Inastronautics andaerospace engineering, thebi-elliptic transfer is anorbital maneuver that moves aspacecraft from oneorbit to another and may, in certain situations, require lessdelta-v than aHohmann transfer maneuver.

Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

The bi-elliptic transfer consists of two half-elliptic orbits. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with anapoapsis at some point${\displaystyle r_b}$ away from thecentral body. At this point a second burn sends the spacecraft into the second elliptical orbit withperiapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit.^[1]

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 initialsemi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen.^[2]

The idea of the bi-elliptical transfer trajectory was first^{[citation needed]} published byAry Sternfeld in 1934.^[3]

The three required changes in velocity can be obtained directly from thevis-viva equation$${\displaystyle v^2=\mu \left(\frac{2}{r}-\frac{1}{a}\right),}$$where

• ${\displaystyle v}$ is the speed of an orbiting body,
• ${\displaystyle \mu =GM}$ is thestandard gravitational parameter of the primary body,
• ${\displaystyle r}$ is the distance of the orbiting body from the primary, i.e., the radius,
• ${\displaystyle a}$ is thesemi-major axis of the body's orbit.

In what follows,

• ${\displaystyle r_1}$ is the radius of the initial circular orbit,
• ${\displaystyle r_2}$ is the radius of the final circular orbit,
• ${\displaystyle r_b}$ is the common apoapsis radius of the two transfer ellipses and is a free parameter of the maneuver,
• ${\displaystyle a_1}$ and${\displaystyle a_2}$ are the semimajor axes of the two elliptical transfer orbits, which are given by$${\displaystyle a_1=\frac{r_1+r_b}{2},}$$ and$${\displaystyle a_2=\frac{r_2+r_b}{2}.}$$

Starting from the initialcircular orbit with radius${\displaystyle r_1}$ (dark blue circle in the figure to the right), aprograde 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$${\displaystyle \mathrm{\Delta }{v}_1=\sqrt{\frac{2\mu }{r_1}-\frac{\mu }{a_1}}-\sqrt{\frac{\mu }{r_1}}.}$$

When the apoapsis of the first transfer ellipse is reached at a distance${\displaystyle r_b}$ from the primary, a second prograde burn (mark 2) raises the periapsis to match the radius of the target circular orbit, putting the spacecraft on a second elliptic trajectory (orange half-ellipse). The magnitude of the required delta-v for the second burn is$${\displaystyle \mathrm{\Delta }{v}_2=\sqrt{\frac{2\mu }{r_b}-\frac{\mu }{a_2}}-\sqrt{\frac{2\mu }{r_b}-\frac{\mu }{a_1}}.}$$

Lastly, when the final circular orbit with radius${\displaystyle r_2}$ is reached, aretrograde burn (mark 3) circularizes the trajectory into the final target orbit (red circle). The final retrograde burn requires a delta-v of magnitude$${\displaystyle \mathrm{\Delta }{v}_3=\sqrt{\frac{2\mu }{r_2}-\frac{\mu }{a_2}}-\sqrt{\frac{\mu }{r_2}}.}$$

If${\displaystyle r_b=r_2}$, then the maneuver reduces to a Hohmann transfer (in that case${\displaystyle \mathrm{\Delta }{v}_3}$ 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.

The maximal possible savings can be computed by assuming that${\displaystyle r_b=\mathrm{\infty }}$, in which case the total${\displaystyle \mathrm{\Delta }v}$ simplifies to$\sqrt{\mu /r_1}\left(\sqrt{2}-1\right)\left(1+\sqrt{r_1/r_2}\right)$. In this case, one also speaks of abi-parabolic transfer because the two transfer trajectories are no longer ellipses butparabolas. The transfer time increases to infinity too.

Like the Hohmann transfer, both transfer orbits used in the bi-elliptic transfer constitute exactly one half of an elliptic orbit. This means that the time required to execute each phase of the transfer is half the orbital period of each transfer ellipse.

Using the equation for theorbital period and the notation from above,$${\displaystyle T=2\pi \sqrt{\frac{a^3}{\mu }}.}$$

The total transfer time${\displaystyle t}$ is the sum of the times required for each half-orbit. Therefore:$${\displaystyle t_1=\pi \sqrt{\frac{a_1^3}{\mu }}\text{and}{t}_2=\pi \sqrt{\frac{a_2^3}{\mu }},}$$and finally:$${\displaystyle t=t_1+t_2.}$$

The figure shows the total${\displaystyle \mathrm{\Delta }v}$ required to transfer from a circular orbit of radius${\displaystyle r_1}$ to another circular orbit of radius${\displaystyle r_2}$. The${\displaystyle \mathrm{\Delta }v}$ is shown normalized to the orbital speed in the initial orbit,${\displaystyle v_1}$, and is plotted as a function of the ratio of the radii of the final and initial orbits,${\displaystyle R\equiv r_2/r_1}$; this is done so that the comparison is general (i.e. not dependent of the specific values of${\displaystyle r_1}$ and${\displaystyle r_2}$, only on their ratio).^[2]

The thick black curve indicates the${\displaystyle \mathrm{\Delta }v}$ for the Hohmann transfer, while the thinner colored curves correspond to bi-elliptic transfers with varying values of the parameter${\displaystyle \alpha \equiv r_b/r_1}$, defined as the apoapsis radius${\displaystyle r_b}$ of the elliptic auxiliary orbit normalized to the radius of the initial orbit, and indicated next to the curves. The inset shows a close-up of the region where the bi-elliptic curves cross the Hohmann curve for the first time.

