Geoscience Reference
In-Depth Information
ω
. This can be seen as follows. We write the spherical-harmonic expansion
of the disturbing potential in the form
+
∞
n
a
nm
R
nm
(
ϑ, λ
)+
b
nm
S
nm
(
ϑ, λ
)
.
T
=
GδM
R
(5-154)
n
=1
m
=0
Then, according to Sect. 2.17, Eqs. (2-351) and (2-359) or (2-363), we have
N
=
GδM
∞
n
a
nm
R
nm
(
ϑ, λ
)+
b
nm
S
nm
(
ϑ, λ
)
(5-155)
δW
γ
0
1
γ
0
Rγ
0
−
+
n
=1
m
=0
and
GδM
R
2
+
2
δW
R
∆
g
=
−
∞
n
(
n −
1)
a
nm
R
nm
(
ϑ, λ
)+(
n −
1)
b
nm
S
nm
(
ϑ, λ
)
;
1
R
+
n
=1
m
=0
(5-156)
remember that
γ
0
denotes a global mean value of gravity. The condition of
equal masses,
δM
= 0, is very natural and will be assumed. If we square
the formulas for
N
and ∆
g
and integrate over the whole earth, then all
the integrals of products of different harmonics
R
nm
and
S
nm
will be zero,
according to the orthogonality property (1-83), and the remaining integrals
will be given by (1-84). Thus, we find
N
2
dσ
=
4
π
δW
2
γ
0
σ
a
2
nm
,
∞
n
m
)!
a
2
+
4
π
γ
0
1
2
n
+1
(
n
+
m
)!
2(
n
nm
+
b
2
n
0
+
−
n
=1
m
=1
(5-157)
(∆
g
)
2
dσ
=
16
π
R
2
δW
2
σ
a
2
.
∞
n
1)
2
2
n
+1
m
)!
a
2
nm
+
4
π
R
2
(
n
−
(
n
+
m
)!
2(
n
nm
+
b
2
n
0
+
−
n
=1
m
=1
(5-158)
By a more complicated derivation, which we omit here but which can be
found in Molodenskii et al. (1962: p. 87), one gets the similar formula
a
2
.
(
ξ
2
+
η
2
)
dσ
=
∞
n
m
)!
a
2
nm
4
π
R
2
γ
0
n
(
n
+1)
2
n
+1
(
n
+
m
)!
2(
n
nm
+
b
2
n
0
+
−
n
=1
m
=1
σ
(5-159)
