Geology Reference
In-Depth Information
into account, the sixth-order spheroidal governing system, for
n
≥
1, expressed by
equations (3.102) through (3.107), becomes
d
y
1
dr
=−
2
λ
r
y
1
+
βy
2
+
n
1
λ
r
y
3
,
(3.130)
r
2
n
1
r
2
−
ρ
0
4γ
+
2
d
y
2
dr
=
2
δ
4
μ
r
y
2
2
Ω
+
y
1
−
+
ρ
0
γ
−
y
3
n
1
r
y
4
2
+
−
ρ
0
y
6
−
ω
ρ
0
y
1
+
2
m
ω
Ω
ρ
0
y
3
,
(3.131)
d
y
3
dr
=−
1
r
y
1
1
r
y
3
1
μ
y
4
,
+
+
(3.132)
ρ
0
γ
−
r
2
d
y
4
dr
=
y
1
−
λ
r
y
2
+
r
2
y
3
−
3
r
y
4
−
ρ
0
r
y
5
2
m
ω
Ω
ρ
0
n
1
2
−
ω
ρ
0
y
3
+
(y
1
+
y
3
),
(3.133)
d
y
5
dr
=
4π
G
ρ
0
y
1
+
y
6
,
(3.134)
d
y
6
dr
=−
4π
G
ρ
0
n
1
n
1
r
2
y
5
−
2
r
y
6
.
r
y
3
+
(3.135)
Taking only self-coupling into account, the second-order torsional governing sys-
tem, (3.119) and (3.120), for arbitrary radius, has the form
dz
1
dr
=
1
r
z
1
1
μ
+
z
2
,
(3.136)
dz
2
dr
=
2
)
r
2
z
1
−
3
r
z
2
−
ω
2
m
ω
Ω
ρ
0
n
1
2
(
n
1
−
ρ
0
z
1
+
z
1
.
(3.137)
Solutions of the spheroidal system, regular at the geocentre, can be expanded in
power series as
r
α
∞
A
i
,ν
r
ν
,
y
i
(
r
)
=
(3.138)
ν
=
0
with α non-negative. The coe
cients
A
i
,ν
vanish for ν<0. A series of indicial
equations, determining the admissible values of α, are found on substituting these
expansions in the governing equations and equating like powers of the radius.
Substitution of the expansions (3.138) in equations (3.130) through (3.133), and
equating the coe
cients of like powers of the radius, yields the system
M
n
+
η
I
x
ν
=
b
ν
,
(3.139)
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