Back to the front page  Figures AF Kepler and the First Law of Planetary Motion Jenny Hwang Johannes Kepler was an astronomer, mathematician, theologian and philosopher. His many achievements are commendable but it is one particular triumph which is familiar to many. That is the discovery of Kepler's Laws of Planetary Motion. I will concentrate on his first law: the orbits of the planets are ellipses with the sun at one focus. Kepler was born in WeilderStadt, Germany in December 1571 to Heinrich Kepler and Katharina Guldenmann. His education started at the German Schreibschule in Leonberg, Germany as a result of his family's relocation. Later, he transferred to the Latin school, building a foundation for the language to be used in his future publications. In 1584, he entered the Adelburg Monastery school and Maulbronn, preparatory school for the University of Tübingen, in 1586. During his stay at Tübingen, Kepler was especially influenced by his professor of astronomy, Michael Mästlin. Mästlin was a quiet proponent of Copernican astronomy, which supports the idea of a heliocentric (suncentered) universe. His thorough knowledge of the ideas of Copernicus sparked Kepler's interest in astronomy and created another Copernican advocate. Ironically, Kepler was a reluctant astronomer. Of his interest in the subject, Kepler states, "these were prescribed studies and nothing indicated to me a particular bent for astronomy." Nevertheless, he was a "straight A student" and when he went to renew his scholarship at Tübingen, the senate spoke of Kepler as having "such a superior and magnificent mind that something special may be expected of him." [2] After receiving his masters degree from Tübingen in 1591, Kepler continued his studies in theology. But in his last year of courses, the professor of mathematics at the Lutheran school in Graz died and the University of Tübingen was asked to recommend a replacement. The school chose Kepler. Reluctantly, he abandoned his initial calling to be a clergyman and, at the age of 22, began his duties as a math instructor in Graz. Just after a year of residency at Graz, Kepler published his first work, Mysterium cosmographicum (1596). In it, he presents his imaginative theories on the "secrets of the universe", the quest for which was inspired by his religious convictions. In the name of unveiling God's mathematical plan of the universe, Kepler concludes that the existance of six planets is due to the existance of five perfect polyhedrons. Of course this was based on the "fact" that there were only six planets in the universe and only five perfect polyhedrons. Around Earth's orbit, Kepler circumscribes a perfect dodecahedron and the sphere containing this is Mars' orbit. Similarly, around Mars' sphere, a tetrahedron is circumscribed and the sphere containing this is Jupiter's orbit. Inscribing an icosahedron in Earth's orbital sphere, the resulting inscribed sphere is Venus. This is done with the rest of the planets using the remainder of the perfect polyhedrons. Astonishingly, the ratios of adjacent planetary orbits represented in Kepler's nested spheres model coincides with Copernicus' calculations. Of course, Kepler relied mostly on divine inspiration for this theory. There was no real basis for his argument. The importance of Mysteruim cosmographicum comes mostly from the fact that it was the first publicly Copernican treatise since Copernicus himself. Kepler was quite a revolutionary. Even his search for reasons why the planets moved the way they did was a bold endeavor, contrasting with the medieval tradition of simply accepting information without question. In September 1598, the Protestants in Graz, including Kepler, were ousted from the city by the Catholic rulers. Although Kepler was allowed to return, conditions were tense. Looking for relief, he turned to Tycho Brahe. Brahe, Imperial Mathematician for the Emperor Rudolph II and renowned for his impressive collection of planetary observations, knew of Kepler from Mysterium cosmographicum. When Kepler arrived at his Benatky Castle Observatory outside of Prague, in 1600, Brahe welcomed him as another aid to his own work, instead of an equal. A tumultuous relationship between them unsued until Brahe's death a year later. It was then that Brahe finally entrusted Kepler with his jealously guarded treasure of observations. As a final request, Brahe relied upon Kepler to finish the Rudophine Tables of planetary