Kepler's Third

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Epreuve orale spécifique SECTION EUROPEENNE Baccalauréat général Sciences Physiques Académie d’Amiens Langue anglaise durée : 20min Kepler’s third law Kepler's third law, which is often called the harmonic law, is a mathematical relationship between the time it takes the planet to orbit the Sun and the distance between the planet and the Sun. The time it takes for a planet to orbit the Sun is its orbital period, which is often simply called its period. For the average distance between the planet and the Sun, Kepler used what we call the semi-major axis of the ellipse. Think of it as the longest radius of the ellipse. Kepler's third law states that the square of the period, P, is proportional to the cube of the semi-major axis, a. In equation form Kepler expressed the third law as: P²=ka3. k is the proportionality constant. To Kepler it was just a number that he determined from the data. Significance of Kepler's Third Law Kepler's third law is extremely important to astronomers. Because it involves the mass it allows astronomers to find the mass of any astronomical object with something orbiting it. Astronomers find the masses of all astronomical objects by applying Kepler's third law to orbits. They measure the mass of the Sun by studying the orbits of the planets. They measure the mass of the planets by studying the orbits of their moons. Moons have nothing orbiting them, so to find the mass of the moons astronomers need to send a probe to be affected by their gravity. In all these cases astronomers use Kepler's third law. Extrait du site: http://www.suite101.com/content/johannes-kepler-1571-1630-a84571, le 23/01/11 Thanks to Kepler’s laws you will explain how the astronomers can have a better knowledge of the universe. .

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The Astronomers Tycho Brahe and Johannes Kepler
Ice Core Records – From Volcanoes to Supernovas The Astronomers Tycho Brahe and Johannes Kepler Tycho Brahe (1546-1601, shown at left) was a nobleman from Denmark who made astronomy his life's work because he was so impressed when, as a boy, he saw an eclipse of the Sun take place at exactly the time it was predicted. Tycho's life's work in astronomy consisted of measuring the positions of the stars, planets, Moon, and Sun, every night and day possible, and carefully recording these measurements, year after year. Johannes Kepler (1571-1630, below right) came from a poor German family. He did not have it easy growing Tycho Brahe up. His father was a soldier, who was killed in a war, and his mother (who was once accused of witchcraft) did not treat him well. Kepler was taken out of school when he was a boy so that he could make money for the family by working as a waiter in an inn. As a young man Kepler studied theology and science, and discovered that he liked science better. He became an accomplished mathematician and a persistent and determined calculator. He was driven to find an explanation for order in the universe. He was convinced that the order of the planets and their movement through the sky could be explained through mathematical calculation and careful thinking. Johannes Kepler Tycho wanted to study science so that he could learn how to predict eclipses. He studied mathematics and astronomy in Germany. Then, in 1571, when he was 25, Tycho built his own observatory on an island (the King of Denmark gave him the island and some additional money just for that purpose).
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Thinking Outside the Sphere Views of the Stars from Aristotle to Herschel Thinking Outside the Sphere
