Kepler's Second Law Simulator

Kepler's Second Law

Eccentricity 0.55

Orbital period 10 s

Sweep interval 1/12 orbit

Orbital speed

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Radius (r)

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Wedge area

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Position

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How it works

A planet on an elliptical orbit does not move at constant speed: it speeds up near perihelion (closest approach to the focus, where the star sits) and slows down near aphelion (farthest point). Kepler's second law says the line joining planet and star sweeps out equal areas in equal times — so a thin, long wedge swept quickly near perihelion has the same area as a short, fat wedge swept slowly near aphelion. This simulator computes the true anomaly by solving Kepler's equation each frame, so the planet's angular position — not just its speed — is physically exact for the chosen eccentricity.

M = 2π·t/T           (mean anomaly)
M = E − e·sin(E)      (Kepler's equation, solved for E)
tan(ν/2) = √((1+e)/(1−e))·tan(E/2)
r = a(1 − e·cos(E))
dA/dt = ½·r²·(dν/dt) = constant

Eccentricity (e) — 0 is a perfect circle; higher values stretch the ellipse and make the speed difference between perihelion and aphelion more dramatic.
Orbital period — how long one full revolution takes; the sweep rate dA/dt scales with 1/period so total area per orbit stays π·a·b.
Sweep interval — how finely the orbit is divided into wedges; each wedge is shaded and its area is measured live to show they are all equal despite the varying speed.
Area wedges — toggle the shaded equal-area sectors to see the geometric proof of the law directly on the ellipse.

Real-world relevance: this is the same equal-area law Kepler derived from Tycho Brahe's naked-eye observations of Mars in 1609, and it applies unchanged to every gravitationally bound orbit — comets, moons, exoplanets and satellites alike — because it follows directly from conservation of angular momentum, not from any property specific to planets.

Planet Star (focus) Swept wedge

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