Kepler's laws of planetary motion
Before anyone understood why planets move, Johannes Kepler stared at decades of painstaking observations of Mars and asked a simpler question: what shape and rhythm do the planets actually trace? Working in the early 1600s, with no telescope of the modern kind and no theory of gravity, he found three plain patterns that the planets obey with stubborn precision. They are the rules of the cosmic dance, written down before anyone knew the music.
The first law says each planet moves on an ellipse — a gently squashed circle — with the Sun sitting at one of the two focus points, not at the center. The second law says a planet sweeps out equal areas in equal times, which is just a vivid way of saying it speeds up when it is near the Sun and slows down when it is far. The third law links a planet's orbital period P to the size of its orbit (its semi-major axis a): P squared is proportional to a cubed. In tidy units, if you measure a in astronomical units (the Earth–Sun distance) and P in years, then P^2 = a^3. Earth: 1^2 = 1^3. Mars, at a = 1.52 AU, takes about 1.88 years, and indeed 1.88^2 is close to 1.52^3.
Kepler's laws are purely empirical — descriptions, not explanations. They matter because Isaac Newton later showed they all fall out of a single deeper idea, his law of universal gravitation, turning three observed coincidences into one law of nature. The same three laws govern moons around planets, stars in binary systems, and exoplanets around distant suns, which is exactly how we weigh those distant worlds.
Halley's Comet swings in on a very stretched ellipse: near the Sun it whips around in days, but at its far point beyond Neptune it crawls — equal areas in equal times made visible by an object that takes about 76 years per loop.
The same area-sweeping rule that holds for sedate planets is dramatic for a comet on a steep ellipse.
A common slip is to picture the Sun at the center of the ellipse; it sits at a focus, which is offset from the center, and for nearly-circular orbits the two foci are close enough that the ellipse looks almost round.