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| In [[astronautics]] and [[aerospace engineering]], the '''bi-elliptic transfer''' is an [[orbital maneuver]] that moves a [[spacecraft]] from one [[orbit]] to another and sometimes requires less [[delta-v]] than a [[Hohmann transfer orbit|Hohmann transfer]] maneuver.
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| 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|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.{{Citation needed|date=December 2011}} | |
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| While they require one more engine burn than a Hohmann transfer and generally requires 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>{{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 = http://books.google.com/books?id=PJLlWzMBKjkC&printsec}}</ref>
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| [[Ary Sternfeld]] in 1934 first{{cn|date=January 2014}}<!-- a primary source of Sternfeld's paper does not establish that he was the "first" --> published the idea of the bi-elliptical transfer trajectory.<ref name=sternfeld1934>
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| {{Citation |last=Sternfeld |first=Ary {{sic|J.|expected=A.}} |title=Sur les trajectoires permettant d'approcher d'un corps attractif central à partir d'une orbite keplérienne donnée |language=French |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 |year=1934 |month=January–June |pages=711–713}}</ref>
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| ==Calculation==
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| ===Delta-v===
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| [[File:Bi-elliptic transfer.svg|thumb|250px|A bi-elliptic transfer from a low circular starting orbit (dark blue), to a higher circular orbit (red).]]
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| The three required changes in velocity can be obtained directly from the [[Specific orbital energy|''vis viva'' equation]],
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| :<math> v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right) </math>
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| * <math> v \,\!</math> is the speed of an orbiting body
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| * <math>\mu = GM\,\!</math> is the [[standard gravitational parameter]] of the primary body
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| * <math> r \,\!</math> is the distance from the orbiting body to the primary
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| * <math> a \,\!</math> is the [[semi-major axis]] of the body's orbit
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| * <math>r_b</math> is the common apoapsis distance of the two transfer ellipses and is a free parameter of the maneuver.
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| * <math>a_1</math> and <math>a_2</math> are the semimajor axes of the two elliptical transfer orbits, which are given by
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| *: <math>a_1 = \frac{r_0+r_b}{2}</math>
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| *: <math>a_2 = \frac{r_f+r_b}{2}</math>
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| Starting from the initial [[circular orbit]] with radius <math>r_0</math> (dark blue circle in the figure to the right) a [[Retrograde and prograde motion|prograde]] burn (mark 1 in the figure) puts the spacecraft on the first elliptical transfer orbit (aqua half ellipse). The magnitude of the required delta-v for this burn is:
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| :<math>\Delta v_1 = \sqrt{ \frac{2 \mu}{r_0} - \frac{\mu}{a_1}} - \sqrt{\frac{\mu}{r_0}} </math>
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| When the apoapsis of the first transfer ellipse is reached at a distance <math>r_b</math> from the primary, a second prograde burn (mark 2) so raises the periapsis to the radius of the target circular orbit, thus putting the spacecraft on a second elliptic trajectory (orange half ellipse). The magnitude of the required delta-v for the second burn is:
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| :<math>\Delta v_2 = \sqrt{ \frac{2 \mu}{r_b} - \frac{\mu}{a_2}} - \sqrt{ \frac{2 \mu}{r_b} - \frac{\mu}{a_1}} </math>
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| Lastly, when the final circular orbit with radius <math>r_f</math> is reached, a [[Retrograde and prograde motion|''retrograde'']] burn (mark 3) circularizes the trajectory into the final target orbit (red circle). The final retograde burn requires a delta-v of magnitude:
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| :<math>\Delta v_3 = \sqrt{ \frac{2 \mu}{r_f} - \frac{\mu}{a_2}} - \sqrt{\frac{\mu}{r_f}} </math>
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| If <math>r_b=r_f</math>, then the maneuver reduces to a Hohmann transfer (in that case <math>\Delta v_3</math> can be verified to become zero). Thus the bi-elliptic transfer constitutes a more general class of orbital transfers, of which the Hohmann transfer is a special two-impulse case.
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| The maximum savings possible can be computed by assuming that <math>r_b=\infty</math>, in which case the total <math>\Delta v</math> simplifies to <math>\left(\sqrt 2 - 1\right) \left(\sqrt{\mu/r_0} + \sqrt{\mu/r_f}\right)</math>.
