Hohmann transfer orbit

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two (typically coplanar) circular orbits.

The orbital maneuver to perform the Hohmann transfer uses two engine impulses which, under standard assumptions, move a spacecraft onto and off of the transfer orbit. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in 1925. (See also interplanetary travel.) Hohmann was influenced in part by the German science fiction author Kurd Laßwitz and his 1897 book Two Planets.

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.^{[citation needed]}

The Hohmann transfer orbit is theoretically based on two impulsive (i.e. instantaneous) velocity changes. Extra fuel is required to compensate for the fact that in reality the bursts take time; this is minimized by minimizing the duration of the bursts, i.e., by using high thrust engines. 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 2 impulse transfer orbit (see also below), and takes longer to complete.^{[citation needed]}

Calculation

For a small body orbiting another (such as a satellite orbiting the earth), the total energy of the body is just 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}}mv^{2}-{\frac {GMm}{r}}={\frac {-GMm}{2a}}\,

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

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

where:

$v\,\!$ is the speed of an orbiting body
$\mu =GM\,\!$ is the standard gravitational parameter of the primary body
$r\,\!$ is the distance of the orbiting body from the primary focus
$a\,\!$ is the semi-major axis of the body's orbit

Therefore the delta-v required for the Hohmann transfer can be computed as follows (this is only valid for instantaneous burns):

\Delta v_{1}={\sqrt {\frac {\mu }{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right)

,

\Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{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}+\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}{\frac {1}{2}}\end{matrix}}{\sqrt {\frac {4\pi ^{2}a_{H}^{3}}{\mu }}}=\pi {\sqrt {\frac {(r_{1}+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 0.68 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).

Worst case, maximum delta-v

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 orbit is 15.6 (positive root of $x^{3}-15x^{2}-9x-1=0$ ) times as large as that of the original orbit. For higher target orbits 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.^{[citation needed]}

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 "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.^{[citation needed]}

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 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 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).

Interplanetary Transport Network

In 1997,^[1] 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.

References

^ 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
Bate, R.R., Mueller, D.D., White, J.E., (1971). Fundamentals of Astrodynamics. Dover Publications, New York. ISBN 978-0486600611.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
Vallado, D. A. (2001). Fundamentals of Astrodynamics and Applications, 2nd Edition. Springer. ISBN 978-0792369035.
Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics & Ast, Washington, DC. ISBN 978-1563473425.

External links

ORBITAL MECHANICS (Rocket and Space Technology)

[1] Lo, M., S. Ross, Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System, JPL IOM 312/97, 1997.

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