Geology Reference
In-Depth Information
can be written as follows:
from the Earth. The
n
D
0 term is evidently the
potential (
14.19
) generated by a homogeneous
sphere. Regarding the term for
n
D
1, noting that
r
0
cos§
D
r
r
0
,wehave:
2
2x—
—
2
C
8
2x—
—
2
2
1
3
G.x;—/
D
1
C
16
2x—
—
2
3
5
Z
C
C
:::
¡
r
0
r
0
cos §dx
0
dy
0
d
z
0
G
r
2
V
1
.r/
D
3
2
x
2
—
2
1
2
D
1
C
x—
C
R
Z
¡
r
0
r
0
dx
0
dy
0
d
z
0
D
0
G
r
2
5
2
x
3
2
x
—
3
D
r
3
C
C
:::
R
(14.44)
X
—
n
P
n
.x/
D
In fact, the last integral in (
14.44
)represents
the centre of mass of the Earth. Finally, let us
consider the contribution for
n
D
3. Recalling
(14.41)
n
D
0
Consequently, the function
G
seems to have
the capability to “generate” Legendre polynomi-
als when it is expanded in power series. In this
sense, it is called the
generating function
for these
polynomials. Now, using (
14.40
) we see that at
sufficiently large distance from the body it is
possible to write:
2r
3
Z
R
¡
r
0
r
0
2
3cos
2
§
1
dx
0
dy
0
d
z
0
G
V
2
.r/
D
r
3
Z
R
¡
r
0
r
0
2
dx
0
dy
0
d
z
0
G
D
r
2
2rr
0
cos §
1=2
2r
3
Z
R
j
r
r
0
j
1
C
r
0
2
¡
r
0
r
0
2
sin
2
§dx
0
dy
0
d
z
0
D
3G
1
C
r
0
=r
2
2
r
0
=r
cos§
1=2
1
r
D
r
3
Z
R
¡
r
0
r
0
2
dx
0
dy
0
d
z
0
G
D
r
G
cos§;r=r
0
1
D
(14.42)
Z
¡
r
0
r
0
2
1
cos
2
§
dx
0
dy
0
d
z
0
3G
2r
3
Therefore, from (
14.41
) we have that the Pois-
son integral (
14.38
) can be expressed as follows:
R
Z
2r
3
I
xx
C
I
yy
C
I
zz
¡
r
0
G
3G
2r
3
D
V.r/
D
G
Z
¡.r
0
/
j
r
r
0
j
dx
0
dy
0
d
z
0
R
h
r
0
2
r
r
0
2
i
dx
0
dy
0
d
z
0
R
r
0
r
0
r
n
Z
X
G
r
G
2r
3
Œ.A
C
B
C
C/
3I .
D
D
r/
nD0
R
(14.45)
P
n
.cos §/dx
0
dy
0
d
z
0
(14.43)
where
A
I
xx
,
B
I
yy
,and
C
I
zz
are the mo-
ments of inertia about the three coordinate axes
and I.
Now let us apply these concepts to the Earth
and assume that the reference frame is a geocen-
tric frame, so that the origin coincides with the
first three terms of the series (
14.43
) give a good
approximation of the real field at large distances
r/ is the moment of inertia about the
axis of
r
. Consequently, considering only the first
three terms of the expansion (
14.43
)wehavethe
following approximate
formula of MacCullagh
for the potential:
