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When Kepler went to Tycho Brahe, he learned that Tycho Brahe was using the mean position of the sun to note down the position of Mars. Well, Kepler needed to convert these datas of Brahe to do his own calculations for the orbit of Mars.

In this link, the difference between the real and mean position is shown. http://www.keplersdiscovery.com/MeanSun.html http://www.keplersdiscovery.com/CorrectedTable.html

But how did Kepler calculate the real position of the Sun? I don't get it. And due to my first language is not English it might be possible that I didn't understand some important aspects of the given links.

It depends what you mean by "real position". Also, just to clarify something that should be gotten out of the way, the Earth rotates, obviously, and the night sky appears to rotate around the Earth a complete 360 degrees in a 24 hours. That's why the geocentric model lasted as long as it did. That's what it looks like, and that's what everyone believed prior to Copernicus, but that's not really relevant to your question. I mention it, because what is relevant is that the stars are so far away that they appear fixed. They rotate together, they don't move past each other so the stars operate as a fixed background against which the sun, moon and planets move. So, when we talk about "movement" in the sky or position, we're talking about against the fixed stars in the background, not the rotation of the Earth.

After the fixed stars, the Sun is the simplest object in the sky, it moves a little bit against the background every day and it returns to the same place every year, the Sun's position against the fixed background stars was known long before Kepler, so Kepler always knew the position or direction of the Sun, even at night because the Sun held it's position against the background stars, moving only a little bit each day. He didn't know the distance to the sun (more on that later), but he knew the position.

Kepler also knew that Mars' orbit around the sun was 687 days. How did he know that? Well (these words aren't my own, but it's discussed here)

It's a simple matter to measure that the synodic period of Mars is about 780 days. This is the time between successive oppositions of Mars, meaning that Mars is due South at midnight. Given the heliocentric model, it is easy to see, as shown in the Wikipedia article, that 1/Synodic Period = 1/Earth's orbital period - 1/Mars orbital period. So, once the heliocentric model was understood, it all becomes easy.

Kepler was a brilliant Mathematician. Brahe wasn't so much. Now, I'm not saying Brahe was dumb, cause that's not accurate either, but he wasn't skilled in math the way Kepler was. Brahe was very patient and dedicated and he made the most accurate observations to date, some 10 times more accurate than anyone before him. He had the best equipment for making accurate observations in history, so to a person like Kepler, Brahe's notes were pure gold, containing decades of great accuracy. There was even a rumor that Kepler killed Brahe to get access to his observatory and his work, but most people think that's just a rumor, that it didn't actually happen, but there was enough uncertainty that Brahe's body was exhumed and tested for poison. (none was found). But I digress.

A 3rd bit of information is that by using parallax and the maximum angle that Mars varies in the sky, you can get a rough estimate of the distance to the planets relative to each other. Copernicus did that so Kepler didn't have to. A bit more on that here. and here.

So Kepler had a lot of information at his fingertips when he got access to Brahe's notes. He knew that Mars was about 1.5 times as far from the Sun as the Earth (from Copernicus) and he knew that Mars orbit was 687 days (not difficult given the heliocentric model), which means, every year and 322 days, Mars is in the same spot relative to the Sun, but the Earth has moved. - So you have 2 still objects and one moving object that's some "43 days" earlier in it's orbit each time Mars is in the same place and by taking that measurement every 687 days, you can learn a lot about the relative position of Mars, Earth and the Sun. And that's what Kepler did.

His two major breakthroughs (in my opinion) were observing that Mars wasn't on the same plane as the Earth (He did that observing Copernicus' observations, not Brahe's - I read that, I can't find the notation right now). And his other breakthrough was to figure out that the Sun wasn't at the center of the Earth's orbit. He did that by measuring the angle between the Earth and Sun and Earth and Mars every 687 days. More on that here.

His measurements telling him that the sun wasn't at the center of the Earth's orbit led him to his ellipse model. After that it was all clarification and more detailed measurement that led him to his 3 laws, but I think, recognizing that the Sun wasn't at the center of the Earth's orbit was his big "aha" moment that and his discovery of Mars being on a different orbital plane. Kepler's measurement was quite accurate. Mars' inclination to Earth is 1.87 degrees. Kepler had it measured at 1 degree, 50 minutes, or 1.83 degrees.

## Astronomical work of Johannes Kepler

The ideas that Kepler would pursue for the rest of his life were already present in his first work, *Mysterium cosmographicum* (1596 “Cosmographic Mystery”). Kepler had become a professor of mathematics at the Protestant seminary in Graz, Austria, in 1594, while also serving as the district mathematician and calendar maker. In 1595, while teaching a class, Kepler experienced a moment of illumination. It struck him suddenly that the spacing among the six Copernican planets might be explained by circumscribing and inscribing each orbit with one of the five regular polyhedrons. Since Kepler knew Euclid’s proof that there can be five and only five such mathematical objects made up of congruent faces, he decided that such self-sufficiency must betoken a perfect idea. If now the ratios of the mean orbital distances agreed with the ratios obtained from circumscribing and inscribing the polyhedrons, then, Kepler felt confidently, he would have discovered the architecture of the universe. Remarkably, Kepler did find agreement within 5 percent, with the exception of Jupiter, at which, he said, “no one will wonder, considering such a great distance.” He wrote to Maestlin at once: “I wanted to become a theologian for a long time I was restless. Now, however, behold how through my effort God is being celebrated in astronomy.”

