Hohmann transfer orbit

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits, typically both in the same plane.

The orbital maneuver to perform the Hohmann transfer uses two engine impulses which move a spacecraft onto and off the transfer orbit. 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).[1] Hohmann was influenced in part by the German science fiction author Kurd Laßwitz and his 1897 book Two Planets.

Contents

Explanation

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 (labeled 1 on diagram) and the higher circular 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; 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, decelerating 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 decelerate the spacecraft into the lower circular orbit.

The Hohmann transfer orbit is theoretically based on two instantaneous velocity changes. Extra fuel is required to compensate for the fact that in reality 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.

Calculation

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 a, (the semi-major axis):

E=\begin{matrix}\frac{1}{2}\end{matrix} m v^2 - \frac{G(M%2Bm)}{r} = \frac{-G M m}{2 a} \,

Solving this equation for velocity results in the Vis-viva equation,

v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)
where:

Therefore the delta-v required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:

\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1%2Br_2}} - 1 \right),
\Delta v_2 = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1%2Br_2}}\,\! \right) ,

where r_1 and r_2 are, respectively, the radii of the departure and arrival circular orbits; the smaller (greater) of r_1 and r_2 corresponds to the periapsis distance (apoapsis distance) of the Hohmann elliptical transfer orbit. The total \Delta v is then:

\Delta v_{total} = \Delta v_1 %2B \Delta v_2

Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is:

t_H = \begin{matrix}\frac12\end{matrix} \sqrt{\frac{4\pi^2 a^3_H}{\mu}} = \pi \sqrt{\frac {(r_1 %2B r_2)^3}{8\mu}}

(one half of the orbital period for the whole ellipse), where a_H\,\! is length of semi-major axis of the Hohmann transfer orbit.

Example

For the geostationary transfer orbit we have r_2 = 42,164 km and e.g. r_1 = 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]

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 ratio of the radius of the larger orbit is 15.58 times that of the smaller orbit.[2] This number is the positive root of x^3-15x^2-9x-1=0. For higher orbit ratios the delta-v decreases again, and tends to \sqrt{2}-1 times the original orbital speed (41.4%). The first burst tends to accelerate to the escape speed, which is \sqrt{2} times the circular orbit speed, and the second tends to zero.

Low-thrust transfer

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 very high-efficiency, low-thrust engine with a high effective exhaust velocity might generate this higher delta-v using less total mass than a high-thrust engine using a nominally more efficient Hohmann transfer maneuver. This is more efficient for a small satellite because the additional mass of the propellant, especially for electric propulsion systems, is lower than the added mass would be for a separate high-thrust system.

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,[3] a set of orbits known as the Interplanetary Transport Network was published, providing even lower-energy (though much slower) paths between different orbits than Hohmann transfer orbits.

See also

References

  1. ^ Walter Hohmann, The Attainability of Heavenly Bodies (Washington: NASA Technical Translation F-44, 1960) Internet Archive.
  2. ^ Vallado, David Anthony (2001). Fundamentals of Astrodynamics and Applications. Springer. p. 317. ISBN 0792369033. http://books.google.com/books?id=PJLlWzMBKjkC&printsec. 
  3. ^ Lo, M., S. Ross, Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System, JPL IOM 312/97, 1997.

External links