Geology Reference
In-Depth Information
Returning to expression (5.114) for the gravitational potential we have
G
D
d
V
(
r
)
=−
V
,
(5.127)
V
where the integral is carried out throughout the entire volume
V
of the Earth. We
can then write the integral
4π
G
D
d
V
(5.128)
V
i
over the volume
V
i
enclosed by the internal equipotential
S
i
as
4π
G
4π
G
4π
G
D
d
D
d
D
d
V=
V−
V
,
(5.129)
V
i
V
V−V
i
where the integral
D
d
V
(5.130)
V−V
i
is over the volume bounded on the inside by
S
i
, and on the outside by the reference
surface equipotential
S
. We can then write
4π
G
4π
G
D
d
D
d
V=−
4π
V
(
r
)
−
V
.
(5.131)
V
i
V−V
i
The integral in the transformed expression (5.131) is then over a region exterior to
S
i
.
Of the integrals in (5.3), only the second now involves integration over the region
interior to
S
i
. We can write it as
1
D
d
V=
1
D
d
V+
1
D
d
V
,
(5.132)
V
i
V
i
−V
0
V
o
where
V
o
is the region bounded by the sphere
S
o
with radius
r
o
, equal to the
equivolumetric radius of the surface
S
i
. The last integral in (5.132) can be evaluated
exactly using the geometry shown in Figure 5.2.
From the law of cosines,
r
0
=
r
2
E
2
+
−
2
rE
cos(π
−
t
)
(5.133)
or
E
2
r
2
r
0
=
+
2
rE
cos
t
+
−
0.
(5.134)
Since
E
must be positive, the only admissible root of this quadratic equation is
r
2
cos
2
t
r
0
−
r
0
−
r
2
sin
2
t
.
=−
+
+
r
2
=−
+
E
r
cos
t
r
cos
t
(5.135)
Search WWH ::
Custom Search