Had Kepler’s investigation ended with the establishment of this architectonic principle, he might have continued to search for other sorts of harmonies but his work would not have broken with the ancient Greek notion of uniform circular planetary motion. Kepler’s God, however, was not only orderly but also active. In place of the tradition that individual incorporeal souls push the planets and instead of Copernicus’s passive, resting Sun, Kepler posited the hypothesis that a single force from the Sun accounts for the increasingly long periods of motion as the planetary distances increase. Kepler did not yet have an exact mathematical description for this relation, but he intuited a connection. A few years later he acquired William Gilbert’s groundbreaking book *De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure* (1600 “On the Magnet, Magnetic Bodies, and the Great Magnet, the Earth”), and he immediately adopted Gilbert’s theory that Earth is a magnet. From this Kepler generalized to the view that the universe is a system of magnetic bodies in which, with corresponding like poles repelling and unlike poles attracting, the rotating Sun sweeps the planets around. The solar force, attenuating inversely with distance in the planes of the orbits, was the major physical principle that guided Kepler’s struggle to construct a better orbital theory for Mars.

But there was something more: the standard of empirical precision that Kepler held for himself was unprecedented for his time. The great Danish astronomer Tycho Brahe (1546–1601) had set himself the task of amassing a completely new set of planetary observations—a reform of the foundations of practical astronomy. In 1600 Tycho invited Kepler to join his court at Castle Benátky near Prague. When Tycho died suddenly in 1601, Kepler quickly succeeded him as imperial mathematician to Holy Roman Emperor Rudolf II. Kepler’s first publication as imperial mathematician was a work that broke with the theoretical principles of Ptolemaic astrology. Called *De Fundamentis Astrologiae Certioribus* (1601 *Concerning the More Certain Fundamentals of Astrology*), this work proposed to make astrology “more certain” by basing it on new physical and harmonic principles. It showed both the importance of astrological practice at the imperial court and Kepler’s intellectual independence in rejecting much of what was claimed to be known about stellar influence. The relatively great intellectual freedom possible at Rudolf’s court was now augmented by Kepler’s unexpected inheritance of a critical resource: Tycho’s observations. In his lifetime Tycho had been stingy in sharing his observations. After his death, although there was a political struggle with Tycho’s heirs, Kepler was ultimately able to work with data accurate to within 2′ of arc. Without data of such precision to back up his solar hypothesis, Kepler would have been unable to discover his “first law” (1605), that Mars moves in an elliptical orbit. At one point, for example, as he tried to balance the demand for the correct heliocentric distances predicted by his physical model with a circular orbit, an error of 6′ or 8′ appeared in the octants (assuming a circle divided into eight equal parts). Kepler exclaimed, “Because these 8′ could not be ignored, they alone have led to a total reformation of astronomy.” Kepler’s reformation of astronomy was of a piece with his reform of astrology’s principles and Tycho’s radical improvement of the celestial observations. Just as the spacing of the planets bore a close relation to the polyhedral forms, so, too, Kepler regarded only those rays hitting Earth at the right harmonic angles to be efficacious.

During the creative burst of the early Prague period (1601–05) when Kepler won his “war on Mars” (he did not publish his discoveries until 1609 in the *Astronomia Nova* [*New Astronomy*], which contained the first two laws of planetary motion), he also wrote important treatises on the nature of light and on the sudden appearance of a new star (1606 *De Stella Nova*, “On the New Star”). Kepler first noticed the star—now known to have been a supernova—in October 1604, not long after a conjunction of Jupiter and Saturn in 1603. The astrological importance of the long-anticipated conjunction (such configurations take place every 20 years) was heightened by the unexpected appearance of the supernova. Typically, Kepler used the occasion both to render practical predictions (e.g., the collapse of Islam and the return of Christ) and to speculate theoretically about the universe—for example, that the star was not the result of chance combinations of atoms and that stars are not suns.

Kepler’s interest in light was directly related to his astronomical concerns: how a ray of light, coming from a distant heavenly body located in the outer regions of space, deflects when entering the denser atmosphere surrounding Earth and then, in turn, what happens to light as it enters the relatively denser medium of the human eye. These problems had some medieval precedent, but, as usual, Kepler treated them in his own individual way. Although a court astronomer, Kepler chose a traditional academic form in which to compose his ideas on light. He called it *Ad Vitellionem Paralipomena, Quibus Astronomiae Pars Optica Traditur* (1604 “Supplement to Witelo, in Which Is Expounded the Optical Part of Astronomy”). Witelo (Latin: Vitellio) had written the most important medieval treatise on optics. But Kepler’s analysis of vision changed the framework for understanding the behaviour of light. Kepler wrote that every point on a luminous body in the field of vision emits rays of light in all directions but that the only rays that can enter the eye are those that impinge on the pupil, which functions as a diaphragm. He also reversed the traditional visual cone. Kepler offered a punctiform analysis, stating that the rays emanating from a single luminous point form a cone the circular base of which is in the pupil. All the rays are then refracted within the normal eye to meet again at a single point on the retina. For the first time the retina, or the sensitive receptor of the eye, was regarded as the place where “pencils of light” compose upside-down images. If the eye is not normal, the second short interior cone comes to a point not on the retina but in front of it or behind it, causing blurred vision. For more than three centuries eyeglasses had helped people see better. But nobody before Kepler was able to offer a good theory for why these little pieces of curved glass had worked.

After Galileo built a telescope in 1609 and announced hitherto-unknown objects in the heavens (e.g., moons revolving around Jupiter) and imperfections of the lunar surface, he sent Kepler his account in *Siderius Nuncius* (1610 *The Sidereal Messenger*). Kepler responded with three important treatises. The first was his *Dissertatio cum Nuncio Sidereo* (1610 “Conversation with the Sidereal Messenger”), in which, among other things, he speculated that the distances of the newly discovered Jovian moons might agree with the ratios of the rhombic dodecahedron, triacontahedron, and cube. The second was a theoretical work on the optics of the telescope, *Dioptrice* (1611 “Dioptrics”), including a description of a new type of telescope using two convex lenses. The third was based upon his own observations of Jupiter, made between August 30 and September 9, 1610, and published as *Narratio de Jovis Satellitibus* (1611 “Narration Concerning the Jovian Satellites”). These works provided strong support for Galileo’s discoveries, and Galileo, who had never been especially generous to Kepler, wrote to him, “I thank you because you were the first one, and practically the only one, to have complete faith in my assertions.”