motion. Two days later, Kepler was named Imperial Mathematician. Without his strenuous relationship with Brahe, Kepler was free to pursue his own areas of interest in astronomy. He decided to continue on his observation of Mars. Kepler was interested in documenting its orbit because he believed that if he could discover the particulartly confusing path of this planet it would be possible to uncover the orbits of all the other planets. Contrary to popular belief, he did not simply use Brahe's observations and pick a geometrical figure that fit the description. He was interested in discovering a "physical theory of planetary motion, from which the orbits could be deduced." [1] Before proceeding to explain Kepler's study of the orbits, a few astronomical terms should be clarified. eccentric circle: Planet, P, moves uniformily around a circle with center, C, which is eccentric to the sun, S. equant point: Introduced by Ptolemy, the planets do not move with uniform velocity but with constant angular velocity about another point, the equant point. Thus, the planet, P, moves about the eccentric circle with uniform velocity about the equant point, E. According to Ptolemy, best approximation of the position of E was EC=CS (Figure A) [1]. aphelion: The point on the orbit where the planet is farthest from the sun. perihelion: The point on the orbit where the planet is closest to the sun. Kepler began his quest to understand the orbit of Mars by, first, describing Earth's orbit. After all, Mars was observed from a moving Earth. It was traditionally believed that the orbits were circlular and Kepler didn't stray from the idea for Earth's orbit. Also, he already noted that the linear speed of the planets decreased with distance from the sun in Mysterium cosmographicum and with Brahe's observations supporting the idea, Kepler incorporated the equant point in the orbit. He located the position of the equant point in relation to the center of the orbit and the position of the sun and concluded that Earth's velocity at aphelion and perihelion varied inversely with distance from the sun. This relation was generalized for the rest of the orbit (which was false) and was labeled the distance law. Without the conveniences of calculus available to him, Kepler expressed this relation as:
Where is called the eccentric anomaly, t is time, is the distance for eccentric anomaly i, and i is measured (Figure B). This calculation is, no doubt a tedious task. Therefore, Kepler opted to approximate the distance value using area. This approximation proved to be reliable in predicting the positions of the Earth. Thus, the planet sweeps out equal areas at equal times. The relation was: [1] With Earth's orbit defined (it's orbit is very close to a circle so Kepler's model fit with observed data), Kepler moved on to Mars. Again, he started with a circular orbit but when the area law was applied to this model, Kepler found that his model produced discrepancies of up to 8'. At degrees, the error was +8'. At , the error was 8'. This meant that the planet on Kepler's model was moving too quickly at aphelion and perihelion and too slowly at the sides ( or ) (Figure C). To resolve this, Kepler decided to change the shape of the orbit. Bringing the sides of the orbit inside the circle would result in a more accurate representation of Mars' actual position. Indeed, further investigation on the position of Mars at different times of the year confirmed that its orbit did bow in from a circle. Thus, Kepler felt justified in making the change. The orbit that followed was more similar to the shape of an oval. But how could Kepler use the area law on this strange shape? For purposes of calculation, he chose to approximate the oval by an ellipse. The area law could be applied to an ellipse because the areas of ellipse sectors were known through the work of Archimedes. Through trial and error, Kepler found that making the semiminor axis and the semimajor axis equal to 1 was the correct ellipse (where e is the distance CS. Inscribing the ellipse in the unit circle (Figure D): Given: PC=1. For , , . BUT Kepler realized: He also knew that , the angle between PC and PS. Thus, Then, Kepler became inspired. What if this relation were true for all angles, ? Then, for any angle, , between CA and CP, the ratio of the distance, SP, to the actual sunMars distance would equal the ratio of SP with the perpendicular projection, PT, on a diameter of the circle. Thus, the