Geology Reference
In-Depth Information
the expansions (5.118) provides a uniformly convergent series for 1/
D
, which can
be integrated term by term in each of the integrals of expression (5.137). Equating
like powers of
r
in the result yields
4π
G
2
ρ
P
n
(cos
Θ
)
P
n
(cos
Θ
)
g
P
n
(cos
Θ
)
d
V+
2
Ω
d
V+
dS
(
r
)
n
+
1
(
r
)
n
+
1
(
r
)
n
+
1
V−V
i
V
i
−V
o
S
i
4π
GM
−
2
r
0
,
n
=−
⎩
2/3
Ω
1,
4π
U
i
+Ω
2
r
0
,
−
n
=
0,
=
(5.138)
2
P
2
(cosθ),
n
(4/3)π
Ω
=
2,
0,
n
=
1,3,4,5,
···
.
The relation for
n
= −
1 is found by defining
P
−
1
(cos
Θ
)
=
1 and integrating the
main gravity equation (5.5) throughout
V
i
to obtain
2
Ud
V
i
∇
V=
S
i
∇
U
·
ν
dS
=
S
i
g
dS
4π
G
4π
G
4
2
3
π
r
0
,
=
ρ
d
V−
V−V
i
ρ
d
V−
2
Ω
·
(5.139)
V
since by definition the volume
V
i
is the same as that of a sphere of radius
r
0
,
denoted by
V
0
. This then yields
4π
G
4π
GM
2
r
0
.
V−V
i
ρ
d
V+
S
i
g
dS
=
−
2/3
Ω
(5.140)
Hence, if we agree to make
P
−
1
unity, (5.140) can be incorporated into the scheme
(5.138) with
n
=−
1.
By the same sequence of arguments used in Section 5.2 to deduce equation (5.8)
for the external equipotential surface, the equation for the internal surfaces is found
to be
r
0
1
+···
,
m
2
R
2
(
r
0
,θ)
R
(
r
0
,θ)
=
+
mR
1
(
r
0
,θ)
+
(5.141)
where
R
1
and
R
2
have identical functional forms to those given before, except that
the quantities
f
andκ are now allowed to depend on
r
0
. Thus, expression (5.75) for
R
(θ) now takes the form
r
0
1
2
3
f
(
r
0
)
P
2
(cosθ)
4
23
45
f
2
(
r
0
)
63
f
2
(
r
0
)
P
2
(cosθ)
R
(
r
0
,θ)
=
−
−
−
8
21
κ(
r
0
)
P
2
(cosθ)
12
35
f
2
(
r
0
)
P
4
(cosθ)
−
+
32
35
κ(
r
0
)
P
4
(cosθ)
+
+···
.
(5.142)
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