Oberth Effect

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In the English-speaking world, Oberth Effect describes the dependence of the efficiency of a rocket engine on the location in a gravitational field: The deeper the gravitational potential (the closer to a heavy celestial body) the fuel is used, the greater the energy gain of the rocket. The effect is named after Hermann Oberth , who was the first to describe it.

The effect explains why the transfer to a high orbit via a Hohmann-Bahn , in which most of the fuel is used close to the periapsis , the low starting orbit, is cheaper than via a spiral-shaped path with continuous consumption. The same applies when pivoting from a hyperbolic orbit into an orbit: The braking maneuver should be carried out close to the celestial body. Often the orbit is then still very stretched because in this situation no powerful engine is available. The track is then lowered in several rounds by igniting in the periapsis.

Explanation

When a rocket drive is activated, the speed of the spacecraft changes by an amount ( Delta v ) that - and this is the crucial point - does not depend on the current speed ( Galileo transformation ). However, the kinetic energy of the spacecraft is proportional to the square of the speed, so it changes more, the greater is:

The increase in kinetic energy is proportional to the mean speed during the burning time of the engine. If the change in speed is small, the following applies approximately simply:

and in the infinitesimal limit one gets:

where is the thrust of the rocket. This corresponds to the well-known formula: work = force × distance.

A spacecraft in orbit around a celestial body has the greatest speed at the moment of closest approach. If the engine is ignited at this point, the spacecraft receives the greatest possible proportion of the chemical energy stored in the fuel as additional kinetic energy; accordingly, the (unusable) kinetic energy of the ejected rocket gases (supporting mass) is then lowest. At high speed of the spacecraft one can also argue that the support mass loses speed and thus kinetic energy by being ejected against the direction of flight, which the spacecraft benefits.

Example on a parabola

If, during a parabolic flyby of a spacecraft past Jupiter at a speed of 50 km / s, an engine ignition of 5 km / s is carried out at the periapsis , it is shown that the resulting final speed of the vehicle after the long-distance flyby is around 22, 9 km / s increases, i.e. 4.6 times that used . This can be derived as follows:

A spacecraft describes a parabolic orbit when it is just about to escape speed . By definition, this means that it can move as far away from the central body and that its speed approaches zero in the extreme case of greater distance. The kinetic energy that was there at closest approach (periapsis) , it loses completely when leaving the gravitational field: .

If it gets a boost there , its energy is:

.

If it has already received this thrust at the periapsis (optimal use of the Oberth effect), its energy when it leaves the gravitational field is derived as above:

The ratio of these energies is:

and thus applies to the speeds

If one uses the values ​​of the example of the flyby of Jupiter, one obtains the factor for an escape speed of v esc = 50 km / s and an ignition with a speed change of Δv = 5 km / s .

Corresponding results are obtained for closed and hyperbolic orbits .

See also

Vis-Viva equation

Individual evidence

  1. Ways to spaceflight. NASA TT F-622, translation of ways to space travel. R. Oldenbourg, Munich / Berlin 1929.

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