In 1611 Kepler’s life took a turn for the worse. His wife, Barbara, became ill, and his three children contracted smallpox one of his sons died. Emperor Rudolf soon abdicated his throne. Although Kepler hoped to return to an academic post at Tübingen, there was resistance from the theology faculty Kepler’s irenic theological views and his friendships with Calvinists and Catholics were characteristic of his independence in all matters, and in this case it did not help his cause. Meanwhile, Kepler was appointed to the position (created for him) of district mathematician in Linz. He continued to hold the position of imperial mathematician under the new emperor, Matthias, although he was physically removed from the court in Prague. Kepler stayed in Linz until 1626, during which time creative productions continued amid personal troubles—the death of his wife and his exclusion from the Lutheran communion. Although he was married again in 1613 (to Susanna Reuttinger), five of his seven children from that marriage died in childhood. After the Counter-Reformation came in 1625, Catholic authorities temporarily removed his library and ordered his children to attend mass.

In 1615 Kepler used the occasion of a practical problem to produce a theoretical treatise on the volumes of wine barrels. His *Stereometria Doliorum Vinariorum* (“The Stereometry of Wine Barrels”) was the first book published in Linz. Kepler objected to the rule-of-thumb methods of wine merchants to estimate the liquid contents of a barrel. He also refused to be bound strictly by Archimedean methods eventually he extended the range of cases in which a surface is generated by a conic section—a curve formed by the intersection of a plane and a cone rotating about its principal axis—by adding solids generated by rotation about lines in the plane of the conic section other than the principal axis.

The Linz authorities had anticipated that Kepler would use most of his time to work on and complete the astronomical tables begun by Tycho. But the work was tedious, and Kepler continued his search for the world harmonies that had inspired him since his youth. In 1619 his *Harmonice Mundi* ( *Harmonies of the World*, which contained Kepler’s third law) brought together more than two decades of investigations into the archetypal principles of the world: geometrical, musical, metaphysical, astrological, astronomical, and those principles pertaining to the soul. All harmonies were geometrical, including musical ones that derived from divisions of polygons to create “just” ratios (1/2, 2/3, 3/4, 4/5, 5/6, 3/5, 5/8) rather than the irrational ratios of the Pythagorean scale. When the planets figured themselves into angles demarcated by regular polygons, a harmonic influence was impressed on the soul. And the planets themselves fell into an arrangement whereby their extreme velocity ratios conformed with the harmonies of the just tuning system, a celestial music without sound.

Finally, Kepler published the first textbook of Copernican astronomy, *Epitome Astronomiae Copernicanae* (1618–21 *Epitome of Copernican Astronomy*). The title mimicked Maestlin’s traditional-style textbook, but the content could not have been more different. The *Epitome* began with the elements of astronomy but then gathered together all the arguments for Copernicus’s theory and added to them Kepler’s harmonics and new rules of planetary motion. This work would prove to be the most important theoretical resource for the Copernicans in the 17th century. Galileo and Descartes were probably influenced by it. It was capped by the appearance of *Tabulae Rudolphinae* (1627 “ Rudolphine Tables”). The *Epitome* and the Rudolphine Tables cast heliocentric astronomy and astrology into a form where detailed and extensive counterargument would force opponents to engage with its claims or silently ignore them to their disadvantage. Eventually Newton would simply take over Kepler’s laws while ignoring all reference to their original theological and philosophical framework.

The last decade of Kepler’s life was filled with personal anguish. His mother fell victim to a charge of witchcraft that resulted in a protracted battle with her accusers, lasting from 1615 until her exoneration in 1621 she died a few months later. Kepler used all means at his disposal to save his mother’s life and honour, but the travels, legal briefs, and maneuvers that this support required seriously disrupted his work. In 1627 Kepler found a new patron in the imperial general Albrecht von Wallenstein. Wallenstein sent Kepler to Sagan in Silesia and supported the construction of a printing press for him. In return Wallenstein expected horoscopes from Kepler—and he accurately predicted “horrible disorders” for March 1634, close to the actual date of Wallenstein’s murder on February 25, 1634. Kepler was less successful in his ever-continuing struggle to collect monies owed him. In August 1630 Wallenstein lost his position as commander in chief in October Kepler left for Regensburg in hopes of collecting interest on some Austrian bonds. But soon after arriving he became seriously ill with fever, and on November 15 he died. His grave was swept away in the Thirty Years’ War, but the epitaph that he composed for himself survived:

At times referred to as the law of equal areas – describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun.

Kepler’s third law provides an accurate description of the period and distance for a planet’s orbits about the sun. His first and second laws describe the motion characteristics of a single planet. The third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.

“For once I firmly believed that the motive force of a planet was a soul.”

What we are seeing here is a gradual emergence from medieval concepts. At first, Kepler thought the planets moved because they had souls-they were alive, magical, not just lumps of matter. Later, he realized a more mechanical approach was more appropriate.

## Galileo and the International Year of Astronomy

Four hundred years ago this year, two events marked what scientists and historians today regard as the birth of modern astronomy. The first of them, the beginning of Galileo's telescopic observations, has been immortalized by playwrights and authors and widely publicized as the cornerstone anniversary for the International Year of Astronomy. Through his looking glass, the Italian astronomer saw the mountains and valleys of the moon, the satellites of Jupiter, and sunspots&mdashobservations that would play a huge role in discrediting the prevailing, church-endorsed view of an Earth-centered cosmos.

