Hohmann transfer orbit
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In orbital mechanics, the Hohmann transfer orbit (/ˈhoʊmən/) is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane. In general a Hohmann transfer orbit uses the lowest possible amount of energy in traveling between two objects orbiting at these radii, and so is used to send the maximum amount of mission payload with the fixed amount of energy that can be imparted by a particular rocket. NonHohmann transfer paths may have other advantages for a particular mission such as shorter transfer times, but will necessarily require a reduction in payload mass and/or use of a more powerful rocket.^{[1]}^{[2]}^{[3]}^{[4]}
A Hohmann transfer requires that the starting and destination points be at particular locations in their orbits relative to each other. Space missions using a Hohmann transfer must wait for this required alignment to occur, which opens a socalled launch window. For a space mission between Earth and Mars, for example, these launch windows occur every 26 months. A Hohmann transfer orbit also determines a fixed time required to travel between the starting and destination points; for an EarthMars journey this travel time is 9 months.
The orbital maneuver to perform the Hohmann transfer uses two engine impulses, one to move a spacecraft onto the transfer orbit and a second to move off it. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in his 1925 book Die Erreichbarkeit der Himmelskörper (The Attainability of Celestial Bodies).^{[5]} Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets.
Contents
Explanation
The diagram shows a Hohmann transfer orbit to bring a spacecraft from a lower circular orbit into a higher one. It is one half of an elliptic orbit that touches both the lower circular orbit the spacecraft wishes to leave (green and labeled 1 on diagram) and the higher circular orbit that it wishes to reach (red and labeled 3 on diagram). The transfer (yellow and labeled 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 (and hence its orbital energy) must be increased again in order to change the elliptic orbit to the larger circular one.
Due to the reversibility of orbits, 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, slowing the spacecraft and causing it to drop into the lowerenergy elliptical transfer orbit. The engine is then fired again at the lower distance to slow the spacecraft into the lower circular orbit.
The Hohmann transfer orbit is based on two instantaneous velocity changes. Extra fuel is required to compensate for the fact that the bursts take time; this is minimized by using highthrust engines to minimize the duration of the bursts. For transfers in Earth orbit, the two burns are labelled the perigee burn and the apogee burn (or ''apogee kick^{[6]}); more generally, they are labelled periapsis and apoapsis burns. Alternately, the second burn to circularize the orbit may be referred to as a circularization burn.
Type I and Type II
An ideal Hohmann transfer orbit transfers between two circular orbits in the same plane and traverses exactly 180° around the primary. In the real world, the destination orbit may not be circular, and may not be coplanar with the initial orbit. Real world transfer orbits may traverse slightly more, or slightly less, than 180° around the primary. An orbit which traverses less than 180° around the primary is called a "Type I" Hohmann transfer, while an orbit which traverses more than 180° is called a "Type II" Hohmann transfer.^{[7]}^{[8]}
Calculation
For a small body orbiting another much larger body, such as a satellite orbiting Earth, the total energy of the smaller body is the sum of its kinetic energy and potential energy, and this total energy also equals half the potential at the average distance (the semimajor axis):
Solving this equation for velocity results in the visviva equation,
where:
 is the speed of an orbiting body,
 is the standard gravitational parameter of the primary body, assuming is not significantly bigger than (which makes ),
 is the distance of the orbiting body from the primary focus,
 is the semimajor axis of the body's orbit.
Therefore, the deltav required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:
to enter the elliptical orbit at from the circular orbit
to leave the elliptical orbit at to the circular orbit, where and are respectively the radii of the departure and arrival circular orbits; the smaller (greater) of and corresponds to the periapsis distance (apoapsis distance) of the Hohmann elliptical transfer orbit. Typically, is given in units of m^{3}/s^{2}, as such be sure to use meters, not kilometers, for and . The total is then:
Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is
(one half of the orbital period for the whole ellipse), where is length of semimajor axis of the Hohmann transfer orbit.
In application to traveling from one celestial body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. Considering the target angular velocity being
angular alignment α (in radians) at the time of start between the source object and the target object shall be
Example
Consider a geostationary transfer orbit, beginning at r_{1} = 6,678 km (altitude 300 km) and ending in a geostationary orbit with r_{2} = 42,164 km (altitude 35,786 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 Δv for the two burns are thus 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s.
