Hohmann transfer orbit

In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that, under common astrodynamic assumptions, transfers a spacecraft from one circular orbit to another using two impulses from your engine. The name comes from the German scientist Walter Hohmann who published his theory in 1925.

Explanation

Hohmann transfer orbit (in yellow).

The Hohmann transfer orbit is one half of an elliptical orbit that touches both the initial orbit you want to leave (in green on the diagram) and the final orbit you want to reach (in red on the diagram). The transfer orbit (in yellow in the diagram) is initiated by firing the spacecraft's engine to accelerate it creating an elliptical orbit; this adds energy to the orbit of the spacecraft. When the ship reaches the final orbit, its orbital speed must be increased again to make a new circular orbit; the motor accelerates again to reach the necessary speed.

Hohmann transfer orbit theory relies on instantaneous velocity changes to create circular orbits, so spacecraft using a Hohmann transfer orbit will generally use high-thrust engines to reduce the amount of fuel additional. Low thrust engines can approximate a Hohmann transfer orbit, creating a gradually elongating circular orbit using the engine in a controlled manner. This requires up to 141% higher delta-v than the two-pulse system and takes longer to complete.

The Hohmann transfer orbit also works to take a ship from a higher to a lower orbit. In this case, the ship's engines work in the opposite direction of its trajectory, slowing the ship and causing it to fall into a lower-energy elliptical orbit. The engine then runs a second time to slow the ship's acceleration toward a circular orbit.

Although the Hohmann transfer orbit is almost always the cheapest method of getting from one circular orbit to another, in some situations where the semimajor axis of the final orbit is larger than the semimajor axis of the initial orbit in an order of twelve, it may be more advantageous to use a bi-elliptic transfer.

In Soviet works, such as Pionery Raketnoi Tekhniki, the term Hohmann-Vetchinkin transfer orbit is sometimes used, citing the mathematician Vladimir Vetchinkin who introduced the concept of elliptic transfer in lectures on the interplanetary journey between 1921 and 1925.

Calculation

For a small body m{displaystyle m} orbiting around another much greater M{displaystyle M}, like for example a satellite orbiting the Earth, the total energy of the orbiting body is simply the sum of its kinetic energy and its potential energy, and this total energy E{displaystyle E} equals half of the potential energy at the mid distance point in orbit a{displaystyle a} = larger semage:

E=12mv2− − GMmr=− − GMm2a{displaystyle E={frac {1}{2}}mv^{2}-{frac {GMm}{r}}{frac {-GMm}{2a}}}}{,}

Solving the equation for velocity in the orbital energy conservation equation,

v2=μ μ (2r− − 1a){displaystyle v^{2}=mu left({frac {2}{r}}}-{frac {1}{a}}}right)}

Where v{displaystyle v,!} is the speed of an orbiting body,
μ μ =GM{displaystyle mu =GM,!} is the standard gravitational parameter of the main body,
r{displaystyle r,!} is the distance from the orbiting body to the main and
a{displaystyle a,!} is the major semieje of the orbit m{displaystyle m} around M{displaystyle M}.

Therefore, the delta-v needed for a Hohmann transfer is,

Δ Δ vP=μ μ r1(2r2r1+r2− − 1){displaystyle Delta v_{P}={sqrt {frac {mu }{r_{1}}}{left({sqrt {frac {2r_{2}}}{r_{1}}+r_{2}}}}}}}}{1right)} (for the delta-v in periaster).

Δ Δ vA=μ μ r2(1− − 2r1r1+r2){displaystyle Delta v_{A}={sqrt {frac {mu }{r_{2}}}}left(1-{sqrt {frac {2r_{1}}}{r_{1}+r_{2}}}}}}}{,right)}}} (for the delta-v in apoastro).

Where r1{displaystyle r_{1},} is the radius of the minor orbit and the periaster distance of the Hohmann transfer orbit and
r2{displaystyle r_{2},} is the radius of the larger orbit and the apoastro distance of the Hohmann transfer orbit.

Whether it is moving to a higher or lower orbit, by Kepler's third law, the time to make the transfer is:

tH=124π π 2aH3μ μ =π π (r1+r2)38μ μ {displaystyle t_{H}={begin{matrix}{frac {1}{2}}}{matrix}}}{sqrt {frac {4pi ^{2}{2}{3}{3}}{mu }}}}}}{pi {sqrt {frac {(r_{1}{1}{8}{3}{3}{c}{c}{3}{c}{3}{c}{c}{c}{c}{cd}{cd}{cd}{cd}{cd}}{cd}{cd}{cd}{cd}{cd}{cd}}{cd}{cd}{cd}{c)}{cd}{cd}{cd}{cd}{cd}{cd}{cd}{

Where aH{displaystyle a_{H},!} is the length of the larger semieje of the transfer orbit of Hohmann.

Example

For geostationary transfer orbit, r2{displaystyle r_{2}} = 42,164 km and as an example, r1{displaystyle r_{1}} = 6,678 km (an altitude of 300 km).

The velocity in the smaller circular orbit is 7.73 km/s and in the larger one it is 3.07 km/s. In the elliptical orbit the velocity varies from 10.15 km/s at the perigee and 1.61 km/s at the apogee.

The delta-v's are 10.15 - 7.73 = 2.42 km/s and 3.07 - 1.61 = 1.46 km/s, or a total of 3.88 km/s.

Compared to the delta-v of an escape orbit: 10.93 - 7.73 = 3.20 km/s. Applying a low Earth orbit delta-v of only 0.78 km/s more than would give the rocket at escape velocity, while a geostationary orbit delta-v of 1.46 km/s to reach escape velocity of this circular orbit. This illustrates that at higher speeds the same delta-v provides more specific orbital energy and the energy increment is maximized by expending the delta-v as soon as possible rather than using it twice.

Maximum delta-v

In a transfer orbit of Hohmann from a circular orbit to a larger one, in the case of a single central body, it costs a greater delta-v (53.6% of the original orbital velocity) if the radius of the final orbit is 15.6 (the positive root of the equation x3− − 15x2− − 9x− − 1=0{displaystyle x^{3}-15x^{2}-9x-1=0}Sometimes bigger than the initial orbit. For larger final orbits, delta-v decreases again and tends to 2− − 1{displaystyle {sqrt {2}}-1} sometimes the original orbital velocity (41.4%).

Use in interplanetary travel

When a spacecraft moves from one planet's orbit to another's, the situation becomes more complex. On a journey between Earth and Mars, the spacecraft would already have a certain speed associated with its orbit around the Earth, which is not needed when in transfer orbit around the Sun. At the other extreme, the spacecraft would need a speed to to orbit Mars, which will be less than the velocity needed to continue orbiting the Sun. Thus, the spacecraft must slow down to be caught by Martian gravity, and small amounts of thrust will be needed during its journey to fix the transfer. However, it is essential to know the alignment of the planets in their orbits, since the destination planet and the ship must be at the same point in their respective orbits around the Sun at the same moment.

A Hohmann transfer orbit will take a spacecraft from low Earth orbit (LEO) to geosynchronous orbit in about five hours (geostationary transfer orbit), from LEO to the Moon in five days, and from Earth to Mars in about 260 days. However, Hohmann transfers are very slow for longer distances, so gravity assist is often used to increase speed.

Interplanetary Transport Network

In 1997, a group of orbits known as the Interplanetary Transportation Network was published, providing low-energy, albeit slower, paths between different orbits that are not Hohmann transfer orbits.