Geology Reference
In-Depth Information
Using the addition theorem (B.9) for
P
n
(cos
) as well as expression (5.148) for
the product of the density and the element of volume, the contribution of region
➂
to the gravitational potential becomes
Θ
−
G
∞
n
(
−
1)
m
r
n
P
n
(cosθ)
e
im
φ
×
m
=−
n
n
=
0
(5.228)
2π
π
d
R
n
−
1
∂
R
1
P
−
n
(cosθ
)
e
−
im
φ
ρ(
r
0
)
∂
r
0
sinθ
d
θ
d
φ
dr
0
.
0
0
2
3
f
i
)
a
i
(1
+
The inner boundary of region
➂
is the equipotential
r
0
1
2
3
f
(
r
0
)
P
2
(cosθ)
R
=
−
+···
.
(5.229)
Thus,
1
1
R
n
−
1
=
1
r
n
−
1
2
3
(
n
+
−
1)
f
(
r
0
)
P
2
(cosθ)
+···
,
(5.230)
0
and
∂
R
∂
r
0
=
2
3
f
(
r
0
)
r
0
f
(
r
0
)
P
2
(cosθ)
1
−
+
+···
.
(5.231)
Then,
1
R
n
−
1
∂
R
1
1
r
n
−
1
2
3
(
n
∂
r
0
=
+
−
2)
f
(
r
0
)
P
2
(cosθ)
0
2
3
r
0
f
(
r
0
)
P
2
(cosθ)
−
+···
.
(5.232)
Substituting this expression in (5.228), and using the orthogonality of spherical har-
monics (B.7), the contribution of region
➂
to the gravitational potential is reduced to
d
d
8
15
π
Gr
2
P
2
(cosθ)
ρ(
r
0
)
f
(
r
0
)
dr
0
+···
.
−
4π
G
r
0
ρ(
r
0
)
dr
0
+
(5.233)
2
3
f
i
)
2
3
f
i
)
a
i
(1
+
a
i
(1
+
To interpret this result, we use the uniform method of Wavre, as described in
Section 5.3. The mean equivolumetric radius of the equipotential forming the
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