A Variational Calculus Solution to the Optimum Orbital Escape Problem, and Comparison with Several Steering Programs of Simple Analytical Form

摘要

The calculus of variations is used to determine the power level, thrust magntiude, and steering program which will minimize the propellant necessary for a vehicle to effect an escape from an orbit around the Earth. The trajectory is shown to consist of a number of thrusting sub-arcs proportional to the value of the time constraint. If the time constraint is less than one revolution of the initial orbit, the trajectory consists of one thrusting-arc. For a thrust-limited vehicle it is shown on physical grounds that the thrusting-arc consists of maximum thrust. For a power-limited vehicle, the variational apparatus yields a maximum power, maximum specific impulse thrusting-arc. The optimum steering program determined numerically through the use of a digital computer, is found to begin slightly below the horizontal, and increases until the thrust direction is exactly aligned with the velocity vector at the terminal point. Several steering programs of a simple analytical form are also computed numerically for a comparison with the optimum program. (Author)