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Johannes Kepler The Harmonies of the World “By the study of the orbit of Mars, we must either arrive at the secrets of astronomy or forever remain in ignorance of them.â€ Johannes Kepler Biography Kepler, Johannes (1571-1630) who was a German astronomer and natural philosopher is noted for coming up with and verifying the three laws of planetary motion. These laws are now known as Kepler's laws of Planetary Motion. Johanne was born on December 27, 1571. Childhood Johanne was born in Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576. His father was a mercenary soldier and his mother was the daughter of an innkeeper. He was their first child. His father left home for the last time when Johannes was five, and is believed to have died in a war in the Netherlands. Whenever he was a child, Kepler lived with his mother in his grandfather's inn. Sources said that he used to help by serving in the inn. Customers were amused by the child's unusual competence at math. 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 enroll at the University of TÑŒbingen, a bastion of Lutheran orthodoxy. Johannes Kepler Leaving Prague for Linz Johanne’s years in Prague were peaceful, and scientifically productive. In fact, even when things went badly, he never seemed to have allowed external circumstances to prevent him from getting on with his work. Things began to go very badly in late 1611. His seven year old son died. Kepler wrote to a friend that this death was particularly hard because the child reminded him so much of himself at that age. Then his wife died. Emperor Rudolf, whose health was failing, was forced to abdicate in favor 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 Johanne seemed 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. A dedicatory letter to the resultant book explains that at the wedding celebrations he noticed that the barrels of wine barrels were estimated by means of a rod slipped in diagonally through a hole, and he began to wonder how that could work. The result was a study of the volumes of solids of revolution (New Stereometry of wine barrels, 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 - 1547) and is part of the ancestry of the infinitesima. Development of Laws He was influenced by a mathematics professor, Michael Maestlin, an advocate of the heliocentric theory of planetary motion first developed by Nicolaus Copernicus. Kepler accepted the Copernican Theory immediately. He believed that the simplicity of Copernican planetary ordering must have been God's plan. In 1594, when Kepler left TÑŒbingen for Graz, Austria, he worked out a complex geometric hypothesis to account for distances between the planetary orbits. Orbits that he mistakenly assumed were circular. (Kepler later found that planetary orbits are elliptic; nevertheless, these preliminary calculations agreed with observations to within 5 percent.) Kepler then proposed that the sun emits a force that diminishes inversely with distance and forces the planets around in their orbits. Kepler published his account in a treatise entitled Mysterium Cosmographicum (Cosmographic Mystery) in 1596. This work is significant because it presented the first comprehensive and cogent account of the geometrical advantages of Copernican theory. Kepler held the chair of astronomy and mathematics at the University of Graz from 1594 until 1600, when he became assistant to the Danish astronomer Tycho Brahe in the observatory near Prague. Kepler assumed his position as imperial mathematician and court astronomer to Rudolf II, Holy Roman emperor on the death of Brahe in 1601. One of his major works during this period was Astronomia Nova (New Astronomy, 1609), the great culmination of his painstaking efforts to calculate the orbit of Mars. This treatise contains statements of two of Kepler's so-called laws of planetary motion. The first is that the planets move in elliptic orbits with the sun at one focus. The second states that a hypothetical line from the sun to a planet sweeps out equal areas of an ellipse during equal intervals of time; in other words, the closer a planet comes to the sun, the more rapidly it moves. In 1612 Kepler became mathematician to the states of Oberцsterreich (Upper Austria). While living in Linz, he published his Harmonice Mundi (Harmony of the World, 1619), the final section of which contained another discovery about planetary motion: The ratio of the cube of a planet's distance from the sun and the square of the planet's orbital period is a constant and is the same for all planets. Ta2 / Tb2 = Ra3 / Rb3 At about