Hohmann transfer orbit

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Hohmann transfer orbit, labelled 2, from a low orbit (1) to a higher orbit (3).

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

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).[1] Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets.[citation needed]

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 (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.

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):

$E=\begin{matrix}\frac{1}{2}\end{matrix} m v^2 - \frac{G M m}{r} = \frac{-G M m}{2 a} .\,$

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

$v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)$
where:
• $v \,\!$ is the speed of an orbiting body
• $\mu = GM\,\!$ is the standard gravitational parameter of the primary body, assuming $M+m$ is not significantly bigger than $M$ (which makes $v_M \ll v$)
• $r \,\!$ 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, under the assumption of instantaneous impulses:

$\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1 \right)$,

to enter the elliptical orbit at $r=r_1$ from the $r_1$ circular orbit

$\Delta v_2 = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1+r_2}}\,\,\right)$,

to leave the elliptical orbit at $r=r_2$ to the $r_2$ circular orbit 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:

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

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

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

$\omega_2 = \sqrt{\frac{\mu}{r_2^3}}$

angular alignment α (in radians) at the time of start between the source object and the target object shall be

$\alpha = \pi - \omega_2 t_H = \pi\left(1 - \frac{1}{2\sqrt{2}}\sqrt{\left(\frac{r_1}{r_2}+1\right)^3}\,\right)$

Example

Total energy balance during a Hohmann transfer between two circular orbits with first radius $r_p$ and second radius $r_a$

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.

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 Δv for the two burns are thus 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s.[1]

It is interesting to note that this is greater than the Δv required for an escape orbit: 10.93 − 7.73 = 3.20 km/s. Applying a Δv at the LEO of only 0.78 km/s more (3.20−2.42) would give the rocket the escape speed, which is less than the Δv of 1.46 km/s required to circularize the geosynchronous orbit. This illustrates that at large speeds the same Δv provides more specific orbital energy, and energy increase is maximized if one spends the Δv 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

As the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not maximized when the destination is at infinity. Escape speed is √2 times orbital speed, so the Δv required to escape is √2−1 (41.4%) of the orbital speed.

The Δv required is greatest (53.0% of smaller orbital speed) if the radius of the larger orbit is 15.58 times that of the smaller orbit.[2] This number is the positive root of x³−15x²-9x−1=0. For higher orbit ratios the Δv required for the second burn decreases faster than the first increases.[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, but fortuitously, much less delta-v is required due to Oberth effect.

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 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 reach the Hohmann transfer orbit (around the Sun). Because the rocket engine is able 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.

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. This capture burn should optimally be done at low altitude to also make best use of 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.

Hohmann transfer versus low thrust orbits

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 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.[citation needed]

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.[3] 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. The gravity assist provides "free" delta-v without the use of the propulsion systems.

Application

One of the most successful application of this maneuver was in The Mars Orbiter Mission (MOM), also called Mangalyaan,[4] which is a spacecraft orbiting Mars since 24 September 2014. It was launched on 5 November 2013 by the Indian Space Research Organisation (ISRO). It is India's first interplanetary mission and ISRO has become the fourth space agency to reach Mars, after the Soviet space program, NASA, and the European Space Agency. It is also the first nation to reach Mars orbit on its first attempt, and the first Asian nation to do so. The efficient application of this orbital maneuver has also contributed in significantly reducing the cost of the overall mission.