Thinking Outside the Sphere Views of the Stars from Aristotle to Herschel Thinking Outside the Sphere A Constellation of Rare Books from the History of Science Collection The exhibition was made possible by generous support from Mr. & Mrs. James B. Hebenstreit and Mrs. Lathrop M. Gates. CATALOG OF THE EXHIBITION Linda Hall Library Linda Hall Library of Science, Engineering and Technology Cynthia J. Rogers, Curator 5109 Cherry Street Kansas City MO 64110 1 Thinking Outside the Sphere is held in copyright by the Linda Hall Library, 2010, and any reproduction of text or images requires permission. The Linda Hall Library is an independently funded library devoted to science, engineering and technology which is used extensively by The exhibition opened at the Linda Hall Library April 22 and closed companies, academic institutions and individuals throughout the world. September 18, 2010. The Library was established by the wills of Herbert and Linda Hall and opened in 1946. It is located on a 14 acre arboretum in Kansas City, Missouri, the site of the former home of Herbert and Linda Hall. Sources of images on preliminary pages: Page 1, cover left: Peter Apian. Cosmographia, 1550. We invite you to visit the Library or our website at www.lindahlll.org. Page 1, right: Camille Flammarion. L'atmosphère météorologie populaire, 1888. Page 3, Table of contents: Leonhard Euler. Theoria motuum planetarum et cometarum, 1744. 2 Table of Contents Introduction Section1 The Ancient Universe Section2 The Enduring Earth-Centered System Section3 The Sun Takes
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Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich hätt sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
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Chapter 1 Chapter 2 Chapter 3
Notes CHAPTER 1 1. Herbert Westren Turnbull, The Great Mathematicians in The World of Mathematics. James R. Newrnan, ed. New York: Sirnon & Schuster, 1956. 2. Will Durant, The Story of Philosophy. New York: Sirnon & Schuster, 1961, p. 41. 3. lbid., p. 44. 4. G. E. L. Owen, "Aristotle," Dictionary of Scientific Biography. New York: Char1es Scribner's Sons, Vol. 1, 1970, p. 250. 5. Durant, op. cit., p. 44. 6. Owen, op. cit., p. 251. 7. Durant, op. cit., p. 53. CHAPTER 2 1. Williarn H. Stahl, '' Aristarchus of Samos,'' Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 1, 1970, p. 246. 2. Jbid., p. 247. 3. G. J. Toorner, "Ptolerny," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 11, 1975, p. 187. CHAPTER 3 1. Stephen F. Mason, A History of the Sciences. New York: Abelard-Schurnan Ltd., 1962, p. 127. 2. Edward Rosen, "Nicolaus Copernicus," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 3, 1971, pp. 401-402. 3. Mason, op. cit., p. 128. 4. Rosen, op. cit., p. 403. 391 392 NOTES 5. David Pingree, "Tycho Brahe," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 2, 1970, p. 401. 6. lbid.. p. 402. 7. Jbid., pp. 402-403. 8. lbid., p. 413. 9. Owen Gingerich, "Johannes Kepler," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 7, 1970, p. 289. 10. lbid.• p. 290. 11. Mason, op. cit., p. 135. 12. Jbid .. p. 136. 13. Gingerich, op. cit., p. 305. CHAPTER 4 1.
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Deriving Kepler's Laws of Planetary Motion
DERIVING KEPLER’S LAWS OF PLANETARY MOTION By: Emily Davis WHO IS JOHANNES KEPLER? German mathematician, physicist, and astronomer Worked under Tycho Brahe Observation alone Founder of celestial mechanics WHAT ABOUT ISAAC NEWTON? “If I have seen further it is by standing on the shoulders of Giants.” Laws of Motion Universal Gravitation Explained Kepler’s laws The laws could be explained mathematically if his laws of motion and universal gravitation were true. Developed calculus KEPLER’S LAWS OF PLANETARY MOTION 1. Planets move around the Sun in ellipses, with the Sun at one focus. 2. The line connecting the Sun to a planet sweeps equal areas in equal times. 3. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of the ellipse. INITIAL VALUES AND EQUATIONS Unit vectors of polar coordinates (1) INITIAL VALUES AND EQUATIONS From (1), (2) Differentiate with respect to time t (3) INITIAL VALUES AND EQUATIONS CONTINUED… Vectors follow the right-hand rule (8) INITIAL VALUES AND EQUATIONS CONTINUED… Force between the sun and a planet (9) F-force G-universal gravitational constant M-mass of sun Newton’s 2nd law of motion: F=ma m-mass of planet r-radius from sun to planet (10) INITIAL VALUES AND EQUATIONS CONTINUED… Planets accelerate toward the sun, and a is a scalar multiple of r. (11) INITIAL VALUES AND EQUATIONS CONTINUED… Derivative of (12) (11) and (12) together (13) INITIAL VALUES AND EQUATIONS CONTINUED… Integrates to a constant (14) INITIAL VALUES AND EQUATIONS CONTINUED… When t=0, 1.
