Kepler's laws

Before Newton, Kepler had discovered three empirical laws of planetary motion, derived from Tycho Brahe’s observational data without knowing their cause. Newton’s triumph was to show that all three follow from a single law, that of universal gravitation with its $1/r^2$ dependence.

Kepler's three laws

1. First law: planets trace out elliptical orbits, with the Sun at one of the two foci.
2. Second law (of areas): the radius vector from planet to Sun sweeps out equal areas in equal times (a consequence of conserved $L$).
3. Third law (harmonic): for all planets in the Solar System, $\dfrac{T^2}{a^3} = \text{constant}$.

The third law, in explicit form, relates the period and semi-major axis to the mass of the central body:

$\frac{T^2}{a^3} = \frac{4\pi^2}{GM_\star} = \text{constant}$

where $a$ is the semi-major axis of the ellipse and $T$ the orbital period.

Newton's test

The third law $T^2 \propto a^3$ is the relation Newton verified by comparing the motion of the Moon with the fall of an apple. If the force scales as $1/r^2$, then $T^2/a^3$ must be constant for all planets: and that is exactly what Kepler had observed. The fact that a theoretically derived law reproduced data gathered decades earlier was one of the strongest pieces of evidence in favour of universal gravitation.

The third law is also a measuring tool: knowing $T$ and $a$ of a satellite gives the mass $M_\star$ of the body it orbits. This is how the Sun, the planets, and even galaxies are “weighed”.

Links

Topics: Gravitazione Concepts: Leggi di Keplero · Orbite

Related exercises: Altitude of the geostationary orbit · True or false on energy and orbits · Ranking the periods of four satellites