math

Kepler's Laws

According to Aristotle, the planets must revolve around the Earth in circles.
When circles didn't quite work, circular motion on top of circles was added, much like a Spirograph. Circular motion was a difficult concept to shake loose from.

Johannes Kepler (1571 - 1630) was the person to successfully challenge this with elliptical orbits. He was successful at this because he provided a simpler model, and it fit the existing data better than the Spirograph model of circles on circles.

The Three Laws

Kepler was able to describe the motions of the planet quite succinctly in three laws:

1) The planets travel about the Sun in elliptical paths.
2) The speed of the planet depends on its distance to the Sun. An arc drawn from the Sun to any planet will trace out equal areas in equal periods of time.
3) The square of the period of revolution (the planet's year) is proportional to the cube of its mean distance from the Sun.

There is great beauty in the simplicity of these laws, but the simplicity hides the great deal of toil that went into these laws.

A Metaphysical Start

Kepler learned of the Copernican model from a professor in college, and he felt that this must be part of the master plan. He hit on the idea of nesting a sequence of the five regular polyhedra, the so-called Platonic solids.

Kepler found that if the orbit of the Earth was on a sphere, and a dodecahedron is circumscribed about the Earth's celestial sphere, then that same dodecahedron would inscribe the orbital sphere of Mars. To his immense joy, Kepler found that the five regular polyhedra could each be used exactly once to similarly nest the orbital spheres of all six of the known planets. Not only did Kepler have an explanation for size of the orbits, but an explanation for why there were six planets.

It was a beautiful model, but it was unfortunately wrong.

Getting More Data

Kepler came to work as an assistant to Tycho Brahe, who was then head of an observatory at Prague. Brahe was rich, vigorous, gifted in the use of his hands, and had the patience to gaze for hours through a telescope. Kepler, on the other hand, was sickly, poor, and mystical, but possessed the gift of mathematical intuition in great measure. It is from the conjunction of these complimentary talents that great work could be done.

Brahe provided Kepler with detailed and precise observations of the positions of the planets, and Kepler set to work trying to make sense of them.

From 1601 to 1609, Kepler took many approaches on matching curves to Brahe's data. He chose the positions of Mars in the night sky because its relatively rapid movement provided for much data. Kepler labored for years, stuck in Aristotle's paradigm of circular orbits.

When he had finally exhausted all arrangements of circles he could imagine, he reluctantly started working with oval shapes. By no means was the ellipse his first try, since it was too simple. Certainly the ancient Greeks would have tried this one already.

In 1609, Kepler published Astronomia Nova which described Kepler's first two laws. It was not until 1619 that Kepler found his long sought after third law that brought order to the size of the orbits by relating the planet's year and mean distance from the Sun.

And the Church Responds

Kepler's model was based on putting the Sun at the center of the universe. Since this contradicted the official arrangement with the Earth at the special place in the center of the universe, it was not surprising that the Catholic Church quickly placed Kepler's masterpiece on the list of banned books, along with Copernicus' masterpiece De Revolutionibus Orbium Coelestium Libri Sex (Six Books on the Revolutions of the Celestial Spheres).

Old paradigms are hard to break. Kepler struggled to break the paradigm of circular orbits for the planets. He provided a wonderful model based on ellipses, but unfortunately, the world was not quite ready.

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