Page 74 - Satellite Communications, Fourth Edition

54 Chapter Two

The perifocal system is convenient for describing the motion of the
satellite in the orbital plane. If the earth were uniformly spherical, the
perifocal coordinates would be fixed in space, that is, inertial. However,
the equatorial bulge causes rotations of the perifocal coordinate system,
as described in Sec. 2.8.1. These rotations are taken into account when
the satellite position is transferred from perifocal coordinates to geo-
centric-equatorial coordinates, described in the next section.

Example 2.15 Using the values r 7257.5 km and 204.81° obtained in
the previous example, express r in vector form in the perifocal coordinate
system.
Solution

r P 7257.5 cos 204.81
6587.7 km
r Q 7257.5 sin 204.81
3045.3 km

Hence

r 6587.7P 3045.3Q km

2.9.6 The geocentric-equatorial
coordinate system
The geocentric-equatorial coordinate system is an inertial system of
axes, the reference line being fixed by the fixed stars. The reference
line is the line of Aries described in Sec. 2.5. (The phenomenon known
as the precession of the equinoxes is ignored here. This is a very slow
rotation of this reference frame, amounting to approximately
1.396971° per Julian century, where a Julian century consists of
36,525 mean solar days.) The fundamental plane is the earth’s equa-
torial plane. Figure 2.9 shows the part of the ellipse above the equato-
rial plane and the orbital angles Ω, w, and i. It should be kept in mind
that Ω and w may be slowly varying with time, as shown by Eqs. (2.12)
and (2.13).
The unit vectors in this system are labeled I, J, and K, and the coor-
dinate system is referred to as the IJK frame, with positive I pointing
along the line of Aries. The transformation of vector r from the PQW
frame to the IJK frame is most easily expressed by matrix multiplication.
If A is an m n matrix and B is an n p matrix, the product AB is an
m p matrix (details of matrix multiplication will be found in most good