Geoscience Reference
In-Depth Information
The corresponding fully normalized coecients
1
C
n
0
√
2
n
+1
C
n
0
,
=
⎫
⎬
⎭
(
n
+
m
)!
2(2
n
+1)(
n
C
nm
=
m
)!
C
nm
(2-80)
−
(
m
=0)
(
n
+
m
)!
2(2
n
+1)(
n
S
nm
=
m
)!
S
nm
−
are also used.
It is obvious that the nonzonal terms (
m
= 0) would be missing in
all these expansions if the earth had complete rotational symmetry, since
the terms mentioned depend on the longitude
λ
. In rotationally symmetrical
bodies there is no dependence on
λ
because all longitudes are equivalent. The
tesseral and sectorial harmonics will be small, however, since the deviations
from rotational symmetry are slight.
Finally, we discuss the convergence of (2-71), or of the equivalent series
expansions, of the earth's potential. This series is an expansion in powers
of 1
/r
. Therefore, the larger
r
is, the better the convergence. For smaller
r
it is not necessarily convergent. For an arbitrary body, the expansion of
V
into spherical harmonics can be shown to converge always outside the small-
est sphere
r
=
r
0
that completely encloses the body (Fig. 2.10). Inside this
sphere, the series is usually divergent. In certain cases it can converge partly
inside the sphere
r
=
r
0
. If the earth were a homogeneous ellipsoid of about
r=
0
r
0
O
Fig. 2.10. Spherical-harmonic expansion of
V
converges outside the sphere
r
=
r
0
