Geoscience Reference
In-Depth Information
where, by (1-89),
∆
gdσ.
1
4
π
∆
g
0
=
(2-362)
σ
Expressing
T
0
by (2-331) in terms of
δM
,weobtain
1
R
2
GδM
+
2
R
δW .
∆
g
0
=
−
(2-363)
The two equations (2-356) for
N
0
and (2-363) for ∆
g
0
can now be solved
for
δM
and
δW
:
GδM
=
R
(
R
∆
g
0
+2
γ
0
N
0
)
,
(2-364)
δW
=
R
∆
g
0
+
γ
0
N
0
.
The constant
N
0
may be expressed by either of these equations:
2
γ
0
∆
g
0
+
GδM
R
N
0
=
−
,
2
γ
0
R
(2-365)
γ
0
∆
g
0
+
δW
R
N
0
=
−
.
γ
0
A final note
A direct consequence of Eq. (2-356) is that
N
0
has an immediate geometrical
meaning: if
a
is the equatorial radius (semimajor axis) of the given reference
ellipsoid, then
a
E
=
a
+
N
0
(2-366)
is the equatorial radius of an ellipsoid whose normal potential
U
0
is equal to
the actual potential
W
0
of the geoid, and which encloses the same mass as
that of the earth, the flattening
f
remaining the same. The reason is that
for such a new ellipsoid
E
the new
N
0
= 0 by (2-356) with
δM
=0and
δW
=0.
Asmall
additive
constant
N
0
is equivalent to a change of
scale
for a nearly
spherical earth. To see this, imagine a nearly spherical orange. Increasing the
thickness of the peel of an orange everywhere by 1 mm (say) is equivalent
to a similarity transformation (uniform increase of the size) of the orange's
surface.
So, the usual Stokes formula, without
N
0
, gives a global geoid that is
determined
only up to the scale
which implicitly is contained in
N
0
.Itis,
however,
geocentric
, at least in theory, because it contains no spherical har-
monic of first degree,
T
1
(
ϑ, λ
). It would be exactly geocentric if the earth
were covered uniformly by gravity measurements. The scale was formerly de-
termined astrogeodetically, historically by grade measurements dating back
