Geology Reference
In-Depth Information
In spherical polar co-ordinates, at the surface of the Earth (
r
=
d
), this potential
becomes
2
ω
3
d
2
1
2
1
2
2
2
U
=
+
ω
+
ω
(4.37)
2
ω
1
sinθcosφ
+
ω
2
sinθsinφ
+
ω
3
cosθ
1
2
d
2
−
.
In terms of the Legendre functions,
2
3cos
2
1
,
1
P
2
(cosθ)
=
θ
−
(4.38)
P
2
(cosθ)
=−
3cosθsinθ,
(4.39)
P
2
(cosθ)
3sin
2
=
θ,
(4.40)
the potential is expressible as
6
ω
3
d
2
P
2
(cosθ)
1
3
ω
1
2
d
2
1
2
U
=
+
+
ω
−
2ω
1
1
3
ω
1
ω
3
d
2
P
2
(cosθ)cosφ
+
3
ω
2
ω
3
d
2
P
2
(cosθ)sinφ
(4.41)
+
12
ω
2
d
2
P
2
(cosθ)cos2φ
−
1
1
2
1
2
6
ω
1
ω
2
d
2
P
2
(cosθ) sin 2φ,
−
−
ω
using the identities
1
2
sin 2φ,
sinφcosφ
=
(4.42)
1
−
cos 2
2
, cos
2
1
+
cos 2
2
.
sin
2
φ
=
φ
=
A similar expansion can be made for the external gravitational potential,
G
ρ(
r
)
V
(
r
)
=−
d
V
.
(4.43)
r
|
|
r
−
V
r
|
in a three-dimensional Taylor series around
r
=
Expanding 1/
|
r
−
0gives
x
j
x
j
r
3
+
2
r
5
3
x
i
x
j
−
r
2
j
x
i
x
j
+···
,
1
1
r
+
1
r
|
=
δ
(4.44)
|
r
−
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