Geoscience Reference
In-Depth Information
where
M, N
are the principal radii of curvature of the ellipsoid, defined by
(2-149). Since
a
2
1+
e
2
cos
2
ϕ
3
/
2
=
a
2
1+
2
e
2
cos
2
ϕ
···
,
1
M
b
b
=
(2-210)
a
2
1+
e
2
cos
2
ϕ
1
/
2
=
a
2
1+
2
e
2
cos
2
ϕ
···
,
1
N
b
b
=
we have
a
2
2+2
e
2
cos
2
ϕ
=
2
b
1
M
1
N
b
a
2
(1 + 2
f
cos
2
ϕ
)
.
+
=
(2-211)
Here we have limited ourselves to terms linear in
f
, since the elevation
h
is
already a small quantity. Thus, we find from (2-209) after simple manipula-
tions:
∂γ
∂h
=
−
2
γ
a
(1 +
f
+
m −
2
f
sin
2
ϕ
)
.
(2-212)
The second derivative
∂
2
γ/∂h
2
may be taken from the spherical approxima-
tion, obtained by neglecting
e
2
or
f
:
∂
2
γ
∂h
2
=
∂
2
γ
∂a
2
γ
=
GM
a
2
∂γ
∂h
=
∂γ
2
GM
a
3
=
6
GM
a
4
,
∂a
=
−
,
,
(2-213)
so that
∂
2
γ
∂h
2
=
6
γ
a
2
.
(2-214)
Thus we obtain
γ
h
=
γ
1
a
2
h
2
.
2
a
(1 +
f
+
m
3
2
f
sin
2
ϕ
)
h
+
−
−
(2-215)
Using Eq. (2-198) for
γ
, we may also write the difference
γ
h
−
γ
in the form
1+
f
+
m
+
−
3
f
+
2
m
sin
2
ϕ
)
h
+
3
γ
a
2
γ
a
a
h
2
.
γ
h
−
γ
=
−
(2-216)
a
2
The symbol
γ
h
denotes the normal gravity for a point at latitude
ϕ
,situated
at height
h
above the ellipsoid;
γ
is the gravity at the ellipsoid itself, for the
same latitude
ϕ
, as given by (2-202) or equivalent formulas.
Second-order series developments for the
inner
gravity field are found in
Moritz (1990: Chap. 4); this is the main reason for such a development here,
because today one uses the closed formulas wherever possible.
