Graphical analysis of singular trajectories for optimal navigation

It is well-known that the thrust level used to optimize a rocket trajectory depends on the value of a "switching function" K(t), and that the thrust can have an intermediate value only where K(t) is zero. In this paper the necessary conditions for a drag-free intermediate-thrust suharc (in a plane containing an inverse-square-law central force field) are pursued to the point of graphical representation. A coordinate system is presented on which can be defined regions where optimal intermediate-thrust trajectories are impossible. For this same coordinate system (the coordinates are functions of thrust direction and velocity direction) it is shown how analytical results permit plotting several families of contours, including lines of constant instantaneous eccentricity, instantaneous perigee angle, and acceleration ratio (ratio of thrust acceleration to local gravitational acceleration). It is further shown how numerical integration permits construction of other families of curves including a family of intermediate-thrust curves, one of which must be followed during an intermediate-thrust subarc.