No Access Submitted: 18 May 2020 Accepted: 26 August 2020 Published Online: 23 December 2020
American Journal of Physics 89, 72 (2021);
You have a rocket in a high circular orbit around a massive central body (a planet or the Sun) and wish to escape with the fastest possible speed at infinity for a given amount of fuel. In 1929, Hermann Oberth showed that firing two separate impulses (one retrograde and one prograde) can be more effective than a direct transfer that expends all the fuel at once. This is due to the Oberth effect, whereby a small impulse applied at periapsis can produce a large change in the rocket's orbital mechanical energy, without violating energy conservation. In 1959, Theodore Edelbaum showed that this effect could be exploited further by using up to three separate impulses: prograde, retrograde, and then prograde. The use of more than one impulse to escape can produce a final speed even faster than that of a fictional spacecraft that is unaffected by gravity. We compare the three escape strategies in terms of their final speeds attainable, and the time required to reach a given distance from the central body. To do so, in the Appendix we use conservation laws to derive a “radial Kepler equation” for hyperbolic trajectories, which provides a direct relationship between travel time and distance from the central body. The 3-impulse Edelbaum maneuver can be applied to interplanetary transfers, exploration of the outer solar system and beyond, and (in time reverse) efficient arrival and orbital capture. The physics principles employed are appropriate for an undergraduate mechanics course.
The authors thank San Diego State University Library staff for assistance with locating aerospace journals published before 1970, an anonymous referee for suggestions, and Donna and Laura Edelbaum for encouraging this work.
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