A Hohmann transfer calculator that shows its working

The problem

Orbital mechanics is taught largely through textbook equations and decades-old charts, with little to see. A Hohmann transfer, the cheapest two-burn move between coplanar circular orbits, is one of the first maneuvers students learn and a good test case for an interactive tool.

The tool

The calculator runs in the browser in HTML and JavaScript. A student chooses the inputs and outputs, so the same page answers different homework questions: initial and final orbit altitudes, spacecraft mass with and without propellant, specific impulse or exhaust velocity, and the central body. It returns the burns and the propellant needed, plots the transfer, and lays out each intermediate step.

The mechanics

For circular orbits of radius $r_1$ and $r_2$ around a body with gravitational parameter $\mu$, the transfer ellipse touches both, so its semi-major axis is their average, and the vis-viva equation gives the speed anywhere on it:

$$a_t = \frac{r_1 + r_2}{2}, \qquad v = \sqrt{\mu \left( \frac{2}{r} - \frac{1}{a} \right)}$$

Each burn is the difference between the circular speed and the transfer speed at that radius:

$$\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right), \qquad \Delta v_2 = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1 + r_2}} \right)$$

The coast takes half the transfer orbit’s period, and the rocket equation turns the total into propellant for an engine of specific impulse $I_{sp}$:

$$t_{\text{transfer}} = \pi \sqrt{\frac{a_t^3}{\mu}}, \qquad m_{\text{prop}} = m_0 \left( 1 - e^{-\Delta v / (I_{sp} g_0)} \right)$$

Try it

I rebuilt the core of the calculator for this site. Ticks along the transfer mark equal intervals of time, so they crowd together near apoapsis, where the spacecraft is slowest.

Rings: Earth radii Ticks every 30 min

Circular orbits, altitude in km

From

To

LEO to GEO LEO to GPS GEO to LEO

Δv₁

2.397 km/s

Δv₂

1.456 km/s

Total

3.854 km/s

Time of flight

5.29 h

Transfer a

24,471 km

Transfer e

0.7230

The study

We ran a formative usability study with three undergraduate and three graduate aerospace students. An “Experiment Mode” in the page logged time on task and clicks; afterwards each participant completed the NASA-TLX workload survey and the System Usability Scale.

Click heat maps showed the graduate students spending more of their time in the step-by-step panel than the undergraduates. Participants’ feedback pointed most often to clearer labels on the plot.