The second event is not as well known, but is arguably equally important. It was the publication of Johannes Kepler's *Astronomia Nova* (*The New Astronomy*) in 1609, a treatise in which the German astronomer introduced the first two of his laws describing planetary motion.

The first law states that the planets travel in elliptical orbits around the sun and describes the sun's position as the focal point in that ellipse. The second law states that an imaginary line connecting a planet to the sun will sweep out a region of equal size in a given time period, wherever in the orbit that time period falls.

Kepler would later go on to introduce yet another law, this one relating the dimensions of an orbit to the time it takes to complete that orbit. He also made fundamental contributions to optics, working out how images are formed by pinhole cameras, a telescope and the human eye as well as developed the principles for corrective lenses for near- and farsightedness. He coined the terms "orbit" and "satellite" and explained how ocean tides are caused by the moon.

"He was an astronomer's astronomer," says Owen Gingerich, a professor emeritus of astronomy and science history at the Harvard&ndashSmithsonian Center for Astrophysics.

Kepler was born in the town of Weil der Stadt in modern-day Baden&ndashWürttemberg, Germany, in 1571. His parents were not very well off: His father was a mercenary and his mother a housewife who would later be accused of witchery. Kepler was introspective and sickly as a child but he excelled in mathematics. He decided to join the clergy and won a scholarship to the University of Tübingen, where he was first acquainted with the work of Polish astronomer Nicholaus Copernicus.

In 1609 the accepted view was that Earth was stationary at the center of the universe and the moon, sun and planets moved around it. The stars lay beyond, encircling Earth in a sphere. This view of the heavens had originated with the Greeks, and it was formalized as an astronomical system by Claudius Ptolemy in the second century A.D. Ptolemaic astronomy was not simple&mdashto model the motion of the planets, it made use of a complicated system of circles and epicycles&mdashbut it had been accepted as truth for almost a millennium and a half.

In the middle of the 16th century Copernicus had put forward an alternate, heliocentric system in which the sun was the center of the universe, with Earth and the other planets encircling it. Copernicus's treatise on heliocentrism, *De Revolutionibus Orbium Coelestium* (*On the Revolutions of Heavenly Spheres*), was published in 1543. Kepler discovered it as a student in Tübingen and was much taken by Copernicus's views.

But most people were not similarly enamored with the concept of a heliocentric universe. First of all, Copernicus's ideas were not widely disseminated, as they ran counter to the teachings of the Roman Catholic Church (although 73 years would pass before *De Revolutionibus* was listed as a forbidden work by the Church). It was only in universities and their surroundings that they found a following. Secondly, even for those who heard of it, Copernicus's heliocentric astronomy was by some measures hardly more accurate than Ptolemaic astronomy.

**The Great Martian Catastrophe**

As Gingerich explains, the Ptolemaic system predicted positions of Mars approximately every 32 years that were in error by about 5 degrees in longitude for a short time. Copernicus's system wasn't much better: it was off by about 4 degrees longitudinally. Gingerich calls this "the great Martian catastrophe"&mdasha problem that observers such as Danish astronomer Tycho Brahe knew about, but that Kepler would solve.

After Tübingen Kepler worked as a mathematics teacher in Graz (in modern-day Austria), where he continued his interest in astronomy. It was in Graz that he wrote the *Mysterium Cosmographicum* (*The Cosmographic Mystery*), published in 1596, in support of Copernicus. He sent copies of his book to leading astronomers, including Brahe&mdashthe greatest observational astronomer of the day. Brahe and Kepler started a correspondence in which they talked about Copernicanism and other astronomical issues. By then, Kepler had realized the need for raw data&mdashobservations that would help him understand the underlying laws of nature.

In 1600, as a consequence of the religious and political unrest during the Protestant Reformation, Kepler lost his job at Graz. He made his way to Prague, where Brahe was the court astronomer to Emperor Rudolph II. Prague was where Kepler would spend some of his most productive years. Brahe died suddenly in 1601, and Kepler succeeded him as court astronomer. In addition to his royal duties, Kepler tried to resolve the motion of Mars. He found that his initial model, which assumed that Mars revolved around the sun in a circular orbit, failed to match his predecessor's observations. He reluctantly altered the orbit and made it more egg-shaped.

"There is a myth that Kepler, fitting a curve through Tycho Brahe's records of Mars, discovered that planetary orbits are elliptical," Gingerich says. "The fact is that Tycho's observations showed that the orbit was not a circle, but the choice of an ellipse was largely theoretical."

It was one of those intellectual leaps that would change the course of science. Kepler found that not only did an elliptical orbit with the sun at one focus explain the movement of Mars, but also of the other planets. In fact, as Gingerich points out, Kepler realized the momentous nature of his discovery. In the *Astronomia Nova,* the typeface suddenly becomes larger to account for the significance as Kepler explains the motion of Mars and puts forward his first two planetary laws. (The third would come later.)

But the underlying physical reason for the planetary motion eluded Kepler, who thought a sort of magnetism was responsible. That puzzle would have to wait for another revolutionary thinker, Isaac Newton, whose law of gravity appeared on the scientific stage and explained orbital behavior eight decades later.

## Music of the spheres

Kepler found that the planets move in ellipses, not circles, around the sun. He also found that when the planets are closer to the sun, they move faster than when they're farther away.