the same time he began publishing a book that took three years to appear, the Epitome Astronomiae Copernicanae (Epitome of Copernican Astronomy, 1618-1621), which brought all of Kepler's discoveries together in a single book. Equally important, it became the first textbook of astronomy to be based on the Copernican theory, and for the next three decades it was a major influence in converting many astronomers to Keplerian Copernicanism. The last major work to appear in Kepler's lifetime was the Tabulae Rudolfinae (Rudolfine Tables, 1625). Based on Brahe's data, the new tables of planetary motion reduced the mean errors from 5° to within 10ў of the actual position of a planet. The English mathematician Sir Isaac Newton relied heavily on Kepler's theories and observations in formulating his theory of gravity. Johanne also made contributions in the field of optics and developed a system of infinitesimals in mathematics, which was a forerunner of calculus. Death 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. Copernican System The Copernican System is the systematic explanation of the movement of the planets around the sun; advanced in 1543 by the Polish astronomer Nicolaus Copernicus. The Copernican system advanced the theories that the earth and the planets are all revolving in orbits around the sun, and that the earth is spinning on its north-south axis from west to east at the rate of one rotation per day. These two hypotheses superseded the Ptolemaic system, which had been the basis of astronomical theory until that time. The Copernican system first described the precession of the equinoxes but did not explain it. Publication of the Copernican system stimulated the study of astronomy and mathematics and laid the basis for the discoveries of Johannes Kepler and Sir Isaac Newton. Kepler's Laws of Planetary Motion The planets orbit the sun in elliptical paths, with the sun at one focus of the ellipse. Equation for ellipse: x2/a2=y2/b2 The orbit of a planet around the Sun is an ellipse with the Sun's center of mass at one focus. The areas described in a planetary orbit by the straight line joining the center of the planet and the center of the sun are equal for equal time intervals; that is, the closer a planet comes to the sun, the more rapidly it moves. A line joining a planet and the Sun sweeps out equal areas in equal intervals of time. The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time. The ratio of the cube of a planet's mean distance, "d", from the sun to the square of its orbital period, "t", is a constant-that is, d3/t2 is the same for all planets. The squares of the periods of the planets are proportional to the cubes of their semimajor axes. Ta2 / Tb2 = Ra3 / Rb3 Mathematical statement: T = kR3/2 , where T = sidereal period, and R = semi-major axis Example - If a is measured in astronomical units (AU = semi-major axis of Earth's orbit) and sidereal period in years (Earth's sidereal period), then the constant k in mathematical expression for Kepler's third law is equal to 1, and the mathematical relation becomes T2 =R3 Bibliography Robert S. Westman Encarta 97 CD-Rom http://www.cvc.org/science/kepler.htm Internet-Laws of Planetary Motion http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html Internet-Biography http://observe.ivv.nasa.gov/nasa/education/reference/orbits/orbit_sim.html Internet- Laws of Planetary Motion http://csep10.phys.utk.edu/astr161/lect/history/kepler.html Internet- Laws of Motiion
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Johannes Kepler Johannes Kepler (1571. – 1630) bio je siromasni protestantski svestenik. 1593. godine postao profesor matematike i filozofije na ucilistu u Grazu, a osim predavanjima i naucnim radom bavio se sastavljanjem kalendara te, kako bi zaradio za zivot, izradom horoskopa. Kepler je poznat kao jedan od najsistematicnijih umova u nauci uopste. Godinama je strpljivo izradjivao stotine stranica proracuna i nije se zadovoljavao osrednjim slaganjem teorije i promatranja. Zanimljiva je cinjenica da se Kepler uvijek pokoravao promatranjima Njegove rijeci su: Geometrija je sam Bog. Bio je pod uticajem pitagorejaca. Upustio se u potragu za brojenom harmonijom medju nebeskim sferama, pocevsi od pracenja proporcija brojeva i njihovih kvadrata, trazeci podudarnosti oblika i njihovih kombinacija, metodom poznatom kao pitagoriziranje. U to je vrijeme, uz Zemlju, bilo poznato 6 planeta za koje je pokusao pronaci brojni odnos izmedju njihovih udaljenosti od Sunca. Isprobao je razne kombinacije i dosao da geometrijske seme koju je izlozio u svom radu Kozmografska tajna (Mysterium cosmographicum). Nakon