Hohmann Transfer Orbits Simulation

About this experiment

A spacecraft cannot fly straight to Mars. It coasts around the Sun on an ellipse that just touches the orbit of Earth at one end and the orbit of Mars at the other: a Hohmann transfer, the cheapest route in fuel between two circular orbits. The craft takes months to get there, so it must be launched when Mars is the right distance ahead, otherwise the planet will not be there when the craft arrives. Pick two planets, run time forward, find the next launch window and launch.

Learning goals

Describe a Hohmann transfer orbit as half of an ellipse touching both planetary orbits
Use Kepler's third law to find the transfer time
Explain why launch windows exist and why they repeat after one synodic period
Calculate the two velocity changes needed for a transfer
Compare transfers to inner and outer planets

How to use it

1Launch to Mars straight away on 1 Jan 2026. Where is Mars when the craft reaches its orbit?
2Press Reset, then Next launch window. Compare the target lead angle with the needed angle, then launch.
3Read the flight time and check it with Kepler: half the period of an orbit whose semi-major axis is the mean of the two radii.
4Jump to the next window after the one you used. How many days later is it? Compare with the readout Windows repeat every.
5Choose Venus as the target. Is the needed lead angle positive or negative? Why?
6Try Jupiter and Saturn. How do the flight time and the Δv change?

The model

a_{t} = \frac{r _{1} + r _{2}}{2}, t = π \frac{a _{t}^{3}}{GM}

Transfer time: half an orbit on the ellipse, from Kepler's third law

Δ v_{1} = \frac{GM}{r _{1}} (\frac{2 r _{2}}{r _{1} + r _{2}} - 1)

First burn: speed up from the circular orbit of the departure planet onto the ellipse

Δ v_{2} = \frac{GM}{r _{2}} (1 - \frac{2 r _{1}}{r _{1} + r _{2}})

Second burn: match the circular orbit of the target

θ = π - ω_{2} t

Lead angle at launch: the target moves ω₂t while the craft sweeps half a turn

\frac{1}{T _{syn}} = \frac{1}{T _{1}} - \frac{1}{T _{2}}

Synodic period: how often the planets return to the same arrangement

Assumptions

All planets move on circular orbits in the same plane at constant speed. Real orbits are slightly elliptical and tilted, so real windows and flight times differ by a few weeks.
Planet positions are found from their mean positions on 1 Jan 2000 and their periods, so launch windows fall near the real dates (Earth to Mars: April 2001, July 2005, August 2007).
The gravity of the planets themselves is ignored. Δv is the change from the orbital speed of the planet, not from the ground, so escaping from the planet and landing are not included.
The burns are instant, and the craft feels only the gravity of the Sun between them.

Try this in class

Q1Why does a Mars mission have to wait about 26 months for the next launch window?
Q2A Hohmann transfer to Saturn takes about 6 years. Why do real missions use flybys of other planets instead?
Q3Why is the transfer to an inner planet such as Venus also a Hohmann transfer, even though the craft has to slow down?
Q4After arriving at Mars by a Hohmann transfer, astronauts want to come home. Why must they wait at Mars for more than a year?
Q5Why is the transfer time to Mars much shorter than half the orbital period of Mars?

Frequently asked questions

Is the Hohmann transfer the fastest way to reach another planet?

No, it is the cheapest in fuel for two circular orbits, but not the fastest. A bigger first burn gives a faster, more elongated orbit that crosses the target orbit earlier, at the cost of more fuel.

Why does the craft not fly in a straight line?

Between the burns, the engine is off and the only force is the gravity of the Sun. Any object moving only under the gravity of the Sun follows an ellipse, so the craft coasts along one, just like a planet.

Why is the needed lead angle negative for Venus?

Venus moves faster than Earth. During the transfer it travels more than half an orbit, so at launch Venus has to be behind Earth, not ahead of it.