Geology Reference
In-Depth Information
r 0
0 ρ( r 0 ) π
R
r 0 R 2 sinθ d θ dr 0
V i ρ dV
=
0
r 0 ρ( r 0 ) 1
m 2
r 0
8 r 0
8 r 0
45 m 2 ff +
45 m 2 ff
=
+
0
15 m 2 f 2
dr 0
4
4
15 m 2 f 2
+
+···
r 0
r 0 ρ( r 0 ) dr 0 ,
=
(5.155)
0
where the expansion (5.152) has been used with n
= −
1, and the prime indicates
erentiation. Division of (5.155) by (4/3)π r 0 , the volume of
di
ff
V
i , yields the mean
density of material enclosed by U i as
r 0
3
r 0
r 0 ρ( r 0 ) dr 0 .
¯ρ( r 0 )
=
(5.156)
0
For n
=−
1, we then have that
dU i
dr 0
g 0 ( r 0 )
=
1
45 r 0 f 2
5 f 2
GM i
r 0
2
3 m ( r 0 )
4
2 r 0 ff +
=
+
+···
,
(5.157)
2 r 0 / GM i .
where g 0 ( r 0 ) is introduced as a mean gravity intensity and m ( r 0 )
After substitution from (5.157), the relation for n
=
0 becomes
1
45 r 0 ff +
f 2
GM i
r 0
1
3 m ( r 0 )
4
U i =
V (0)
+
+
+···
,
(5.158)
V (0) being the gravitational potential at the geocentre, due to mass external to S i ,
which by (5.127) is
G
G d
r 0 ρ( r 0 ) π
r
R
r 0 R sinθ d θ dr 0
V (0)
=−
dV
=−
V V i
0
r 0 ρ( r 0 ) 1
dr 0 .
G d
r 0
45 r 0 ff +
f 2
4
=−
+···
(5.159)
With substitution for g 0 ( r 0 ) from (5.157), the relations for n
=
2and n
=
4 become,
respectively,
 
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