Global Positioning System Reference
In-Depth Information
a
r
n
∞
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
m
−
C
nm
sin
m
λ
P
nm
(
cos
GM
λ +
S
nm
cos
m
η =−
θ
)
(2.75)
γ
r
2
sin
θ
n
=
2
m
=
0
Ge
oid and deflection of the vertical maps specifically adjusted to the NAD83 datum
ca
n be viewed at NGS (2002). NGS also provides software for convenient computa-
tio
n of these gravity functions.
The ellipsoid of revolution provides a simple model for the geometric shape of the
ea
rth. It is the reference for geometric computations in two and three dimensions.
As
signing a gravitational field that approximates the actual gravitational field of
th
e earth extends the functionality of the ellipsoid. Merely a few specifications are
ne
eded to fix the gravity and potential of the ellipsoid of revolution. We need an
ap
propriate mass for the ellipsoid and assume that the ellipsoid rotates with the earth.
Fu
rthermore, by means of mathematical conditions, the surface of the ellipsoid is
de
fined to be an equipotential surface of its own gravity field. Therefore, the plumb
lin
es of this gravity field intersect the ellipsoid perpendicularly. Because of this
pr
operty, this gravity field is called the normal gravity field, and the ellipsoid itself is
so
metimes referred to as the level ellipsoid.
It can be shown that the normal gravity potential
U
is completely specified by four
de
fining constants, which are symbolically expressed by
[39
Lin
—
*
1
——
No
*PgE
=
ω
U
f(a,J
2
,GM,
)
(2.76)
In
addition to
a
and
GM
, which have already been introduced above, we need the
dy
namical form factor
J
2
and the angular velocity of the earth
. The dynamic
fo
rm factor is a function of the principal moments of inertia of the earth (polar
an
d equatorial moment of inertia) and is functionally related to the flattening of
th
e ellipsoid. One important definition of the four constants in (2.76) comprises the
Ge
odetic Reference System of 1980 (GRS80). The defining constants are listed in
Ta
ble 2.3. A full documentation on this reference system is available in Moritz (1984).
The normal gravitational potential does not depend on the longitude and is given
by a series of zonal spherical harmonics
ω
[39
TABLE 2.3 Constants for GRS80
Defining Constants
Derived Constants
a
=
6378138 m
b
=
6356752
.
3141 m
=
×
10
8
m
3
/
s
2
=
GM
3986005
1
/f
298
.
257222101
10
−
8
J
2
=
108263
×
m
=
0
.
00344978600308
10
−
11
9
.
7803267715 m
/
s
2
ω =
7292115
×
rad
/
s
γ
e
=
9
.
8321863685 m
/
s
2
γ
p
=