Geoscience Reference
In-Depth Information
Relations between
T
,
N
,and
∆
g
By (2-347), we have
T
=
γ
0
N
+
δW .
(2-354)
Substituting this into (2-333) and dividing by
γ
0
,weobtain
∆
gS
(
ψ
)
dσ .
N
=
GδM
δW
γ
0
R
4
πγ
0
Rγ
0
−
+
(2-355)
σ
This is the generalization of Stokes' formula for
N
. It applies for an arbitrary
reference ellipsoid whose center coincides with the center of the earth.
While formula (2-333) for
T
contains only the effect of a mass difference
δM
, the formula (2-355) for
N
contains, in addition, the potential differ-
ence
δW
. These formulas also show clearly that the simple Stokes integrals
(2-304) and (2-307) hold only if
δM
=
δW
= 0, that is, if the reference el-
lipsoid has the same potential as the geoid and the same mass as the earth.
Otherwise, they give
N
and
T
only up to additive constants: putting
N
0
=
GδM
δW
γ
0
Rγ
0
−
(2-356)
and taking into account (2-331), we have
∆
gS
(
ψ
)
dσ ,
R
4
π
T
=
T
0
+
(2-357)
σ
∆
gS
(
ψ
)
dσ .
R
4
πγ
0
N
=
N
0
+
(2-358)
σ
Alternative forms of (2-355), which are sometimes useful, are obtained in
the following way. Substituting the series (2-268) and (2-270) into (2-352),
we get
∞
∆
g
(
ϑ, λ
)=
1
R
1)
T
n
(
ϑ, λ
)+
2
(
n
−
R
δW
(2-359)
n
=0
as the generalization of (2-273). Expanding the function ∆
g
(
ϑ, λ
)intothe
usual series of Laplace surface spherical harmonics,
∆
g
(
ϑ, λ
)=
∞
∆
g
n
(
ϑ, λ
)
,
(2-360)
n
=0
and comparing the constant terms (
n
= 0) of these two equations, we get
1
R
T
0
+
2
R
δW
=∆
g
0
,
−
(2-361)
