In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that, under standard assumption, moves a spacecraft from one circular orbit to another using two engine impulses. This maneuver was named after Walter Hohmann, the German scientist who published it in 1925. (See also interplanetary travel.) Astronautics is the branch of engineering that deals with machines designed to work outside of Earths atmosphere, whether manned or unmanned. ... Aerospace engineering is the branch of engineering that concerns aircraft, spacecraft, and related topics. ... An orbital maneuver is a change from one orbit to another, accomplished by applying thrust. ... For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... A spacecraft is a vessel, craft or device designed to operate beyond the surface of the Earth in outer space. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... Walter Hohmann (March 18, 1880 - March 11, 1945) was a German engineer who made an important contribution to the understanding of orbital dynamics. ... By definition, interplanetary travel is travel between bodies in a given star system. ...

Hohmann Transfer Orbit

Hohmann transfer orbit File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

[edit] Explanation

The Hohmann transfer orbit is one half of an elliptic orbit that touches both the orbit that one wishes to leave (labeled 1 on diagram) and the orbit that one wishes to reach (3 on diagram). The transfer (2 on diagram) is initiated by firing the spacecraft's engine in order to accelerate it so that it will follow the elliptical orbit. When the spacecraft has reached its destination orbit, it has slowed down to a speed not only lower than the speed in the original circular orbit, but even lower than required for the new circular orbit; the engine is fired again to accelerate it again, to that required speed. Two bodies with similar mass orbiting around a common barycenter with elliptic orbits. ... In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ...

The Hohmann transfer orbit is theoretically based on impulsive velocity changes to create the circular orbits, therefore a spacecraft using a Hohmann transfer orbit will typically use high thrust engines to minimize the amount of extra fuel required to compensate for the non-impulsive maneuver. Low thrust engines can perform an approximation of a Hohmann transfer orbit, by creating a gradual enlargement of the initial circular orbit through carefully timed engine firings. This requires a delta-v that is up to 141% greater than the 2 impulse transfer orbit (see also below), and takes longer to complete.

Hohmann transfer orbits also work to bring a spacecraft from a higher orbit into a lower one – in this case, the spacecraft's engine is fired in the opposite direction to its current path, decellerating the spacecraft and causing it to drop into the lower energy elliptical transfer orbit. The engine is then fired again in the lower orbit to decellerate the spacecraft into a circular orbit.

Although the Hohmann transfer orbit is almost always the most economical way to get from one circular orbit to another, in certain situations where the semi-major axis of the final orbit is greater than the semimajor axis of the initial orbit by a factor of about 12, it may be more advantageous to use a bi-elliptic transfer. The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... 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. ...

In Soviet literature, the term Hohmann-Vetchinkin transfer orbit is sometimes used, citing the presentation of the ellitpical transfer concept by mathematician V.P. Vetchinkin in public lectures on interplanary travel given 1921-1925.

[edit] Calculation

For a small body orbiting another (such as a satellite orbiting the earth), the total energy of the body is just the sum of its kinetic energy and potential energy, and this total energy also equals half the potential at the farthest point, 'a' (the semi-major axis): Kinetic energy is the energy by virtue of the motion of an object. ... In the physical sciences, potential energy is energy which is captured within a physical system by virtue of the relative positions or configurations of objects, and which has the potential to be released when the system is allowed to attain a configuration with a lower energy state. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...

Solving this equation for velocity results in the Vis-viva equation, In astrodynamics, the vis-viva equation, also referred to as orbital energy conservation equation, is one of the fundamental and useful equations that govern the motion of orbiting bodies. ...

where:

• is the speed of an orbiting body
• is the standard gravitational parameter of the primary body
• is the distance of the orbiting body from the primary
• is the semi-major axis of the body's orbit

Therefore the delta-v required for the Hohmann transfer can be computed as follows: In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... General In general physics delta-v is simply the change in velocity. ...

, Delta-v required at periapsis.

, Delta-v required at apoapsis.

where: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

• is radius of lower orbit, and periapsis distance of Hohmann transfer orbit,
• is radius of higher orbit, and apoapsis distance of Hohmann transfer orbit.

Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... This article does not cite its references or sources. ...

(one half of the orbital period for the whole ellipse) The orbital period is the time it takes a planet (or another object) to make one full orbit. ...

where:

• is length of semi-major axis of the Hohmann transfer orbit.

The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...

[edit] Example

For the geostationary transfer orbit we have r₂ = 42,164 km and e.g. r₁ = 6,678 km (altitude 300 km). A geostationary transfer orbit (GTO) is a Hohmann transfer orbit around the Earth between a low Earth orbit (LEO) and a geostationary orbit (GEO). ...

In the smaller circular orbit the speed is 7.73 km/s, in the larger one 3.07 km/s. In the elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee.

