Geoscience Reference
In-Depth Information
In rectangular coordinates this assumes the symmetrical form
2
r
5
(
B
+
C
V
=
GM
r
G
2
A
)
x
2
+(
C
+
A
2
B
)
y
2
+
+
−
−
(2-93)
(
A
+
B −
2
C
)
z
2
+
O
(1
/r
4
)
,
which is obtained by taking into account the relations (1-26) between rect-
angular and spherical coordinates.
Terms of order higher than 1
/r
3
may be neglected for larger distances
(say, for the distance to the moon), so that (2-92) or (2-93), omitting the
higher-order terms 0(1
/r
4
), are sucient for many astronomical purposes,
cf. Moritz and Mueller (1987). Note that the notation 0(1
/r
4
)meansterms
of the order of 1
/r
4
. For planetary distances even the first term,
V
=
GM
r
,
(2-94)
is generally sucient; it represents the potential of a point mass. Thus, for
very large distances, every body acts like a point mass.
Using the form (2-78) of the spherical-harmonic expansion of
V
,then
the coecients of lower degree are obtained from (2-79) and (2-91). We find
C
10
=
C
11
=
S
11
=0
,
C
−
(
A
+
B
)
/
2
Ma
2
C
20
=
−
,
C
21
=
S
21
=0
,
(2-95)
C
22
=
B
A
4
Ma
2
,
−
D
2
Ma
2
.
S
22
=
The first of these formulas shows that the summation in (2-78) actually
begins with
n
= 2; the others relate the coecients of second degree to the
mass and the moments and products of inertia of the earth.
2.7
The gravity field of the level ellipsoid
As a first approximation, the earth is a sphere; as a second approximation,
it may be considered an ellipsoid of revolution. Although the earth is not an
