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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, an Austro-Hungarian-born German physicist and a founder of modern rocketry.^{ [2] }

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

The Oberth effect is strongest at the periapsis, where the gravitational potential is lowest, and the speed is highest. This is because a given firing of a rocket engine at high speed causes a greater change in kinetic energy than when fired otherwise similarly at lower speed.

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, adding 1 m/s increases the kinetic energy from 1 J to 4 J, for a gain of 3 J;
- at 10 m/s, starting with a kinetic energy of 100 J, the rocket ends with 121 J, for a net 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 get

- ≈

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.

A **rocket** is a spacecraft, aircraft, vehicle or projectile that obtains thrust from a rocket engine. Rocket engine exhaust is formed entirely from propellant carried within the rocket. Rocket engines work by action and reaction and push rockets forward simply by expelling their exhaust in the opposite direction at high speed, and can therefore work in the vacuum of space.

In celestial mechanics, **escape velocity** or **escape speed** is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

**Specific impulse** is a measure of how efficiently a reaction mass engine creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity.

In fluid dynamics, **Bernoulli's principle** states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book *Hydrodynamica* in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived **Bernoulli's equation** in its usual form. The principle is only applicable for isentropic flows: when the effects of irreversible processes and non-adiabatic processes are small and can be neglected.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if 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 orbital mechanics and aerospace engineering, a **gravitational slingshot**, **gravity assist maneuver**, or **swing-by** is the use of the relative movement and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

In orbital mechanics, the **Hohmann transfer orbit** is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane that is sometimes tangential to both. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can use less in some cases.

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

**Delta- v**, symbolized as

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 **Tsiolkovsky rocket equation**, **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.

**Verlet integration** is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by Cowell and Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field . The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method.

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** 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."

A **gravity turn** or **zero-lift turn** is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory controlled solely through the vehicle's own thrust. First, the thrust is not used to change the spacecraft's direction, so more of it is used to accelerate the vehicle into orbit. Second, and more importantly, during the initial ascent phase the vehicle can maintain low or even zero angle of attack. This minimizes transverse aerodynamic stress on the launch vehicle, allowing for a lighter launch vehicle.

- 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 12 June 2014,
*www.grc.nasa.gov*, accessed 9 January 2021. - ↑ 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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