Hohmann transfer orbit
The orbital maneuver to perform the Hohmann transfer uses two 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 Accessibility of Celestial Bodies). Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets.
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 that one wishes to leave (green and labeled 1 on diagram) and the higher circular orbit that one 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 in order 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 in order 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. 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 up to 141% greater than the two impulse transfer orbit (see also below), and takes longer to complete.
For a small body orbiting another, very much larger body (such as a satellite orbiting the earth), the total energy of the 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 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
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. 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:
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
For the geostationary transfer orbit we have = 42,164 km and e.g. = 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.
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 (3.20-2.42) 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 escape speed of this circular orbit. This illustrates that at large speeds the same delta-v provides more specific orbital energy, and 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).
Worst case, maximum delta-v
A Hohmann transfer orbit between a given circular orbit and 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 larger orbit is 15.58 times that of the smaller orbit. This number is the positive root of . For higher orbit ratios the delta-v decreases again, and tends to times the original orbital speed (41.4%). The first burst tends to accelerate to the escape speed, which is times the circular orbit speed, and the second tends to zero.
It can be shown 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.
Such a low-thrust maneuver requires more delta-v than a 2-burn Hohmann transfer maneuver, requiring more fuel for a given engine design. However, if only low-thrust maneuvers are required on a mission, then continuously firing a low-thrust, but very high-efficiency (high effective exhaust velocity) engine might generate this higher delta-v using less propellant mass than a high-thrust engine using an otherwise more efficient Hohmann transfer maneuver.
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 vastly compensates for the inability to make the more efficient Hohmann maneuver.
Application to interplanetary travel
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 the 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 a Mars-like orbit. Therefore, the spacecraft will have to decelerate in order for Mars' gravity to capture it. Therefore, relatively small amounts of thrust at either end of the trip are needed to arrange the transfer. However, 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.
Interplanetary Transport Network
In 1997, a set of orbits known as the Interplanetary Transport Network was published, providing even lower propulsive delta-v (though much slower and longer) paths between different orbits than Hohmann transfer orbits. The Interplanetary Transport Network is different in nature than Hohmann transfers because Hohmann transfers assume only one large body while 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. The gravity assist provides "free" delta-v without the use of the propulsion systems.
- Bi-elliptic transfer
- Delta-v budget
- Geostationary transfer orbit
- Halo orbit
- Lissajous orbit
- List of orbits
- Orbital mechanics
||This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (April 2009)|
- Walter Hohmann, The Attainability of Heavenly Bodies (Washington: NASA Technical Translation F-44, 1960) Internet Archive.
- Vallado, David Anthony (2001). Fundamentals of Astrodynamics and Applications. Springer. p. 317. ISBN 0-7923-6903-3.
- Lo, M., S. Ross, Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System, JPL IOM 312/97, 1997.
- Walter Hohmann (1925). Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg in München. ISBN 3-486-23106-5.
- Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0-534-40896-6.
- Bate, R.R., Mueller, D.D., White, J.E., (1971). Fundamentals of Astrodynamics. Dover Publications, New York. ISBN 978-0-486-60061-1.
- Vallado, D. A. (2001). Fundamentals of Astrodynamics and Applications, 2nd Edition. Springer. ISBN 978-0-7923-6903-5.
- Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics & Ast, Washington, DC. ISBN 978-1-56347-342-5.
- ORBITAL MECHANICS (Rocket and Space Technology)
- Basics of Spaceflight - Chapter 4. Interplanetary Trajectories