gravity field. But the principle remains the same.
