Bi-elliptic transfer

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Hohmann transfer orbit, labelled 2, from an orbit (1) to a higher orbit (3). This is comparable to a bi-elliptic transfer orbit. Hohmann transfer orbit.svg
Hohmann transfer orbit, labelled 2, from an orbit (1) to a higher orbit (3). This is comparable to a bi-elliptic transfer orbit.
A bi-elliptic transfer from a low circular starting orbit (blue) to a higher circular orbit (red). A boost at 1 makes the craft follow the green half-ellipse. Another boost at 2 brings it to the orange half-ellipse. A negative boost at 3 makes it follow the red orbit. Bi-elliptic transfer.svg
A bi-elliptic transfer from a low circular starting orbit (blue) to a higher circular orbit (red). A boost at 1 makes the craft follow the green half-ellipse. Another boost at 2 brings it to the orange half-ellipse. A negative boost at 3 makes it follow the red orbit.

In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

Contents

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 an apoapsis at some point 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. [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 initial semi-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 by Ary Sternfeld in 1934. [3]

Calculation

Delta-v

The three required changes in velocity can be obtained directly from the vis-viva equation

where

In what follows,

Starting from the initial circular orbit with radius (dark blue circle in the figure to the right), a 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

When the apoapsis of the first transfer ellipse is reached at a distance 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

Lastly, when the final circular orbit with radius is reached, a retrograde burn (mark 3) circularizes the trajectory into the final target orbit (red circle). The final retrograde burn requires a delta-v of magnitude

If , then the maneuver reduces to a Hohmann transfer (in that case 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.

A bi-parabolic transfer from a low circular starting orbit (dark blue) to a higher circular orbit (red) Bi-parabolicTransfer.png
A bi-parabolic transfer from a low circular starting orbit (dark blue) to a higher circular orbit (red)

The maximal possible savings can be computed by assuming that , in which case the total simplifies to . In this case, one also speaks of a bi-parabolic transfer because the two transfer trajectories are no longer ellipses but parabolas. The transfer time increases to infinity too.

Transfer time

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 the orbital period and the notation from above,

The total transfer time is the sum of the times required for each half-orbit. Therefore:

and finally:

Comparison with the Hohmann transfer

Delta-v

Delta-v required for Hohmann (thick black curve) and bi-elliptic transfers (colored curves) between two circular orbits as a function of the ratio of their radii Bielliptic transfers comparison.svg
Delta-v required for Hohmann (thick black curve) and bi-elliptic transfers (colored curves) between two circular orbits as a function of the ratio of their radii

The figure shows the total required to transfer from a circular orbit of radius to another circular orbit of radius . The is shown normalized to the orbital speed in the initial orbit, , and is plotted as a function of the ratio of the radii of the final and initial orbits, ; this is done so that the comparison is general (i.e. not dependent of the specific values of and , only on their ratio). [2]

The thick black curve indicates the for the Hohmann transfer, while the thinner colored curves correspond to bi-elliptic transfers with varying values of the parameter , defined as the apoapsis radius 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 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 than a Hohmann transfer. Between the ratios of 11.94 and 15.58, which transfer is best depends on the apoapsis distance . For any given in this range, there is a value of above which the bi-elliptic transfer is superior and below which the Hohmann transfer is better. The following table lists the value of that results in the bi-elliptic transfer being better for some selected cases. [4]

Minimal such that a bi-elliptic transfer needs less [5]
Ratio of radii, Minimal Comments
<11.94N/AHohmann transfer is always better
11.94Bi-parabolic transfer
12815.81
1348.90
1426.10
1518.19
15.5815.58
>15.58Any bi-elliptic transfer is better

Transfer time

The long transfer time of the bi-elliptic transfer,

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,

Versatility in combination maneuvers

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.

Example

To transfer from a circular low Earth orbit with r0 = 6700 km to a new circular orbit with r1 = 93 800 km using a Hohmann transfer orbit requires a Δv of 2825.02 + 1308.70 = 4133.72 m/s. However, because r1 = 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 at r2 = 40r0 = 268 000 km, then at apogee accelerated another 608.825 m/s to a new orbit with perigee at r1 = 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 of 75.8r0 = 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 of 1757r0 = 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
TypeHohmannBi-elliptic
Apogee (km)93 800268 000507 68811 770 000
Burn
(m/s)
1Increase2.svg 2825.02Increase2.svg 3061.04Increase2.svg 3123.62Increase2.svg 3191.79Increase2.svg 3194.89
2Increase2.svg 1308.70Increase2.svg 608.825Increase2.svg 351.836Increase2.svg 16.9336Steady2.svg 0
3Steady2.svg 0Decrease2.svg 447.662Decrease2.svg 616.926Decrease2.svg 842.322Decrease2.svg 853.870
Total (m/s)4133.724117.534092.384051.044048.76
Of Hohmann100%99.6%99.0%98.0%97.94%

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

See also

Related Research Articles

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Hohmann transfer orbit Elliptical orbit used to transfer between two orbits of different altitudes, in the same plane

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane that is sometimes tangential to both. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can use less in some cases.

Orbital mechanics Field of classical mechanics concerned with the motion of spacecraft

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Orbital speed Speed at which a body orbits around the barycenter of a system

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Sub-orbital spaceflight Spaceflight that does not complete an orbit around the earth

A sub-orbital spaceflight is a spaceflight in which the spacecraft reaches outer space, but its trajectory intersects the atmosphere or surface of the gravitating body from which it was launched, so that it will not complete one orbital revolution or reach escape velocity.

Delta-<i>v</i> budget

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Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

Hyperbolic trajectory

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

Elliptic orbit Kepler orbit with an eccentricity of less than one

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Circular orbit Orbit with a fixed distance from the barycenter

A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.

Specific orbital energy

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In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or the focus.

Vis-viva equation

In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight.

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Semi-major and semi-minor axes Term in geometry; longest and shortest semidiameters of an ellipse

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References

  1. Curtis, Howard (2005). Orbital Mechanics for Engineering Students. Elsevier. p. 264. ISBN   0-7506-6169-0.
  2. 1 2 Vallado, David Anthony (2001). Fundamentals of Astrodynamics and Applications. Springer. p. 318. ISBN   0-7923-6903-3.
  3. Sternfeld, Ary J. (1934-02-12), "Sur les trajectoires permettant d'approcher d'un corps attractif central à partir d'une orbite keplérienne donnée" [On the allowed trajectories for approaching a central attractive body from a given Keplerian orbit], Comptes rendus de l'Académie des sciences (in French), Paris, 198 (1): 711–713.
  4. Gobetz, F. W.; Doll, J. R. (May 1969). "A Survey of Impulsive Trajectories". AIAA Journal. American Institute of Aeronautics and Astronautics. 7 (5): 801–834. Bibcode:1969AIAAJ...7..801D. doi:10.2514/3.5231.
  5. Escobal, Pedro R. (1968). Methods of Astrodynamics. New York: John Wiley & Sons. ISBN   978-0-471-24528-5.