Page 51 - Satellite Communications, Fourth Edition

Orbits and Launching Methods 31

consequence of this is that the satellite takes longer to travel a given
distance when it is farther away from earth. Use is made of this property
to increase the length of time a satellite can be seen from particular
geographic regions of the earth.

2.4 Kepler’s Third Law
Kepler’s third law states that the square of the periodic time of orbit
is proportional to the cube of the mean distance between the two
bodies. The mean distance is equal to the semimajor axis a. For the
artificial satellites orbiting the earth, Kepler’s third law can be written
in the form

3
a (2.2)
n 2
where n is the mean motion of the satellite in radians per second and is
the earth’s geocentric gravitational constant. Its value is (see Wertz, 1984,
Table L3).

14 3 2
3.986005 10 m /s (2.3)
Equation (2.2) applies only to the ideal situation of a satellite orbit-
ing a perfectly spherical earth of uniform mass, with no perturbing
forces acting, such as atmospheric drag. Later, in Sec. 2.8, the effects
of the earth’s oblateness and atmospheric drag will be taken into
account.
With n in radians per second, the orbital period in seconds is given by

2
P (2.4)
n
The importance of Kepler’s third law is that it shows there is a fixed
relationship between period and semimajor axis. One very important
orbit in particular, known as the geostationary orbit, is determined by
the rotational period of the earth and is described in Chap. 3. In antic-
ipation of this, the approximate radius of the geostationary orbit is
determined in the following example.

Example 2.1 Calculate the radius of a circular orbit for which the period is 1 day.

Solution There are 86,400 seconds in 1 day, and therefore the mean motion is
2
n
86400
5
7.272 10 rad/s