Geoscience Reference
In-Depth Information
the actual evaluation of the integrals requires that the density
be expressed
as a function of
r
,ϑ
,λ
. Although no such expression is available at present,
this fact does not diminish the theoretical and practical significance of spher-
ical harmonics, since the coe
cients
A
nm
,B
nm
can be determined from the
boundary values of gravity at the earth's surface. This is a boundary-value
problem (see Sect. 1.13) and will be elaborated later.
Recalling the relations (1-91) and (1-98) between conventional and fully
normalized spherical harmonics, we can also write equations (2-71) and (2-
72) in terms of conventional harmonics, readily obtaining
A
nm
R
nm
(
ϑ, λ
)
r
n
+1
,
V
=
∞
n
+
B
nm
S
nm
(
ϑ, λ
)
r
n
+1
(2-76)
n
=0
m
=0
where
A
n
0
=
G
r
n
P
n
(cos
ϑ
)
dM
;
earth
(
n
+
m
)!
G
earth
⎫
⎬
A
nm
=2
(
n
−
m
)!
r
n
R
nm
(
ϑ
,λ
)
dM
(2-77)
(
m
=0)
.
(
n
+
m
)!
G
earth
⎭
B
nm
=2
(
n − m
)!
r
n
S
nm
(
ϑ
,λ
)
dM
In connection with satellite dynamics, the potential
V
is often written in
the form
1+
∞
r
n
C
nm
R
nm
(
ϑ, λ
)+
S
nm
S
nm
(
ϑ, λ
)
,
(2-78)
a
n
V
=
GM
r
n
=1
m
=0
where
a
is the equatorial radius of the earth, so that
(
n
A
nm
=
GM a
n
C
nm
=0)
.
(2-79)
B
nm
=
GM a
n
S
nm
Distinguish the coecient
S
nm
and the function
S
nm
! The coecient
C
n
0
has formerly been denoted by
−
J
n
.Notethat
C
is related to cosine and
S
is related to sine.
