# Comparison of a closed from approximate solution with a variational solution for optimal lunar launch

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## Abstract

In this study, optimal solutions are generated for a launch of a spacecraft from the lunar surface using the same physical parameters for the vehicle as used in Apollo 11. A calculus of variations solution is generated and compared with an approximate closed form solution. The problem involves minimizing the fuel for a lunar launch to obtain a desired orbital state vector. The control variable is the flight path angle. The problem is solved using two different methods. The first method is the method of perturbation functions (MPF) which was used by Lewallen to solve a nonlinear two-point boundary value problem. The second method is an approximate closed form solution which was developed by Jezewski. Since the MPF solution satisfies the classical optimality conditions, it is used as a reference by which to evaluate the results of the approximate closed form solution, thus providing a basis for evaluating the accuracy of the closed form solution. Solutions are obtained for two different sets of terminal constraints. The first set corresponds to the range free case in which only the terminal radius and velocities are specified. The second set corresponds to the fixed range case in which the terminal radius velocities, and range are specified. The terminal range solutions investigated are in the neighborhood of the range free solutions. The solutions are illustrated in terms of graphs of minimum fuel (or time) versus range angle. The graphs generated by MPF are then compared with those of the approximate closed form solution. It is found that the solutions for the closed form techniqure adequately match MPF only if the range angle is greater than or equal to the range free angle.