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| ===Transfer time===
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| Like the Hohmann transfer, both transfer orbits of the bi-elliptic transfer constitute half an elliptic orbit. The time required to execute each phase of the transfer therefore is half the orbital period of each transfer ellipse.
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| Using the equation for the [[orbital period]] and the notation from above:
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| :<math>T = 2 \pi \sqrt{\frac{a^3}{\mu}}</math>
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| The total transfer time <math>t</math> is the sum of the time that each half orbit requires:
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| :<math>t_1 = \pi \sqrt{\frac{a_1^3}{\mu}} \quad and \quad t_2 = \pi \sqrt{\frac{a_2^3}{\mu}}</math>
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| :<math>t = t_1 + t_2 \;</math>
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| ==Example==
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| To transfer from circular low earth orbit with ''r''<sub>0</sub>=6700 km to a new circular orbit with ''r''<sub>1</sub>=93800 km using [[Hohmann transfer orbit]] requires delta-v of 2825.02+1308.70=4133.72 m/s. However, because ''r''<sub>1</sub>=14''r''<sub>0</sub> >11.94''r''<sub>0</sub>, a bi-elliptic transfer is better. If the spaceship first accelerated 3061.04 m/s, thus achieving an elliptic orbit with apogee at ''r''<sub>2</sub>=40''r''<sub>0</sub>=268000 km, then in apogee accelerated another 608.825 m/s to a new orbit with perigee at ''r''<sub>1</sub>=93800 km, and finally in perigee decelerated by 447.662 m/s, entering final circular orbit, then the total delta-v would be only 4117.53, which is 16.19 m/s (0.4%) less.
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| The Δv ratio could be further improved by increasing the intermediate apogee, at the expense of longer transfer time. For example, an apogee of 75.8''r''<sub>0</sub>=507,688 km (1.3 times the distance to the moon) would result in a 1% Δv saving over a Hohmann transfer, but a transit time of 17 days. As an impractical extreme example, 1757''r''<sub>0</sub>=11,770,000 km (30 times the distance to the moon) would, result in a 2% Δv saving over a Hohmann transfer, but the transfer would require 4.5 years (and, in practice, be perturbed by the gravitational effects of other solar system bodies). For comparison. the Hohmann transfer requires 15 hours 34 minutes.
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| {| class='wikitable'
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| |+ Δv for various orbital transfers
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| ! Type !! Hohmann ||colspan=4| Bi-elliptic
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| |-
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| ! Apogee (km)
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| | 93800 || 268000 || 507688 || 11770000 || ∞ <!-- The infinite limit -->
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| |-
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| ! Burn 1 (m/s)
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| | {{green|2825.02}} || {{green|3061.04}} || {{green|3123.62}} || {{green|3191.79}} || {{green|3194.89}}
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| |-
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| ! Burn 2 (m/s)
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| | {{green|1308.70}} || {{green|608.825}} || {{green|351.836}} || {{green|16.9336}} || 0
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| |-
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| ! Burn 3 (m/s)
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| | 0 || {{red|447.662}} || {{red|616.926}} || {{red|842.322}} || {{red|853.870}}
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| |-
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| ! Total (m/s)
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| | ''4133.72'' || ''4117.53'' || ''4092.38'' || ''4051.04'' || ''4048.76''
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| |-
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| ! Percentage
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| | ''100%'' || ''99.6%'' || ''99.0%'' || ''98.0%'' || ''97.94%''
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| |}
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| *''{{green|Δv applied prograde}}''
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| *''{{red|Δv applied retrograde}}''
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| Evidently, the bi-elliptic orbit spends more of its delta-v early on (in the first burn) thus yielding a higher contribution to the [[specific orbital energy]] and, due to the [[Oberth effect]], reduces required delta-v.
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| ==See also==
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| {{Portal|Spaceflight}}
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| *[[Delta-v budget]]
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| *[[Oberth effect]]
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| ==References==
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| {{Reflist}}
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| {{orbits}}
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| {{DEFAULTSORT:Bi-Elliptic Transfer}}
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| [[Category:Astrodynamics]]
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| [[Category:Spacecraft propulsion]]
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| [[Category:Orbital maneuvers]]
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