# Hohmann transfer orbit

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 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, $\Delta v_{1}$ and $\Delta v_{2}$ , can be solved for as:

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

• $M$ is the mass of the planet,
• $G$ is the universal gravitational constant, and
• $r_{1}$ and $r_{2}$ 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.