This is greater than the Δv required for an escape orbit: 10.93 − 7.73 = 3.20 km/s. Applying a Δv at the Low Earth orbit (LEO) of only 0.78 km/s more (3.20−2.42) would give the rocket the escape speed, which is less than the Δv of 1.46 km/s required to circularize the geosynchronous orbit. This illustrates the Oberth effect that at large speeds the same Δv provides more specific orbital energy, and energy increase is maximized if one spends the Δv as quickly as possible, rather than spending some, being decelerated by gravity, and then spending some more to overcome the deceleration (of course, the objective of a Hohmann transfer orbit is different).
Worst case, maximum deltav
As the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not the greatest when the destination radius is infinite. (Escape speed is √2 times orbital speed, so the Δv required to escape is √2 − 1 (41.4%) of the orbital speed.) The Δv required is greatest (53.0% of smaller orbital speed) when the radius of the larger orbit is 15.5817... times that of the smaller orbit.^{[9]} This number is the positive root of x^{3} − 15 x^{2} − 9 x − 1 = 0, which is . For higher orbit ratios the Δv required for the second burn decreases faster than the first increases.
Application to interplanetary travel
When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex, but much less deltav is required, due to the Oberth effect, than the sum of the deltav required to escape the first planet plus the deltav required for a Hohmann transfer to the second planet.
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 and kinetic energy associated with its orbit around Earth. During the burn the rocket engine applies its deltav, but the kinetic energy increases as a square law, until it is sufficient to escape the planet's gravitational potential, and then burns more so as to gain enough energy to get into the Hohmann transfer orbit (around the Sun). Because the rocket engine is able to make use of the initial kinetic energy of the propellant, far less deltav is required over and above that needed to reach escape velocity, and the optimum situation is when the transfer burn is made at minimum altitude (low periapsis) above the planet. The deltav needed is only 3.6 km/s, only about 0.4 km/s more than needed to escape Earth, even though this results in the spacecraft going 2.9 km/s faster than the Earth as it heads off for Mars (see table below).
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 Marslike orbit. Therefore, the spacecraft will have to decelerate in order for the gravity of Mars to capture it. This capture burn should optimally be done at low altitude to also make best use of Oberth effect. Therefore, relatively small amounts of thrust at either end of the trip are needed to arrange the transfer compared to the free space situation.
However, with any Hohmann transfer, 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. This requirement for alignment gives rise to the concept of launch windows.
The term lunar transfer orbit (LTO) is used for the Moon.
It is possible to apply the formula given above to calculate the Δv in km/s needed to enter a Hohmann transfer orbit to arrive at various destinations from Earth (assuming circular orbits for the planets). In this table, the column labeled "Δv to enter Hohmann orbit from Earth's orbit" gives the change from Earth's velocity to the velocity needed to get on a Hohmann ellipse whose other end will be at the desired distance from the Sun. The column labeled "v exiting LEO" gives the velocity needed (in a nonrotating frame of reference centered on the earth) when 300 km above the Earth's surface. This is obtained by adding to the specific kinetic energy the square of the speed (7.73 km/s) of this low Earth orbit (that is, the depth of Earth's gravity well at this LEO). The column "Δv from LEO" is simply the previous speed minus 7.73 km/s.
Destination  Orbital radius (AU) 
Δv (km/s)  

to enter Hohmann orbit from Earth's orbit 
exiting LEO 
from LEO 

Sun  0  29.8  31.7  24.0 
Mercury  0.39  7.5  13.3  5.5 
Venus  0.72  2.5  11.2  3.5 
Mars  1.52  2.9  11.3  3.6 
Jupiter  5.2  8.8  14.0  6.3 
Saturn  9.54  10.3  15.0  7.3 
Uranus  19.19  11.3  15.7  8.0 
Neptune  30.07  11.7  16.0  8.2 
Pluto  39.48  11.8  16.1  8.4 
Infinity  ∞  12.3  16.5  8.8 
Note that in most cases, Δv from LEO is less than the Δv to enter Hohmann orbit from Earth's orbit.
To get to the Sun, it is actually not necessary to use a Δv of 24 km/s. One can use 8.8 km/s to go very far away from the Sun, then use a negligible Δv to bring the angular momentum to zero, and then fall into the Sun. This can be considered a sequence of two Hohmann transfers, one up and one down. Also, the table does not give the values that would apply when using the Moon for a gravity assist. There are also possibilities of using one planet, like Venus which is the easiest to get to, to assist getting to other planets or the Sun.
Hohmann transfer versus low thrust orbits
Lowthrust transfer
Lowthrust 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 change in velocity (deltav) that is greater than the twoimpulse transfer orbit^{[10]} and takes longer to complete.
