Geoscience Reference
In-Depth Information
we see by comparison that the Laplace surface spherical harmonic
Y
n
(
ϑ, λ
)
is given by
Y
n
(
ϑ, λ
)=
G
r
n
P
n
(cos
ψ
)
dM ,
(2-68)
earth
the dependence on
ϑ
and
λ
arises from the angle
ψ
since
cos
ψ
=cos
ϑ
cos
ϑ
+sin
ϑ
sin
ϑ
cos(
λ
−
λ
)
.
(2-69)
The spherical coordinates
ϑ, λ
have been defined in Sect. 1.4.
A more explicit form is obtained by using the decomposition formula
(1-108):
¯
S
nm
(
ϑ
,λ
)
.
=
∞
n
¯
R
nm
(
ϑ, λ
)
r
n
+1
S
nm
(
ϑ, λ
)
r
n
+1
1
l
1
2
n
+1
r
n
¯
r
n
¯
R
nm
(
ϑ
,λ
)+
n
=0
m
=0
(2-70)
Substituting this relation into the integral (2-64), we obtain
A
nm
,
V
=
∞
n
¯
¯
R
nm
(
ϑ, λ
)
r
n
+1
S
nm
(
ϑ, λ
)
r
n
+1
B
nm
+
(2-71)
n
=0
m
=0
A
nm
and
B
nm
are given by
where the constant coe
cients
(2
n
+1)
A
nm
=
G
¯
r
n
R
nm
(
ϑ
,λ
)
dM ,
earth
(2-72)
(2
n
+1)
B
nm
=
G
¯
r
n
S
nm
(
ϑ
,λ
)
dM .
earth
These formulas are very symmetrical and easy to remember: the coecient,
multiplied by 2
n
+ 1, of the solid harmonic
¯
R
nm
(
ϑ, λ
)
r
n
+1
(2-73)
is the integral of the solid harmonic
r
n
¯
R
nm
(
ϑ
,λ
)
.
(2-74)
An analogous relation results for
¯
S
nm
.
Note the nice analogy:
V
is a
sum
and the coecients are
integrals
!
Since the mass element is
dM
=
dx
dy
dz
=
r
2
sin
ϑ
dr
dϑ
dλ
,
(2-75)
