Geoscience Reference
In-Depth Information
As a starting point, we use Eq. 3.115 rewritten in terms of the dimensionless
variables:
Z
e u
2
".a/ D
. u /d u :
(3.151)
0
It is possible to prove that . . u // has no zeros at u > 0and . .0// D 0.
Equation 3.149 confirms this statement.
At small a (large ), small u / 1= contributes to the integral on the right-
hand side of Eq. 3.151 . We thus must seek the roots of Eq. 3.143 in the limit of
small u . As is seen from Eq. 3.148 ,small u corresponds to large . The following
approximations are used:
1
" C 1 S./
1
" C 2
1
4 C
2
2" C 3
1
6
and
1
1
4 C
2
6 :
1/ 2
. 2
Now we get
" C 1
" C 2
1
4 C 4
" C 1
2" C 3
1
6
u D
(3.152)
In the lowest approximation we have
s " C 1
." C 2/ u
2
. u /
To find the next approximation, we rearrange Eq. 3.152 :
s " C 1
." C 2/ u
s 1 C 4
" C 2
2" C 3
1
2
2
D
The last term under the square root on the right-hand side of this equation is
small. Expanding the square root and replacing 2 with its lowest approximation
give
s " C 1
." C 2/ u
" C 2
2" C 3 C
2
. u / D 2

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