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In astronautics, a **powered flyby**, or **Oberth maneuver**, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed.^{ [1] } 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 a reaction engine at higher speeds 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.^{ [1] } 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.^{ [1] } The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, a Transylvanian Saxon physicist and a founder of modern rocketry.^{ [2] }

- Explanation in terms of momentum
- Explanation in terms of work and kinetic energy
- Description in terms of work
- Impulsive burn
- Oberth calculation for a parabolic orbit
- Parabolic example
- Paradox
- See also
- References
- External links

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 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.^{ [2] }

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 in addition to its chemical potential energy.^{ [2] }^{: 204 } 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.^{ [2] }^{: 204 }

A rocket works by transferring momentum to its propellant.^{ [3] } At a fixed exhaust velocity, this will be a fixed amount of momentum per unit of propellant.^{ [4] } For a given mass of rocket (including remaining propellant), this implies a fixed change in velocity per unit of propellant.

Because kinetic energy equals *mv*^{2}/2, this change in velocity imparts a greater increase in kinetic energy at a high velocity than it would at a low velocity. For example, considering a 2 kg rocket:

- at 1 m/s, the rocket starts with 1
^{2}= 1 J of kinetic energy. Adding 1 m/s increases the kinetic energy to 2^{2}= 4 J, for a gain of 3 J; - at 10 m/s, the rocket starts with 10
^{2}= 100 J of kinetic energy. Adding 1 m/s increases the kinetic energy to 11^{2}= 121 J, for a gain of 21 J.

This greater change in kinetic energy can then carry the rocket higher in the gravity well than if the propellant were burned at a lower speed.

Rocket engines produce the same force regardless of their velocity. A rocket acting on a fixed object, as in a static firing, does no useful work at all; the rocket's stored energy is entirely expended on accelerating its propellant in the form of exhaust. But when the rocket moves, its thrust acts through the distance it moves. Force multiplied by distance is the definition of mechanical energy or work. So the farther the rocket and payload move during the burn (i.e. the faster they move), the greater the kinetic energy imparted to the rocket and its payload and the less to its exhaust.

This is shown as follows. The mechanical work done on the rocket () is defined as the dot product of the force of the engine's thrust () and the displacement it travels during the burn ():

If the burn is made in the prograde direction, . The work results in a change in kinetic energy

Differentiating with respect to time, we obtain

or

where is the velocity. Dividing by the instantaneous mass to express this in terms of specific energy (), we get

where is the acceleration vector.

Thus it can be readily seen that the rate of gain of specific energy of every part of the rocket is proportional to speed and, given this, the equation can be integrated (numerically or otherwise) to calculate the overall increase in specific energy of the rocket.

Integrating the above energy equation is often unnecessary if the burn duration is short. Short burns of chemical rocket engines close to periapsis or elsewhere are usually mathematically modeled as impulsive burns, where the force of the engine dominates any other forces that might change the vehicle's energy over the burn.

For example, as a vehicle falls toward periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Briefly burning the engine (an “impulsive burn”) prograde at periapsis increases the velocity by the same increment as at any other time (). However, since the vehicle's kinetic energy is related to the *square* of its velocity, this increase in velocity has a non-linear effect on the vehicle's kinetic energy, leaving it with higher energy than if the burn were achieved at any other time.^{ [5] }

If an impulsive burn of Δ*v* is performed at periapsis in a parabolic orbit, then the velocity at periapsis before the burn is equal to the escape velocity (*V*_{esc}), and the specific kinetic energy after the burn is^{ [6] }

where .

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 traveling at a speed

For the case where the added impulse Δ*v* is small compared to escape velocity, the 1 can be ignored, and the effective Δ*v* of the impulsive burn can be seen to be multiplied by a factor of simply

and one gets

- ≈

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 Δ*v*, then the change in specific orbital energy (SOE) due to the new orbit 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 in which 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 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.

It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's kinetic energy is balanced by a relative decrease in the kinetic energy the exhaust is left with (the kinetic energy of the exhaust may still increase, but it does not increase as much).^{ [2] }^{: 204 } Contrast this to the situation of static firing, where the speed of the engine is fixed at zero. This means that its kinetic energy does not increase at all, and all the chemical energy released by the fuel is converted to the exhaust's kinetic energy (and heat).

At very high speeds the mechanical power imparted to the rocket can exceed the total power liberated in the combustion of the propellant; this may also seem to violate conservation of energy. But the propellants in a fast-moving rocket carry energy not only chemically, but also in their own kinetic energy, which at speeds above a few kilometres per second exceed the chemical component. When these propellants are burned, some of this kinetic energy is transferred to the rocket along with the chemical energy released by burning.^{ [7] }

The Oberth effect can therefore partly make up for what is extremely low efficiency early in the rocket's flight when it is moving only slowly. Most of the work done by a rocket early in flight is "invested" in the kinetic energy of the propellant not yet burned, part of which they will release later when they are burned.

In physics, the **Maxwell–Boltzmann distribution**, or **Maxwell(ian) distribution**, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.

A **rocket** is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirely from propellant carried within the vehicle; therefore a rocket can fly in the vacuum of space. Rockets work more efficiently in a vacuum and incur a loss of thrust due to the opposing pressure of the atmosphere.

