Geoscience Reference
In-Depth Information
8.5
The spherical case
As we have agreed, we work formally with a sphere (the reference ellipsoid
stays at its geometric place!). This means putting
r
=
R
= constant. Fur-
thermore, we assume (fictitiously!) that
S
is a level surface.
Expanding
T
and ∆
g
into a series of Laplace spherical harmonics, see
(2-322) and (2-320), we find
T
(
ϑ, λ
)=
∞
T
n
(
ϑ, λ
)
,
(8-40)
2
∆
g
(
ϑ, λ
)=
∞
∆
g
n
(
ϑ, λ
)
(8-41)
2
on the surface of the sphere, whence by (8-38) and (2-321) with
r
=
R
,
T
=
R
∞
∆
g
n
n
1
.
(8-42)
−
n
=2
The summation starts conventionally with
n
= 2, rather than
n
=0,forsev-
eral reasons, one of them being that
n
= 1 would lead to a zero denominator
in (8-42).
Using (2-325) and (2-326) leads to the well-known Stokes' formula
S
(
ψ
)∆
gdσ,
R
4
π
T
=
(8-43)
σ
where
S
(
ψ
)=
∞
2
n
+1
n
P
n
(cos
ψ
)
,
(8-44)
−
1
n
=2
where
P
(cos
ψ
) are Legendre polynomials. Here
ψ
denotes the spherical dis-
tance from the point at which
T
is to be computed.
In exactly the same way, we obtain for the gravity disturbance with the
boundary condition (8-39), summarizing the derivation in Sect. 2.18,
δg
(
ϑ, λ
)=
∞
δg
n
(
ϑ, λ
)
,
(8-45)
0
T
(
ϑ, λ
)=
R
∞
δg
n
n
+1
,
(8-46)
n
=0
