1. Real‑world motivation

Orbit changes are expensive in terms of propellant, so mission designers seek maneuvers that minimize the required delta‑v. The Hohmann transfer, proposed in 1925 by Walter Hohmann, is the most fuel‑efficient way to move between two coplanar circular orbits using two impulsive burns. The first burn lifts the spacecraft onto an elliptical path whose apogee touches the destination orbit, and a second burn at apogee circularizes the orbit. This calculator retains the classic purpose of determining the delta‑v and transfer time, but it also animates the journey on an HTML5 canvas. As parameters change, you watch a tiny spacecraft leave a lower blue circle, coast along an orange ellipse, and finally settle onto a higher ring. Energy bars display the tug of gravity versus the craft’s kinetic energy, while captions describe each phase for screen‑reader users. The result is an intuitive portrayal of orbital mechanics beyond static tables.

2. Variables and assumptions

The scenario assumes two circular, coplanar orbits about a much heavier central body. The initial orbit radius is $r_1$ and the target radius is $r_2$, both measured from the center of the primary. The gravitational parameter $\mu$ equals $\mathrm{GM}$ of the central body. The spacecraft mass is negligible, so the body remains fixed at the origin. Burns are treated as instantaneous velocity changes (impulses) with no finite thrust duration. Motion is confined to two dimensions and modeled with Newton’s inverse‑square law. Inputs are given in kilometers and seconds; the script converts to meters internally to maintain SI units. The time step $\mathrm{\Delta t}$ governs numerical integration accuracy and is clamped between 0.1 and 100 s.

3. Governing equations

The orbital speed in a circular orbit of radius $r$ is $\mathrm{v\_c}=\sqrt{\frac{\mu }{r}}$. For a Hohmann transfer the required burns are

$${\mathrm{\Delta v}}_1=\mathrm{v\_c}(\sqrt{\frac{2}{r_2}}-1)$$

$${\mathrm{\Delta v}}_2=\mathrm{v\_c\text{'}}(1-\sqrt{\frac{2}{r_1}})$$

where $\mathrm{v\_c}$ uses $r_1$ and $\mathrm{v\_c\text{'}}$ uses $r_2$. The transfer time equals half the period of the elliptical orbit with semi‑major axis $\frac{r_1}{r_2}2$:

$$t_{\text{transfer}}=\frac{\pi }{\sqrt{8}}\sqrt{\frac{{r_1+r_2}^3}{\mu }}$$

The canvas uses Newton’s law to compute acceleration during coast:

$$\frac{d\overrightarrow{v}}{dt}=-\frac{\mu \overrightarrow{r}}{r^3}$$

with position update $\frac{d\overrightarrow{r}}{dt}=\overrightarrow{v}$.

4. Numerical scheme

Although the analytic solution for a Keplerian orbit is well known, integrating the equations numerically emphasizes the link between forces and motion. The simulator employs a fourth‑order Runge–Kutta (RK4) method. Each step computes four acceleration estimates: $\mathrm{k\_1}$ at the start, $\mathrm{k\_2}$ and $\mathrm{k\_3}$ at midpoints, and $\mathrm{k\_4}$ at the end. The weighted average advances position and velocity. RK4 is stable for the modest time steps allowed, but extremely large $\mathrm{\Delta t}$ increases energy error. After the first burn is applied at $t=0$, the integrator evolves the spacecraft until the simulation time exceeds the transfer time, whereupon the second burn is applied. Energy bars display kinetic $\frac{1}{2}{v}^2$ and potential $-\frac{\mu }{r}$ components to monitor conservation.

5. Worked example

Consider raising a satellite from a 300 km‑altitude low Earth orbit to geostationary orbit. The initial radius is $r_1=6 771$ km, and the target radius is $r_2=42 164$ km. Earth’s gravitational parameter is $\mu =398 600$ km³/s². Enter these numbers and press Play. The output reports ${\mathrm{\Delta v}}_1\approx 2.42 km/s,$ {\mathrm{\Delta v}}_2\approx 1.46 km/s,\; total$ \mathrm{\Delta v}\approx 3.88$ km/s,\; and\; a\; transfer\; time\; of\; about\; 5.3 hours.\; On\; the\; canvas,\; the\; spacecraft\; leaves\; the\; lower\; circular\; path,\; follows\; the\; elliptical\; arc\; for\; half\; an\; orbit,\; then\; a\; second\; impulse\; adjusts\; the\; speed\; to\; match\; the\; high\; circular\; track.\; The\; energy\; bars\; rise\; during\; the\; first\; burn,\; remain\; nearly\; constant\; during\; coast\; (aside\; from\; numerical\; wiggles),\; and\; change\; again\; during\; the\; second\; burn.$$

6. Comparison table

The table below compares the baseline Earth maneuver with two alternatives.

r₁ (km)	r₂ (km)	Total Δv (km/s)	Transfer time (h)
6 771	42 164	3.88	5.3
6 771	20 000	2.42	2.3
10 000	20 000	1.41	3.6

Increasing the target radius increases both delta‑v and travel time. Starting from a higher initial orbit reduces the first burn and overall cost. The simulator reproduces each entry; you can verify by entering the values.

7. How to read the animation

The central yellow dot represents the massive body. Blue and gray circles show the initial and final orbits, while the orange ellipse traces the transfer path. The spacecraft is a small white triangle moving counter‑clockwise. When a burn occurs, the craft briefly flashes to indicate an impulse. The energy bars beneath the canvas use orange stripes for kinetic energy and blue stripes for the magnitude of potential energy. The caption updates with current radius, speed, and elapsed time; the hidden text fallback echoes this for accessibility. Keyboard users can focus the canvas and press the space bar to play or pause.

8. Limitations

The Hohmann transfer assumes coplanar, circular orbits and instantaneous burns. Real spacecraft experience finite burn times, gravitational perturbations, and often require plane changes. The simulation ignores shadowing, atmospheric drag near low altitudes, and non‑spherical gravity fields. For very low orbits the time step must be small to resolve rapid motion, while extremely high orbits may drift numerically if the step is too large. Despite these simplifications, the model captures the essential energy trade‑offs and timing of Hohmann transfers.

9. Suggested extensions

Potential enhancements include bi‑elliptic transfers, continuous low‑thrust spirals, or visualizing plane‑change burns. Adding a phase‑space diagram of radial versus tangential velocity would expose orbital invariants. Users might also toggle a grid of the vis‑viva energy curve or compare Hohmann delta‑v with the rocket equation using the Rocket Equation Calculator. Because the script exports trajectories, you can import the CSV into plotting software for further analysis.

10. References and related tools

Classical derivations appear in D. A. Vallado’s Fundamentals of Astrodynamics and Applications. Walter Hohmann’s original pamphlet, Die Erreichbarkeit der Himmelskörper, laid the groundwork for modern mission design. For further orbital exploration try the Orbital Period Simulator, the Orbital Velocity Calculator, or the Rocket Equation Calculator.