Kepler's three laws: statement and physical meaning

Historical context

Johannes Kepler (1571-1630) established his laws from Tycho Brahe's very precise observations of the motion of Mars. He showed that planetary orbits are not perfect circles, contrary to the Copernican model that dominated at the time.

First law (law of orbits)

Each planet describes an ellipse with the Sun at one of the two foci. In polar coordinates, with the origin at the focus occupied by the Sun: r(theta) = p / (1 + e*cos(theta)) where p is the parameter of the ellipse and e the eccentricity (0 <= e < 1; e = 0 corresponds to a circle).

Second law (law of areas)

The segment connecting the Sun to the planet sweeps out equal areas in equal times. Direct consequence: the speed is maximum at perihelion (the closest point to the Sun) and minimum at aphelion (the farthest point). This law reflects the conservation of the planet's angular momentum.

Third law (law of periods)

The ratio T^2/a^3 is the same for all planets of a given system, where T is the orbital period and a the semi-major axis of the ellipse.

Summary table

Law	Statement	Associated quantity
1st	Elliptical orbit, Sun at a focus	eccentricity e
2nd	Equal areas swept in equal times	constant angular momentum
3rd	T^2/a^3 = constant	mass of the attracting body

Classic pitfall

Do not confuse the focus (position of the Sun) with the center of the ellipse: they coincide only in the special case of a circle (e = 0). Likewise, the law of areas does not say that speed is constant, only that the area swept per unit of time is.