When it comes to the Earth, the ratio between its fastest speed and slowest speed reduces to 16/15, which is the same ratio between the notes *fa* and *mi*. Needless to say, Kepler thought this was fantastically important:

*&ldquoThe Earth sings Mi, Fa, Mi: you may infer even from the syllables that in this our home misery and famine hold sway.&rdquo*

To Kepler, this was the clincher. Why were the heavens so perfect but the Earth so full of wretchedness? The music of the spheres tells us - it fit so perfectly! His new system wasn't just a mathematical convenience, but a window into the mind of God and the hidden order of the universe.

## Ellipses!

The ellipse, the shape of a flattened circle, was well known to the ancient Greeks. It belonged to the family of "conic sections," of curves produced by the intersections of a plane and a cone.

The curves generated as "conic sections" when flat planes are cut across a cone. |

As the drawing above on the left shows, **when that plane is.**

--perpendicular to the axis of the cone, the result is a

circle.--moderately inclined, an

ellipse.--inclined so much that it is parallel to one side of the cone, a

parabola.--inclined even more, a

hyperbola.

(and when the plane includes the tip of the cone, it becomes two intersecting straight lines. That is the shape of a hyperbola under extreme magnification, when its curving tip becomes insignifantly small.)

All these intersections are easily produced by a flashlight in a moderately dark room (**drawing** below). The flashlight creates a cone of light and when that cone hits a wall, the shape produced is a conic section--the intersection of the cone of light with the flat wall.

The axis of the flashlight is also the axis of the cone of light. Aim the beam perpendicular to the wall to get a **circle** of light. Slant the beam: an **ellipse**. Slant further, to where the closing point of the ellipse is very, very far: a **parabola**. Slant even more, to where the two edges of the patch of light not only fail to meet again, but seem to head in completely different directions: a **hyperbola**.

## Kepler's Laws

Although he had an eventful life, Kepler is most remembered for "cracking the code" that describes the orbits of the planets.

Prior to Kepler's discoveries, the predominate theory of the solar system was an Earth-centered geometry as described by Ptolemy. A Sun-centered theory had been proposed by Copernicus, but its predictions were plagued with inaccuracies.

Working in Prague at the Royal Observatory of Denmark, Kepler succeeded by using the notes of his predecessor, Tycho Brahe, which recorded the precise position of Mars relative to the Sun and Earth.

Kepler developed his laws empirically from observation, as opposed to deriving them from some fundamental theoretical principles. About 30 years after Kepler died, Isaac Newton was able to derive Kepler's Laws from basic laws of gravity.

**Law 1. The orbits of the planets are ellipses, with the Sun at one focus.**

Any ellipse has two geometrical points called the foci (focus for singular). There is no physical significance of the focus without the Sun but it does have mathematical significance. The total distance from a planet to each of the foci added together is always the same regardless of where the planet is in its orbit.

The importance of this is that by not assuming the orbits are perfect circles, the accuracy of predictions in the Sun-centered theory was (for the first time) greater than those of the Earth-centered theory.

**Law 2. The line joining a planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.**

In any given amount of time, 30 days for example, the planet sweeps out the same amount of area regardless of which 30 day period you choose. Therefore the planet moves faster when it is nearer the Sun and slower when it is farther from the Sun. A planet moves with constantly changing speed as it moves about its orbit. The fastest a planet moves is at perihelion (closest) and the slowest is at aphelion (farthest).

**Law 3. The square of the total time period (T) of the orbit is proportional to the cube of the average distance of the planet to the Sun (R).**

This law is sometimes referred to as the law of harmonies. It compares the orbital time period and radius of an orbit of any planet, to those of the other planets. The discovery Kepler made is that the *ratio* of the squares of the revolutionary time periods to the cubes of the average distances from the Sun, is the same for every planet.

## Education

Through the Duke’s continued generosity, Johannes Kepler was able to begin attending the University of Tübingen in 1587. His studies included Latin, Hebrew, Greek, the Bible, mathematics, and astronomy. Kepler was taught mathematics and astronomy by Michael Mästlin, one of the few astronomy professors of that time who had accepted Copernicus’ idea that the planets, including the earth, revolved around the sun. Almost all scholars of that era still believed that the earth was the centre of the solar system.

In 1594, Kepler was asked to go to the Lutheran high school in Graz, Austria, to replace the mathematics teacher who had just died. Although close to finishing his theological training, Kepler felt led by God to take up this teaching position.

## Kepler’s Law

[/caption]

There are actually three, Kepler’s laws that is, of planetary motion: 1) every planet’s orbit is an ellipse with the Sun at a focus 2) a line joining the Sun and a planet sweeps out equal areas in equal times and 3) the square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit. As it’s the third which is most often used, Kepler’s law usually means Kepler’s third law (of planetary motion).

Tycho Brahe’s decades-long, meticulous observations of the stars and planets provided Kepler with what today we’d call a robust, well-controlled dataset to test his hypotheses concerning planetary motion (this way of describing it is, dear reader, a deliberate anachronism). In particular, Tycho’s observations of the position of Mars in the Uraniborg night sky were the primary source of hard data Kepler used to derive, and test, his three laws.

Kepler’s laws have an important place in the history of astronomy, cosmology, and science in general. They marked a key step in the revolution which moved the center of the universe from the Earth (geocentric cosmology) to the Sun (heliocentric), and they laid the foundation for the unification of heaven and earth, by Newton, a century later (before Newton the rules, or laws, which governed celestial phenomena were widely believed to be disconnected with those controlling things which happened on Earth Newton showed – with his universal law of gravitation – that the same law rules both heaven and earth).

Although Kepler’s laws are only an approximation – they are exact, in classical physics, only for a planetary system of just one planet (and then the focus is the baricenter, not the Sun) – for systems in which one object dominates, mass-wise, they are a good approximation.

