Geology Reference
In-Depth Information
P
r
-
r
r
Q
r
O
Figure 5.4 Field point
P
and source point
Q
.
5.4 Gravity coupling
When the symmetry axis of the inner core is inclined with respect to that of the rest
of the Earth, a large gravity restoring torque results.
The evaluation of such coupling begins with elementary Newtonian gravitational
relations. With reference to Figure 5.4, an element of mass ρ(
r
)
d
V
at the source
point
Q
at radius
r
from the origin
O
, produces a gravitational force per unit mass
at the field point
P
at radius
r
from the origin
O
,givenby
G
r
−
r
|
V
.
−
3
d
(5.213)
r
|
r
−
If
S
i
is the surface of the inner core, bounding the volume
V
i
, the rest of the Earth
is contained in the volume
V
0
, bounded by the surface
S
i
on the inside and the
Earth's surface
S
0
on the outside. The total gravitational torque acting on the inner
core is then
G
×
ρ(
r
)
r
r
−
3
ρ(
r
)
d
V
d
−
r
V
(5.214)
r
|
|
r
−
V
i
V
i
G
×
ρ(
r
)
r
r
−
3
ρ(
r
)
d
V
d
−
r
V
.
r
|
|
r
−
V
i
V
0
The first integral (5.214) represents the torque generated on the inner core by
self-gravitation. With the exchange of dummy integration variables, it can be
written
G
×
ρ(
r
)
r
1
2
r
−
3
ρ(
r
)
d
V
d
−
r
V
r
|
|
r
−
V
i
V
i
V
G
r
×
ρ(
r
)
r
−
r
−
3
ρ(
r
)
d
V
d
.
(5.215)
r
−
|
r
|
V
i
V
i
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