sto je upoznao Keplerovo djelo, danski plemic Tycho Brahe, vlasnik tada najbogatije i najkvalitetnije zbirke astronomskih mjerenja, pozvao ga je u Prag. Kepler pise: `Tycho mi ne pruza priliku da se upoznam s njegovim iskustvom. On bi mi samo spomenuo, za vrijeme jela ili povodom drugih stvari, onako usput, danas vrijednost apogeja jednoga planeta, sutra presjeciste nekoga drugog. Tycho raspolaze najboljim podacima promatranja. Ima i suradnike. Jedino mu nedostaje graditelj koji bi sve to znao iskoristiti`. Tycho je bio najveci promatracki genij toga doba, a Kepler najveci teoreticar. I jedan i drugi su znali da pojedinacno ni jedan od njih ne moze ostvariti sintezu tocnoga i suvisloga kozmickog sistema, ali osjecali su da je sinteza tu na vidiku. No, Tycho nije htio da svoje zivotno djelo pokloni jednom znatno mladjem potencijalnom suparniku. Osim toga, iz nekog razloga, nije dolazilo u obzir ni zajednicko autorstvo nad rezultatima. Rodjenje moderne znanosti – djeteta teorije i promatranja, kolebalo se nad provalijom uzajamnoga nepovjerenja dvoje ljudi. Na samrtnoj je postelji Brahe zaklinjao Keplera da ne zaboravi sistem koji je on zastupao te ga savjetovao da svoje istrazivanje usredotoci na opazacke podatke Marsa. U svojim se kasnijim radovima Kepler savjesno drzao obecanja, a neizmjerno postovanje koje je gajio za Tycha Brahea dokazuje i cinjenica da je 25 godina kasnije svoje velike planetarne tablice Tabuli Rudolphinae posvetio njegovoj uspomeni. Dva je dana nakon Braheove smrti 1601. godine Kepler, tada tridesetogodisnjak, imenovan carskim matematicarom. Proucavanjem je astronomskih mjerenja naslijedjenim od Brahea dosao do spoznaje da su planetne putanje elipse, a ne kruznice kako se do tada mislilo i u sto je isprva bio i on uvjeren. Buduci da je ekscentricitet planetnih elipsa u Suncevu sustavu mali, do toga otkrica nije bilo lako doci. O napustanju ideje kruznica Kepler pise „...istina prirode, koju sam stalno odbacivao i tjerao, tajno se provukla na mala vrata prerusivsi se da bi bila prihvacena. Ah, kakva sam glupa tikva bio.“ Svoje je proracune Kepler izlozio u knjizi Nova astronomija ili nebeska fizika s komentarima gibanja planeta Marsa prema motrenjima Tycha Brahea koja je tiskana u Pragu 1609. godine. Tu su izlozena prva dva zakona koja danas zovemo Keplerovim zakonima. Planetne su putanje elipse kojima se Sunce nalazi u jednom zaristu. Planeti se po elipsama gibaju nejednoliko, brze kada su blize Suncu, a sporije kada su dalje, odnosno gibaju se tako da spojnica Sunce – planet u jednakim vremenskim razmacima prebrise jednake povrsine. Trazeci matematicku vezu izmedju udaljenosti planeta oko Sunca i vremena njihova ophoda, Kepler dolazi do trecega zakona, odnosno do spoznaje da su kvadrati vremena obilaska planeta razmjerni kubovima njihovih srednjih udaljenosti od Sunca. Taj je zakon objavio 1619. godine u radu Harmonija svijeta (Harmonice mundi). Pod uticajem platonizma Kepler naucava da se sve u svijetu privlaci. Ne samo Zemlja privlaci kamen, nego i kamen privlaci Zemlju. Plima i oseka su posljedice privlacnoga djelovanja Sunca i Mjeseca. Kepler je poopstio problem privlacenja. Ipak, Keplerova je zasluga da je pojam sile preuredio iz platonistickoga oblika u relacijski pojam, tj. u matematicki iskaz. Keplerovi zakoni opisuju kretanje planeta oko Sunca. Keplerova zakoni - Prvi Keplerov zakon. Svi planeti krecu se po elipsama kojima je u jednom od fokusa Sunce. - Drugi Keplerov zakon. Radijus-vektor Sunce-planet prelazi u jednakim vremenskim razmacima jednake povrsine. - Treci Keplerov zakon. Kvadrati ophodnih vremena planeta proporcionalni su kubovima njihovih srednjih udaljenosti od Sunca. Kretanje satelita oko maticnog planeta i svaki drugi slican sistem takodjer se opisuju Keplerovim zakonima. I keplerov zakon Planeti se oko Sunca krecu po elipticnim putanjama; u zajednickom fokusu tih elipsa nalazi se Sunce. Geometrija planetarnog kretanja: planet (M) obilazi oko Sunca (S) po elipsi (P-perihel, A-afel) Na prikazanoj slici elipsa predstavlja putanju nekog planeta. Linearni ekscentricitet je: Ovo je matematski izraz prvoga Keplerova zakona. Iz te se jednacine e vidi zavisnost udaljenosti planeta od Sunca o uglu v koji se naziva pravom anomalijom planeta. Kad je v=π/2 , tada je r=p , pa iz toga proizlazi definicija parametra planetske putanje kao radijus-vektora planeta koji je normalan na glavnu os elipse. Karakteristicne tocke planetske putanje su afel i perihel ). Afel je tacka u kojoj je planet najudaljeniji od Sunca, a perihel je tocka u kojoj je planet najblizi Suncu. Ako je a srednja udaljenost planeta od Sunca, onda proizlazi da su: