Equation for Optimal Power-Limited Spacecraft Trajectories

For a power-limited spacecraft the necessary conditions for an optimal (minimum-propellant) trajectory and the equations of motion can be combined into a single fourth-order differential equation for the position vector. Every solution to this equation then represents an optimal trajectory through the specified gravitational field. By imposing the appropriate boundary conditions on the solution of this equation, the desired optimal trajectory can be determined. A significant increase in payload can be attained by an optimal trajectory.