Geoscience Reference
In-Depth Information
Substituting these functions into the expression (2-77) for the coecients
A
nm
and
B
nm
yields for the zero-degree term
A
00
=
G
dM
=
GM
;
(2-84)
earth
that is, the product of the mass of the earth times the gravitational constant.
For the first-degree coecients, we get
A
10
=
G
A
11
=
G
B
11
=
G
z
dM ,
x
dM ,
y
dM
;
earth
earth
earth
(2-85)
and for the second-degree coecients
2
G
earth
A
20
=
1
x
2
y
2
+2
z
2
)
dM ,
(
−
−
A
21
=
G
B
21
=
G
x
z
dM ,
y
z
dM ,
(2-86)
earth
earth
4
G
earth
2
G
earth
A
22
=
1
B
22
=
1
(
x
2
y
2
)
dM ,
x
y
dM .
−
It is known from mechanics that
x
dM ,
y
dM ,
z
dM
1
M
1
M
1
M
x
c
=
y
c
=
z
c
=
(2-87)
are the rectangular coordinates of the center of gravity (center of mass,
geocenter). If the origin of the coordinate system coincides with the center
of gravity, then these coordinates and, hence, the integrals (2-85) are zero.
If the origin r
=0
is the center of gravity of the earth, then there will be
no first-degree terms in the spherical-harmonic expansion of the potential V
.
Therefore, this is true for our geocentric coordinate system.
The integrals
x
y
dM ,
y
z
dM ,
z
x
dM
(2-88)
are the
products of inertia
. They are zero if the coordinate axes coincide with
the principal axes of inertia. If the
z
-axis is identical with the mean rotational
axis of the earth, which coincides with the axis of maximum inertia, at least
the second and third of these products of inertia must vanish. Hence,
A
21
and
B
21
will be zero, but not so
B
22
, which is proportional to the first product of