A **single-stage-to-orbit** (**SSTO**) vehicle reaches orbit from the surface of a body using only propellants and fluids and without expending tanks, engines, or other major hardware. The term usually, but not exclusively, refers to reusable vehicles. To date, no Earth-launched SSTO launch vehicles have ever been flown; orbital launches from Earth have been performed by either fully or partially expendable multi-stage rockets.

A **pulsed plasma thruster** (**PPT**), also known as a plasma jet engine, is a form of electric spacecraft propulsion. PPTs are generally considered the simplest form of electric spacecraft propulsion and were the first form of electric propulsion to be flown in space, having flown on two Soviet probes starting in 1964. PPTs are generally flown on spacecraft with a surplus of electricity from abundantly available solar energy.

In celestial mechanics, **escape velocity** or **escape speed** is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming:

**Specific impulse** is a measure of how efficiently a reaction mass engine, such as a rocket using propellant or a jet engine using fuel, generates thrust. For engines like cold gas thrusters whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do *positive work* if when applied it has a component in the direction of the displacement of the point of application. A force does *negative work* if it has a component opposite to the direction of the displacement at the point of application of the force.

In astronautics, the **Hohmann transfer orbit** is an orbital maneuver used to transfer a spacecraft between two orbits of different altitudes around a central body. For example, a Hohmann transfer could be used to raise a satellite's orbit from low Earth orbit to geostationary orbit. In the idealized case, the initial and target orbits are both circular and coplanar. The maneuver is accomplished by placing the craft into an elliptical transfer orbit that is tangential to both the initial and target orbits. The maneuver uses two impulsive engine burns: the first establishes the transfer orbit, and the second adjusts the orbit to match the target.

**Orbital mechanics** or **astrodynamics** is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

**Delta- v**, symbolized as and pronounced

In aerospace engineering, the **propellant mass fraction** is the portion of a vehicle's mass which does not reach the destination, usually used as a measure of the vehicle's performance. In other words, the **propellant mass fraction** is the ratio between the propellant mass and the initial mass of the vehicle. In a spacecraft, the destination is usually an orbit, while for aircraft it is their landing location. A higher mass fraction represents less weight in a design. Another related measure is the payload fraction, which is the fraction of initial weight that is payload. It can be applied to a vehicle, a stage of a vehicle or to a rocket propulsion system.

A **multistage rocket** or **step rocket** is a launch vehicle that uses two or more rocket *stages*, each of which contains its own engines and propellant. A *tandem* or *serial* stage is mounted on top of another stage; a *parallel* stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched.

The **classical rocket equation**, or **ideal rocket equation** is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum. It is credited to Konstantin Tsiolkovsky, who independently derived it and published it in 1903, although it had been independently derived and published by William Moore in 1810, and later published in a separate book in 1813. Robert Goddard also developed it independently in 1912, and Hermann Oberth derived it independently about 1920.

In astrodynamics and aerospace, a **delta-v budget** is an estimate of the total change in velocity (delta-*v*) required for a space mission. It is calculated as the sum of the delta-v required to perform each propulsive maneuver needed during the mission. As input to the Tsiolkovsky rocket equation, it determines how much propellant is required for a vehicle of given empty mass and propulsion system.

In spaceflight, an **orbital maneuver** is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth an orbital maneuver is called a *deep-space maneuver (DSM)*.

In astrodynamics or celestial mechanics, a **hyperbolic trajectory** or **hyperbolic orbit** is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In the gravitational two-body problem, the **specific orbital energy** of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass. According to the orbital energy conservation equation, it does not vary with time:

**Spacecraft flight dynamics** is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

In astronautics and aerospace engineering, the **bi-elliptic transfer** is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

A **reaction engine** is an engine or motor that produces thrust by expelling reaction mass, in accordance with Newton's third law of motion. This law of motion is commonly paraphrased as: "For every action force there is an equal, but opposite, reaction force."

- 1 2 3 Robert B. Adams, Georgia A. Richardson (25 July 2010). Using the Two-Burn Escape Maneuver for Fast Transfers in the Solar System and Beyond (PDF) (Report). NASA. Archived (PDF) from the original on 11 February 2022. Retrieved 15 May 2015.
- 1 2 3 4 5 Hermann Oberth (1970). "Ways to spaceflight". Translation of the German language original "Wege zur Raumschiffahrt," (1920). Tunis, Tunisia: Agence Tunisienne de Public-Relations.
- ↑ What Is a Rocket? 13 July 2011/ 7 August 2017
*www.nasa.gov*, accessed 9 January 2021. - ↑ Rocket thrust. NASA Glenn Research Center. Accessed 16 April, 2024.
- ↑ Atomic Rockets web site: nyrath@projectrho.com. Archived July 1, 2007, at the Wayback Machine
- ↑ Following the calculation on rec.arts.sf.science.
- ↑ Blanco, Philip; Mungan, Carl (October 2019). "Rocket propulsion, classical relativity, and the Oberth effect".
*The Physics Teacher*.**57**(7): 439–441. Bibcode:2019PhTea..57..439B. doi: 10.1119/1.5126818 .

- Animation (MP4) of the Oberth effect in orbit from the Blanco and Mungan paper cited above.

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