Geoscience Reference
In-Depth Information
and (2-160) takes the form
+
∞
V
=
GM
r
+
A
2
r
3
+
A
4
r
5
+
···
=
GM
r
1
r
2
n
+1
.
A
2
n
(2-162)
n
=1
Here we have used the fact that for all values of
n
P
n
(1) = 1
(2-163)
(see also Fig. 1.4). Comparing the coecients in both expressions for
V
,we
find
1
−
.
A
2
n
=(
−
1)
n
GM E
2
n
2
n
+1
me
3
q
0
2
n
2
n
+3
(2-164)
Equations (2-160) and (2-164) give the desired expression for the potential
of the level ellipsoid as a series of spherical harmonics.
The second-degree coecient
A
2
is
A
2
=
G
(
A
−
C
)
.
(2-165)
This follows from (2-91) by using
A
=
B
for reasons of symmetry. The
C
is the moment of inertia with respect to the axis of rotation, and
A
is the
moment of inertia with respect to any axis in the equatorial plane. By letting
n
= 1 in (2-164), we obtain
3
GM E
2
1
.
me
q
0
1
2
15
A
2
=
−
−
(2-166)
Comparing this with the preceding Eq. (2-165), we find
3
GM E
2
1
−
.
me
q
0
G
(
C − A
)=
1
2
15
(2-167)
Thus, the difference between the principal moments of inertia is expressed
in terms of “Stokes' constants”
a, b, M
,and
ω
.
It is possible to eliminate
q
0
from Eqs. (2-164) and (2-167), obtaining
1
ME
2
.
3
GM E
2
n
(2
n
+ 1)(2
n
+3)
n
+5
n
C
−
A
1)
n
A
2
n
=(
−
−
(2-168)
If we write the potential
V
in the form
1+
C
2
a
r
2
P
2
(cos
ϑ
)+
C
4
a
r
4
V
=
GM
r
P
4
(cos
ϑ
)+
···
1+
∞
r
2
n
P
2
n
(cos
ϑ
)
,
(2-169)
C
2
n
a
=
GM
r
n
=1
