Geology Reference
In-Depth Information
r
0
ρ(
r
0
)
r
0
κ
−
1
2κ
−
1
6
r
0
¯ρ(
r
0
)
r
0
f
2
16κ
2
r
0
ff
−
4
r
0
κ
−
−
+
−
ρ/¯ρ)
r
0
f
2
16κ
1
4
r
0
2
r
0
ff
−
4
r
0
κ
−
−
¯ρ(1
+
¯ρ(
r
0
)
2
r
0
f
2
r
0
f
f
2
ff
+
r
0
ff
1
12
r
0
r
0
f
2
+
+
+
+
16κ
4
κ
+
r
0
κ
−
−
=
0.
(5.164)
Correct to first order, equation (5.163) reduces to
r
0
ρ
¯ρ
6
6
r
0
f
+
f
−
(1
−
ρ/¯ρ)
f
=
0,
(5.165)
the first-order
Clairaut equation
. Now, equation (5.164) is entirely of second order.
Thus, we can use the first-order expression (5.165) to replace
f
and commit, at
most, a third-order error in small quantities. With this substitution, equation (5.164)
produces
10
3
−
ρ/¯ρ
r
0
ρ
¯ρ
κ
−
6
6
r
0
κ
+
κ
1
9
+
ρ/¯ρ
f
2
2
9
−
ρ/¯ρ
ff
+
(5.166)
9
4
9
2
1
r
0
3
r
0
−
ρ/¯ρ)
f
2
=−
+
(1
.
Equation (5.166) governs the departure of the internal equipotentials from ellips-
oidal shape, as measured by the second-order quantity κ.
Equation (5.166) can be used to eliminate the derivatives of κ from relation
(5.163), and, once again, the first-order expression (5.165) can be used to replace
f
in the second-order terms of relation (5.163). After collection of terms,
we obtain
r
0
ρ
¯ρ
6
6
r
0
f
+
f
−
(1
−
ρ/¯ρ)
f
3
4
f
2
4
r
0
4
r
0
(
1
−
ρ/¯ρ
)
mf
−
=
(
1
−
ρ/¯ρ
)
mf
+
9
−
ρ/¯ρ
2
3
−
ρ/¯ρ
ff
−
6
r
0
5
r
0
8
r
0
κ.
−
ρ/¯ρ)
f
2
−
(1
−
(5.167)
Expression (5.167) is the
Clairaut di
ff
erential equation
carried to second-order
accuracy.
Following Darwin (1899) and de Sitter (1924) it is usual to replace the flattening,
f
, by the new variable
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