Kepler's Law

Of the planets known to Kepler, Mercury has the most elliptical orbit.

Even it deviates only slightly from a circle.

As these five solids were the only regular solids, he supposed they were the 'spacers' between the planetary orbits.

He advanced astrological, numerological, and even musical arguments for his theory.

In spite of family disadvantages and chronic poor health, Kepler did well in school—

winning promotion to a Latin school and eventually a scholarship to the university at Tübingen, where he studied to become a Lutheran pastor.

• Tycho saw no parallax in the position of the new star. •So, he concluded that it must lie above the sphere of the moon and was probably on the starry sphere itself.

This contradicted Aristotle's belief that the starry sphere was perfect and unchanging.

Kepler used ellipses to describe the motion of the planets in three fundamental rules that have been tested and confirmed so many times that astronomers now refer to them as natural laws.

They are commonly called Kepler's laws of planetary motion.

He was surprised to find that it displayed no parallax.

To understand the significance of this observation, you should note that Tycho believed that the heavens rotated westward around Earth

Not long after his university days, in 1572, a 'new star' (now called Tycho's supernova) appeared in the sky.

Tycho rushed to measure its position.

•Life was unsettled for Kepler because of the persecution of Protestants in the region. •So, when Tycho Brahe invited him to Prague in 1600, Kepler went readily—eager to work with the famous Danish astronomer.

Tycho's sudden death in 1601 left Kepler in a position to use the observations from Hveen to analyze the motions of the planets and complete the Rudolphine Tables.

The reason for Kepler's evident deference was Tycho's family—still powerful and still intent on protecting the memory of Tycho.

-They even demanded a share of the profits and the right to censor the book before publication. -However, they changed nothing but a few words on the title page and added an elaborate dedication to the emperor.

Tycho was interested in astronomy even during his days at university.

He expressed amazement that neither The Alfonsine Tables nor The Prutenic Tables properly described the motions of the planets along the ecliptic.

Kepler's third law states that the orbital period squared is proportional to the semimajor axis cubed.

Measuring P in years and a in astronomical units, you can summarize the third law as: P2y = a3 AU

The puzzle of planetary movement was solved during the century following the death of Copernicus—through the work of two men.

One compiled the observations and one made the analysis.

Tycho's family—recognizing that Kepler was a Copernican and guessing that he would not follow the Tychonic system in completing the Rudolphine Tables—sued to recover the instruments and books of observations.

The legal wrangle went on for years.

Kepler's second law states that a line from the planet to the sun sweeps over equal areas in equal intervals of time.

Thus, when the planet is closer to the sun and the line connecting it to the sun is shorter, the planet moves more rapidly to sweep over the same area that is swept over when the planet is farther from the sun.

Kepler felt that he had found the underlying architecture of the universe in the sphere plus the five regular solids

cube, tetrahedron, dodecahedron, icosahedron, and octahedron.

The eccentricity of an ellipse informs you about its shape.

-If e is nearly equal to one, the ellipse is very elongated. -If e is closer to zero, the ellipse is more circular

•The second half of the book is no better than the first. •However, it has one virtue.

-As Kepler tried to fit the five solids to the planetary orbits, he demonstrated that he was a talented mathematician and that he was well versed in astronomy. -He sent copies to Tycho on Hveen and to Galileo in Rome.

In spite of Kepler's recurrent involvement with astrology and numerology, he continued to work on the Rudolphine Tables.

-At last, in 1627, they were ready and he financed their printing himself, dedicating them to the memory of Tycho Brahe. -In fact, Tycho's name appears in larger type on the title page than Kepler's own. -This is especially surprising when you recall that the tables were based on the heliocentric model of Copernicus and the elliptical orbits of Kepler and not on the Tychonic system

In November 1601, Tycho collapsed at a nobleman's home.

-Before he died, 11 days later, he asked Rudolph II to make Kepler imperial mathematician. -Thus, the newcomer, Kepler, became Tycho's replacement—though at one-sixth Tycho's salary

In spite of the abdication of Rudolph II in 1611, Kepler continued his astronomical studies.

-He wrote about a supernova that had appeared in 1604 (now known as Kepler's supernova) and about comets. -He also wrote a textbook about Copernican astronomy.

He announced his discovery in a small book, De Stella Nova (The New Star), published in 1573.

-His family took the opportunity to introduce him to the court of the Danish King Frederik II. -Soon, the king offered him funds to build an observatory on the island of Hveen, just off the Danish coast.

•The new star was a change in the heavens. •Therefore, according to classical astronomy, it had to lie below the sphere of the moon.

-In that case, Tycho reasoned, the new star should show parallax. -That is, it would appear slightly too far east as it rose and slightly too far west as it set

His goal was to revise the Alfonsine Tables and publish the revision as a monument to his new patron.

-It would be called the Rudolphine Tables. -Tycho did not intend to base the Rudolphine Tables on the Ptolemaic system but rather on his own Tychonic system—proving once and for all the validity of his hypothesis. -To assist him, he hired a few mathematicians and astronomers—including one Johannes Kepler

•Kepler's three laws are empirical. •That is, they describe a phenomenon without explaining why it occurs.

-Kepler derived them from Tycho's extensive observations—not from any first principle, fundamental assumption, or theory. -In fact, Kepler never knew what held the planets in their orbits or why they continued to move around the sun

This makes it easy to draw ellipses with two thumbtacks and a loop of string.

-Press the thumbtacks into a board, loop the string about the tacks, and place a pencil in the loop. -If you keep the string taut as you move the pencil, it traces out an ellipse

Tycho lived before the invention of the telescope.

-So, his observatory contained none. -Rather he built large, carefully designed instruments for the measurement of the positions of the sun, moon, and planets. •His observations were more accurate and more extensive than any that had been made before.