One sees that the Hohmann transfer is always more efficient if the ratio of radii${\displaystyle R}$ is smaller than 11.94. On the other hand, if the radius of the final orbit is more than 15.58 times larger than the radius of the initial orbit, then any bi-elliptic transfer, regardless of its apoapsis radius (as long as it's larger than the radius of the final orbit), requires less${\displaystyle \mathrm{\Delta }v}$ than a Hohmann transfer. Between the ratios of 11.94 and 15.58, which transfer is best depends on the apoapsis distance${\displaystyle r_b}$. For any given${\displaystyle R}$ in this range, there is a value of${\displaystyle r_b}$ above which the bi-elliptic transfer is superior and below which the Hohmann transfer is better. The following table lists the value of${\displaystyle \alpha \equiv r_b/r_1}$ that results in the bi-elliptic transfer being better for some selected cases.^[4]

Minimal${\displaystyle \alpha \equiv r_b/r_1}$ such that a bi-elliptic transfer needs less${\displaystyle \mathrm{\Delta }v}$^[5]

Ratio of radii,${\displaystyle \frac{r_2}{r_1}}$	Minimal${\displaystyle \alpha \equiv \frac{r_b}{r_1}}$	Comments
<11.94	—	Hohmann transfer is always better
11.94	${\displaystyle \mathrm{\infty }}$	Bi-parabolic transfer
12	815.81
13	48.90
14	26.10
15	18.19
15.58	15.58
>15.58	${\displaystyle >\frac{r_2}{r_1}}$	Any bi-elliptic transfer is better

The long transfer time of the bi-elliptic transfer,$${\displaystyle t=\pi \sqrt{\frac{a_1^3}{\mu }}+\pi \sqrt{\frac{a_2^3}{\mu }},}$$is a major drawback for this maneuver. It even becomes infinite for the bi-parabolic transfer limiting case.

The Hohmann transfer takes less than half of the time because there is just one transfer half-ellipse. To be precise,$${\displaystyle t=\pi \sqrt{\frac{a^3}{\mu }}.}$$

While a bi-elliptic transfer has a small parameter window where it's strictly superior to a Hohmann Transfer in terms of delta V for a planar transfer between circular orbits, the savings is fairly small, and a bi-elliptic transfer is a far greater aid when used in combination with certain other maneuvers.

At apoapsis, the spacecraft is travelling at low orbital velocity, and significant changes in periapsis can be achieved for small delta V cost. Transfers that resemble a bi-elliptic but which incorporate a plane-change maneuver at apoapsis can dramatically save delta-V on missions where the plane needs to be adjusted as well as the altitude, versus making the plane change in low circular orbit on top of a Hohmann transfer.

Likewise, dropping periapsis all the way into the atmosphere of a planetary body for aerobraking is inexpensive in velocity at apoapsis, but permits the use of "free" drag to aid in the final circularization burn to drop apoapsis; though it adds an extra mission stage of periapsis-raising back out of the atmosphere, this may, under some parameters, cost significantly less delta V than simply dropping periapsis in one burn from circular orbit.

To transfer from a circular low Earth orbit withr₀ = 6700 km to a new circular orbit withr₁ = 93 800 km using aHohmann transfer orbit requires a Δv of2825.02 + 1308.70 = 4133.72 m/s. However, becauser₁ = 14r₀ > 11.94r₀, it is possible to do better with a bi-elliptic transfer. If the spaceship first accelerated 3061.04 m/s, thus achieving an elliptic orbit with apogee atr₂ = 40r₀ = 268 000 km, then at apogee accelerated another 608.825 m/s to a new orbit with perigee atr₁ = 93 800 km, and finally at perigee of this second transfer orbit decelerated by 447.662 m/s, entering the final circular orbit, then the total Δv would be only 4117.53 m/s, which is 16.19 m/s (0.4%) less.

The Δv saving could be further improved by increasing the intermediate apogee, at the expense of longer transfer time. For example, an apogee of75.8r₀ = 507 688 km (1.3 times the distance to the Moon) would result in a 1% Δv saving over a Hohmann transfer, but require a transit time of 17 days. As an impractical extreme example, an apogee of1757r₀ = 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 and 34 minutes.

Δv for various orbital transfers

Type		Hohmann	Bi-elliptic
Apogee (km)		93 800	268 000	507 688	11 770 000	∞
Burn (m/s)	1	2825.02	3061.04	3123.62	3191.79	3194.89
	2	1308.70	608.825	351.836	16.9336	0
	3	0	447.662	616.926	842.322	853.870
Total (m/s)		4133.72	4117.53	4092.38	4051.04	4048.76
Of Hohmann		100%	99.6%	99.0%	98.0%	97.94%

Evidently, the bi-elliptic orbit spends more of its delta-v closer to the planet (in the first burn). This yields a higher contribution to thespecific orbital energy and, due to theOberth effect, is responsible for the net reduction in required delta-v.

• Delta-v budget
• Oberth effect

• Astrodynamics
• Spacecraft propulsion
• Orbital maneuvers

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