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Gravity, Orbital Motion, and Relativity
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The Mass of the Sun
Name Date The Mass of the Sun Orbits The planets in our Solar System all orbit the Sun in elliptical orbits. Most planetary orbits are, however, almost circular. Since the planets are moving in circles they must be experiencing a centripetal acceleration. We know that the centripetal acceleration on an orbiting body (e.g. a planet) is given by: = where r is the distance from the central body (e.g. the Sun), and v is the velocity it is moving at. The centripetal acceleration is caused by the gravitational attraction of the two bodies, which is given by: = where M is the mass of the central body, m is the mass of the orbiting body, and G is Newton’s Gravitational constant and is equal to 6.67 x 10-11 m3 s-2 kg-1 Since the centripetal acceleration is caused by the gravitational force, we use Newton's 2nd law: = Putting the first two equations into the third one we therefore know that: = We can then work out a relationship between the velocity, the mass of the central object, and the orbital radius: = The Mass of the Sun Name Date Kepler's Laws of Planetary Motion In 1605 Johannes Kepler wrote down three laws of planetary orbits: 1. The orbit of every planet is an ellipse with the Sun at a focus. 2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3. The square of the orbital period of a planet is directly proportional to the cube of the semi- major axis of its orbit.
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The Golden Ratio
IOSR Journal Of Applied Physics (IOSR-JAP) e-ISSN: 2278-4861.Volume 12, Issue 2 Ser. II (Mar. – Apr. 2020), PP 36-57 www.Iosrjournals.Org The Golden Ratio Nafish Sarwar Islam Abstract:This paper is all about golden ratio Phi = 1.61803398874989484820458683436563811772030917980576286213544862270526046281890 244970720720418939113748475408807538689175212663386222353693179318006076672635 443338908659593958290563832266131992829026788067520876689250171169620703222104 321626954862629631361443814975870122034080588795445474924618569536486444924104 432077134494704956584678850987433944221254487706647809158846074998871240076521 705751797883416625624940758906970400028121042762177111777805315317141011704666 599146697987317613560067087480710131795236894275219484353056783002287856997829 778347845878228911097625003026961561700250464338243776486102838312683303724292 675263116533924731671112115881863851331620384005222165791286675294654906811317 159934323597349498509040947621322298101726107059611645629909816290555208524790 352406020172799747175342777592778625619432082750513121815628551222480939471234 145170223735805772786160086883829523045926478780178899219902707769038953219681 986151437803149974110692608867429622675756052317277752035361393621076738937645 560606059216589466759551900400555908950229530942312482355212212415444006470340 565734797663972394949946584578873039623090375033993856210242369025138680414577 995698122445747178034173126453220416397232134044449487302315417676893752103068 737880344170093954409627955898678723209512426893557309704509595684401755519881
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4. Central Forces
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Kepler's Laws for Planets and Satellites Learning Objectives
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The Short History of Science
PHYSICS FOUNDATIONS SOCIETY THE FINNISH SOCIETY FOR NATURAL PHILOSOPHY PHYSICS FOUNDATIONS SOCIETY THE FINNISH SOCIETY FOR www.physicsfoundations.org NATURAL PHILOSOPHY www.lfs.fi Dr. Suntola’s “The Short History of Science” shows fascinating competence in its constructively critical in-depth exploration of the long path that the pioneers of metaphysics and empirical science have followed in building up our present understanding of physical reality. The book is made unique by the author’s perspective. He reflects the historical path to his Dynamic Universe theory that opens an unparalleled perspective to a deeper understanding of the harmony in nature – to click the pieces of the puzzle into their places. The book opens a unique possibility for the reader to make his own evaluation of the postulates behind our present understanding of reality. – Tarja Kallio-Tamminen, PhD, theoretical philosophy, MSc, high energy physics The book gives an exceptionally interesting perspective on the history of science and the development paths that have led to our scientific picture of physical reality. As a philosophical question, the reader may conclude how much the development has been directed by coincidences, and whether the picture of reality would have been different if another path had been chosen. – Heikki Sipilä, PhD, nuclear physics Would other routes have been chosen, if all modern experiments had been available to the early scientists? This is an excellent book for a guided scientific tour challenging the reader to an in-depth consideration of the choices made. – Ari Lehto, PhD, physics Tuomo Suntola, PhD in Electron Physics at Helsinki University of Technology (1971).
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