Engines such as ion thrusters are more difficult to analyze with the deltav model. These engines offer a very low thrust and at the same time, much higher deltav budget, much higher specific impulse, lower mass of fuel and engine. A 2burn Hohmann transfer maneuver would be impractical with such a low thrust; the maneuver mainly optimizes the use of fuel, but in this situation there is relatively plenty of it.
If only lowthrust maneuvers are planned on a mission, then continuously firing a lowthrust, but very highefficiency engine might generate a higher deltav and at the same time use less propellant than a rocket engine.
Going from one circular orbit to another by gradually changing the radius simply requires the same deltav as the difference between the two speeds.^{[10]} Such maneuver requires more deltav than a 2burn Hohmann transfer maneuver, but does so with continuous low thrust rather than the short applications of high thrust.
The amount of propellant mass used measures the efficiency of the maneuver plus the hardware employed for it. The total deltav used measures the efficiency of the maneuver only. For electric propulsion systems, which tend to be lowthrust, the high efficiency of the propulsive system usually compensates for the higher deltaV compared to the more efficient Hohmann maneuver.
Transfer orbit using electrical propulsion or lowthrust engines optimize the transfer time to reach the final orbit and not the deltav as in the Hohmann transfer orbit. For geostationary orbit, the initial orbit is set to be supersynchronous and by thrusting continuously in the direction of the velocity at apogee, the transfer orbit transforms to a circular geosynchronous one. This method however takes much longer to achieve due to the low thrust injected into the orbit ^{[11]}
Interplanetary Transport Network
In 1997, a set of orbits known as the Interplanetary Transport Network (ITN) was published, providing even lower propulsive deltav (though much slower and longer) paths between different orbits than Hohmann transfer orbits.^{[12]} The Interplanetary Transport Network is different in nature than Hohmann transfers because Hohmann transfers assume only one large body whereas the Interplanetary Transport Network does not. The Interplanetary Transport Network is able to achieve the use of less propulsive deltav by employing gravity assist from the planets.^{[citation needed]}
See also
 Bielliptic transfer
 Deltav budget
 Geostationary transfer orbit
 Halo orbit
 Lissajous orbit
 List of orbits
 Orbital mechanics
References
 ^ "Hohmann transfer orbit diagram". www.planetary.org. Retrieved 20180327.
 ^ "Multimedia". www.planetary.org. Retrieved 20180327.
 ^ "Homann Transfers". jwilson.coe.uga.edu. Retrieved 20180327.
 ^ "Calculating a Hohmann Transfer". Instructables.com. Retrieved 20180327.
 ^ Walter Hohmann, The Attainability of Heavenly Bodies (Washington: NASA Technical Translation F44, 1960) Internet Archive.
 ^ Jonathan McDowell, "Kick In the Apogee: 40 years of upper stage applications for solid rocket motors, 19571997", 33rd AIAA Joint Propulsion Conference, July 4, 1997. abstract. Retrieved 18 July 2017.
 ^ NASA, Basics of Space Flight, Section 1, Chapter 4, "Trajectories". Retrieved 26 July 2017. Also available spaceodyssey.dmns.org.
 ^ Tyson Sparks, Trajectories to Mars, Colorado Center for Astrodynamics Research, 12/14/2012. Retrieved 25 July 2017.
 ^ Vallado, David Anthony (2001). Fundamentals of Astrodynamics and Applications. Springer. p. 317. ISBN 0792369033.
 ^ ^{a} ^{b} MIT, 16.522: Space Propulsion, Session 6, "Analytical Approximations for Low Thrust Maneuvers", Spring 2015 (retrieved 26 July 2017)
 ^ Spitzer, Arnon (1997). Optimal Transfer Orbit Trajectory using Electric Propulsion. USPTO.
 ^ Lo, M. W.; Ross, S. D. (1997). "Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System". Technical Report. IOM. JPL. pp. 2–4. 312/97.
Sources
 Walter Hohmann (1925). Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg in München. ISBN 3486231065.
 Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0534408966.
 Bate, R.R., Mueller, D.D., White, J.E., (1971). Fundamentals of Astrodynamics. Dover Publications, New York. ISBN 9780486600611.
 Vallado, D. A. (2001). Fundamentals of Astrodynamics and Applications, 2nd Edition. Springer. ISBN 9780792369035.
 Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics & Ast, Washington, DC. ISBN 9781563473425.
External links
 "Orbital Mechanics". Rocket and Space Technology. Robert A. Braeunig.
 "4. Interplanetary Trajectories". Basics of Spaceflight. JPL: NASA.