In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravity well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravity well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds (relative to any reference frame) generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity (and so, its kinetic energy) is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry, who first described them in 1927. Because the vehicle remains near periapsis only for a short time, for the Oberth maneuver to be most effective the vehicle must be able to generate as much impulse as possible in the shortest possible time. As a result the Oberth maneuver is much more useful for high-thrust rocket engines such as liquid-propellant rockets, and less useful for low-thrust reaction engines such as ion drives, which take a long time to gain speed. Low-thrust rockets can use the Oberth effect by splitting a long departure burn into several short burns near the periapsis. The Oberth effect also can be used to understand the behavior of multi-stage rockets: the upper stage can generate much more usable kinetic energy than the total chemical energy of the propellant it carries. In terms of the energies involved, the Oberth effect is more effective at higher speeds because at high speed the propellant has significant kinetic energy due to its mass in addition to its chemical potential energy. At higher speed the vehicle is able to employ the greater change (reduction) in kinetic energy of the propellant (as it is exhausted backward and hence at reduced speed and hence reduced kinetic energy) to generate a greater increase in kinetic energy of the vehicle.
Explanation in terms of work and kinetic energy Because the kinetic energy of a mass varies in proportion to the square of change in its velocity ( E k = 1 2 m v 2 {\textstyle E_{\text{k}}={\frac {1}{2}}mv^{2}} ), an increase in velocity imparts a greater increase in kinetic energy at a high velocity than it would at a low velocity. For example, considering a rocket with a mass of 2 kg:
the rocket travelling at 1 metre per second has 12 = 1 joule of kinetic energy; increasing its velocity by 1 m/s increases the kinetic energy to 22 = 4 J, a gain of 3 J; at a velocity of 10 m/s, the rocket has 102 = 100 J of kinetic energy; increasing its velocity by 1 m/s increases the kinetic energy to 112 = 121 J, a gain of 21 J. at a velocity of 100 m/s, the rocket has 1002 = 10,000 J of kinetic energy; increasing its velocity by 1 m/s increases the kinetic energy to 1012 = 10,201 J, a gain of 201 J. This greater change in kinetic energy can then carry the rocket higher in the gravity well than if the propellant was burned at a lower speed.
Description in terms of work The thrust produced by a rocket engine is independent of the rocket’s velocity relative to the surrounding atmosphere. A rocket acting on a fixed object, as in a static firing, does no useful work; the rocket's chemical energy is progressively converted to kinetic energy of the exhaust, plus heat. But when the rocket moves, its thrust acts through the distance it moves. Force multiplied by displacement is the definition of mechanical work. The greater the velocity of the rocket and payload during the burn, the greater the displacement and the work done, and the greater the increase in kinetic energy of the rocket and its payload. As the velocity of the rocket increases, progressively more of the available kinetic energy goes to the rocket and its payload, and less to the exhaust. This is shown as follows. The mechanical work done on the rocket ( W {\displaystyle W} ) is defined as the dot product of the force of the engine's thrust ( F → {\displaystyle {\vec {F}}} ) and the displacement it travels during the burn ( s → {\displaystyle {\vec {s}}} ):
W = F → ⋅ s → . {\displaystyle W={\vec {F}}\cdot {\vec {s}}.}
If the burn is made in the prograde direction, F → ⋅ s → = ‖ F ‖ ⋅ ‖ s ‖ = F ⋅ s {\displaystyle {\vec {F}}\cdot {\vec {s}}=\|F\|\cdot \|s\|=F\cdot s} . The work results in a change in kinetic energy
Δ E k = F ⋅ s . {\displaystyle \Delta E_{k}=F\cdot s.}
Differentiating with respect to time, we obtain
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