Geology Reference
In-Depth Information
Using (5.179) with
F
(η)
=
1, for small
r
0
we find that
4
ρ
(0)
1
η(
r
0
)
=−
ρ(0)
r
0
+···
.
(5.197)
Performing the integration in expression (5.188) and expanding the exponential for
small exponent, we obtain
e
(0)
1
4
ρ
(0)
1
e
(
r
0
)
=
−
ρ(0)
r
0
+···
.
(5.198)
Recalculating
F
(η) from (5.180), we now find that
10
ρ
(0)
1
F
(η)
=
1
−
ρ(0)
mr
0
+···
.
(5.199)
Thus, η(
r
0
) is linear in
r
0
for small
r
0
but has a coe
ers fractionally
from that given in (5.197), by a term of order
m
. The integrand in the exponent of
(5.188) is then obviously proper.
From relation (5.195), the homogeneous form of theκ equation (5.192) for small
r
0
becomes
cient which di
ff
6
14
r
0
κ
=
κ
+
r
0
κ
−
0.
(5.200)
This equation has the two complementary functions
r
0
and 1/
r
0
with Wronskian
−
9/
r
0
.Forsmall
r
0
, the non-homogeneous right side of the κ equation (5.192) is
8
ρ
(0)
e
2
(0)
r
0
+···
.
1
=
I
(
r
0
)
(5.201)
ρ(0)
By the usual formula for linear di
ff
erential equations (Kreyszig, 1967, pp. 147-
149), the particular integral is
r
0
I
(
r
0
)
dr
0
r
0
9
64
ρ
(0)
1
r
0
I
(
r
0
)
dr
0
1
9
r
0
1
ρ(0)
e
2
(0)
r
0
−
=−
+···
.
(5.202)
Thus, we may write the full solution for κ, regular at the geocentre, as the sum of
the regular complementary function and the particular integral,
64
ρ
(0)
1
Dr
0
+···−
ρ(0)
e
2
(0)
r
0
κ(
r
0
)
=
+···
,
(5.203)
valid for small
r
0
, with
D
an arbitrary constant to be determined.
The numerical solution of theκ equation (5.192) follows the Runge-Kutta meth-
ods outlined in Section 3.6. The κ equation is first converted to a pair of first-order,
Search WWH ::
Custom Search