Why is there not an opportunity to launch a satellite if the launch-site latitude is greater than the inclination?
The Angle between the projection of the initial orbit plane and the north direction is referred to as the Launch Azimuth, (Dux et al. 2011, pp. 1-16). The Azimuth has several implications to the activities of launching a satellite. It creates three possible windows of launching. The launch window refers to the time during which a launch can be done directly from a specific site into an orbital plane. The window duration is affected by factors like the fuel requirements, safety, and the desired points of injection. The launch window is always at the center of the optimal launch time. The launch window provides an opportunity to launch into the specified orbital plane. There are two possible launch opportunities if the orbit’s inclination angle is greater than the latitude of the launch site, (Zhang, Zheng, & Tang, 2012 pp. 35-41). However, when the angle of inclination equals the latitude of the launch site, the opportunities are reduced to one. If the latitude is greater than the inclination, then there is no opportunity to launch.
The reason can be explained mathematically using spherical trigonometry by calculating the launch Azimuth. If the desired orbit inclination angle is a, and the launch site latitude is b, assuming the launch azimuth is c, then
Cos (a) = cos (b) sin (c)
arcsincos(a)cos(b)=c
Mathematically speaking, the value of a must be greater than that of b otherwise the equation above will give an unrealistic solution. If the value of a is greater, the equation gives two possible solutions but when a and b are equal, the equation produces only one solution.
Bibliography:
Dux, I. J., Huwaldt, J. A., McKamey, R. S., & Dankanich, J. W. (2011, March). Mars ascent vehicle gross lift-off mass sensitivities for robotic Mars sample return. In Aerospace Conference, 2011 IEEE (pp. 1-16). IEEE.
Zhang, H., Zheng, W., & Tang, G. (2012). Stellar/inertial integrated guidance for responsive launch vehicles. Aerospace Science and Technology, 18(1), 35-41.