Several Universe Today articles cover one aspect of Kepler’s Law or another, among them Let’s Study Law: Kepler Would Be So Proud!, and Happy Birthday Johannes Kepler

Astronomy Cast has three episode relevant to Kepler’s law: Gravity, and two Questions Shows Jan 27th, 2009, and May 19th, 2009 check them out!

## Johannes Kepler

**Johannes Kepler** is now chiefly remembered for discovering the three laws of planetary motion that bear his name published in 1609 and 1619) . He also did important work in optics (1604 , 1611) , discovered two new regular polyhedra (1619) , gave the first mathematical treatment of close packing of equal spheres ( leading to an explanation of the shape of the cells of a honeycomb, 1611) , gave the first proof of how logarithms worked (1624) , and devised a method of finding the volumes of solids of revolution that ( with hindsight! ) can be seen as contributing to the development of calculus (1615 , 1616) . Moreover, he calculated the most exact astronomical tables hitherto known, whose continued accuracy did much to establish the truth of heliocentric astronomy ( *Rudolphine Tables*, Ulm, 1627) .

A large quantity of Kepler's correspondence survives. Many of his letters are almost the equivalent of a scientific paper ( there were as yet no scientific journals ) , and correspondents seem to have kept them because they were interesting. In consequence, we know rather a lot about Kepler's life, and indeed about his character. It is partly because of this that Kepler has had something of a career as a more or less fictional character ( see historiographic note below ) .

Kepler was born in the small town of Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576 . His father was a mercenary soldier and his mother the daughter of an innkeeper. Johannes was their first child. His father left home for the last time when Johannes was five, and is believed to have died in the war in the Netherlands. As a child, Kepler lived with his mother in his grandfather's inn. He tells us that he used to help by serving in the inn. One imagines customers were sometimes bemused by the child's unusual competence at arithmetic.

Kepler's early education was in a local school and then at a nearby seminary, from which, intending to be ordained, he went on to enrol at the University of Tübingen, then ( as now ) a bastion of Lutheran orthodoxy.

**Kepler's opinions**

Throughout his life, Kepler was a profoundly religious man. All his writings contain numerous references to God, and he saw his work as a fulfilment of his Christian duty to understand the works of God. Man being, as Kepler believed, made in the image of God, was clearly capable of understanding the Universe that He had created. Moreover, Kepler was convinced that God had made the Universe according to a mathematical plan ( a belief found in the works of Plato and associated with Pythagoras ) . Since it was generally accepted at the time that mathematics provided a secure method of arriving at truths about the world ( Euclid's common notions and postulates being regarded as actually true ) , we have here a strategy for understanding the Universe. Since some authors have given Kepler a name for irrationality, it is worth noting that this rather hopeful epistemology is very far indeed from the mystic's conviction that things can only be understood in an imprecise way that relies upon insights that are not subject to reason. Kepler does indeed repeatedly thank God for granting him insights, but the insights are presented as rational.

**University education**

At this time, it was usual for all students at a university to attend courses on "mathematics". In principle this included the four mathematical sciences: arithmetic, geometry, astronomy and music. It seems, however, that what was taught depended on the particular university. At Tübingen Kepler was taught astronomy by one of the leading astronomers of the day, Michael Mästlin (1550 - 1631) . The astronomy of the curriculum was, of course, geocentric astronomy, that is the current version of the Ptolemaic system, in which all seven planets - Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn - moved round the Earth, their positions against the fixed stars being calculated by combining circular motions. This system was more or less in accord with current ( Aristotelian ) notions of physics, though there were certain difficulties, such as whether one might consider as 'uniform' ( and therefore acceptable as obviously eternal ) a circular motion that was not uniform about its own centre but about another point ( called an 'equant' ) . However, it seems that on the whole astronomers ( who saw themselves as 'mathematicians' ) were content to carry on calculating positions of planets and leave it to natural philosophers to worry about whether the mathematical models corresponded to physical mechanisms. Kepler did not take this attitude. His earliest published work (1596) proposes to consider the actual paths of the planets, not the circles used to construct them.

At Tübingen, Kepler studied not only mathematics but also Greek and Hebrew ( both necessary for reading the scriptures in their original languages ) . Teaching was in Latin. At the end of his first year Kepler got 'A's for everything except mathematics. Probably Mästlin was trying to tell him he could do better, because Kepler was in fact one of the select pupils to whom he chose to teach more advanced astronomy by introducing them to the new, heliocentric cosmological system of Copernicus. It was from Mästlin that Kepler learned that the preface to *On the revolutions*, explaining that this was 'only mathematics', was not by Copernicus. Kepler seems to have accepted almost instantly that the Copernican system was physically true his reasons for accepting it will be discussed in connection with his first cosmological model ( see below ) .

It seems that even in Kepler's student days there were indications that his religious beliefs were not entirely in accord with the orthodox Lutheranism current in Tübingen and formulated in the *Confessio Augustana* Ⓣ . Kepler's problems with this Protestant orthodoxy concerned the supposed relation between matter and 'spirit' ( a non-material entity ) in the doctrine of the Eucharist. This ties up with Kepler's astronomy to the extent that he apparently found somewhat similar intellectual difficulties in explaining how 'force' [ See the History Topic on Kepler's planetary laws ] from the Sun could affect the planets. In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians. In real life, it seems likely that a similar tendency to openness led the authorities at Tübingen to entertain well-founded doubts about his religious orthodoxy. These may explain why Mästlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz. Religious intolerance sharpened in the following years. Kepler was excommunicated in 1612 . This caused him much pain, but despite his ( by then ) relatively high social standing, as Imperial Mathematician, he never succeeded in getting the ban lifted.

