Geology Reference
In-Depth Information
Substituting relations (5.174) and (5.175) into the transformed Clairaut equation
(5.172), we obtain
2
r
0
¯ρ
d
¯ρ
dr
0
(1
r
0
η
+
2
5η
+
η
+
+
η)
(5.176)
3η
+
η)
21
r
0
e
4
d
¯ρ
dr
0
7
m
2
−
+
6η
+
7
−
e
(1
=
0.
¯ρ
Making use of the easily established identity
2
1
2
5
(
1
dr
0
¯ρ
r
0
1
+
η
+
η
¯ρ
r
0
d
r
0
¯ρ
d
¯ρ
dr
0
r
0
η
+
η
)
−
+
=
(5.177)
to replace
r
0
η
, and using the second of relations (5.175), this equation becomes
1
dr
0
¯ρ
r
0
1
+
η
5ρ
r
0
d
1
2
η
−
1
10
η
2
=
1
+
(5.178)
+
η
4
e
−
ρ/¯ρ)
7
m
(1
2
35
(1
+
η)
2
+
+
η)
−
3
e
(1
−
.
With one integration, (5.172) is converted into the integral equation for η,
1
r
0
5
¯ρ
r
0
¯ρ
r
0
F
(η)
dr
0
,
+
η
=
(5.179)
0
with
1
1
1
1
2
η
−
1
10
η
2
F
(η)
=
+
+
η
4
e
−
ρ/¯ρ)
7
m
(1
2
35
(1
+
η)
2
+
+
η)
−
3
e
(1
−
.
(5.180)
F
(η) is a very weak function of η and is nearly unity for any reasonable value
of η. By its definition (5.173), η is itself of order unity.
F
(η) derives its nearly
stationary property from the fact that the terms in curly brackets in (5.180) are of
order
m
and the rest of the function for η<1 has the expansion 1
2
.
Prior to the advent of computers, the Radau transformation reduced enormously
the labour involved in solving the Clairaut equation, as a good first approximation
to the solution could be obtained by taking
F
(η)
+
(1/40)η
+···
1.
Equations (5.172) and (5.166) are second-order ordinary di
=
erential equations,
and if solutions regular at the geocentre are sought, only one surface condition is
required for each equation. These are provided by (5.160) and (5.161), evaluated
at the surface
r
0
ff
=
d
. From (5.160), we obtain
21
d
2
e
2
13
e
2
3
m
de
+
5
e
=
5
2
m
4
2
de
+
10
dee
+
2
e
−
+
+
−
0,
(5.181)
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