Geology Reference
In-Depth Information
coupled, ordinary di
ff
erential equations by the dependent variable changes, y
1
=
κ
and y
2
=
d
κ/
dr
0
. We then have the pair of equations
d
y
1
dr
0
=
y
2
,
d
y
2
dr
0
=−
1
r
0
r
0
ρ
¯ρ
y
2
6
(6ρ/¯ρ
−
20)y
1
−
−
ρ/¯ρ)
4
2
3
e
2
4
r
0
e
2
2
r
0
+
(1
+
6η
+
3η
−
(7
+
5η)η.
(5.204)
Two separate pairs of numerical solutions are required, one for the regular com-
plementary function of the homogeneous equation,
y
c
1
,y
c
2
, and a second for the
particular integral of the non-homogeneous equation,
y
p
1
,y
p
2
. From expression
(5.203), at the geocentre, the complementary function obeys
y
c
1
=
0,
y
c
2
=
0,
(5.205)
d
y
c
1
d
y
c
2
dr
0
=
0,
dr
0
=
2
D
,
(5.206)
while the particular integral obeys
64
ρ
(0)
1
ρ(0)
e
2
(0),
y
p
1
=
0,
y
p
2
=−
(5.207)
d
y
p
1
dr
0
=−
64
ρ
(0)
d
y
p
2
dr
0
=
1
ρ(0)
e
2
(0),
0.
(5.208)
Since the complementary function is a solution of the linear homogeneous equa-
tion, it is proportional to the constant
D
.If
y
c
1
,y
c
2
is the complementary function
found for
D
1, then the solution for
D
1is
D
y
c
1
,
D
y
c
2
. The full solution of the
=
kappa equation (5.192) is then
D
y
c
1
+
y
p
1
,
D
y
c
2
+
y
p
2
. The constant
D
is determ-
ined by the surface condition (5.187), which becomes
d
D
y
c
2
+
y
p
2
4
D
y
c
1
+
y
p
1
5
4
me
25
16
m
2
+
+
−
=
0.
(5.209)
Thus,
D
is given by
25/16
m
2
d
y
p
2
+
4y
p
1
+
5/4
me
−
D
=−
.
(5.210)
d
y
c
2
+
4y
c
1
The programme FIGURE.FOR computes the mean density ¯ρ, the reciprocal of
the flattening 1/
f
, and the departure of the internal equipotentials from ellipsoidal
shape, κ. Input to the programme is the Earth model file, the name of which is
typed in on request. This can be one of the four models listed in Appendix C, or
one in a file with the appropriate format. The Clairaut equation is solved using the
Radau transformation, which reduces its solution to two successive quadratures.
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