Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Inastronautics, apowered flyby, orOberth maneuver, is a maneuver in which aspacecraft falls into agravitational 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 gainkinetic energy than applying the sameimpulse outside of a gravitational well. The gain in efficiency is explained by theOberth effect, wherein the use of areaction 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 toburn its fuel is at the lowest possibleorbital 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 theTransylvanian Saxonphysicist and a founder of modernrocketryHermann Oberth, who first described them in 1927.^[2]

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 likeliquid-propellant rockets, and less useful for low-thrust reaction engines such asion 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 ofmulti-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 thepropellant 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

Because kinetic energy equalsmv²/2, 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 1² = 1 J of kinetic energy. Adding 1 m/s increases the kinetic energy to 2² = 4 J, for a gain of 3 J;
• at 10 m/s, the rocket starts with 10² = 100 J of kinetic energy. Adding 1 m/s increases the kinetic energy to 11² = 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.

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 on the rocket; 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 ofmechanical work. The greater the velocity of the rocket and payload during the burn the greater is 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(${\displaystyle W}$) is defined as thedot product of the force of the engine's thrust(${\displaystyle \overrightarrow{F}}$) and the displacement it travels during the burn(${\displaystyle \overrightarrow{s}}$):

${\displaystyle W=\overrightarrow{F}\cdot \overrightarrow{s}.}$

If the burn is made in theprograde direction,${\displaystyle \overrightarrow{F}\cdot \overrightarrow{s}=\Vert F\Vert \cdot \Vert s\Vert =F\cdot s}$. The work results in a change in kinetic energy

${\displaystyle \mathrm{\Delta }{E}_k=F\cdot s.}$

Differentiating with respect to time, we obtain

${\displaystyle \frac{\mathrm{d}{E}_k}{\mathrm{d}t}=F\cdot \frac{\mathrm{d}s}{\mathrm{d}t},}$

or

${\displaystyle \frac{\mathrm{d}{E}_k}{\mathrm{d}t}=F\cdot v,}$

where${\displaystyle v}$ is the velocity. Dividing by the instantaneous mass${\displaystyle m}$ to express this in terms ofspecific energy(${\displaystyle e_k}$), we get

${\displaystyle \frac{\mathrm{d}{e}_k}{\mathrm{d}t}=\frac{F}{m}\cdot v=a\cdot v,}$

where${\displaystyle a}$ is theacceleration 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 towardperiapsis 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 (${\displaystyle \mathrm{\Delta }v}$). However, since the vehicle's kinetic energy is related to thesquare 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.^[3]

If an impulsive burn ofΔv is performed at periapsis in aparabolic orbit, then the velocity at periapsis before the burn is equal to theescape velocity (V_esc), and the specific kinetic energy after the burn is^[4]

where${\displaystyle V=V_{\text{esc}}+\mathrm{\Delta }v}$.

When the vehicle leaves the gravity field, the loss of specific kinetic energy is

${\displaystyle \frac{1}{2}{V}_{\text{esc}}^2,}$

so it retains the energy

${\displaystyle \mathrm{\Delta }v{V}_{\text{esc}}+\frac{1}{2}\mathrm{\Delta }{v}^2,}$

which is larger than the energy from a burn outside the gravitational field (${\displaystyle \frac{1}{2}\mathrm{\Delta }{v}^2}$) by

${\displaystyle \mathrm{\Delta }v{V}_{\text{esc}}.}$

When the vehicle has left the gravity well, it is traveling at a speed

${\displaystyle V=\mathrm{\Delta }v\sqrt{1+\frac{2V_{\text{esc}}}{\mathrm{\Delta }v}}.}$

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

${\displaystyle \sqrt{\frac{2V_{\text{esc}}}{\mathrm{\Delta }v}}}$

and one gets

${\displaystyle V}$ ≈${\displaystyle \sqrt{2V_{\text{esc}}\mathrm{\Delta }v}.}$

Similar effects happen in closed andhyperbolic orbits.

If the vehicle travels at velocityv at the start of a burn that changes the velocity by Δv, then the change inspecific orbital energy (SOE) due to the new orbit is

${\displaystyle v\mathrm{\Delta }v+\frac{1}{2}(\mathrm{\Delta }v{)}^2.}$

Once the spacecraft is far from the planet again, the SOE is entirely kinetic, sincegravitational potential energy approaches zero. Therefore, the larger thev 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 aparabolic flyby of Jupiter with aperiapsis 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 violateconservation 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.^[5]

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.

• Bi-elliptic transfer
• Gravity assist
• Propulsive efficiency

• Oberth effect
• Explanation of the effect byGeoffrey Landis.

• Rocket propulsion, classical relativity, and the Oberth effect

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

• Aerospace engineering
• Rocketry
• Astrodynamics

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