Page 147 - Satellite Communications, Fourth Edition

Polarization 127

The coordinates for the local gravity direction, obtained from r R are

5138.48
r ≥ 3210.88 ¥ km
1968.75

From Eq. (5.16), the geocentric-equatorial coordinates for the propagation direc-
tion are

37025.5
R x a GSO
k ≥ R y ¥ ≥ 3210.88 ¥ km
R z 1968.75

For vertical polarization at the satellite, the geocentric-equatorial coordinates for
the polarization vector are x 0, y 0, and z 1:

0
e ≥ 0 ¥
1

The vector cross products can be written in determinant form, where a x , a y , a z , are
the unit vectors along the x, y, z, axes. Thus, Eq. (5.8) is
f k r

a x a y a z
≥ 37025.5 3210.88 1968.75 ¥
5138.48 3210.88 1968.75

7
8
a x 0 a y 8.3 ⋅ 10 a z 1.35 ⋅ 10 km 2
From Eq. (5.11):
g k e

a x a y a z
≥ 37025.5 3210.88 1968.75 ¥
0 0 1

a x 3210.88 a y 37025.5 a z 0 km
From Eq. (5.12):
h g k

a x a y a z
≥ 3210.88 37025.5 0 ¥
37025.5 3210.88 1968.75
7 6 9 2
a x 7.2894 ⋅ 10 a y 6.3214 ⋅ 10 a z 1.3812 ⋅ 10 km