sunMars distance=PT=PC+CT (Figure E). So, sunMars distance= . Knowing this distance, how was it to be portrayed in the orbital model? At first, Kepler decided to place Mars at point Q ,in Figure E, on the radius of the eccentric anomaly. So, the distance would have to correspond to the line connecting the sun to a point on the eccentric anomaly, Q. However, this was false. The curve traced out did not agree with observation. There were errors of up to 8'. Finally, Kepler discovered that the distance, , should apply to the segment connecting the sun with the point, W, on the line perpendicular to CA (Figure F). The curve traced by this model was an ellipse and it predicted the correct positions of Mars. Consequently, Kepler accepted this ellipse, a shape initially used for approximation, as the true orbit of Mars. Also, Kepler was able to prove that the orbit traced out was an ellipse. KEPLER'S PROOF: Given an ellipse with center= , where e=CS, let v= the angle subtended by the arc, WA then, and . Squaring and adding both equations:
(neglecting the term in )
Thus, the ellipse can be written as , which is also the value that Kepler worked out before from his orbital model. Of course to work out the location of the focus:
(neglecting the term in ) Thus, c=e, which is precisely the distance of the sun from the center in Kepler's model. Therefore, the sun is at one focus and Kepler's first law of planetary motion was born. After eight years of calculations, Kepler's first law of planetary motion was incorporated into the Astronomia nova, along with his second law: a planet sweeps out equal areas in equal times, which was mentioned earlier. Kepler's third law, dealing with the ratios of periods of planetary orbits appears in Harmonice mundi, although it is not stated as clearly as the other two laws. The astronomer also published several minor works, including Stereometria doliorum vinariorum, considered to be a significant work in the prehistory of calculus. Epitome astronomiae Copernicanae is also worth mentioning. It is an eight volume collection on Kepler's heliocentric astronomy and includes his three laws of planetary motion. The Tabulae Rudolphinae that Brahe had wanted Kepler to work on was finally published in 1627. It was a table of planetary positions far more accurate than any other consisting of 119 pages of tables. Besides planetary observation, there were other subjects that entertained Kepler. He took an interest in optics and was noted, by Descartes, as being the most knowledgable of his time on the subject. Even Galileo looked towards Kepler for approval of his publications [2]. Another of Kepler's publications concentrates on the hexagonal trait of snowflakes. Although he disparaged its practice, Kepler also participated in astrological prognostications. He published several calendars while in Graz and Prague with enough accuracy to earn him some early fame. It also provided income when Kepler's financial situation was less than favorable. Despite his success, Kepler was continually plagued with financial trouble. The turbulance of the times and Kepler's constant relocation, in search of religious tolerance, contributed to positions with witheld income. Several times, he tried to secure a teaching position at Tübingen but, he was perceived as a renegade by fellow Protestants and refused. Johannes Kepler died on November 15, 1630. He was on his way to collecting his backpay when stricken by an acute fever and was buried in a Protestant cemetary. Kepler's epitaph as noted by his soninlaw: Mensus eram coelos, nunc terrae metior umbras: Mens coelestis erat, corporis umbra jacet. I used to measure the heavens, now I shall measure the shadows of the earth. Although my soul was from heaven, the shadow of my body lies here.
BIBLIOGRAPHY 1. Aiton, Eric J, How Kepler Discovered the Elliptical Orbit, The Mathematical Gazette, v.59, 1975 2. Gillispie, Charles Coulston, Dictionary of Scientific Biography, v.7, Charles Scribner's Sons, New York, 1981 3. Katz, Victor J, A History of Mathematics, AddisonWesley, 1998 4. Kepler, Johannes, Astronomia nova, Heidelberg, 1609 5. Kepler, Johannes, Harmonice mundi, Linz, 1618 6. Kepler, Johannes, Mysterium cosmographicum, Graz, 1596 7. Kepler, Johannes, New Astronomy, translated by Donahue, William H, Cambridge University Press, Great Britain, 1992 8. Wilson, Curtis, How Did Kepler Discover His First Two Laws?, Scientific American, v.226, 1972
