Hohmann transfer orbit
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In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that moves a spacecraft from one orbit to another using a fairly low delta-v. It was named after Walter Hohmann, the German scientist who published it in 1925. (See also interplanetary travel.) A Hohmann transfer orbit will take a spacecraft from Low Earth Orbit (LEO) to geosynchronous orbit (GEO) in just over five hours (geostationary transfer orbit), from LEO to the Moon in about 5 days and from the Earth to Mars in about 260 days. However, Hohmann transfers are very slow for trips to more distant points, so when visiting the outer planets it is common to use a gravitational slingshot to increase speed in-flight.
The Hohmann transfer orbit is one half of an elliptic orbit that touches both the orbit that one wishes to leave (labelled 1 on diagram) and the orbit that one wishes to reach (3 on diagram). The transfer (2 on diagram) is initiated by firing the spacecraft's engine in order to accelerate it so that it will follow the elliptical orbit. When the spacecraft has reached its destination orbit, it has slowed down to a speed not only lower than the speed in the original circular orbit, but even lower than required for the new circular orbit; the engine is fired again to accelerate it again, to that required speed.
Since this transfer requires two powerful bursts, it can not be applied with a low-thrust engine. With that, the circular orbit can be gradually enlarged. Together this requires a delta-v that is up to 141% more (see also below), and, of course, the lower the thrust the longer it takes.
When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex. For example, consider a spacecraft travelling from the Earth to Mars. At the beginning of its journey, the spacecraft will already have a certain velocity associated with its orbit around Earth – this is velocity that will not need to be found when the spacecraft enters the transfer orbit (around the Sun). 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 an Mars-like orbit. Therefore, the spacecraft will have to decelerate and allow Mars' gravity to capture it. Therefore, relatively small amounts of thrust at either end of the trip are all that are needed to arrange the transfer. Note, however, that 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, see launch window.
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, causing it to drop into the elliptical transfer orbit, and fired again in the lower orbit to brake it to the correct speed for that lower orbit.
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Calculation
Ignoring the delta-v needed to get to/from orbits around planets at either end of the journey, and just calculating the delta-v needed to get from one circular orbit to another coplanar circular orbit around a primary body, e.g. the Sun, the vis viva equation says,
- <math> v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right) <math>
where:
- <math> v \,\!<math> is the speed of an orbiting body
- <math>\mu = GM\,\!<math> is the standard gravitational parameter of the primary body
- <math> r \,\!<math> is the distance of the orbiting body from the primary
- <math> a \,\!<math> is the semi-major axis of the body's orbit
Therefore the Delta-v required for the Hohmann transfer can be computed as follows:
- <math>\Delta v_P
= \sqrt{\frac{\mu}{r_1}}
\left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1 \right)<math>, Delta-v required at periapsis.
- <math>\Delta v_A
= \sqrt{\frac{\mu}{r_2}}
\left( 1 - \sqrt{\frac{2 r_1}{r_1+r_2}}\,\! \right) <math>, Delta-v required at apoapsis.
where:
- <math>r_1\,\!<math> is radius of lower orbit, and periapsis distance of Hohmann transfer orbit,
- <math>r_2\,\!<math> is radius of higher orbit, and apoapsis distance of Hohmann transfer orbit.
Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is:
- <math> t_H
= \begin{matrix}\frac12\end{matrix} \sqrt{\frac{4\pi^2 a^3_H}{\mu}} = \pi \sqrt{\frac {(r_1 + r_2)^3}{8\mu}} <math>
(one half of the orbital period for the whole ellipse)
where:
- <math> a_H\,\!<math> is length of semi-major axis of the Hohmann transfer orbit.
Example; maximum delta-v
For the geostationary transfer orbit we have <math>r_2<math> = 42,164 km and e.g. <math>r_1<math> = 6,678 km (altitude 300 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.
The delta-v's are 10.15 - 7.73 = 2.42 and 3.07 - 1.61 = 1.46 km/s, together 3.88 km/s. [1] (http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=sqrt%28398600%2F6678%29*sqrt%282%2F%286678%2F42164%2B1%29%29)
Compare with the delta-v for an escape orbit: 10.93 - 7.73 = 3.20 km/s. Applying a delta-v at the LEO of only 0.78 km/s more would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1.46 km/s is needed for reaching the sub-escape speed of this circular orbit. This illustrates that at large speeds the same delta-v provides more specific orbital energy, and, as explained in gravity drag, energy increase is maximized if one spends the delta-v as soon as possible, rather than spending some, being decelerated by gravity, and then spending some more (of course, the objective of a Hohmann transfer orbit is different).
A Hohmann transfer orbit from a given circular orbit to a larger circular orbit, in the case of a single central body, costs the largest delta-v (53.6 % of the original orbital speed) if the radius of the target circle is 15.5 times as large as that of the original circle. For larger target circles the delta-v decreases again, and tends to <math>\sqrt{2}-1<math> times the original orbital speed (41.4%). (The first burst tends to acceleration to the escape speed, the second tends to zero.)
Low-thrust transfer
It can be derived that going from one circular orbit to another by gradually changing the radius costs a delta-v of simply the absolute value of the difference between the two speeds. Thus for the geostationary transfer orbit 7.73 - 3.07 = 4.66 km/s, the same as, in the absence of gravity, the deceleration would cost. In fact, acceleration is applied to compensate half of the deceleration due to moving outward. Therefore the acceleration due to thrust is equal to the deceleration due to the combined effect of thrust and gravity. For escaping the Earth from LEO the required delta-v is 7.73 km/s, compare with the 3.20 km/s in the case of a single burst.
Interplanetary Superhighway
In 1997, a set of orbits known as the Interplanetary Superhighway was published, providing even lower-energy (though much slower) paths between different orbits than Hohmann transfer orbits.