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 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, $\Delta v_{1}$ 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.