The Other “Rocket Equation”: Re-introducing a classic result for the early-stage, approximate design of power-limited, variable-exhaust-velocity rockets and missions

C. P. S. Swanson

jbis-078-08-0271

DOI https://doi.org/10.59332/jbis-078-08-0271

In this paper I describe the utility of, and present an accessible derivation of, the classic Leitmann equation. This is an analytic, approximate equation which relates propulsion system requirements to space mission requirements for a power-limited or variable-specific-impulse rocket. It is valid in the limiting case that the rocket thrust acceleration is much larger than that of gravity, which is true for certain categories of interplanetary transfers. This equation has been known since at least 1959, but its utility to early-stage mission design and scoping studies may no longer be universally appreciated. The accessible derivation is an elementary application of the Calculus of Variations and does not require detailed knowledge of that field, making it suitable for application by students and for building understanding and physical insight. I show that this equation can be used to produce approximate designs and scoping studies for crewed missions to Mars and the outer planets, and to robotic flybys of other star systems. Finally I summarize a few other scenarios approximately susceptible to this accessible approach: Ascending to orbit from a circular, airless world; starting and ending an interplanetary transfer in a low orbit; and accomplishing higher-energy missions by jettisoning (staging) propulsion units.

Keywords: Spacecraft Trajectory Optimization, Variable Specific Impulse, Mars