Geoscience Reference
In-Depth Information
Since the normal to the sphere is the direction of the radius vector
r
,we
have to the same approximation
∂
∂n
=
∂
∂h
=
∂
∂r
.
(2-260)
In Bruns' theorem (2-237) we may replace
γ
by
γ
0
, and Eqs. (2-246)
through (2-250) and (2-251) become
∂T
∂h
=∆
g
+
2
γ
0
R
−
N,
(2-261)
∂T
∂r
−
2
γ
0
R
∆
g
=
−
N,
(2-262)
∂T
∂r
−
2
R
T,
∆
g
=
−
(2-263)
δg
=∆
g
+
2
γ
0
R
N,
(2-264)
2
R
T,
δg
=∆
g
+
(2-265)
∂T
∂r
2
R
T
+∆
g
=0
.
+
(2-266)
The last equation is the spherical approximation of the fundamental bound-
ary condition.
Remark
The meaning of this spherical approximation should be carefully kept in
mind. It is used only in equations relating the small quantities
T, N,
∆
g, δg
,
etc. The reference surface is
never
a sphere in any geometrical sense, but
always an ellipsoid. As the flattening
f
is very small, the ellipsoidal formulas
canbeexpandedintopowerseriesintermsof
f
, and then all terms containing
f, f
2
, etc., are neglected. In this way one obtains formulas that are rigorously
valid for the sphere, but approximately valid for the actual reference ellipsoid
as well. However, normal gravity
γ
in the gravity anomaly ∆
g
=
g
γ
must be computed for the ellipsoid to a high degree of accuracy. To speak
of a “reference sphere” in space, in any geometric sense, may be highly
misleading.
−
Since the anomalous potential
T
=
W − U
is a harmonic function, it can be
expanded into a series of spherical harmonics:
R
r
n
+1
T
(
r, ϑ, λ
)=
∞
T
n
(
ϑ, λ
)
.
(2-267)
n
=0
