Geology Reference
In-Depth Information
cients of the expansions are found easily from the generating free con-
stant
A
1,1
. Solving (3.247), we find that
The coe
ρ
0
4γ
+
ω
2
2
+
2
Ω
A
1,3
=−
A
1,1
,
(3.255)
10 (λ
+
2μ)
A
2,2
=
(5λ
+
6μ)
A
1,3
.
(3.256)
Then (3.249) gives
A
5,4
=
π
G
ρ
0
A
1,3
.
(3.257)
Solving (3.248), we find that
ρ
0
4γ
+
ω
2
2
+
2
Ω
A
1,5
=−
A
1,3
,
(3.258)
28 (λ
+
2μ)
A
2,4
=
(7λ
+
10μ)
A
1,5
.
(3.259)
Then (3.250) gives
2
3
π
G
ρ
0
A
1,5
.
A
5,6
=
(3.260)
3.6 Numerical integration of the radial equations
The power series solutions regular at the geocentre, considered in the previous sec-
tion, are useful only for small radii. In general, the radial equations for the spher-
oidal and torsional deformations, summarised in Section 3.4 for small harmonic
oscillations, require integration for arbitrary radii. All of the governing systems are
ordinary di
ff
erential equations.
There are many methods for the numerical integration of ordinary di
ff
erential
equations (see, for example, Press
et al.
(1992), Ch. 6). We focus our attention on
one of the most commonly used, the Runge-Kutta method, and implement it to
integrate the radial equations.
Let us first consider the numerical integration of a single first-order di
ff
erential
equation,
d
dr
=
f
(
r
,y),
(3.261)
where
f
(
r
,y) is a continuous, di
erentiable function of radius
r
and the solution
y(
r
). The basic step in numerical integration is to extend the solution y(
r
) from
radius
r
ff
=
r
i
to radius
r
=
r
i
+
1
. The integration stepsize is then
h
=
r
i
+
1
−
r
i
. Taylor
series expansion yields
∞
(
n
)
i
/
n
!,
h
n
y
i
+
1
−
y
i
=
y
(3.262)
n
=
1
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