How Many Impulses Redux
A central problem in orbit transfer optimization is to determine the number, time, direction and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum's question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum's question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the $N$-impulse solution. It is also shown that a \textit{fundamental} minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. We find the answer to Edelbaum's question is not generally unique, but is frequently a set of equal-$Δv$ extremals. We further find, when Edelbaum's question is refined to seek the number of finite duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of \textit{optimal switching surfaces}