Hohmann Transfer

What
The Hohmann Transfer describes the most energy efficient way for interplanetary travel.

In this example, we are trying to reach Mars from Earth. We will accomplish this using two rocket stages. Once, at the point labeled "Earth at departure" and another at "Mars at arrival"

The first rocket boost wil bring the rocketship into the transfer ellipse and the second will bring the rocketship into mar's orbit for landing

First Transfer
The velocity change required to propell the rocketship into the transfer ellipse is calculated as follows:

Velocity of the rocketship in Earth's orbit:

$$\frac {1}{2} m v _1 ^2 - \frac {GMm}{r_1} = - \frac {GMm}{2r_1}~,\textrm{total~energy~of~initial~orbit}$$

$$v_1 = \sqrt {\frac {GM}{r_1}}$$

Velocity of the rocketship at perihelion assuming elliptical orbit:

$$\frac {1}{2} m v _2 ^2 - \frac {GMm}{r_1} = - \frac {GMm}{2a}$$

From the diagram, we see that the semi major axis of the transfer ellipse can be defined as:

$$a = \frac {r_1 + r_2}{2}$$

Therefore, we can update our energy equation to:

$$\frac {1}{2} m v_2 ^2 - \frac {GMm}{r_1} = - \frac {GMm}{r_1 + r_2}$$

$$v_2 = \sqrt {\frac {2GM}{r_1} \frac {r_2}{r_1 + r_2}}$$

$$\Delta v = v_2 - v_1$$

Second Transfer
The second transfer is computed in exactly the same way, so the calculations will be ommitted for brevity.