Kepler (1571–1630)

Kepler

The model we use today is not the one introduced by Copernicus, but is essentially the one introduced later by Johannes Kepler. Kepler was a protege of Tycho Brahe and used Tycho's measurements to arrive at his model.

Kepler

The model we use today is not the one introduced by Copernicus, but is essentially the one introduced later by Johannes Kepler. Kepler was a protege of Tycho Brahe and used Tycho's measurements to arrive at his model.

Kepler's model is based on the following three laws:

  • Kepler's First Law:
    A planet moves in a plane along an elliptical orbit with the sun at one focus.
  • Kepler's Second Law:
    The position vector from the sun to a planet sweeps out area at a constant rate.
  • Kepler's Third Law:
    The square of the period of a planet around the sun is proportional to the cube of the average distance between the planet and the sun.

Note on Circular Motion and uniform speed

All the models prior to Kepler's model involved objects moving in circular orbits at uniform speed. You might wonder why this restriction to circular motion and uniform motion lasted so long. The idea originated with Aristotle. He believed that all the heavenly objects that were farther from the earth than the moon were composed of a special material called aether or quintessence whose essence was to move in circular orbits at uniform speed. For a long time this was accepted as a fact of nature and wasn't questioned. It was an amazing achievement for Kepler to set aside this long held belief.

Kepler's laws were not derived from some theory, but were obtained by carefully examining empirical data. Kepler's laws were the first description of planetary orbits that didn't require epicycles to produce accurate results. Let us look briefly at the man behind these laws.

Johannes Kepler was born in the German city of Weil der Stadt on December 27, 1571. His family could be described as dysfunctional. His father was abusive and frequently left for extended periods of time to fight as a mercenary. Kepler described his mother as “sharp-tongued, quarrelsome, and possessing a bad spirit.” Johannes almost died of smallpox when he was four years old and had poor health throughout his life. He also suffered from poor eyesight, a severe handicap in astronomy. Kepler was educated in Lutheran schools and was a very good student. His family couldn't afford to send him to a university, but he received a scholarship to the University of Tübingen (see Figure 15) in order to pursue his goal of becoming a Lutheran pastor.

Figure 15: University of Tübingen

At the university he excelled in mathematics and astronomy in addition to theology. It was here that he first heard of the theory proposed by Copernicus and he was very intrigued. In 1594, shortly before completing his studies at the university, his career path changed abruptly. The university was asked to supply someone to fill a vacant position of a mathematics and astronomy teacher at a protestant boys school in Graz, Styria (a district of Austria). Kepler was chosen by the university to fill this position. He was not pleased by this move, but gradually he realized that this was part of God's plan. Here is what he later wrote to his former teacher Michael Maestlin,

I had the intention of becoming a theologian. For a long time I was restless: but now see how God is, by my endeavors, also glorified in astronomy.

His teaching position in Graz gave him the opportunity to further explore his interest in astronomy and the Copernican theory. In 1596 he published his first paper in astronomy entitled Mysterium Cosmographicum. In this paper he tried to show that the spacing between the various planets could be explained geometrically in terms of the five regular polyhedra. This was not very successful, but in this paper he made an observation that would later have great importance. He observed that the planets moved faster when they were close to the sun and slower when they were farther from the sun. He postulated that there was some unknown force emanating from the sun that influenced the motion of the planets. This idea was later pursued by Newton.

In 1600 Kepler was forced to leave Graz as the Catholic leadership there decided to expel all protestants. Kepler had previously collaborated informally with Tycho Brahe, but he now accepted a position as Tycho's assistant. Tycho was well known for his accurate astronomical measurements. Kepler was assigned the task of coming up with a better description of the orbit of Mars. One of the first things that Kepler did was to actually plot the motion of Mars as predicted by the Ptolemaic model. This motion is shown in Figure 16.

Figure 16: Plot of Mars orbit as predicted by Ptolemaic model [Astrnomia nova, chapter 1 (1609)].

Kepler found it hard to believe that Mars would follow such a strange path with all the loops. He was attracted to the Copernican model, but he was not satisfied with the accuracy of its results.

When Tycho died in 1601, Kepler became his successor. In 1603 Kepler interrupted his research on mars orbit to produce a book on optics called Astronomiae Pars Optica. In this book he describes such things as the inverse square law for the intensity of light, atmospheric refraction, and parallax. He also describes the optics of the human eye.

When Kepler returned to the study of Mars orbit he used Tycho's measurements to come up with his first two laws of planetary motion. He actually came up with the second law first and later determined that the orbit must be elliptical with the sun at a focus. In 1609 he published his results in a paper entitled Astronomia nova. The third law was not discovered until 1618. A more complete exposition of his laws was entitled Epitome astronomiae copernicanae. It was published in three parts between 1618 and 1621.

In 1611 Kepler published a paper entitled dioptrice that explained the optics of a telescope. In preparing this paper he came up with a better design for the telescope. The telescopes at this time used a convex lens to capture the light and a concave eyepiece to focus the image. Kepler designed a telescope in which both of the lens were convex. This had a slight disadvantage in that the image was inverted. However, this was overshadowed by the fact that Kepler's telescope had a much larger field of view.

In 1621 Kepler published a set of tables called the Rudolphine tables that were based on his three laws. These tables allowed for the future prediction of the position of the stars and planets. Kepler died rather suddenly in 1630 during a trip to Regensburg. This was about two years before Galileo's trial. He was buried somewhere outside the city, but all the graves there were destroyed a couple years later during the 30 years war.

It should be noted that Galileo and Kepler were contemporaries and corresponded frequently. Although Galileo was aware of Kepler's work he never abandoned his belief in circular orbits. Even though Kepler's model was a major advancement, it was slow to catch on. For one thing, his publications were overshadowed by those of Galileo that were much more widely distributed. In addition, Galileo's advocacy of the Copernican model prompted the Catholic church to take a hard line on publications promoting a sun-centered view. Thus, Kepler's Astronomia nova as well as some of his later works were placed on the index of banned books. It was the success of his Rudolphine tables that sparked interest in his model. Some astronomers accepted his arguments for elliptical orbits, but there was no general acceptance of his full model. For one thing, Kepler's second law proved to be very difficult to apply in practice since it doesn't provide a direct relationship between the position of a planet and time.

In 1631 the French astronomer Pierre Gassendi observed the transit of Mercury across the face of the sun that was predicted by Kepler's Rudolphine tables. This was the first time this transit had been seen. In 1639 Jeremiah Horrox was able to observe the transit of Venus. These observations were evidence for the validity of Kepler's laws. In 1687 Kepler's laws were derived by Newton from his laws of motion and his law of universal gravitation. This proved to be the key to the general acceptance of Kepler's model. The derivation of Kepler's laws from Newton's laws is contained in the Appendix. The final nail in the coffin of the earth-centered view was the measurement of stellar parallax. In 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory. Over 6 months he detected a slight angular shift of 0.3 seconds or 83 millionths of a degree.

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