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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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