Global Positioning System Reference
In-Depth Information
[
XYZ
]
T
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
the Kepler elements but by the vector
r
=
=
X
and the velocity
[
X
Y
Z
]
T
X
, expressed in the true celestial coordinate system (X). Figure
r
=
=
3.2 shows that
q
=
R
3
(
ω
)
R
1
(i)
R
3
(
Ω
)
X
(3.44)
=
R
qX
(
Ω
,i,
ω
)
X
where
R
i
denotes a rotation around axis
i
. The inverse transformation is
R
−
1
X
=
qX
(
Ω
,i,
ω
)
q
(3.45)
Differentiating (3.45) once gives
[61
X
R
−
1
=
qX
(
Ω
,i,
ω
)
q
(3.46)
Note that the elements of
R
qX
are constants, because the orbital ellipse does not
change its position in space. Using relations (3.25), (3.31), and (3.34), it follows that
Lin
—
1.3
——
No
*PgE
a(
cos
E
−
e)
r
cos
f
r
sin
f
0
=
a
√
1
q
=
(3.47)
−
e
2
sin
E
0
Th
e velocity becomes
[61
=
−
sin
E
−
sin
f
na
√
1
na
√
1
q
=
e
+
cos
f
0
(3.48)
−
e
2
cos
E
0
1
−
e
cos
E
−
e
2
Th
e first part of (3.48) follows from (3.39), and the second part can be verified using
kn
own relations between the anomalies
E
and
f
. Equations (3.45) to (3.48) transform
th
e Kepler elements into Cartesian coordinates and their velocities
(
X
,
X
)
.
The transformation from
(
X
,
X
)
to Kepler elements starts with the computation of
th
e magnitude and direction of the angular momentum vector
X
h
Z
]
T
h
=
X
×
=
[
h
X
h
Y
(3.49)
wh
ich is the vector form of Equation (3.15). The various components of
h
are shown
in
Figure 3.3. The right ascension of the ascending node and the inclination of the
or
bital plane are, according to Figure 3.3,
tan
−
1
h
X
−
Ω =
(3.50)
h
Y
