Global Positioning System Reference
In-Depth Information
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J
2
n
a
r
2
n
∞
GM
r
V
=
−
P
2
n
(
cos
θ
(2.77)
n
=
1
N
ote that the subscript 2
n
is to be read “2 times
n
.”
P
2
n
denotes Legendre polyno-
m
ials. The coefficients
J
2
n
are a function of
J
2
that can be readily computed. Sev-
er
al useful expressions can be derived from (2.77). For example, the normal gravity,
de
fined as the magnitude of the gradient of the normal gravity field (normal gravita-
tio
nal potential plus centrifugal potential), is given by Somigliana's closed formula
(H
eiskanen and Moritz, 1967, p. 70):
γ
p
sin
2
ϕ
a
γ
e
cos
2
ϕ
+
b
γ =
a
2
cos
2
ϕ
(2.78)
b
2
sin
2
ϕ
+
[40
Th
e normal gravity at height
h
above the ellipsoid is given by (Heiskanen and Moritz,
19
67, p. 70)
Lin
—
6.2
——
Nor
PgE
1
−
2
m
sin
2
ϕ
h
2
γ
e
a
5
3
γ
e
a
2
h
2
γ
h
− γ =−
+
f
+
m
+
3
f
+
+
(2.79)
Eq
uations (2.78) and (2.79) are often useful approximations of the actual gravity. The
va
lue for the auxiliary quantity
m
in (2.79) is given in Table 2.3. The normal gravity
va
lues for the poles and the equator,
γ
p
and
γ
e
are also listed in Table 2.3.
[40
2.3.4 Reductions to the Ellipsoid
Th
e relationship between the ellipsoidal height
h
, the orthometric height
H
, and the
ge
oid undulation is
h
=
H
+
N
(2.80)
Figure 2.13
Orthometric versus ellipsoidal heights.
