Page 59 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 39

2
K is a constant which evaluates to 66,063.1704 km . The earth’s
1
oblateness has negligible effect on the semimajor axis a, and if a is
known, the mean motion is readily calculated. The orbital period taking
into account the earth’s oblateness is termed the anomalistic period
(e.g., from perigee to perigee). The mean motion specified in the NASA
bulletins is the reciprocal of the anomalistic period. The anomalistic
period is
2
s (2.9)
P A
n
where n is in radians per second.
If the known quantity is n (e.g., as is given in the NASA bulletins),
one can solve Eq. (2.8) for a, keeping in mind that n is also a function
0
of a. Equation (2.8) may be solved for a by finding the root of the fol-
lowing equation:

2
K (1 1.5 sin i)
1
n c1 d 0 (2.10)
Å a 3 a (1 e )
2 1.5
2
This is illustrated in the following example.
Example 2.4 A satellite is orbiting in the equatorial plane with a period from
perigee to perigee of 12 h. Given that the eccentricity is 0.002, calculate the semi-
major axis. The earth’s equatorial radius is 6378.1414 km.
2
Solution Given data: e 0.002; i 0°; P 12 h; K 1 66063.1704 km ; a E
3
6378.1414 km; 3.986005 1014 m /s 2
The mean motion is:
2
n
P
1.454 10 4 s 1

Assuming this is the same as n 0 , Kepler’s third law gives

1/3
a a b
n 2
26610 km

Solving the root equation yields a value of 26,612 km.
The oblateness of the earth also produces two rotations of the
orbital plane. The first of these, known as regression of the nodes, is
where the nodes appear to slide along the equator. In effect, the line