From Kepler's laws to universal gravitation

From Kepler to Newton

Isaac Newton showed that Kepler's laws follow from the law of universal gravitation F = Gm1m2/r^2. For a circular orbit of radius r, equating the gravitational force and the centripetal force gives: GMm/r^2 = mv^2/r hence v = sqrt(GM/r).

Deriving the third law

Substituting v = 2pir/T into the previous expression: (2pir/T)^2 = GM/r so T^2 = (4pi^2/(G*M)) * r^3

The ratio T^2/r^3 = 4pi^2/(GM) depends only on the mass M of the attracting body (the Sun, or the Earth for a satellite). This is the generalization of Kepler's third law, also valid for an elliptical orbit by replacing r with a, the semi-major axis.

Numerical example

For an Earth satellite, M = 5.97 x 10^24 kg and G = 6.67 x 10^-11 (SI units). For r = 42,164 km (geostationary orbit), we find T = 86,164 s, i.e. about one sidereal day (23h56min).

Classic pitfall

Watch the units: G*M must be combined with r in meters and T in seconds. A common mistake is to keep the radius in kilometers without converting, which throws off the result by a factor of 10^9 on r^3.

Angular momentum and the law of areas

The law of areas is equivalent to dL/dt = 0, where L = mr^2(dtheta/dt) is the angular momentum. This conservation results from the central nature of the gravitational force: the torque relative to the focus is zero because F is always collinear with the position vector.