Johannes Kepler published his book Harmony of the world 400 years ago contains what is now commonly known as “Kepler’s third law”:
Sed res est certissima exactissimaque, quòd proportio quae est inter binorum quorumcunque Planetarum tempora periodica, sit praecisè sesquialtera proportionis mediarum distantiarum, …Johannes Kepler Harmonices Mundi 1619, see p. 302 of the Kepler edition by Max Caspar.
Expressed in equations: the ratio of orbital periods T1 : T2 is proportional to the semimajor axes a11.5 : a21.5 to the power of 1.5 for any two planets 1 and 2. What is more difficult to predict (and still unknown today!) is why the planets in our solar system are at their respective orbit. With the vast statistics of thousands of extra solar planets obtained from the Kepler mission and other surveys we might soon find out if there is an empirical relation hidden in the formation of planetary systems.
Kepler did make several attempts to find a law behind the distances of the 5 known planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Most known is his book Mysterium Cosmographicum published already 1597.
In the first rendering, I show the elliptical orbits of the five planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, in black. The spherical shells have radii corresponding to the perihelion and aphelion distances of the planets. The red tetrahedron has as insphere a ball of radius the aphelion distance of mars and a circumsphere with radius equal to Jupiter’s perihelion distance. The cube has as insphere a ball with radius aphelion distance of Jupiter and circumscribed a sphere with the perihelion distance of Saturn.
The second figure shows the inner planets in more detail. The black ellipses are the actual orbits of Mercury, Venus, Earth, and Mars. The innermost yellow octahedron encloses the orbit of Mercury (inscribed in the square shaped plane), then the construction proceeds by connecting the outer sphere of the yellow octahedron to the Venus shell (perihelion distance), then put around the aphelion shell of Venus the green icosahedron. The sphere circumscribed around the icosahedron has the radius of the Earth’s perihelion distance. The blue dodecahedron harbors an insphere with radius of the Earth’s aphelion distance and its circumsphere yields the perihelion sphere of Mars. The larger eccentricities of the Mars and Mercury orbits yield thicker shells for these planets. Note that the construction takes the eccentricities to be known and does not rely on circular trajectories.
Kepler was well aware of the eccentricity of the planets and did not assume circular orbits. He also noted that the theory agrees only within few percent with observations. In Harmonices Mundi he investigates if alternatively the planetary distances are encoded in musical proportions.