Editing Hohmann transfer orbit (section)

==Comparison to other transfers==

=== Bi-elliptic transfer ===
{{main|Bi-elliptic transfer}}
The bi-elliptic transfer consists of two half-[[elliptic orbit]]s. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an [[apoapsis]] at some point <math>r_b</math> away from the [[central body]]. At this point a second burn sends the spacecraft into the second elliptical orbit with [[periapsis]] at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit.<ref name="Curtis">{{Cite book | last = Curtis | first = Howard | title = Orbital Mechanics for Engineering Students | page = 264 | publisher = [[Elsevier]] | year = 2005 | isbn = 0-7506-6169-0 | url = https://books.google.com/books?id=6aO9aGNBAgIC}}</ref>

While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial [[semi-major axis]] is 11.94 or greater, depending on the intermediate semi-major axis chosen.<ref name="Vallado">{{Cite book | last = Vallado | first = David Anthony | title = Fundamentals of Astrodynamics and Applications | page = 318 | publisher = Springer | year = 2001 | isbn = 0-7923-6903-3 | url = https://books.google.com/books?id=PJLlWzMBKjkC}}</ref>

The idea of the bi-elliptical transfer trajectory was first{{citation needed|date=January 2014}}<!-- a primary source of Sternfeld's paper does not establish that he was the "first" --> published by [[Ary Sternfeld]] in 1934.<ref name=sternfeld1934>
{{Citation |last=Sternfeld |first=Ary J. |title=Sur les trajectoires permettant d'approcher d'un corps attractif central à partir d'une orbite keplérienne donnée |language=fr |trans-title=On the allowed trajectories for approaching a central attractive body from a given Keplerian orbit |url=http://gallica.bnf.fr/ark:/12148/bpt6k31506/f711.image.langEN |journal=Comptes rendus de l'Académie des sciences |location=Paris |volume=198 |number=1 |date=1934-02-12 |pages=711–713}}.</ref>

=== Low-thrust transfer ===
{{main|Low thrust relative orbital transfer}}
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|change in velocity (delta-''v'')]] that is greater than the two-impulse transfer orbit<ref name="MIT_16.522">MIT, ''16.522: Space Propulsion'', Session 6, "[https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-522-space-propulsion-spring-2015/lecture-notes/MIT16_522S15_Lecture6.pdf Analytical Approximations for Low Thrust Maneuvers]", Spring 2015 (retrieved 26 July 2017)
</ref> and takes longer to complete.

Engines such as [[ion thruster]]s are more difficult to analyze with the delta-''v'' model. These engines offer a very low thrust and at the same time, much higher delta-''v'' budget, much higher [[specific impulse]], lower mass of fuel and engine. A 2-burn 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 low-thrust maneuvers are planned on a mission, then continuously firing a low-thrust, but very high-efficiency engine might generate a higher delta-''v'' and at the same time use less propellant than a conventional chemical rocket engine.

Going from one circular orbit to another by gradually changing the radius simply requires the same delta-''v'' as the difference between the two speeds.<ref name="MIT_16.522" /> Such maneuver requires more delta-''v'' than a 2-burn 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 delta-''v'' used measures the efficiency of the maneuver only. For [[Electrically powered spacecraft propulsion|electric propulsion]] systems, which tend to be low-thrust, the high efficiency of the propulsive system usually compensates for the higher delta-V compared to the more efficient Hohmann maneuver.

Transfer orbits using electrical propulsion or low-thrust engines optimize the transfer time to reach the final orbit and not the delta-v 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.<ref>{{Cite book | last = Spitzer | first = Arnon | title = Optimal Transfer Orbit Trajectory using Electric Propulsion | publisher = [[USPTO]] | date = 1997 | url = https://patents.google.com/patent/US5595360}}</ref>

===Interplanetary Transport Network===
{{main article|Interplanetary Transport Network}}
In 1997, a set of orbits known as the Interplanetary Transport Network (ITN) was published, providing even lower propulsive delta-''v'' (though much slower and longer) paths between different orbits than Hohmann transfer orbits.<ref>{{cite web |last1= Lo |first1= M. W. |author1-link= Martin Lo |first2= S. D. |last2= Ross |title= Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System |publisher= [[JPL]] |work= Technical Report |series= IOM |id= 312/97 |pages= 2–4 |date= 1997 |url= http://www.gg.caltech.edu/~mwl/publications/publications2.htm }}</ref>  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 delta-''v'' by employing [[gravity assist]] from the planets.{{citation needed|date=January 2016}}