A paper and pencil method of evaluating trajectories of space launchers

The determination of the trajectory of a space launcher is a complex optimization problem with boundary and path constraints. In particular the trajectory is initially constrained to a vertical launch, then it has to keep a zero incidence angle, the so called gravity turn phase, while several other path constraints can be active, such as: avoiding the flying over inhabited areas, controlling the fall of exhausted stages within safe regions, ensuring the launcher visibility from selected ground stations, etc. In this paper analytic formulas are developed for the gravity turn trajectory of a multistage space launcher. Formulas are given for relative velocity, flight path angle, altitude and range. All the trajectories start from the same initial conditions: vertical launch with zero relative velocity and they differ by the final value of the last stage flight path angle. It is proved in the paper that from these formulas it is possible to derive: i) a good estimate of the gravitational and aerodynamic losses of a space launcher ii) sub-optimal trajectories providing approximate evaluation of launcher performances iii) determination of the pitch manoeuvre to rotate the launcher from the vertical to the gravity turn trajectory iv) characterization of the attitude manoeuvres that the launcher control system has to implement during the atmospheric phase of the flight The two stage launcher Falcon 1 and the four stage Scout launcher are considered as test case to show effectiveness and criticalities of the analytic solutions proposed here