Johannes Kepler developed three laws of planetary motion using Tycho Brahe's accurate data, with the third law, known as the harmonic law, establishing a relationship between a planet's orbital period and its distance from the sun. This law, expressed mathematically as p^2 = ka^3, allows astronomers to determine the mass of celestial bodies by studying their orbits. Newton later provided a framework explaining why Kepler's third law holds, demonstrating its significance in measuring the masses of astronomical objects.

1. Understanding Kepler's Third Law
Johannes Kepler went to work for Tycho Brahe near the end of Tycho's life. When Tycho died, Kepler
used Tycho's data to deduce three laws of planetary motion. Because Tycho's data were far more
accurate than any previously collected data on planetary positions neither Ptolemy's nor
Copernicus's models of the cosmos worked.
Kepler deduced three laws of planetary motion that did agree with Tycho's data but diverged from
traditional views of the cosmos.
Statement of Kepler's Third Law
Kepler's third law, which is often called the harmonic law, is a mathematical relationship between
the time it takes the planet to orbit the Sun and the distance between the planet and the Sun. The
time it takes for a planet to orbit the Sun is its orbital period, which is often simply called its period.
For the average distance between the planet and the Sun, Kepler used what we call the semi-major
axis of the ellipse. The semi-major axis is half the major axis, which is the longest distance across the
ellipse. Think of it as the longest radius of the ellipse.

2. Kepler's third law states that the square of the period, P, is proportional to the cube of the semi-
major axis, a. In equation form Kepler expressed the third law as: P^2=ka^3. k is the
proportionality constant. To Kepler it was just a number that he determined from the data. Kepler
did not know why this law worked. He found it by playing with the numbers.
Newton's Form of Kepler's Third Law
Using Newton's laws it is possible to show why Kepler's third law works. For circular orbits, the
centripetal force required to keep the planet moving in a circular path equals the gravitational force
between the Sun and planet. For elliptical orbits, the idea is similar but a little more complex.
Because the gravitational force depends on the mass it turns out that the proportionality constant in
Kepler's third law involves the mass of the Sun or other object being orbited. See the figure for the
equation for Newton's form of Kepler's third law. In the case of a planet and a star, the mass of the
planet is negligible and can be dropped from the equation. In the case of two stars or other orbiting
objects of similar mass, both masses must be included.
Significance of Kepler's Third Law
Kepler's third law is extremely important to astronomers. Because it involves the mass it allows
astronomers to find the mass of any astronomical object with something orbiting it. Astronomers find
the masses of all astronomical objects by applying Kepler's third law to orbits. They measure the
mass of the Sun by studying the orbits of the planets. They measure the mass of the planets by
studying the orbits of their moons. Moons have nothing orbiting them, so to find the mass of the
moons astronomers need to send a probe to be affected by their gravity. Astronomers find the
masses of stars by studying the orbits of stars in binary systems. They can not measure the masses
of stars that are not in binary systems. In all these cases astronomers use Kepler's third law.
Kepler's third law is the only way to measure the masses of astronomical objects.
Further Reading
Zeilik, M., Astronomy The Evolving Universe 9th ed. Cambridge, 2002.

3. Kepler's First Law
Understanding Kepler's Second Law