In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well, and then accelerates when its fall reaches maximum speed.   The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of an engine in a deeper gravity well generates greater mechanical energy than use higher up. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its engine is at the lowest possible orbital periapsis, when its orbital velocity is greatest and its gravitational potential energy is lowest. 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, the Austro-Hungarian-born German physicist and a founder of modern rocketry, who first described them in 1927. 
The Oberth effect is strongest at periapsis, where the speed is highest, because the fuel expended by firing a rocket engine at this point will lead to the fuel ending up in a lower orbit and therefore with less potential energy after the maneuver. Because the vehicle remains near periapsis only for a short time, the Oberth maneuver is most effective when the vehicle generates as much impulse as possible in the shortest possible time window. Thus, the Oberth maneuver is much more useful for high-thrust rocket engines like liquid-propellant rockets, and less useful for low-thrust reaction engines such as ion drives, which take a long time to gain speed. 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 propellants it carries.
Oberth calculation for parabolic orbit
If an impulsive burn of Δv is performed at periapsis in a parabolic orbit then the velocity at periapsis before the burn is equal to the escape velocity (Vesc), and the specific kinetic energy after the burn is:
When the vehicle leaves the gravity field, the loss of specific kinetic energy is:
so it retains the energy:
which is larger than the energy from a burn outside the gravitational field () by:
When the vehicle has left the gravity well, it is travelling at a speed of
For the case where the added impulse Δv is small compared to escape velocity, the 1 can be ignored and the effective Δv of the impulsive burn can be seen to be multiplied by a factor of simply:
Similar effects happen in closed and hyperbolic orbits.
If the vehicle travels at velocity v at the start of a burn that changes the velocity by Δv, then the change in specific orbital energy (SOE) is
Once the spacecraft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy approaches zero. Therefore, the larger the v at the time of the burn, the greater the final kinetic energy, and the higher the final velocity.
The effect becomes more pronounced the closer to the central body, or more generally, the deeper in the gravitational field potential the burn occurs, since the velocity is higher there.
So if a spacecraft is on a parabolic flyby of Jupiter with a periapsis velocity of 50 km/s, and it performs a 5 km/s burn, it turns out that the final velocity change at great distance is 22.9 km/s, giving a multiplication of the burn by 4.58 times.
- Robert B. Adams, Georgia A. Richardson. "Using the Two-Burn Escape Maneuver for Fast Transfers in the Solar System and Beyond" (PDF). NASA. Retrieved 15 May 2015.
- Robert Adams (25 Feb 2011). "What Would an Interstellar Mission Look Like?". Discovery News. Retrieved 15 May 2015.
- Hermann Oberth (1970). "Ways to spaceflight". Translation of the German language original "Wege zur Raumschiffahrt," (1920). Tunis, Tunisia: Agence Tunisienne de Public-Relations.
- following the calculation on rec.arts.sf.science