Page 118 - Satellite Communications, Fourth Edition

98 Chapter Three

Figure 3.13 (a) Launch site A, showing launch azimuth A z ; (b) enlarged version of the
spherical triangle shown in (a). l is the latitude of the launch site.

slight difference between geodetic and geocentric latitudes may be
ignored here). The dotted line shows the satellite earth track, the satel-
lite having been launched at some azimuth angle A . Angle i is the
z
resulting inclination.
The spherical triangle of interest is shown in more detail in Fig. 3.13b.
This is a right spherical triangle, and Napier’s rule for this gives

(3.23)
cos i cos l sin A z
For a prograde orbit (see Fig. 2.4 and Sec. 2.5), 0 i 90 , and hence
cos i is positive. Also, 90 l 90 , and hence cos λ is also positive.
It follows therefore from Eq. (3.23) that 0 A z 180 , or the launch
azimuth must be easterly in order to obtain a prograde orbit, confirm-
ing what was already known.
For a fixed l, Eq. (3.23) also shows that to minimize the inclination i,
cos i should be a maximum, which requires sin A to be maximum, or
z
A 90°. Equation (3.23) shows that under these conditions
z
cos i min cos l (3.24)
or

i min l (3.25)

Thus the lowest inclination possible on initial launch is equal to the
latitude of the launch site. This result confirms the converse statement
made in Sec. 2.5 under inclination that the greatest latitude north or south
is equal to the inclination. From Cape Kennedy the smallest initial incli-
nation which can be achieved for easterly launches is approximately 28°.