Kepler's Three Laws: Elliptical Orbit and Equal Areas — 3D Lab Simulation
Change semi-major axis and eccentricity to see twelve equal-area sectors and T² = a³
Theoretical Foundation & Historical Context
Using Tycho Brahe's observations of Mars, Kepler published the first two laws in 1609 and the third in 1619. A planet moves on an ellipse with the Sun at one focus; the line from Sun to planet sweeps equal areas in equal times; and the square of the period is proportional to the cube of the semi-major axis. The simulation solves Kepler's equation M = E − e·sin E to place the planet at every instant.
Law: Elliptical orbits, the law of areas and the harmonic law | Scientist: Johannes Kepler | Year: 1619
Key Learning Objectives & Competencies
- ✓ Describe how a and e shape the orbit and locate the Sun at the focus.
- ✓ Verify that equal times sweep equal areas even though the speed changes along the orbit.
- ✓ Use T² = a³ to predict the period of any orbit around the Sun.
Mathematical Formulations & Differential Equations
r = a(1 − e²) / (1 + e·cos θ)
Perihelion a(1 − e), aphelion a(1 + e); e = 0 is a circle
dA/dt = L / 2m = π·a·b / T = const
The planet is fastest at perihelion and slowest at aphelion
T² = a³ (năm, AU); M = E − e·sin E
With GM_sun = 4π² AU³/yr²; speed from vis-viva v² = GM(2/r − 1/a)
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| a | Semi-major axis (AU) | — | — |
| e | Eccentricity of the ellipse (0 to below 1) | — | — |
| T | Orbital period (years) | — | — |
| M | Mean anomaly (rad), grows uniformly with time | — | — |
| E | Eccentric anomaly (rad), solution of Kepler's equation | — | — |
Real-World & Industrial Engineering Applications
Planning spacecraft transfers and satellite orbits.
Estimating star and exoplanet masses from orbital periods.
Predicting comet returns such as Halley's, with e near 0.97.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why is the planet faster at perihelion?
To keep the swept area constant, a smaller radius requires a higher speed; this is equivalent to conservation of angular momentum.
Is the Sun at the centre of the ellipse?
No. The Sun sits at one of the two foci; the other focus is empty. It is offset by a·e from the centre.
Which bodies obey the third law?
Everything orbiting the Sun obeys T² = a³ in years and AU; for another central body the constant of proportionality scales with its mass.
Related Simulations & Experiments
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.