Page 76 - Satellite Communications, Fourth Edition

56 Chapter Two

Example 2.16 Calculate the magnitude of the position vector in the PQW frame
for the orbit specified below. Calculate also the position vector in the IJK frame
and its magnitude. Confirm that this remains unchanged from the value obtained
in the PQW frame.
Solution The given orbital elements are

Ω 300°; w 60°; i 65°; r P 6500 km; r Q 4000 km
The magnitude of the r vector is

2
r 52( 6500) (4000) 2
7632.2 km

Substituting the angle values in Eq. (2.33b) gives

0.567 0.25
~
R ≥ 0.25 0.856 ¥
0.785 0.453

The vector components in the IJK frame are

0.567 0.25
r I
6500
≥r J ¥ ≥ 0.25 0.856 ¥ c d
4000
r K 0.785 0.453
4685.3
≥ 5047.7 ¥ km
3289.1

The magnitude of the r vector in the IJK frame is
2
2
r 52( 4685.3) (5047.7) ( 3289.1) 2
7632.2 km

This is seen to be the same as that obtained from the P and Q components.

2.9.7 Earth station referred to the IJK frame
The earth station’s position is given by the geographic coordinates of lat-
itude l and longitude f . (Unfortunately, there does not seem to be any
E
E
standardization of the symbols used for latitude and longitude. In some
texts, as here, the Greek lambda is used for latitude and the Greek phi
for longitude. In other texts, the reverse of this happens. One minor
advantage of the former is that latitude and lambda both begin with the
same “la” which makes the relationship easy to remember.)