Geology Reference
In-Depth Information
taking the inner core to be axisymmetric. Again, replacing the product of the dens-
ity and the volume element with (5.148), the equatorial moment of inertia becomes
r
0
2π
π
R
2
sin
2
θ
ρ
0
(
r
0
)
∂
R
θ
cosφ
+
R
2
cos
2
∂
r
0
R
2
sinθ
d
θ
d
φ
dr
0
A
=
0
0
0
∂
r
0
(
R
5
)ρ(
r
0
)
1
θ
sinθ
d
θ
dr
0
.
r
0
π
2
5
∂
2
sin
2
θ
+
cos
2
=
(5.262)
0
0
Substituting for
R
5
from (5.259), and carrying out the integration over θ
,wefind
r
0
1
dr
0
.
0
ρ(
r
0
)
r
0
8
15
d
dr
0
1
3
f
(
r
0
)
A
=
−
+···
(5.263)
The di
erence between the axial and equatorial moments of inertia of the inner
core is then
ff
r
0
dr
0
r
0
f
(
r
0
)
dr
0
,
8
15
d
C
−
A
=
ρ(
r
0
)
(5.264)
0
correct to first order in the flattening. The gravity restoring torque on the inner core,
tilted at angle θ
i
with respect to the mantle and crust, is
cosθ
i
sinθ
i
φ
i
,
Γ
=Γ
(5.265)
with the coe
cient of the gravity torque,
Γ
,givenby
5
(
C
−
A
)
⎩
2
⎭
.
3
GM
i
r
0
2
GM
i
r
0
5
2
Ω
(2
f
i
+
r
0
f
i
)
Γ=
+
f
i
+
(5.266)
The programme TORQUE.FOR uses as input the file innerc.dat, generated by
the programme FIGURE.FOR described in Section 5.3, containing profiles across
the inner core for the density, the mean density and the flattening as functions of
mean radius. The input file innerc.dat also provides the number of model points
across the inner core for the given Earth model, as well as the density at the bot-
tom of the outer core required for the torque calculation. The inner core values
are then interpolated onto a new set of equally spaced model points using the sub-
routine SPMAT and the subroutine INTPL from Section 1.6. The number of model
points in the new profile is specified as input at the request of the programme
TORQUE.FOR, which calculates the di
erence between the axial and equatorial
moments of inertia across the inner core using the subroutine QUAD from Sec-
tion 5.3. The radial derivative of the flattening, required in the integration for the
ff
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