Oberth effect

Encyclopedia

In astronautics

, the Oberth effect is where the use of a rocket engine

when travelling at high speed generates much more useful energy than one at low speed. Oberth effect occurs because the propellant

has more usable energy (due to its kinetic energy on top of its chemical potential energy) and it turns out that the vehicle is able to employ this kinetic energy to generate more mechanical power. It is named after Hermann Oberth

, the Hungarian

-born, German

physicist

and a founder of modern rocket

ry, who apparently first described the effect.

Oberth effect is used in a powered flyby or Oberth maneuver where the application of an impulse, typically from the use of a rocket engine, close to a gravitational body (where the gravity potential is low, and the speed is high) can give much more change in kinetic energy

and final speed (i.e. higher specific energy

) than the same impulse applied further from the body for the same initial orbit. For the Oberth effect to be most effective, the vehicle must be able to generate as much impulse as possible at the lowest possible altitude; thus the Oberth effect is often far less useful for low-thrust reaction engines such as ion drives, which have a low propellant flow rate.

Oberth effect also can be used to understand the behaviour of multi-stage rockets; the upper stage can generate much more usable kinetic energy than might be expected from simply considering the chemical energy of the propellants it carries.

Historically, a lack of understanding of this effect led early investigators to conclude that interplanetary travel would require completely impractical amounts of propellant, as without it, enormous amounts of energy are needed.

. 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.

This can be easily shown. The definition of mechanical work:

Where: is the energy (specifically the kinetic energy

), is the force- the thrust of the rocket which is considered constant, and is the distance

Differentiating with respect to time:

Or:

where is the velocity.

dividing by the instantaneous mass m to express this in terms of specific energy (S)

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 to calculate the overall increase in specific energy of the rocket.

However, integrating this is often unnecessary, if the burn is short. For example as a vehicle falls towards 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

It may seem that the rocket is getting energy for free, which would violate conservation of energy

. However, any gain to the rocket's energy is balanced by an equal decrease in the energy the exhaust is left with. When expended lower in the gravitational field, even if the exhaust is left with more kinetic energy, it is left with less total energy. The effect would be even stronger if the exhaust speed could be made equal to the speed of the rocket, then the exhaust would be left without kinetic energy, so the total energy of the exhaust would be as low as its potential energy. Contrast this to the situation of static firing: as the speed of the engine is zero its specific energy does not increase at all, with all chemical energy of the fuel being converted to the exhaust's kinetic energy.

At very high speed the mechanical power imparted to the rocket can even exceed the total power liberated in the combustion of the propellants, and 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 km/s actually 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. This can make up for what seems like an 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.

(SOE) is

Once the space craft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy tends to zero. Therefore, the larger the

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 4.6 times.

then the velocity at periapsis before the burn is equal to the escape velocity

(

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:

It can then be easily shown that the impulse is multiplied by a factor of:

Plugging in 50 km/s escape velocity and 5 km/s burn we get a multiplier of 4.6.

Similar effects happen in closed and hyperbolic orbits

.

Astronautics

Astronautics, and related astronautical engineering, is the theory and practice of navigation beyond the Earth's atmosphere. In other words, it is the science and technology of space flight....

, the Oberth effect is where the use of a rocket engine

Rocket engine

A rocket engine, or simply "rocket", is a jet engineRocket Propulsion Elements; 7th edition- chapter 1 that uses only propellant mass for forming its high speed propulsive jet. Rocket engines are reaction engines and obtain thrust in accordance with Newton's third law...

when travelling at high speed generates much more useful energy than one at low speed. Oberth effect occurs because the propellant

Propellant

A propellant is a material that produces pressurized gas that:* can be directed through a nozzle, thereby producing thrust ;...

has more usable energy (due to its kinetic energy on top of its chemical potential energy) and it turns out that the vehicle is able to employ this kinetic energy to generate more mechanical power. It is named after Hermann Oberth

Hermann Oberth

Hermann Julius Oberth was an Austro-Hungarian-born German physicist and engineer. He is considered one of the founding fathers of rocketry and astronautics.- Early life :...

, the Hungarian

Hungary

Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

-born, German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

physicist

Physicist

A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

and a founder of modern rocket

Rocket

A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...

ry, who apparently first described the effect.

Oberth effect is used in a powered flyby or Oberth maneuver where the application of an impulse, typically from the use of a rocket engine, close to a gravitational body (where the gravity potential is low, and the speed is high) can give much more change in kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

and final speed (i.e. higher specific energy

Specific energy

Specific energy is defined as the energy per unit mass. Common metric units are J/kg. It is an intensive property. Contrast this with energy, which is an extensive property. There are two main types of specific energy: potential energy and specific kinetic energy. Others are the gray and sievert,...

