Kepler's Laws of Planetary Motion

Johannes Kepler analysed decades of the astronomer Tycho Brahe's painstakingly accurate naked-eye observations of the planet Mars and, from them, worked out three precise geometrical laws that every planet's motion around the Sun obeys.

Kepler's first law (the law of orbits). Every planet moves around the Sun in an elliptical orbit, with the Sun located at one of the two foci of the ellipse -- not at its centre. This alone overturned the older belief, held since antiquity, that heavenly bodies must move in perfect circles; an ellipse only reduces to a circle in the special case where both foci coincide.

Kepler's second law (the law of areas). The line joining a planet to the Sun (its radius vector) sweeps out equal areas in equal intervals of time, wherever the planet happens to be along its orbit. Because the orbit is an ellipse, the planet is sometimes much closer to the Sun (at perihelion) and sometimes much farther away (at aphelion); for the swept-out area in a given time interval to stay the same at both extremes, a planet close to the Sun must sweep through a much longer arc of its orbit than one far from the Sun does in the same time -- so the planet must be moving distinctly faster near perihelion and distinctly slower near aphelion (see the accompanying figure). As later sections of this chapter will show, this law is, in fact, nothing but the conservation of angular momentum for a planet moving under a force that always points exactly along the line joining it to the Sun -- a central force, which produces no torque about the Sun and therefore cannot change the planet's angular momentum about it.

Kepler's third law (the law of periods). The square of a planet's orbital period $T$ is directly proportional to the cube of the semi-major axis $a$ of its elliptical orbit:

$T^2 \propto a^3$ …