Geology Reference
In-Depth Information
Using this result, the total torque, the sum of (5.253) and (5.252), is found to be
⎩
2
r
0
⎭
×
3
GM
i
r
0
16
75
r
0
π
GM
i
r
0
5
2
Ω
r
0
f
i
)
Γ
=
(2
f
i
+
+
f
i
+
(5.255)
0
ρ
0
r
0
dr
0
r
0
f
(
r
0
)
dr
0
cosθ
i
sinθ
i
φ
i
.
d
ff
erence
between the axial and equatorial moments of inertia of the inner core. In the inner
core frame of reference, the axial moment of inertia is
The integral in expression (5.255) for the torque can be related to the di
V
i
ρ(
r
)
x
2
x
2
d
C
=
+
V
.
(5.256)
1
The product of the density and the volume element, in terms of the equipotential
radius
R
, is given by (5.148), and the axial moment of inertia then becomes
r
0
2π
π
θ
ρ(
r
0
)
∂
R
R
2
sin
2
∂
r
0
R
2
sinθ
d
θ
d
φ
dr
0
=
C
0
0
0
r
0
π
2
5
∂
∂
r
0
(
R
5
)ρ(
r
0
)sin
3
θ
d
θ
dr
0
.
=
(5.257)
0
0
To first order, the internal equipotentials (5.142) are
r
0
1
2
R
(
r
0
,θ
)
3
f
(
r
0
)
P
2
(cosθ
)
=
−
+···
.
(5.258)
Then,
r
0
1
10
3
R
5
f
(
r
0
)
P
2
(cosθ
)
=
−
+···
,
(5.259)
and carrying out the integration in (5.257) over θ
, we obtain
0
ρ(
r
0
)
r
0
1
dr
0
.
r
0
8
15
d
dr
0
2
3
f
(
r
0
)
C
=
+
+···
(5.260)
In the inner core frame of reference, the equatorial moment of inertia is
V
i
ρ(
r
)
x
2
x
3
d
V
i
ρ(
r
)
x
2
x
3
d
A
=
+
V=
+
V
,
(5.261)
1
2
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