Geoscience Reference
In-Depth Information
The component
γ
λ
is zero because
U
does not contain
λ
. This is also evident
from the rotational symmetry.
Performing the partial differentiations, we find
ω
2
a
2
E
u
2
+
E
2
q
q
0
2
sin
2
β
6
−
GM
u
2
+
E
2
+
1
ω
2
u
cos
2
β,
−
wγ
u
=
−
(2-132)
wγ
β
=
q
0
+
ω
2
u
2
+
E
2
sin
β
cos
β,
ω
2
a
2
√
u
2
+
E
2
q
−
−
where we have set
du
=3
1+
u
2
E
2
1
u
2
+
E
2
E
dq
u
E
tan
−
1
E
u
q
=
−
−
−
1
.
(2-133)
Note that
q
does not mean
dq/du
; this notation has been borrowed from Hir-
vonen (1960), where
q
is the derivative with respect to another independent
variable which we are not using here.
For the level ellipsoid
S
0
itself, we have
u
=
b
and get
γ
β,
0
=0
.
(2-134)
(Note that we will often mark quantities referred to
S
0
by the subscript
0.) This is also evident because on
S
0
the gravity vector is normal to the
level surface
S
0
. Hence, in addition to the
λ
-component, the
β
-component
is also zero on the reference ellipsoid
u
=
b
. Note that the other coordinate
ellipsoids
u
= constant are not equipotential surfaces
U
= constant, so that
the
β
-component will not in general be zero.
Thus, the total gravity on the ellipsoid
S
0
, which we simply denote by
γ
,isgivenby
GM
a
a
2
sin
2
β
+
b
2
cos
2
β
·
γ
=
|
γ
u,
0
|
=
(2-135)
1+
ω
2
a
2
E
GM
cos
2
β
,
q
0
q
0
2
6
−
ω
2
a
2
b
GM
1
sin
2
β
·
−
since on
S
0
we get the relations
√
u
2
+
E
2
=
√
b
2
+
E
2
=
a,
b
2
+
E
2
sin
2
β
=
1
a
a
2
sin
2
β
+
b
2
cos
2
β.
(2-136)
w
0
=
1
a
Now we introduce the abbreviation
m
=
ω
2
a
2
b
GM
(2-137)