Kepler's Laws of Planetary Motion
In 1609, Johannes Kepler published his first two laws of planetary motion in Astronomia Nova, revolutionizing our understanding of the Solar System. His third law followed in 1619. Together, they describe how planets move in elliptical orbits around the Sun, with elegant mathematical precision that Isaac Newton later derived from his law of universal gravitation.
The orbit of any planet is fully described by two numbers: the semi-major axis a (the average distance) and the eccentricity e (how elongated the ellipse is). From these, we can compute everything — the orbital period, the distance at closest approach (perihelion), the farthest point (aphelion), and the velocity at any position.
The Equal-Area Law
Kepler's second law states that a line drawn from the Sun to a planet sweeps out equal areas in equal intervals of time. This has a profound physical consequence: planets move faster when they are closer to the Sun and slower when they are farther away. The simulator visualizes this by shading two wedge-shaped areas — one near perihelion, one near aphelion — that cover equal areas despite their very different shapes.
From Kepler to Newton
Newton showed that Kepler's laws are a natural consequence of an inverse-square gravitational force. The vis-viva equation v = √(GM(2/r − 1/a)) gives the orbital speed at any point, unifying all three of Kepler's laws into a single expression. This equation is fundamental to modern astrodynamics and is used to plan every interplanetary mission.
Eccentricity in the Solar System
Earth's orbit is nearly circular (e ≈ 0.017), while Mercury's is much more eccentric (e ≈ 0.206). Comets like Halley's comet have eccentricities above 0.96, producing dramatically elongated orbits that plunge close to the Sun before retreating to the outer Solar System. The relationship between shape and speed is one of the most beautiful results in classical mechanics.