The delta-v's are 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s. [1]

Compare with the delta-v for an escape orbit: 10.93 − 7.73 = 3.20 km/s. Applying a delta-v at the LEO of only 0.78 km/s more would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1.46 km/s is needed for reaching the sub-escape speed of this circular orbit. This illustrates that at large speeds the same delta-v provides more specific orbital energy, and, as explained in gravity drag, energy increase is maximized if one spends the delta-v as soon as possible, rather than spending some, being decelerated by gravity, and then spending some more (of course, the objective of a Hohmann transfer orbit is different). An escape orbit (also known as C3 = 0 orbit) is the high-energy parabolic orbit around the central body. ... For the video game title, see Escape Velocity (computer game). ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ... In astrodynamics, gravity drag is inefficiency encountered by a spacecraft thrusting while moving against a gravitational field. ...

[edit] Worst case, maximum delta-v

A Hohmann transfer orbit from a given circular orbit to a larger circular orbit, in the case of a single central body, costs the largest delta-v (53.6 % of the original orbital speed) if the radius of the target orbit is 15.6 (positive root of x³ − 15x² − 9x − 1 = 0) times as large as that of the original orbit. For higher target orbits the delta-v decreases again, and tends to times the original orbital speed (41.4%). (The first burst tends to acceleration to the escape speed, the second tends to zero.)

[edit] Low-thrust transfer

It can be derived that going from one circular orbit to another by gradually changing the radius costs a delta-v of simply the absolute value of the difference between the two speeds. Thus for the geostationary transfer orbit 7.73 - 3.07 = 4.66 km/s, the same as, in the absence of gravity, the deceleration would cost. In fact, acceleration is applied to compensate half of the deceleration due to moving outward. Therefore the acceleration due to thrust is equal to the deceleration due to the combined effect of thrust and gravity.

[edit] Application to interplanetary travel

When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex. For example, consider a spacecraft travelling from the Earth to Mars. At the beginning of its journey, the spacecraft will already have a certain velocity associated with its orbit around Earth – this is velocity that will not need to be found when the spacecraft enters the transfer orbit (around the Sun). At the other end, the spacecraft will need a certain velocity to orbit Mars, which will actually be less than the velocity needed to continue orbiting the Sun in the transfer orbit, let alone attempting to orbit the Sun in an Mars-like orbit. Therefore, the spacecraft will have to decelerate and allow Mars' gravity to capture it. Therefore, relatively small amounts of thrust at either end of the trip are all that are needed to arrange the transfer. Note, however, that the alignment of the two planets in their orbits is crucial – the destination planet and the spacecraft must arrive at the same point in their respective orbits around the Sun at the same time, see launch window. Earth (IPA: , often referred to as the Earth, Terra, the World or Planet Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth largest. ... Note: This article contains special characters. ... Launch window is a term used in aerospace to describe a time period in which a particular rocket must be launched. ...

A Hohmann transfer orbit will take a spacecraft from low Earth orbit (LEO) to geosynchronous orbit (GEO) in just over five hours (geostationary transfer orbit), from LEO to the Moon in about 5 days and from the Earth to Mars in about 260 days. However, Hohmann transfers are very slow for trips to more distant points, so when visiting the outer planets it is common to use a gravitational slingshot to increase speed in-flight. A low Earth orbit (LEO) is an orbit in which objects such as satellites are below intermediate circular orbit (ICO) and far below geostationary orbit, but typically around 350 - 1400 km above the Earths surface. ... It has been suggested that this article or section be merged with geostationary orbit. ... A geostationary transfer orbit (GTO) is a Hohmann transfer orbit around the Earth between a low Earth orbit (LEO) and a geostationary orbit (GEO). ... The eight planets and three dwarf planets of the Solar System. ... It has been suggested that sling effect be merged into this article or section. ...

[edit] Interplanetary Transport Network

In 1997, a set of orbits known as the Interplanetary Transport Network was published, providing even lower‐energy (though much slower) paths between different orbits than Hohmann transfer orbits. 1997 (MCMXCVII) was a common year starting on Wednesday of the Gregorian calendar. ... Artists concept of the Interplanetary Transport Network. ...

[edit] See also

• Delta-v budget
• bi-elliptic transfer
• geostationary transfer orbit

Delta-v budget (or velocity change budget) is a term used in astrodynamics and aerospace industry for velocity change (or delta-v) requirements for the various propulsive tasks and orbital maneuvers over phases of the space mission. ... 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. ... A geostationary transfer orbit (GTO) is a Hohmann transfer orbit around the Earth between a low Earth orbit (LEO) and a geostationary orbit (GEO). ...

[edit] References

• Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0-534-40896-6.

[edit] External links

• http://www.pma.caltech.edu/~chirata/deltav.html
• ORBITAL MECHANICS (Rocket and Space Technology)