Kepler's laws of planetary motion
/ KEP-ler /
Before Newton explained why planets move, Johannes Kepler figured out how they move, purely by staring at decades of superbly careful naked-eye observations left by Tycho Brahe. Out of that mountain of numbers he distilled three simple, exact rules. They are astonishing because they came before any theory of gravity, discovered by pattern-hunting alone.
The first law says each planet travels on an ellipse (a stretched circle) with the Sun at one focus, not at the centre. The second law, the law of equal areas, says the line from the Sun to a planet sweeps out equal areas in equal times; since that line is short and must sweep fast when the planet is near the Sun, planets speed up at their closest approach and slow down when far, which is really just conservation of angular momentum in disguise. The third law says the square of a planet's orbital period is proportional to the cube of its average distance: T^2 = (4 pi^2 / G M) a^3, where T is the period, a is the semi-major axis, and M is the Sun's mass. Bigger orbits take much longer.
Newton later showed all three laws fall straight out of his law of gravitation plus his laws of motion, turning Kepler's patterns into consequences of a single force. The laws work for any two-body orbit, not just planets: moons around planets, and artificial satellites around Earth, obey exactly the same rules with M being the mass of whatever they orbit.
Kepler's third law lets you weigh the Sun: knowing Earth's period (1 year) and average distance (1.5 x 10^11 m) and solving T^2 = 4 pi^2 a^3 / (G M) for M gives about 2 x 10^30 kg, the Sun's mass.
One orbit's period and size, plus Newton's gravity, are enough to measure the mass of the body being orbited.
The Sun sits at one focus of the ellipse, never at the centre; the equal-area second law is simply conservation of angular momentum.