Hohmann transfer orbit

The Hohmann maneuver often uses the lowest possible amount of impulse (which consumes a proportional amount of delta-v, and hence propellant) to accomplish the transfer, but requires a relatively longer travel time than higher-impulse transfers.

In some cases where one orbit is much larger than the other, a bi-elliptic transfer can use even less impulse, at the cost of even greater travel time.

The maneuver was named after Walter Hohmann, the German scientist who published a description of it in his 1925 book Die Erreichbarkeit der Himmelskörper (The Attainability of Celestial Bodies).

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

Space missions using a Hohmann transfer must wait for this required alignment to occur, which opens a launch window.

When transfer is performed between orbits close to celestial bodies with significant gravitation, much less delta-v is usually required, as the Oberth effect may be employed for the burns.

The transfer orbit (yellow, labeled 2 on diagram) is initiated by firing the spacecraft's engine to add energy and raise the apoapsis.

In the context of Earth and the Solar System, this includes any object which orbits the Sun.

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

are respectively the radii of the departure and arrival circular orbits; the smaller (greater) of

This illustrates the Oberth effect that at large speeds the same Δv provides more specific orbital energy, and energy increase is maximized if one spends the Δv as quickly as possible, rather than spending some, being decelerated by gravity, and then spending some more to overcome the deceleration (of course, the objective of a Hohmann transfer orbit is different).

As the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not the greatest when the destination radius is infinite.

For higher orbit ratios the Δv required for the second burn decreases faster than the first increases.

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 get into the Hohmann transfer orbit (around the Sun).

Because the rocket engine is able to 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.

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

This requirement for alignment gives rise to the concept of launch windows.

In this table, the column labeled "Δv to enter Hohmann orbit from Earth's orbit" gives the change from Earth's velocity to the velocity needed to get on a Hohmann ellipse whose other end will be at the desired distance from the Sun.

This is obtained by adding to the specific kinetic energy the square of the escape velocity (10.9 km/s) from this height.

[11] While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial semi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen.

[12] The idea of the bi-elliptical transfer trajectory was first[citation needed] published by Ary Sternfeld in 1934.

[13] 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 greater than the two-impulse transfer orbit[14] and takes longer to complete.

If only low-thrust maneuvers are planned on a mission, then continuously firing a low-thrust, but very high-efficiency engine might generate a higher delta-v and at the same time use less propellant than a conventional chemical rocket engine.

Going from one circular orbit to another by gradually changing the radius simply requires the same delta-v as the difference between the two speeds.

The amount of propellant mass used measures the efficiency of the maneuver plus the hardware employed for it.

This method however takes much longer to achieve due to the low thrust injected into the orbit.

The Interplanetary Transport Network is able to achieve the use of less propulsive delta-v by employing gravity assist from the planets.