) than the same impulse applied further from the body for the same initial orbit. For the Oberth effect to be most effective, the vehicle must be able to generate as much impulse as possible at the lowest possible altitude; thus the Oberth effect is often far less useful for low-thrust reaction engines such as ion drives, which have a low propellant flow rate.

Oberth effect also can be used to understand the behaviour of multi-stage rockets; the upper stage can generate much more usable kinetic energy than might be expected from simply considering the chemical energy of the propellants it carries.

Historically, a lack of understanding of this effect led early investigators to conclude that interplanetary travel would require completely impractical amounts of propellant, as without it, enormous amounts of energy are needed.

## Description

Rocket engines produce the same force regardless of their velocity. A rocket acting on a fixed object or particularly heavy item, as in a static firing, does little or no useful work at all; the rocket's stored energy is entirely expended on accelerating its propellant to hypersonic speed. But when the rocket moves, the thrust of the rocket during any time interval acts through the distance the rocket and payload move during that time. Force acting through a distance is the definition of mechanical energy or workMechanical work

In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does 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.

This can be easily shown. The definition of mechanical work:

Where: is the energy (specifically the kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

), is the force- the thrust of the rocket which is considered constant, and is the distance

Differentiating with respect to time:

Or:

where is the velocity.

dividing by the instantaneous mass m to express this in terms of specific energy (S)

where is the acceleration

Proper acceleration

In relativity theory, proper acceleration is the physical acceleration experienced by an object. It is acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured...

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 to calculate the overall increase in specific energy of the rocket.

However, integrating this is often unnecessary, if the burn is short. For example as a vehicle falls towards 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 (

Delta-v

In astrodynamics a Δv or delta-v is a scalar which takes units of speed. It is a measure of the amount of "effort" that is needed to change from one trajectory to another by making an orbital maneuver....

). However, since the vehicle's kinetic energy is related to the

*square*of its velocity, this increase in velocity has a disproportionate effect on the vehicle's kinetic energy; leaving it with higher energy than if the burn were achieved at any other time.It may seem that the rocket is getting energy for free, which would violate conservation of energy

Conservation of energy

The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

. However, any gain to the rocket's energy is balanced by an equal decrease in the energy the exhaust is left with. When expended lower in the gravitational field, even if the exhaust is left with more kinetic energy, it is left with less total energy. The effect would be even stronger if the exhaust speed could be made equal to the speed of the rocket, then the exhaust would be left without kinetic energy, so the total energy of the exhaust would be as low as its potential energy. Contrast this to the situation of static firing: as the speed of the engine is zero its specific energy does not increase at all, with all chemical energy of the fuel being converted to the exhaust's kinetic energy.

At very high speed the mechanical power imparted to the rocket can even exceed the total power liberated in the combustion of the propellants, and 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 km/s actually 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. This can make up for what seems like an 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.

## Parabolic example

If the ship travels at velocity*v*at the start of a burn that changes the velocity by Δ*v*, then the change in specific orbital energySpecific orbital energy

In the gravitational two-body problem, the specific orbital energy \epsilon\,\! of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy , divided by the reduced mass...

(SOE) is

Once the space craft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy tends to 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

Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

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 4.6 times.

## Detailed proof

If an impulsive burn of Δ*v*is performed at periapsis in a parabolic orbitParabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit...

then the velocity at periapsis before the burn is equal to the escape velocity

Escape velocity

In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

(

*V*_{esc}), and the specific kinetic energy after the burn is: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:

It can then be easily shown that the impulse is multiplied by a factor of:

Plugging in 50 km/s escape velocity and 5 km/s burn we get a multiplier of 4.6.

Similar effects happen in closed and hyperbolic orbits

Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to...

.

## See also

- Bi-elliptic transferBi-elliptic transferIn 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....
- Hermann OberthHermann OberthHermann Julius Oberth was an Austro-Hungarian-born German physicist and engineer. He is considered one of the founding fathers of rocketry and astronautics.- Early life :...
- Gravity assist
- Propulsive efficiencyPropulsive efficiencyIn aircraft and rocket design, overall propulsive efficiency \eta is the efficiency, in percent, with which the energy contained in a vehicle's propellant is converted into useful energy, to replace losses due to air drag, gravity, and acceleration. It can also be stated as the proportion of the...
- Escape velocityEscape velocityIn physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

## External links

- Oberth effect
- explanation of the effect by Geoffrey Landis.
- Mention of the effect