Unhappily for Tycho, King Frederik II died in 1588, and his young son took the throne

-Suddenly, Tycho's temper, vanity, and noble presumptions threw him out of favor. -In 1596, taking most of his instruments and books of observations, he went to Prague, the capital of Bohemia, and became imperial mathematician to the Holy Roman Emperor Rudolph II

•While still a college student, Kepler had become a believer in the Copernican hypothesis and, at Graz, he used his extensive spare time to study astronomy. •By 1596, the same year Tycho arrived in Prague, Kepler was ready to solve the mystery of the universe.

-That year, he published a book called The Forerunner of Dissertations on the Universe, Containing the Mystery of the Universe. -The book, like nearly all scientific works of that age, was written in Latin and is now known as Mysterium Cosmographicum.

In 1619, he published Harmonice Mundi (The Harmony of the World), in which he returned to the cosmic mysteries of Mysterium Cosmographicum.

-The only thing of note in Harmonice Mundi is his discovery that the radii of the planetary orbits are related to the planet's orbital periods. •That and his two previous discoveries have become known as the three most fundamental rules of orbital motion.

•He began by studying the motion of Mars—trying to deduce from the observations how the planet moved. •By 1606, he had solved the mystery—this time correctly.

-The orbit of Mars is an ellipse and not a circle. -With that, he abandoned the 2000-year-old belief in the circular motion of the planets

Kepler's third law relates a planet's orbital period to its average distance from the sun.

-The orbital period, P, is the time it takes to travel around the sun once. -Its average distance from the sun equals the semimajor axis of its orbit, a.

However, he discovered that the mystery was even more complex.

-The planets do not move at a uniform speed along their elliptical orbits. -His analysis showed that they move faster when close to the sun and slower when farther away

For example, Jupiter's average distance from the sun is roughly 5.2 AU.

-The semimajor axis cubed would be about 140.6. -So, the period must be the square root of 140.6—about 11.8 years.

The geometry of an ellipse is described by two simple numbers.

-The semimajor axis, a, is half the longest diameter. -The eccentricity, e, is half the distance between the foci divided by the semimajor axis

The Rudolphine Tables were Kepler's masterpiece.

-They could predict the positions of the planets 10 to 100 times more accurately than previous tables. -Kepler's tables were the precise model of planetary motion that Copernicus had sought but failed to find. -The accuracy of the Rudolphine Tables was strong evidence that both Kepler's model for planetary motion and the Copernican hypothesis for the place of the Earth were correct. -Copernicus would have been pleased

You can see that it is really the Copernican model with Earth held stationary and the sun allowed to move around Earth.

Although Tycho's model was very popular at first, the Copernican model replaced it within a century.

•Kepler died on November 15, 1630. •He had solved the problem of planetary motion, and his Rudolphine Tables demonstrated his solution.

Although he did not understand why the planets moved or why they followed ellipses—insights that had to wait half a century for Isaac Newton—Kepler's three rules worked.

Although Kepler dabbled in the philosophical arguments of the day, he was at heart a mathematician.•His triumph was the solution of the problem of the motion of the planets.•The key to his solution was the ellipse.

An ellipse is a figure drawn around two points—called the foci—in such a way that the distance from one focus to any point on the ellipse and back to the other focus equals a constant.

•His superiors put him to work teaching a few introductory courses and preparing an annual almanac that contained astronomical, astrological, and weather predictions. •Through good luck, some of his weather predictions for the year 1595 were fulfilled, and he gained a reputation as an astrologer and seer.

Even in later life, he was able to earn money from his almanacs.

During his last year of study, Kepler accepted a job in Graz teaching mathematics and astronomy—a job he resented because he knew little about the subjects

Evidently, he was not a good teacher either—he had few students his first year and none at all his second.

As the Ptolemaic model was inaccurate and the Copernican model was heliocentric, Tycho rejected both.

He devised his own geocentric model.

No one could have been more different from Tycho Brahe than Johannes Kepler.

He was born on December 27, 1571, to a poor family in a region now included in southwest Germany.

Tycho Brahe, born on December 14, 1546, was not a churchman like Copernicus, but rather a nobleman from an important family educated at the finest universities.

He was well known for his vanity and his lordly manners and, by all accounts, was a proud and haughty nobleman.

His disposition was not improved by a dueling injury from his university days.

His nose was badly disfigured and, for the rest of his life, he wore false noses made of gold and silver and stuck on with wax

•Thus, the planet would move from point A to point B in one month, sweeping over the area shown. •When the planet is farther from the sun, though, a month's motion would be shorter, from A′ to B′.

However, the areas swept out would be the same.

By modern standards, the book contains almost nothing of value.

It begins with a long appreciation of Copernicanism and then goes on to speculate on the reasons for the spacing of the planetary orbits.

With those two brilliant discoveries, Kepler abandoned both uniform motion and circular motion and finally solved the puzzle of planetary motion.

Kepler published his results in 1609 in a book called Astronomia Nova (New Astronomy).

On Hveen, Tycho constructed a luxurious home with six towers especially equipped for astronomy and populated it with servants, assistants, and a dwarf to act as jester.

Soon, Hveen became an international center of astronomical study and Tycho became the most famous astronomer in Europe.

Kepler's first law states that the orbits of the planets around the sun are ellipses with the sun at one focus.

Thanks to the precision of Tycho's observations and the sophistication of Kepler's mathematics, Kepler was able to recognize the elliptical shape of the orbits even though they are nearly circular.

•Tycho's family did get back the instruments Tycho had brought to Prague. •Kepler, though, had the books and he kept them.

Whether Kepler had any legal right to Tycho's records is debatable, but he put them to good use.