Astronomical units and Kepler’s laws

Johannes Kepler, using Tycho Brahe’s observations, stated the first two laws in 1609 and the third in 1619. First law: each planet moves around the Sun on an ellipse with the Sun at one focus; if the semi-major axis is a and the eccentricity e, the nearest distance (perihelion) is q = a(1 – e) and the farthest (aphelion) is Q = a(1 + e). Second law: the line from the Sun to a planet sweeps out equal areas in equal times, so the planet moves fast near perihelion and slowly near aphelion. Third law: the squares of the planets’ sidereal periods are proportional to the cubes of the semi-major axes of their orbits, T₁² / T₂² = a₁³ / a₂³; with T in years and a in AU, T² = a³ for bodies orbiting the Sun. The Earth’s orbit is nearly circular (e ≈ 0.017): perihelion ≈ 3 January (147 million km), aphelion ≈ 4 July (152 million km), so the seasons come from the tilt of the axis, not from distance.