Interplanetary Mission Design Handbook, Volume I, Part 2

cos 8 -
sin EL LIMIT + COS 0L
(14)
As 5,, gets longer, the sector of unavailable launch azimuths
reaches the safety boundaries of permissible launches, and
planar launch ceases to exist (Fig . 13) .' This subject will be
addressed again in the discussion of "dogleg" ascents .
Figure 14 is a sketch of a typical daily launch geometry
situation, shown upon a Mercator map of the celestial sphere .
The two launch windows exhibit a similar geometry since the
inclinations of the ascent trajectory planes are functions of
launch site latitude OL and azimuth E L only :
cos i = cos 01, X sin E L (15)
The two daily opportunities do differ greatly, however, in
the right ascension of the ascending mode S2 of the orbit and
in the length of the-traversed in-plane arc, the range angle 0 .
The angular equatorial distance between the ascending node
and the launch site meridian is given by
sin X sin EL
sin (a, -- Q) = L
sin i
(16)
Quadrant rules for this equation involve the observation that a
negative cos E L places (UL - 2) into the second or third quadrant,
while the sign of sin (a 1 - 2) determines the choice
between them .
The range angle 0 is measured in the inertial ascent trajectory
plane from the lift-off point at launch all the way to the
departure asymptote direction, and can be computed for a
given launch time t I or a te, - a L (tL ) and an azimuth E L
already known from Eq . (9) as follows :
cos 8 = sin 5 X sin OL +cos 8 - X cos 0I X cos (a,, - al )
(17)