Optimal trajectory for vertical ascent to geosynchronous Earth orbit

In this paper, the optimal trajectory for vertical ascent to the geosynchronous Earth orbit is solved by using the parameter optimization technique. The performance index is to maximize the final mass. In other words, the propellant consumption is to be minimized. The time derivative of the velocity magnitude of the vehicle, called the acceleration profile, is assumed to be a polynomial function of the flight time, with the coefficients as free parameters to be selected. The required thrust vector is then derived as a function of the state variables and the acceleration profile. A first order polynomial function is adopted for the acceleration profile. The two coefficients along with the flight time are selected such that the final condition for geosynchronous Earth orbit insertion is satisfied and the final mass is maximized. When the initial mass is 430,000 kg and the initial flight path angle is 1°, and a laser propulsion system with 2500 s of specific impulse is used, the maximum final mass obtained is 110,965 kg. This best final mass is 25.81% of the initial mass. The ascending flight time is 1.923 h. For vertical ascending trajectory, the relative speed of the vehicle with respect to the atmosphere is the vertical component of the inertial vehicle velocity. Therefore, the dynamic pressure and the aerodynamic drag are reduced to lower levels.