Johannes Kepler - Discovering the Planets' True Motion

Johannes Kepler

Johannes Kepler was a brilliant mathematician who is credited with discovering the true nature of the planets' motion around the Sun. He was born on December 27, 1571, in Weil-der-Stadt, near Württemberg in the Holy Roman Empire, which is present day southwest Germany. The eldest child in a very poor Protestant family of seven children, his childhood was not an easy one. He was a sickly child who was struck by smallpox at the age of six and would suffer ill health throughout his life. His father, having no trade, became a mercenary and went off to fight Protestant insurgents in the Netherlands, rarely returning home.

Kepler's family was so poor that had he lived in any other part of Europe he would have almost certainly not received any education. He was very fortunate, as was the field of astronomy, that a new mode of education had been introduced by the Dukes of Württemberg, who had embraced the Lutheran doctrines. They set up a system of scholarships for poor but gifted children from Protestant families. The goal being to provide a supply of bright young minds who would enter the Priesthood and defend the Protestant church against major religious arguments sweeping throughout Europe. Kepler's obvious brilliance won him scholarships which allowed him education at seminary and university at the great Protestant center of learning, the University of Tübingen, outside of Stuttgart.

He began at a local elementary school, but his attendance was sporadic due to ill health and periods when he was sent out to work by his family. It took him three times longer than the average child to finish his studies, which he finally did at the age of 13. After four years at seminary school, he went to the University of Tübingen, where he graduated from the Faculty of Arts at the age of 20. Being destined for a priesthood in the Protestant Church, he then attended divinity school for another four years before an event happened that ended his religious studies and would alter the course of his life completely.

A Change of Direction

Holy Roman Empire circa 1600 AD The Holy Roman Empire circa 1600 AD

The province of Styria in southeast Austria was then ruled by a Habsburg prince who was an adherent of the Catholic Church. But the province also had a large population of Protestants. The two major centers of learning in it's capital city, Graz, were also divided along these lines. The heavily protestant provincial school in Graz was in search of a mathematics teacher. They approached the University of Tübingen, as they always did, to select for them a suitable candidate.

Tübingen chooses Kepler to fill the position. At first he hesitates, but later accepts on the condition that he may be allowed to resume his divinity studies later in life, although this is something he never did. Kepler is appointed Mathematicus of the Province of Styria in Graz, Austria in April of 1594. His first year at Graz is very difficult, as his training in mathematics was inadequate. However Kepler works hard at his studies and at teaching, and does well.

It is in teaching geometry at Graz that he is struck by his great idea. At Tübingen, he had read about Copernicus' sun centered theory of the solar system. And like Copernicus, Kepler believes the universe to be simple and beautiful in its structure. He desperately wanted to prove Copernicus' theory, but did not have the training in math to do so.

In classical geometry, there are five and only five regular solids. That is, solids bounded by a number of identical sides, proven centuries early by Euclid. These are, the tetrahedron (4 equilateral triangles), cube (6 squares), octahedron (8 equilateral triangles), dodecahedron (12 pentagons), and icosahedron (20 equilateral triangles). One property of the regular solids is that a sphere can perfectly surround them, with all points touching the sphere. Also, a sphere can fit perfectly inside of them.

Kepler's drawing of the five regular solids Kepler's planets within the five regular solids
photo credit: Astronomy by Fred Hoyle

At that time there were six known planets, with five spaces in between them. Could the planets all fit into the five regular solids? This would explain why there were only six planets. Kepler was quite excited about his new idea, and that he, a young mathematician, could be the one to discover it. We know today that this is a delusional idea, and that there are many more than six planets. However, this is the idea that carried Kepler into the field of astronomy, and would lead him to discover the true nature of the planets' motion.

Proving this theory would be mathematically intense, with 120 possible combinations. Kepler intended to write an argument along with his theorem about how Copernicus' heliocentric theory of the solar system does not contradict Holy Scripture. He had trouble proving his theory of the five regular solids, and was convinced that the reason was because the data he was using must be inaccurate, since he was using observational data obtained by Hipparchus in the second century BC.

The Great Tycho Brahe

Tycho Brahe Tycho Brahe, photo credit:
Encyclopedia Britannica

In order to acquire more accurate data, he wrote to the legendary Swedish astronomer, Tycho Brahe, explaining his theory and asking for access to his observational data. Thus began a correspondence between Kepler and the famous Tycho Brahe. Kepler sent Tycho a copy of his work along with a request for all of his data. The correspondence lasts for two years, and ends with Tycho offering Kepler an invitation to come work with him in a new observatory in Benatek outside of Prague.

Kepler's decision to accept was not such a difficult one for him to make. The Archduke Ferdinand II of Habsburg had begun a campaign to cleanse the region of Protestant beliefs, and the school at Graz was shut down, leaving Kepler with no job. He had been married two years earlier and now had to think about the future of his family. He accepts Tycho's offer.

Tycho and Kepler should have made the perfect team. Tycho was the greatest observational astronomer, and Kepler one of the greatest mathematicians. However Tycho viewed Kepler as his subordinate, and was very reluctant to give up all of his data, which represented his life's work. After all, Kepler was there to work under Tycho. Tycho knew that Kepler was probably the only one who could unlock his data and discover the true nature of the planets' motion, and in doing so would likely receive all of the historical credit.