**Kepler's first cosmological model (1596)**

Instead of the seven planets in standard geocentric astronomy the Copernican system had only six, the Moon having become a body of kind previously unknown to astronomy, which Kepler was later to call a 'satellite' ( a name he coined in 1610 to describe the moons that Galileo had discovered were orbiting Jupiter, literally meaning 'attendant' ) . Why six planets?

Moreover, in geocentric astronomy there was no way of using observations to find the relative sizes of the planetary orbs they were simply assumed to be in contact. This seemed to require no explanation, since it fitted nicely with natural philosophers' belief that the whole system was turned from the movement of the outermost sphere, one ( or maybe two ) beyond the sphere of the 'fixed' stars ( the ones whose pattern made the constellations ) , beyond the sphere of Saturn. In the Copernican system, the fact that the annual component of each planetary motion was a reflection of the annual motion of the Earth allowed one to use observations to calculate the size of each planet's path, and it turned out that there were huge spaces between the planets. Why these particular spaces?

Kepler's answer to these questions, described in his *Mysterium cosmographicum* Ⓣ , Tübingen, 1596 , looks bizarre to twentieth-century readers ( see the figure on the right ) . He suggested that if a sphere were drawn to touch the inside of the path of Saturn, and a cube were inscribed in the sphere, then the sphere inscribed in that cube would be the sphere circumscribing the path of Jupiter. Then if a regular tetrahedron were drawn in the sphere inscribing the path of Jupiter, the insphere of the tetrahedron would be the sphere circumscribing the path of Mars, and so inwards, putting the regular dodecahedron between Mars and Earth, the regular icosahedron between Earth and Venus, and the regular octahedron between Venus and Mercury. This explains the number of planets perfectly: there are only five convex regular solids ( as is proved in Euclid's *Elements* , Book 13) . It also gives a convincing fit with the sizes of the paths as deduced by Copernicus, the greatest error being less than 10 % ( which is spectacularly good for a cosmological model even now ) . Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory.

Kepler saw his cosmological theory as providing evidence for the Copernican theory. Before presenting his own theory he gave arguments to establish the plausibility of the Copernican theory itself. Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power. For instance, the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun ( they lie between Earth and the Sun ) whereas in the geocentric theory there is no explanation of this fact. Kepler lists nine such questions in the first chapter of the *Mysterium cosmographicum* Ⓣ .

Kepler carried out this work while he was teaching in Graz, but the book was seen through the press in Tübingen by Mästlin. The agreement with values deduced from observation was not exact, and Kepler hoped that better observations would improve the agreement, so he sent a copy of the Mysterium cosmographicum to one of the foremost observational astronomers of the time, Tycho Brahe (1546 - 1601) . Tycho, then working in Prague ( at that time the capital of the Holy Roman Empire ) , had in fact already written to Mästlin in search of a mathematical assistant. Kepler got the job.

**The 'War with Mars'**

Naturally enough, Tycho's priorities were not the same as Kepler's, and Kepler soon found himself working on the intractable problem of the orbit of Mars [ See the History Topic on Kepler's planetary laws ] . He continued to work on this after Tycho died ( in 1601) and Kepler succeeded him as Imperial Mathematician. Conventionally, orbits were compounded of circles, and rather few observational values were required to fix the relative radii and positions of the circles. Tycho had made a huge number of observations and Kepler determined to make the best possible use of them. Essentially, he had so many observations available that once he had constructed a possible orbit he was able to check it against further observations until satisfactory agreement was reached. Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci ( a result which when extended to all the planets is now called "Kepler's First Law" ) , and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ( "Kepler's Second Law" ) , that is the area is used as a measure of time. After this work was published in *Astronomia nova,* . Ⓣ , Heidelberg, 1609 , Kepler found orbits for the other planets, thus establishing that the two laws held for them too. Both laws relate the motion of the planet to the Sun Kepler's Copernicanism was crucial to his reasoning and to his deductions.

The actual process of calculation for Mars was immensely laborious - there are nearly a thousand surviving folio sheets of arithmetic - and Kepler himself refers to this work as 'my war with Mars', but the result was an orbit which agrees with modern results so exactly that the comparison has to make allowance for secular changes in the orbit since Kepler's time.

**Observational error**

It was crucial to Kepler's method of checking possible orbits against observations that he have an idea of what should be accepted as adequate agreement. From this arises the first explicit use of the concept of observational error. Kepler may have owed this notion at least partly to Tycho, who made detailed checks on the performance of his instruments ( see the biography of Brahe ) .

**Optics, and the New Star of 1604**

The work on Mars was essentially completed by 1605 , but there were delays in getting the book published. Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a *camera obscura*, Kepler did some work on optics, and came up with the first correct mathematical theory of the *camera obscura* and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina. These results were published in *Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur* Ⓣ , Frankfurt, 1604 . He also wrote about the New Star of 1604 , now usually called 'Kepler's supernova', rejecting numerous explanations, and remarking at one point that of course this star could just be a special creation 'but before we come to [ that ] I think we should try everything else' *De stella nova* Ⓣ , Prague, 1606 , Chapter 22 , KGW 1 , p. 257 , line 23 .

Following Galileo's use of the telescope in discovering the moons of Jupiter, published in his *Sidereal Messenger* ( Venice, 1610) , to which Kepler had written an enthusiastic reply (1610) , Kepler wrote a study of the properties of lenses ( the first such work on optics ) in which he presented a new design of telescope, using two convex lenses ( *Dioptrice*, Prague, 1611) . This design, in which the final image is inverted, was so successful that it is now usually known not as a Keplerian telescope but simply as the astronomical telescope.

