Kepler's laws

For thousands of years people watched the planets wander across the sky. The Danish astronomer Tycho Brahe measured their positions very carefully, and Johannes Kepler found three rules hidden in those numbers. Later, Newton showed that his law of gravitation explains all three.

First law (law of orbits): every planet moves in an ellipse, with the Sun at one focus (an ellipse has two). The nearest point to the Sun is the perihelion P, the farthest the aphelion A. The semi-major axis a is half the long distance PA, and it serves as the planet's average distance.

Second law (law of areas): the line from the Sun to the planet sweeps out equal areas in equal times. In other words, its areal velocity is constant.

Why? Gravity always pulls straight towards the Sun, so it exerts no torque about the Sun, and the planet's angular momentum is conserved. The area swept per second is L/2m, so it stays fixed.

The result: a planet moves fastest at perihelion and slowest at aphelion, with r_P v_P = r_A v_A. Example: a comet passes 0.5 AU from the Sun at 60 km/s and goes out to 30 AU. There it crawls along at 60 × 0.5/30 = 1 km/s.

Third law (law of periods): the square of the period is proportional to the cube of the semi-major axis, T² ∝ a³, with the same constant for every planet of the Sun.

Example: a planet four times as far from the Sun as the earth has T² = 4³ = 64 earth-years², so its year lasts 8 earth years. Bigger orbits mean much longer years.