Geoscience Reference
In-Depth Information
Now we can express
U
0
in terms of the mass
M
. For large values of
u
,we
have
tan
−
1
E
u
=
E
u
+
O
(1
/u
3
)
.
(2-115)
From the expressions (1-26) for spherical coordinates and from equations
(1-151) for ellipsoidal-harmonic coordinates, we find
x
2
+
y
2
+
z
2
=
r
2
=
u
2
+
E
2
cos
2
β,
(2-116)
so that for large values of
r
we have
1
u
=
1
r
+
O
(1
/r
3
)
(2-117)
and
tan
−
1
E
u
=
E
r
+
O
(1
/r
3
)
,
(2-118)
where
O
(
x
) means “small of order
x
”, i.e., small of order 1
/r
3
in our case.
For very large distances
r
, the first term in (2-114) is dominant, so that
asymptotically
V
=
U
0
−
3
ω
2
a
2
E
tan
−
1
(
E/b
)
1
r
+
O
(1
/r
3
)
.
1
(2-119)
We know from Sect. 2.6 that
V
=
GM
r
+
O
(1
/r
3
)
.
(2-120)
Substituting this expression for
V
into the left-hand side of (2-119) yields
GM
r
=
U
0
−
3
ω
2
a
2
E
tan
−
1
(
E/b
)
1
1
r
+
O
(1
/r
3
)
.
(2-121)
Now multiply this equation by
r
and let then
r
→
0. The result is (rigor-
ously!)
GM
=
U
0
−
3
ω
2
a
2
E
tan
−
1
(
E/b
)
,
1
(2-122)
which may be rearranged to
U
0
=
GM
E
tan
−
1
E
b
+
3
ω
2
a
2
.
(2-123)
This is the desired relation between mass
M
and potential
U
0
.
Substituting the result for
U
0
obtained in (2-123) into (2-114), simplifies
the expression for
V
to
V
=
GM
E
tan
−
1
E
u
+
3
ω
2
a
2
q
q
0
P
2
(sin
β
)
.
(2-124)
