Page 60 - Satellite Communications, Fourth Edition

40 Chapter Two

of nodes, which is in the equatorial plane, rotates about the center
of the earth. Thus Ω, the right ascension of the ascending node, shifts
its position.
If the orbit is prograde (see Fig. 2.4), the nodes slide westward, and
if retrograde, they slide eastward. As seen from the ascending node, a
satellite in prograde orbit moves eastward, and in a retrograde orbit,
westward. The nodes therefore move in a direction opposite to the direc-
tion of satellite motion, hence the term regression of the nodes. For a
polar orbit (i 90°), the regression is zero.
The second effect is rotation of apsides in the orbital plane, described
below. Both effects depend on the mean motion n, the semimajor axis
a, and the eccentricity e. These factors can be grouped into one factor K
given by

nK 1
K 2 2 2 (2.11)
a (1 e )
K will have the same units as n. Thus, with n in rad/day, K will be in
rad/day, and with n in degrees/day, K will be in degrees/day. An approx-
imate expression for the rate of change of with respect to time is
(Wertz, 1984)

d
K cos i (2.12)
dt
where i is the inclination. The rate of regression of the nodes will have
the same units as n.
When the rate of change given by Eq. (2.12) is negative, the regres-
sion is westward, and when the rate is positive, the regression is east-
ward. It will be seen, therefore that for eastward regression, i must be
greater than 90 , or the orbit must be retrograde. It is possible to choose
o
values of a, e, and i such that the rate of rotation is 0.9856°/day east-
ward. Such an orbit is said to be sun synchronous and is described fur-
ther in Sec. 2.10.
The other major effect produced by the equatorial bulge is a rotation
of the line of apsides. This line rotates in the orbital plane, resulting in
the argument of perigee changing with time. The rate of change is given
by (Wertz, 1984)

d 2
K(2 2.5 sin i) (2.13)
dt
Again, the units for the rate of rotation of the line of apsides will be
the same as those for n (incorporated in K). When the inclination i is
equal to 63.435°, the term within the parentheses is equal to zero, and