In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver using two engine impulses which, under standard assumptions, move a spacecraft between two coplanar circular orbits. Astronautics, or Astronautical Engineering is the branch of Engineering that deals with machines designed to exit or work entirely beyond the Earth's atmosphere. Aerospace engineering is the branch of Engineering behind the design construction and science of Aircraft and Spacecraft. In Spaceflight, an orbital maneuver is the use of propulsion systems to change the Orbit of a Spacecraft. For most of the problems in Astrodynamics involving two bodies m_1\ and m_2\ standard assumptions are usually the following A1 A spacecraft is a Vehicle or machine designed for Spaceflight. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star This maneuver was named after Walter Hohmann, the German scientist who published a description of it in 1925. Walter Hohmann ( March 18 1880 - March 11 1945) was a German engineer who made an important contribution to the understanding of Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Year 1925 ( MCMXXV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar. (See also interplanetary travel. )

In Soviet literature, such as Pionery Raketnoi Tekhniki,[1] the term Hohmann-Vetchinkin transfer orbit is sometimes used, citing the presentation of the elliptical transfer concept by mathematician Vladimir Vetchinkin in public lectures on interplanetary travel given 1921-1925. This article is about literature from Russia For the song by Maxïmo Park, see Our Earthly Pleasures. Vladimir Petrovich Vetchinkin (Владимир Петрович Ветчинкин ( June 17 (29 1888 - March 6, 1950) was a Soviet Year 1921 ( MCMXXI) was a Common year starting on Saturday (link will display full 1921 calendar of the Gregorian calendar Year 1925 ( MCMXXV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar.

Hohmann Transfer Orbit

## 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). In Astrodynamics or Celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1 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; this adds energy to the spacecraft's orbit. When the spacecraft has reached its destination orbit, its orbital speed (i. e. , orbital energy) must be increased again in order to make its new orbit circular; the engine is fired again to accelerate it to the required velocity.

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. In Astrodynamics, the term delta-v, literally "change in velocity" (see symbol delta) has a specific meaning it is a Scalar which takes

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, decelerating 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 decelerate 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 the same plane), there are situations in which a bi-elliptic transfer is even more economical: particularly when the semi-major axis of the final orbit is more than about 12 times greater than that of the initial orbit. In Astronautics and Aerospace engineering, the bi-elliptic transfer is an Orbital maneuver that moves a Spacecraft from one Orbit In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae

## 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):

$E=\begin{matrix}\frac{1}{2}\end{matrix} m v^2 - \frac{GM m}{r} = \frac{-G M m}{2 a} \,$

Solving this equation for velocity results in the Vis-viva equation,

$v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)$
where:
• $v \,\!$ is the speed of an orbiting body
• $\mu = GM\,\!$ is the standard gravitational parameter of the primary body
• $r \,\!$ is the distance of the orbiting body from the primary
• $a \,\!$ is the semi-major axis of the body's orbit

Therefore the delta-v required for the Hohmann transfer can be computed as follows:

$\Delta v_P = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1 \right)$, Delta-v required at periapsis. The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae In Astrodynamics, the vis viva equation, also referred to as orbital energy conservation equation, is one of the fundamental and useful equations that govern Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae In Astrodynamics, the term delta-v, literally "change in velocity" (see symbol delta) has a specific meaning it is a Scalar which takes In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from
$\Delta v_A = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1+r_2}}\,\! \right)$, Delta-v required at apoapsis. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from

where:

• $r_1\,\!$ is radius of lower orbit, and periapsis distance of Hohmann transfer orbit,
• $r_2\,\!$ is radius of higher orbit, and apoapsis distance of Hohmann transfer orbit. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from

Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is:

$t_H = \begin{matrix}\frac12\end{matrix} \sqrt{\frac{4\pi^2 a^3_H}{\mu}}= \pi \sqrt{\frac {(r_1 + r_2)^3}{8\mu}}$

(one half of the orbital period for the whole ellipse)

where:

• $a_H\,\!$ is length of semi-major axis of the Hohmann transfer orbit. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. The orbital period is the time taken for a given object to make one complete Orbit about another object In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae

## Example

For the geostationary transfer orbit we have r2 = 42,164 km and e. A Geosynchronous Transfer Orbit or Geostationary Transfer Orbit ( GTO) is a Hohmann transfer orbit around the Earth between a Low Earth orbit g. r1 = 6,678 km (altitude 300 km).

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. An escape orbit (also known as C 3 = 0 orbit is a high-energy Parabolic orbit around the central body 93 − 7. 73 = 3. 20 km/s. Applying a delta-v at the LEO of only 0. A Low Earth Orbit (LEO is generally defined as an Orbit within the locus extending from the Earth’s surface up to an altitude of 2000 km 78 km/s more would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy 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). In Astrodynamics the specific Orbital energy \epsilon\\! (or vis-viva energy) of an Orbiting body traveling through Space In Astrodynamics and Rocketry, gravity drag (or gravity losses) is the difference between the Delta-v expended and the actual change in speed

## 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 $\sqrt{2}-1$ times the original orbital speed (41. 4%). (The first burst tends to acceleration to the escape speed, the second tends to zero. )

## 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.

Such a low-thrust maneuver requires more delta-v than a 2-burn Hohmann transfer maneuver, requiring more fuel (for a given engine design). However, if only low-thrust maneuvers are required on a mission, then continuously firing a very high-efficiency, low-thrust spacecraft propulsion engines might be able to generate this higher delta-v using less fuel and a smaller engine than a high-thrust engine using a "more efficient" Hohmann transfer maneuver. Spacecraft propulsion is any method used to change the velocity of Spacecraft and artificial Satellites There are many different methods

## 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. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 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). The Sun (Sol is the Star at the center of the Solar System. 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 a 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). 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 (lunar transfer orbit, LTO) in about 5 days and from the Earth to Mars in about 214 days. A Low Earth Orbit (LEO is generally defined as an Orbit within the locus extending from the Earth’s surface up to an altitude of 2000 km A geosynchronous orbit is an Orbit around the Earth with an Orbital period matching the Earth's sidereal rotation period A Geosynchronous Transfer Orbit or Geostationary Transfer Orbit ( GTO) is a Hohmann transfer orbit around the Earth between a Low Earth orbit 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 planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is In Orbital mechanics and Aerospace engineering, a gravitational slingshot, gravity assist or swing-by is the use of the relative movement and

## 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. Year 1997 ( MCMXCVII) was a Common year starting on Wednesday (link will display full 1997 Gregorian calendar The Interplanetary Transport Network (ITN is a collection of Gravitationally determined pathways through the Solar system that require very little Energy