Kepler's Laws - Cosmic Playground

Kepler's Laws Sandbox

Explore orbital mechanics: empirical patterns and physical laws.

Controls

Kepler mode follows an analytic ellipse at fixed period. Newton mode emphasizes force vectors and conservation readouts.

Concept focus

Keyboard shortcuts Space, arrows, presets

Space: Play/Pause
Home / End: Perihelion / Aphelion
Left / Right: Move along orbit (Shift = fine)
K / N: Kepler / Newton mode
1-6: Presets

Semi-major axis $a$

1.00 AU

Log scale from 0.3 AU to 40 AU.

$P \propto a^{3/2}$ (for fixed $M$).

Eccentricity $e$

0.017

0 (circle) to 0.99.

Mean anomaly $M$ (time proxy)

0.00 / 1.00 yr

Speed (yr/s)

Presets

Solar system

Extreme orbits

Kepler 2: Equal areas

Show equal-area slices

Hypothesis: arc length changes, but area swept in equal times stays nearly constant.

Other overlays

Foci Apsides

Readouts

Distance $r$

1.00

Speed $v$

29.8

km/s

Acceleration

5.93

mm/s^2

Period $P$

1.00

Conservation drift (relative to reset)

$\Delta\varepsilon$~0 (analytic)

$\Delta h$~0 (analytic)

Conservation laws energy + angular momentum

Kinetic ($v^2/2$)

AU^2/yr^2

Potential ($-\mu/r$)

AU^2/yr^2

Total $\varepsilon$

AU^2/yr^2

Angular momentum $h$

AU^2/yr

Areal velocity ($dA/dt$)

AU^2/yr

What to notice 3 prompts

When $e$ > 0, the planet moves fastest near perihelion (closest point).
Equal-area slices stay similar in area even when arc lengths differ.
Bigger $a$ means a much longer period (Kepler 3).

Model note (teaching model) AU / yr / M_sun

This simulation is a simplified, planar (2D) two-body model.
It ignores perturbations and treats the planet as negligible-mass in Newton mode.
The animation speed is a teaching scale (years per second), not real time.
Teaching normalization: $G = 4\pi^2\,\frac{\mathrm{AU}^3}{\mathrm{yr}^2\,M_{\odot}}$.
Kepler 3: $P^2 = \frac{a^3}{M}$.
The time slider advances mean anomaly uniformly; position is solved from Kepler's equation.