The Cosmological Distance Ladder

Relative Solar Distances Part II -- Kepler

In 1619 Johannes Kepler succeeded in finding a mathematical relationship between between a planet's orbital period and its distance from the Sun.  This was the third of three laws describing the motion of planets derived by Kepler from his analysis of the extensive observational data collected by Tycho Brahe (1546-1601), and it greatly simplified the task of determining the relative scale of the solar system.  Kepler's third law states:

The cube of a planet's distance from the Sun is directly proportional to the square of the planet's orbital period. 

In equation form, Kepler's third law can be written:

where D is the distance of a planet from the Sun, P is the orbital period of the planet, and k a constant of proportionality which Kepler found to be the same for all planets, (in fact k is not quite the same for all planets, but the difference is very small and need not concern us yet).  If we measure everything relative to the Earth, (i.e. let D be measured in units of AU and P in earth years, then k becomes 1, and Kepler's third law takes the form:

Kepler knew P for each planet.  To determine P, one simply has to observe how long it takes for a planet to circle the celestial sphere once.  Observations show the following orbital periods for each planet:

From this it's easy to determine the distance of each planet from the sun relative to the Earth's distance from the Sun.  We simply solve the above equation for D, and plug in the orbital periods from the above table.  The equation and results are as follows:

Now, as promised, let's move on to steps B and C which will give us the much coveted measure of the astronomical unit.