Geology Reference
In-Depth Information
In view of the expansions (5.244) and (5.248), only the terms in the geopotential,
U
t
, proportional to
P
2
(cosθ) will contribute to the torque integrals (5.219). Adding
the contribution of the centrifugal potential (5.238) to the contributions (5.226) and
(5.237) to the gravitational potential, the total contribution to the geopotential,
U
t
,
proportional to
P
2
(cosθ), becomes
⎩
2
⎭
3
GM
i
r
0
2
15
GM
i
r
0
5
2
Ω
r
0
f
i
)
r
2
P
2
(cosθ),
(2
f
i
+
+
f
i
+
(5.250)
with ρ
i
, the density just outside the inner core, replaced by
3
M
i
ρ
i
=
4π
r
0
.
(5.251)
The surface integral in the torque expression (5.219) can then be evaluated using
(5.244), the expansion (5.249) of
P
2
(cosθ), and the orthogonality relation (B.7) for
spherical harmonics. It reduces to
75
π
r
0
ρ
s
f
i
⎩
2
r
0
⎭
3
GM
i
r
0
16
GM
i
r
0
5
2
Ω
r
0
f
i
)
cosθ
i
sinθ
i
φ
i
,
(2
f
i
+
+
f
i
+
(5.252)
where ρ
s
is the density at the top of the inner core and
φ
i
is the unit vector in
the direction of increasing longitude of the mantle and crust axis. Similarly, the
volume integral in the torque expression (5.219) can be evaluated using (5.248), the
expansion (5.249) of
P
2
(cosθ), and the orthogonality relation (B.7) for spherical
harmonics. In the same way, it reduces to
⎩
2
r
0
⎭
×
3
GM
i
r
0
16
75
r
0
π
GM
i
r
0
5
2
Ω
r
0
f
i
)
−
(2
f
i
+
+
f
i
+
(5.253)
r
0
r
0
d
ρ
0
f
(
r
0
)
dr
0
cosθ
i
sinθ
i
φ
i
.
dr
0
0
Integrating by parts, we find that
r
0
0
ρ
0
r
0
dr
0
r
0
f
(
r
0
)
dr
0
.
r
0
d
ρ
0
d
=
ρ
s
r
0
f
i
−
f
(
r
0
)
dr
0
(5.254)
dr
0
0
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