Geoscience Reference
In-Depth Information
P
±'
0
parallel to
equator
P''
0
Fig. 8.14. Curvature of the plumb line along a north-south profile
These quantities
ξ, η, ε
refer to the earth's surface. Figure 8.13 shows
ε
.
Similarly, we have for the geoid
ξ
0
=Φ
0
−
ϕ,
η
0
=(Λ
0
−
λ
)cos
ϕ,
(8-134)
ε
0
=
ξ
0
cos
α
+
η
0
sin
α.
(8-135)
See again Fig. 8.13 for
ε
0
, noting that we do not distinguish the normals in
Q
0
and
Q
0
as we have mentioned above.
In addition, we need the normal direction of the plumb line at the surface
point
P
; it is defined as the tangent to the normal plumb line at
P
;the
corresponding latitude and longitude will be denoted by
ϕ, λ
. In this “local”
notation, there is no danger of confusion with the spherical coordinate
ϕ
used in earlier chapters. Hence, we have
λ
=
λ
+
δλ
normal
,
ϕ
=
ϕ
+
δϕ
normal
,
(8-136)
where
δϕ, δλ
express the normal plumb line curvature. These equations are
the “normal equivalent” to (8-131): the “normal surface values”
ϕ, λ
cor-
respond to the “actual surface values” Φ
,
Λ and the ellipsoidal values
ϕ, λ
correspond to the geoidal values Φ
0
,
Λ
0
. To make the analogy complete, we
should replace
ϕ
=
ϕ
(
P
0
)by
ϕ
(
P
0
), but we have consistently neglected such
differences.
In contrast to the actual plumb line curvature, it is very easy to compute
the normal curvature of the plumb line: from (5-147) we have
0
.
17
h
[km]
sin 2
ϕ,
δϕ
normal
=
−
δλ
normal
=0
,
(8-137)
where
h
[km]
denotes elevation in kilometers.
Since the ellipsoidal normal and hence
ϕ, λ
are geometrically defined, we
may call the quantities (8-132) “geometric deflections of the vertical” at the
