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
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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