Geology Reference
In-Depth Information
r
0
0
ρ(
r
0
)
π
∂
R
∂
r
0
R
2
sinθ
d
θ
dr
0
V
i
ρ
dV
=
2π
0
r
0
ρ(
r
0
)
1
m
2
r
0
8
r
0
8
r
0
45
m
2
ff
+
45
m
2
ff
=
4π
+
−
0
15
m
2
f
2
dr
0
4
4
15
m
2
f
2
−
+
+···
r
0
r
0
ρ(
r
0
)
dr
0
,
=
4π
(5.155)
0
where the expansion (5.152) has been used with
n
= −
1, and the prime indicates
erentiation. Division of (5.155) by (4/3)π
r
0
, the volume of
di
ff
V
i
, yields the mean
density of material enclosed by
U
i
as
r
0
3
r
0
r
0
ρ(
r
0
)
dr
0
.
¯ρ(
r
0
)
=
(5.156)
0
For
n
=−
1, we then have that
dU
i
dr
0
g
0
(
r
0
)
=
1
45
r
0
f
2
5
f
2
GM
i
r
0
2
3
m
(
r
0
)
4
2
r
0
ff
+
=
−
−
+
+···
,
(5.157)
2
r
0
/
GM
i
.
where g
0
(
r
0
) is introduced as a mean gravity intensity and
m
(
r
0
)
=Ω
After substitution from (5.157), the relation for
n
=
0 becomes
1
45
r
0
ff
+
f
2
GM
i
r
0
1
3
m
(
r
0
)
4
U
i
=
V
(0)
−
+
+
+···
,
(5.158)
V
(0) being the gravitational potential at the geocentre, due to mass external to
S
i
,
which by (5.127) is
G
2π
G
d
r
0
ρ(
r
0
)
π
r
∂
R
∂
r
0
R
sinθ
d
θ
dr
0
V
(0)
=−
dV
=−
V
−
V
i
0
r
0
ρ(
r
0
)
1
dr
0
.
4π
G
d
r
0
45
r
0
ff
+
f
2
4
=−
−
+···
(5.159)
With substitution for g
0
(
r
0
) from (5.157), the relations for
n
=
2and
n
=
4 become,
respectively,
Search WWH ::
Custom Search