Geology Reference
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solution, v 2 ( a )
0every-
where, while y 3 and y 4 do not appear. The system (7.41) then degenerates to
=− y 52 ( a ), a constant, and w 2 ( a )
=
0. In both solutions, y 6
=
= χ 0 ( a + ),
D 1 v 1 ( a )
+ D 2 v 2 ( a )
dV 1 , 0
dr
( a + ).
D 1 w 1 ( a )
=
(7.44)
Without loss of generality, the constant v 2 ( a ) can be absorbed into the linear com-
bination constant D 2 , giving
D 2 = χ 0 ( a + ),
D 1 v 1 ( a )
+
dV 1 , 0
dr
( a + ).
=
D 1 w 1 ( a )
(7.45)
The systems (7.34), (7.36), (7.41) and (7.45) are all seen to depend only on the
boundary values of χ n and the derivatives of V 1, n . Thus, the entire deformation field
in either the shell or the inner core is determined by the values of these two quantit-
ies on the boundaries. Following the traditional definition of the Love number h as
a dimensionless number giving the radial displacement as a proportion of the ratio
of the disturbing potential to gravity, we write the radial coe
cients of the degree- n
parts of the respective radial displacement components on the two boundaries as
h 1 n χ n ( b )
( b )
1
g 0 ( b )
dV 1 , n
dr
bh 2 n
y 1, n ( b )
=
(7.46)
and
h 1 n χ n ( a + )
( a + )
1
g 0 ( a )
dV 1 , n
dr
ah 2 I
y 1, n ( a )
=
.
(7.47)
n
For each spherical harmonic of degree n , the Love numbers h 1 n , h 2 n for the shell,
and the Love numbers h 1 n , h 2 I n for the inner core, are found by numerical integra-
tion of the sixth-order spheroidal system of di
ff
erential equations, (3.102) through
(3.107), for n
1, or the fourth-order spheroidal system, (3.110) through (3.113),
for n
0. Only the conditions of continuity of normal displacement at the two
solid-fluid interfaces remain to be applied.
Application of the subseismic form (6.181) of the equation of continuity provides
an additional constraint. In terms of the radial coe
=
cients of the degree- n vector
spherical harmonics, it becomes
y 2 ( r )
= ρ 0 ( r )g 0 ( r )y 1 ( r ),
(7.48)
for a
r
b . At the core-mantle boundary this yields the constraint
1
ρ 0 ( b ) C 2
g 0 ( b ) C 1
=
0,
(7.49)
 
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