Hohmann transfer orbit

The Hohmann transfer is an energetically favorable transition between two orbits around a dominant celestial body. The transfer ellipse ( Hohmann -Bahn ) touches both the output path and the target path tangentially; there is a force impulse ( kick burn ) is always necessary to adjust the speed (respectively). Such a sketch is found as early as 1911 at Ziolkowski. 1925, this transfer of Walter Hohmann was considered optimal. For coplanar, circular source and target paths with a radius ratio below 11.94 it is also, for more extreme conditions and strongly inclined towards one another, or even opposing lanes is a bi- elliptic transfer energetically favorable.

The idealizing presuppositions comes close to the task of bringing satellite from a low-Earth to geosynchronous orbit, see geostationary transfer orbit. For flights to the moon and neighboring planets, the central field approximation is less good - with swing-by maneuvers and time-consuming detours can be compared to the Hohmann transfer analytically found save fuel.

• 3.1 transfer orbit to Mars

Sample calculation of the transfer to the geostationary orbit

In order to position satellites geostationary, they are often first brought to a circular low orbit, low earth orbit (LEO ), see (1) in the graph. A first impulse () brings the satellite on the elliptic Hohmann- web (2), whose apogee is in the range of the target orbit (3). There is another force impulse () also increases the perigee of the web which is thus again circular.

Speeds

After the Vis -Viva equation the velocity v (r ) is a body at r an elliptical orbit with semi-major axis a around the earth:

With, the Earth's mass and the gravitational constant are. Denoting the perigee or LEO radius, the apogee or GEO radius and the semi-major axis of the transfer ellipse, shall apply to the initial speed vLEO, Perigäumsgeschwindigkeit vP, vA Apogäumsgeschwindigkeit and final speed vGEO the following equations:

Pay

The following values ​​are given:

Then, the computed according to the above equations web speeds:

This results in the two required speed changes.

Energy expenditure as a function of radius ratio

The ellipse of the Hohmann transfer is described by the velocities of the source and target orbit. To move from an initial circular orbit in the ellipse and to return to a circular orbit at the destination, two bursts of pulses or two speed changes necessary. To view the cost of energy may then also the whole difference are considered. Transfer the ellipse is described by the half-axle.

For further discussion it is convenient to consider the dimensionless quantity. With the auxiliary variable is then as follows:

When the Hohmann transfer proves to be useful, can be determined by more detailed discussion of the speed change. By deriving and equating with zero, an extreme value of the formula can be found in the last section:

The only sensible solution is obtained for. The ratio of, for a maximum is therefore by the context: given. Furthermore, the derivation of each is strictly increasing. That for each major relationship between, the energy expenditure is reduced again.

Example

Transfer orbit to Mars

Mars is the planet in opposition position the next. However, a satellite can use only at great expense these geometrical proximity, as it must fly in this case against the orbital motion of the earth.

After Hohmann, however, the lowest energy transfer is the one in which the satellite Mars conjunct reached on the position of the Earth, was launched from which the satellite. In the left figure, the probe starts on Earth in (1) and reached Mars in position ( 3) while the sun is all the time in one of the focal points of the transfer orbit (yellow). The double semi-axis of the transfer ellipse is the sum of the Earth-Sun distance and solar - Mars. Therefore, under Kepler's third law for half an orbital period of eight and a half months.

The picture on the right shows the transfer orbit of Mars Reconnaissance Orbiter, which, although requires a higher energy than the Hohmann -Bahn ( the transfer orbit crosses the orbit of Mars addition ), the probe for this, however, only 7 months on the road is.

Weak Stability Boundary

If the target planet are served with a minimum speed, the so-called Weak Stability Boundary method offers a further energy gain. The probe is braked by being maneuvered along libration. A first useful path calculation was carried out in 1986. ESA's Smart-1 approached by this method the moon.