At this time in history, there was major difficulty in explaining the motion of the planet Mars. Tycho allows Kepler to work on this problem, but even then gives him inadequate data to solve it. Kepler pleads with Tycho for access to his data, but is rejected time and time again until he finally leaves the observatory in frustration.

Kepler returns home, only to find Protestants being badly persecuted. He writes to Tycho explaining his situation. Tycho responds by generously offering him the position of senior assistant, along with the stipulation that he is to be Tycho's assistant, and to only perform the tasks that he is asked to do. Kepler reluctantly accepts and is a good assistant for the time being.

A tragic development

The emperor asks that Tycho be based in Prague, and their work is moved there in October 1599. In October 1601, Tycho is invited to a banquet hosted by a local baron. Tycho, being from a wealthy background, eats and drinks, and has a great time as he is used to. During these years it was tradition for guests to not get up from the table until the one throwing the banquet does. Tycho finds himself having to use the restroom very badly but remains seated as is the custom of the day. When the baron finally gets up from his seat, Tycho attempts to use the restroom, but finds that he has trouble urinating, and is taken home by carriage in quite a bit of pain. He dies at home 11 days later, most likely from a ruptured bladder.

Kepler later describes the incident, "Tycho had held back his water beyond the bounds of courtesy." Tycho Brahe is buried in Prague, and two days after his funeral Kepler is appointed his successor as Imperial Mathematicus.

Breakthrough

Kepler returns to work, and now has access to all of Tychos' observational data. Over time he slowly moves away from his theory of the five regular solids, but continues to work to find a more elegant arrangement of the solar system. He is briefly diverted from his work by the appearance of a supernova. It is interesting to note that the two supernovas in the past 1000 years both occurred around this time, one at the beginning of Tycho Brahe's career and one at the beginning of Johannes Kepler's career.

He returns to the problem of Mars, confidently believing he can solve it. After all, he is only repeating the earlier calculations of Copernicus, but with the much more accurate data of Tycho. He thinks it will only take him eight days. In fact it takes him eight years with thousands of pages of calculations before he finally discovers that the true path of a planet around the sun is that of an ellipse. This leads him to state his first two laws of planetary motion. It took him so long not only because of the enormous amount of mathematical calculations, but also because he had to shed one of the most widely held beliefs within astronomy, that the heavenly bodies moved in perfect circles and at constant velocity. This was the notion that Pythagoras, Aristotle, Ptolimy, the islamic scholars, and even Copernicus had believed.

Kepler finds an orbit and velocity that match Tycho's data perfectly. He did this with only the tools of arithmetic and geometry. In the greatest mathematical analysis ever taken, he publishes his findings in 1609:

A New Astronomy Based on Causation
or
A Physics of the Sky
derived from Investigations of the Motions of the Star Mars
Founded on Observations of
The Noble Tycho Brahe

The fact that Kepler generously gives credit to Tycho directly in the tile of his publication shows the magnitude of the debt he feels he owes to him. This is a quality not present in all scientists or philosophers. He knew that his achievement would not have been possible without the work of Tycho Brahe and goes on to say that his studies began under "Tycho's supreme command."

A New Astronomy gave the arguments and proof of Kepler's first two laws of planetary motion.


Kepler's first law:

The motion of planet Mars, and all other planets,
is an ellipse, with the Sun at one focus.

Kepler's second law:

The velocity of a planet varies in such a way that a line
between it and the Sun sweeps out equal areas in equal time.


These laws imply that when a planet is closest to the Sun, it moves fastest. And when it is farthest from the Sun, it moves slowest. This directly contradicts the belief of so many famous astronomers of the past that the planets move at a constant velocity.

An elliptical orbit around the Sun A planet's elliptical orbit, with the Sun as one focus.

It would take Kepler another nine years to work out his third law of planetary motion. It had to do with the relation between the periods of the planets and their distance from the Sun. Up to this time the understanding of the planets' distance from the Sun had not been advanced since the work of Aristarchus in the third century BC.


Kepler's third law:

The square of a period of a planet is proportional
to the cube of its distance from the Sun.


A period refers to one complete revolution around the Sun. This means that the farther away a planet is from the Sun, the longer it takes to make a complete orbit around the Sun. Not only because it has farther to travel, but also because it moves more slowly. Kepler completed this work in 1618 and published his findings in 1619 in his book: Harmonice Mundo (The Harmony of the World).

Kepler understood that although he had discovered how the planets moved around the Sun, he had yet to discover why they did so. He only suggested that there must be some force attracting the planets over a great distance without actually being in contact with them. He thought this force might be something like a magnet, with the Sun as the source of this force.

Johannes Kepler died on November 15th, 1630, while traveling in Regensburg, Bavaria and was buried there. He leaves behind an enormous legacy within the field of astronomy, that which would lay the ground work for the accomplishments of Isaac Newton. His burial site was destroyed two years later during the Thirty Years War. Only his epitaph remains:

Mensus eram coelos, nunc terrae metior umbras
Mens coelestis erat, corporis umbra iacet.

I used to measure the heavens, now I measure the shadows of Earth.
Although my mind was Heaven-bound, the shadow of my body lies here.



Sources:
  Astronomy Through the Ages by Robert Wilson
  Kepler, a novel by John Banville
  NASA



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