Kepler's laws explained
Kepler spent years fitting the recorded positions of Mars before he abandoned the age-old circle and settled on the ellipse. His three laws tie together an orbit's shape, the way a planet speeds up and slows down along it, and how long one lap takes. The first law fixes the shape, the second the changing speed, and the third the period.
The first law: an orbit is an ellipse with the Sun at one of its two foci. Drag the eccentricity e up from 0 and the gold orbit stretches sideways out of a circle; the Sun stays at the right focus while the empty focus slides off to the left. The green dot nearest the Sun is perihelion, at r = a(1−e), and the orange dot farthest away is aphelion, at r = a(1+e). The semi-major axis a is held fixed here, so the readout is simply the distance shortened or lengthened by e.
The second law: the line from the Sun to the planet sweeps out equal areas in equal times. The gold wedge always spans the same interval of time. Move it with the slider or press play, and the swept-area value never changes while the arc length shown in blue does. Near perihelion the wedge turns short and wide and covers a long arc; near aphelion it grows long and thin over a short one. The equal-area rule is conservation of angular momentum, restated in terms of area.
The third law: the square of the orbital period is proportional to the cube of the semi-major axis. Measure the period in years and the axis in astronomical units and even the constant drops out, leaving T² = a³. Drag a and the gold dot slides along the gray curve while the readout T²/a³ holds the same value everywhere. The green ring marks Earth. Beyond Earth the curve climbs steeply, so the farther out a planet orbits, the far longer one lap takes.
Read in terms of speed, the equal-area law says a planet moves faster the closer it is to the Sun. The slider's true anomaly ν is the planet's angle measured from perihelion, and the length of the blue arrow is its speed at that point. The arrow is longest at perihelion and shortest at aphelion, and the two speeds stand in the ratio (1+e)/(1−e). For an orbit as eccentric as this one, they differ by several times. The equation that links position and speed in a single line, the vis-viva equation v² = μ(2/r − 1/a), comes in the next lesson.
Now all three laws share one screen. Press play and the planet runs its ellipse at true Kepler speed, laying down one wedge of equal area each time an equal interval passes. The wedges near perihelion are short and wide and those near aphelion long and thin, yet every one has the same area. That is why the time and area percentages below never disagree. Speed the playback up or down, or send the planet back to perihelion with Reset, and the two numbers still move together.