# Hohmann transfer orbit

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In orbital mechanics, a Hohmann transfer orbit moves a spacecraft between orbiting heights. It is the most fuel-efficient method to do so, because the spacecraft is not trying to escape the planet's gravity, using an elliptical orbit for the transfer.

A simulation of a Hohmann transfer orbit

A ship using this would have to apply two velocities, one to enter the elliptical orbit, and one to enter the second orbit.

## Calculation

Assuming the mass of the spacecraft is much lower than that of the orbiting planet, the two velocities, ${\displaystyle \Delta v_{1}}$ and ${\displaystyle \Delta v_{2}}$, can be solved for as:

${\displaystyle \Delta v_{1}={\sqrt {\frac {MG}{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right),}$
${\displaystyle \Delta v_{2}={\sqrt {\frac {MG}{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\,\,\right),}$
where

• ${\displaystyle M}$ is the mass of the planet,
• ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are the initial and final distances from the center of the planet.

## Applications

• Satellites can be moved into their proper height using a Hohmann transfer orbit.
• A lunar transfer orbit (LTO) is used to reach the moon.
• The Interplanetary Transport Network uses more than one body and requires lower velocity changes, and thus less fuel.