Planetary Orbits #4: Kepler Finds his Laws

Having gone down the pattern based roads with no success, Johann Kepler was now ready to evaluate Tycho’s data to see what it told, writes Kitty Ferguson in Tycho and Kepler.

Kepler analyzed Tycho’s data on Mars in a very innovative way to infer what the orbit of Earth would like to an observer on Mars! To his surprise, he found that Earth too sped up as it moved closer to the Sun. Hence he concluded:

“Earth, at least in this regard, is nothing unique. It is just a planet.”

Kepler then found the criteria to differentiate between Ptolemy’s and Copernicus’s models:

“If in their geometrical conclusions two hypotheses coincide, nevertheless in physics each will have its own peculiar additional consequences.”

In other words, the “why” question mattered after all. Kepler could see that the “why” seemed to have something to do with the Sun… But Kepler still felt (hoped?) it was not an either/or choice:

“Kepler did not think that mathematical rigor, ideals of symmetry and harmony, and the search for a physical explanation were incompatible.”

Now keep in mind that at that time, calculus hadn’t been invented and logarithms were still a (short) while in the future. Analyzing elliptical orbits without those tools wasn’t easy, which makes what followed an even greater achievement.

He noticed that the speed of the planet was slowest when it was farthest from the Sun; and fastest when it was closest to the Sun. He extrapolated and discovered that the speed was a function of distance from the Sun for all points on the orbit. He had discovered what is called Kepler’s Second law (the First law came later!):

“A straight line drawn from a planet to the Sun… would sweep equal areas in equal times.”

This finding also reinforced his view that the cause must be in the Sun:

“The Sun will melt all this Ptolemaic machinery like butter.”

But didn’t the Moon orbit the Earth? Couldn’t the cause therefore lie with the Earth? Kepler decided to ignore these inconvenient questions for the time being… And what force might the Sun be exerting? Kepler considered magnetism as an option, but never decided.

After a lot of frustration and effort, he then discovered Kepler’s First law, a “feature that was deeply satisfying to one who loved geometric harmony”: the elliptical orbit of planets had the Sun as one of the foci (centers). This further strengthened his view that the orbit’s shape had something to do with the Sun.

Years later, after even more painstaking analysis of the data, he discovered Kepler’s Third law: the relation between the time to orbit and the distance from the Sun.

Kepler was thrilled by the mathematical nature of the laws, and the geometrical aspects of them. He wrote:

“(This is) the sort of thing nature does.”

As Ferguson points out that, beyond the brilliance in discovering these laws, Kepler had now set the tone for all science to come:

“Kepler’s struggle to find those laws would itself become a prototype for what science would be from that time forward. Sir Isaac Newton was referring to Copernicus, Tycho Brahe, Kepler and Galileo when he said he had stood “on the shoulders of giants.”