Page 73 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 53

Example 2.14 For satellite no. 14452 the NASA prediction bulletin for a certain
epoch gives the eccentricity as 9.5981 × 10 and the mean anomaly as 204.9779°.
−3
The mean motion is 14.2171404 rev/day. Calculate the true anomaly and the mag-
nitude of the radius vector 5 s after epoch. The semimajor axis is known to be
7194.9 km.
Solution The rotation in radians per second is

14.2171404 2
n
86400
> 0.001034 rad/s

The mean anomaly of 204.9779°, in radians is 3.57754, and 5 s after epoch the
mean anomaly becomes
M 5 3.57754 0.001034 5

> 3.5827 rad
> 3.5827 2 9.5981 10 3 sin 3.5827

5 3 2
(9.5981 10 ) sin (2 3.5827)
4
3.5746 rad ( 204.81°)

Applying Eq. (2.23) gives r as
2
7194.9 (1 9.5981 ) 10 6
r
1 9.5981 10 3 cos 204.81
7257.5 km
The magnitude r of the position vector r may be calculated by either
Eq. (2.23) or Eq. (2.30). It may be expressed in vector form in the perifocal
coordinate system. Here, the orbital plane is the fundamental plane, and
the origin is at the center of the earth (only earth-orbiting satellites are
being considered). The positive x axis lies in the orbital plane and passes
through the perigee. Unit vector P points along the positive x axis as
shown in Fig. 2.8. The positive y axis is rotated 90° from the x axis in the
orbital plane, in the direction of satellite motion, and the unit vector is
shown as Q. The positive z axis is normal to the orbital plane such that
coordinates xyz form a right-hand set, and the unit vector is shown as W.
The subscript w is used to distinguish the xyz coordinates in this
system, as shown in Fig. 2.8. The position vector in this coordinate
system, which will be referred to as the PQW frame, is given by

r (r cos )P (r sin )Q (2.32)