Page 55 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 35

circular orbit, M gives the angular position of the satellite in the orbit.
For elliptical orbit, the position is much more difficult to calculate, and
M is used as an intermediate step in the calculation as described in
Sec. 2.9.5.
True anomaly. The true anomaly is the angle from perigee to the
satellite position, measured at the earth’s center. This gives the
true angular position of the satellite in the orbit as a function of
time. A method of determining the true anomaly is described in
Sec. 2.9.5.

2.6 Orbital Elements
Earth-orbiting artificial satellites are defined by six orbital elements
referred to as the keplerian element set. Two of these, the semimajor axis
a and the eccentricity e described in Sec. 2.2, give the shape of the
ellipse. A third, the mean anomaly M , gives the position of the satel-
0
lite in its orbit at a reference time known as the epoch. A fourth, the argu-
ment of perigee w, gives the rotation of the orbit’s perigee point relative
to the orbit’s line of nodes in the earth’s equatorial plane. The remain-
ing two elements, the inclination i and the right ascension of the ascend-
ing node Ω, relate the orbital plane’s position to the earth. These four
elements are described in Sec. 2.5.
Because the equatorial bulge causes slow variations in w and Ω, and
because other perturbing forces may alter the orbital elements slightly,
the values are specified for the reference time or epoch, and thus the
epoch also must be specified.
Appendix C lists the two-line elements provided to users by the U.S.
National Aeronautics and Space Administration (NASA). The two-line
elements may be downloaded from Celestrak at http://celestrak.com/
NORAD/elements/. Figure 2.6 shows how to interpret the NASA two-line
elements.
It will be seen that the semimajor axis is not specified, but this can
be calculated from the data given. An example calculation is presented
in Example 2.2.

Example 2.2 Calculate the semimajor axis for the satellite parameters given in
Table 2.1.
Solution The mean motion is given in Table 2.1 as NN 14.23304826 day 1
In rad/s this is

n 0 2 NN
1
0.00104 s