Geology Reference
In-Depth Information
The only solution regular at the geocentre is then generated by the free constant
A
1,1
with α
=
0. Although
A
5,0
can be non-zero, this solution does not propagate
and may be regarded as the arbitrary constant determining the reference level for
gravitational potential, usually set to zero at infinity.
In summary, the power series solutions, near the geocentre, take the forms
y
1
=
A
1,1
r
+
A
1,3
r
3
+
A
1,5
r
5
+···
,
(3.243)
A
2,2
r
2
A
2,4
r
4
=
(
3λ
+
2μ
)
A
1,1
+
+
+···
,
y
2
(3.244)
1
2
(4π
G
ρ
0
)
A
1,1
r
2
A
5,4
r
4
A
5,6
r
6
y
5
=
A
5,0
+
+
+
+···
,
(3.245)
A
6,3
r
3
A
6,5
r
5
A
6,7
r
7
y
6
=
+
+
+···
.
(3.246)
The solutions, regular at the geocentre, are generated from the two free constants,
A
1,1
and
A
5,0
. The latter leads toy
1
0, withy
5
constant and equal to
A
5,0
.
The higher terms in the solutions generated from the free constant
A
1,1
,forα
=
=
y
2
=
y
6
=
0,
are found from the system (3.241) with ν
=
3,
⎝
⎠
⎝
⎠
=
ρ
0
⎝
⎠
,
0
2λβ
+
3
−
β
A
1,3
A
2,2
4γ
+
ω
2
A
1,1
(3.247)
2
−
4μβ
+
−
+
Ω
2δ
2
2
and with ν
=
5,
⎠
=
ρ
0
⎝
⎠
,
⎝
⎠
⎝
0
2λβ
+
5
−
β
A
1,5
A
2,4
4γ
+
ω
2
A
1,3
−
(3.248)
2
−
2δ
4μβ
+
4
−
+
2
Ω
A
6,3
and from the system (3.242) with ν
=
4,
⎝
⎠
⎝
⎠
=
⎝
⎠
,
4
1
05
−
A
5,4
A
6,3
4π
G
ρ
0
A
1,3
0
(3.249)
and with ν
=
6,
⎝
⎠
=
⎝
⎠
.
6
A
5,6
A
6,5
4π
G
ρ
0
A
1,5
0
1
07
−
(3.250)
Thus
A
6,3
0, and the solutions generated from
A
1,1
,again
replacing the y-variables by a system of
z
-variables defined by (3.187) for α
=
=
A
6,5
= ···=
0, y
6
=
n
=
0, become
A
1,3
r
2
A
1,5
r
4
=
y
1
(
r
)/
r
=
+
+
+···
,
z
1
(
r
)
A
1,1
(3.251)
A
2,2
r
2
A
2,4
r
4
=
y
2
(
r
)
=
(
3λ
+
2μ
)
A
1,1
+
+
+···
,
z
2
(
r
)
(3.252)
=
y
5
(
r
)/
r
2
A
5,4
r
2
A
5,6
r
4
z
5
(
r
)
=
2π
G
ρ
0
A
1,1
+
+
+···
,
(3.253)
z
6
(
r
)
=
y
6
(
r
)/
r
=
0.
(3.254)
Search WWH ::
Custom Search