**Leaving Prague for Linz**

Kepler's years in Prague were relatively peaceful, and scientifically extremely productive. In fact, even when things went badly, he seems never to have allowed external circumstances to prevent him from getting on with his work. Things began to go very badly in late 1611 . First, his seven year old son died. Kepler wrote to a friend that this death was particularly hard to bear because the child reminded him so much of himself at that age. Then Kepler's wife died. Then the Emperor Rudolf, whose health was failing, was forced to abdicate in favour of his brother Matthias, who, like Rudolf, was a Catholic but ( unlike Rudolf ) did not believe in tolerance of Protestants. Kepler had to leave Prague. Before he departed he had his wife's body moved into the son's grave, and wrote a Latin epitaph for them. He and his remaining children moved to Linz ( now in Austria ) .

**Marriage and wine barrels**

Kepler seems to have married his first wife, Barbara, for love ( though the marriage was arranged through a broker ) . The second marriage, in 1613 , was a matter of practical necessity he needed someone to look after the children. Kepler's new wife, Susanna, had a crash course in Kepler's character: the dedicatory letter to the resultant book explains that at the wedding celebrations he noticed that the volumes of wine barrels were estimated by means of a rod slipped in diagonally through the bung-hole, and he began to wonder how that could work. The result was a study of the volumes of solids of revolution *Nova stereometria doliorum* . Ⓣ , Linz, 1615 , in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'. This method was later developed by Bonaventura Cavalieri ( c. 1598 - 1647) and is part of the ancestry of the infinitesimal calculus.

*The Harmony of the World*

Kepler's main task as Imperial Mathematician was to write astronomical tables, based on Tycho's observations, but what he really wanted to do was write *The Harmony of the World*, planned since 1599 as a development of his *Mystery of the Cosmos*. This second work on cosmology ( *Harmonices mundi libri V* Ⓣ , Linz, 1619) presents a more elaborate mathematical model than the earlier one, though the polyhedra are still there. The mathematics in this work includes the first systematic treatment of tessellations, a proof that there are only thirteen convex uniform polyhedra ( the Archimedean solids ) and the first account of two non-convex regular polyhedra ( all in Book 2) . *The Harmony of the World* also contains what is now known as 'Kepler's Third Law', that for any two planets the ratio of the squares of their periods will be the same as the ratio of the cubes of the mean radii of their orbits. From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two. In fact, although the Third Law plays an important part in some of the final sections of the printed version of the *Harmony of the World*, it was not actually discovered until the work was in press. Kepler made last-minute revisions. He himself tells the story of the eventual success:

**Witchcraft trial**

While Kepler was working on his *Harmony of the World*, his mother was charged with witchcraft. He enlisted the help of the legal faculty at Tübingen. Katharina Kepler was eventually released, at least partly as a result of technical objections arising from the authorities' failure to follow the correct legal procedures in the use of torture. The surviving documents are chilling. However, Kepler continued to work. In the coach, on his journey to Württemberg to defend his mother, he read a work on music theory by Vincenzo Galilei ( c. 1520 - 1591 , Galileo's father ) , to which there are numerous references in *The Harmony of the World*.

**Astronomical Tables**

Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms ( published in 1614) . However, Mästlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere aid to calculation and second that it was unwise to trust logarithms because no-one understood how they worked. ( Similar comments were made about computers in the early 1960 s. ) Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's *Elements* Book 5 . Kepler calculated tables of eight-figure logarithms, which were published with the *Rudolphine Tables* ( Ulm, 1628) . The astronomical tables used not only Tycho's observations, but also Kepler's first two laws. All astronomical tables that made use of new observations were accurate for the first few years after publication. What was remarkable about the *Rudolphine Tables* was that they proved to be accurate over decades. And as the years mounted up, the continued accuracy of the tables was, naturally, seen as an argument for the correctness of Kepler's laws, and thus for the correctness of the heliocentric astronomy. Kepler's fulfilment of his dull official task as Imperial Mathematician led to the fulfilment of his dearest wish, to help establish Copernicanism.

**Wallenstein**

By the time the *Rudolphine Tables* were published Kepler was, in fact, no longer working for the Emperor ( he had left Linz in 1626) , but for Albrecht von Wallenstein (1583 - 1632) , one of the few successful military leaders in the Thirty Years' War (1618 - 1648) .

Wallenstein, like the emperor Rudolf, expected Kepler to give him advice based on astrology. Kepler naturally had to obey, but repeatedly points out that he does not believe precise predictions can be made. Like most people of the time, Kepler accepted the principle of astrology, that heavenly bodies could influence what happened on Earth ( the clearest examples being the Sun causing the seasons and the Moon the tides ) but as a Copernican he did not believe in the physical reality of the constellations. His astrology was based only on the angles between the positions of heavenly bodies ( 'astrological aspects' ) . He expresses utter contempt for the complicated systems of conventional astrology.

Kepler died in Regensburg, after a short illness. He was staying in the city on his way to collect some money owing to him in connection with the *Rudolphine Tables*. He was buried in the local church, but this was destroyed in the course of the Thirty Years' War and nothing remains of the tomb.

**Historiographic note**

Much has sometimes been made of supposedly non-rational elements in Kepler's scientific activity. Believing astrologers frequently claim his work provides a scientifically respectable antecedent to their own. In his influential *Sleepwalkers* the late Arthur Koestler made Kepler's battle with Mars into an argument for the inherent irrationality of modern science. There have been many tacit followers of these two persuasions. Both are, however, based on very partial reading of Kepler's work. In particular, Koestler seems not to have had the mathematical expertise to understand Kepler's procedures. Closer study shows Koestler was simply mistaken in his assessment.

The truly important non-rational element in Kepler's work is his Christianity. Kepler's extensive and successful use of mathematics makes his work look 'modern', but we are in fact dealing with a Christian Natural Philosopher, for whom understanding the nature of the Universe included understanding the nature of its Creator.