Kepler Orbit Explorer

Set up one orbit and watch all three of Kepler's laws at once. Choose its size (semi-major axis) and how stretched it is (eccentricity), and the body traces an ellipse with the Sun at one focus, not the centre, which is the first law. As it goes, the line from the Sun sweeps out equal areas in equal time: the wedges shaded here all have the same area, so the body must race when it is close to the Sun and crawl when it is far, the second law made visible. And the period follows the size on a precise schedule, period squared proportional to size cubed, the third law, plotted below. The motion is the real inverse-square force integrated with an energy-conserving symplectic scheme. Turn the eccentricity up for a comet-like plunge, or switch the inner planets on for context and to populate the third-law line.

Figure 1. Kepler orbit explorer. Top: a controllable orbit (semi-major axis a, eccentricity e) with the Sun at a focus; the radius vector sweeps equal-area wedges in equal time (second law), and the inner planets can be toggled on as context. Bottom: Kepler's third law, period squared against semi-major axis cubed, on one straight line. Method: symplectic Verlet, GM = 1 units; equal-time wedges from Kepler's equation; T = 2 pi a^(3/2).

semi-major a1.50

eccentricity e0.60

planetshidden

speed1.0

WHAT TO TRY

Crank the eccentricity up: the wedges near the Sun get short and fat, the ones far away long and thin, yet every wedge has the same area. The body must move fast when close and slow when far.
Watch the radius vector and the speed readout: it peaks at perihelion (closest) and bottoms out at aphelion (farthest).
Slide the semi-major axis and watch your point climb the period-size line in the lower plot, always landing on it (third law).
Toggle the inner planets on for context and to fill in the third-law line; toggle them off to focus on the equal-area sweep.