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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.
The orbital maneuver to perform the Hohmann transfer uses two prograde 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).Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets .
The elliptic transfer orbits between different bodies (planets, moons etc.) are often referred to as Hohmann transfer orbits. When used for traveling between celestial bodies, a Hohmann transfer orbit 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 so-called 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 Earth-Mars journey this travel time is about 9 months. When transfer is performed between orbits close to celestial bodies with significant gravitation, much less delta-v is usually required, as the Oberth effect may be employed for the burns.
They are also often used for these situations, but low-energy transfers which take into account the thrust limitations of real engines, and take advantage of the gravity wells of both planets can be more fuel efficient.
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 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 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 lower-energy 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 high-thrust 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); 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.
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.
Transfer orbits can go more than 360° around the sun. These multiple-revolution transfers are sometimes referred to as Type III and Type IV, where a Type III is a Type I plus 360°, and a Type IV is a Type II plus 360°.
A Hohmann transfer orbit can be used to transfer an object's orbit towards another object, as long as they share a common, more massive body which they orbit around. In the context of Earth and the Solar System, this includes any object which orbits the Sun. An example of where a Hohmann transfer orbit could be used is to bring an asteroid, orbiting the Sun, into contact with the Earth.
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 semi-major axis):
Solving this equation for velocity results in the vis-viva equation,
Therefore, the delta-v (Δv) 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, and
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 m3/s2, 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 semi-major 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
Consider a geostationary transfer orbit, beginning at r1 = 6,678 km (altitude 300 km) and ending in a geostationary orbit with r2 = 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.
Therefore the Δv for the first burn is 10.15 − 7.73 = 2.42 km/s, for the second burn 3.07 − 1.61 = 1.46 km/s, and for both 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 velocity, 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).
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 √ times orbital speed, so the Δv required to escape is √ − 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. This number is the positive root of x3 − 15x2 − 9x − 1 = 0, which is . For higher orbit ratios the Δv required for the second burn decreases faster than the first increases.
When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex, but much less delta-v is required, due to the Oberth effect, than the sum of the delta-v required to escape the first planet plus the delta-v required for a Hohmann transfer to the second planet.
For example, consider a spacecraft travelling from 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 delta-v, 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 delta-v 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 delta-v 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 Mars-like 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 the 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 non-rotating 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.
|to enter Hohmann orbit|
from Earth's orbit
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.
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.
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.
The idea of the bi-elliptical transfer trajectory was first[ citation needed ] published by Ary Sternfeld in 1934.
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 change in velocity (delta-v) that is greater than the two-impulse transfer orbitand takes longer to complete.
Engines such as ion thrusters 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.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 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.
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. [ citation needed ]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.
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.
A geosynchronous transfer orbit or geostationary transfer orbit (GTO) is a type of geocentric orbit. Satellites that are destined for geosynchronous (GSO) or geostationary orbit (GEO) are (almost) always put into a GTO as an intermediate step for reaching their final orbit.
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.
Delta-v, symbolized as ∆v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launching from or landing on a planet or moon, or an in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle.
In gravitationally bound systems, the orbital speed of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one object is much more massive than the other bodies in the system, its speed relative to the center of mass of the most massive body.
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.
In astrodynamics and aerospace, a delta-v budget is an estimate of the total change in velocity (delta-v) required for a space mission. It is calculated as the sum of the delta-v required to perform each propulsive maneuver needed during the mission. As input to the Tsiolkovsky rocket equation, it determines how much propellant is required for a vehicle of given empty mass and propulsion system.
In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth an orbital maneuver is called a deep-space maneuver (DSM).
Orbital inclination change is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector at the orbital nodes.
In astrodynamics, orbit phasing is the adjustment of the time-position of spacecraft along its orbit, usually described as adjusting the orbiting spacecraft's true anomaly. Orbital phasing is primarily used in scenarios where a spacecraft in a given orbit must be moved to a different location within the same orbit. The change in position within the orbit is usually defined as the phase angle, ϕ, and is the change in true anomaly required between the spacecraft's current position to the final position.
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.
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
In the gravitational two-body problem, the specific orbital energy of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass. According to the orbital energy conservation equation, it does not vary with time:
Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.
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.
A Mars cycler is a kind of spacecraft trajectory that encounters Earth and Mars regularly. The term Mars cycler may also refer to a spacecraft on a Mars cycler trajectory. The Aldrin cycler is an example of a Mars cycler.
A gravity turn or zero-lift turn is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory controlled solely through the vehicle's own thrust. First, the thrust is not used to change the spacecraft's direction, so more of it is used to accelerate the vehicle into orbit. Second, and more importantly, during the initial ascent phase the vehicle can maintain low or even zero angle of attack. This minimizes transverse aerodynamic stress on the launch vehicle, allowing for a lighter launch vehicle.
In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry.