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Lecture #5 Copernicus, Kepler, Galileo, and Newton 11 June 2012 Eratosthenes of Cyrene (c. 276 BCE – c. 195 BCE) June 19, 240 BCE Hipparchus of Nicaea (c. 190 BCE – c. 120 BCE) z Developed trigonometry. z Probably the best astronomical observer of antiquity. z z z He constructed a very large, reasonably accurate star catalog, and devised the magnitude system for stellar brightnesses. He measured the longitude of Spica (the brightest star in the constellation Virgo) sometime in the period 147 to 127 BCE. Comparing that to the results from Timocharis and Aristillus, roughly a century earlier, he determined that it had moved by 2o. Well, not moved itself; rather the longitude system had moved with respect to Spica. From this, he determined that the Earth's axis of rotation moves slowly, tracing a circle on the sky. The Political Situation z 30 BCE – the Romans under Octavian (later Augustus) annexed Egypt and the Ptolemaic empire. Claudius Ptolemy (c. 90 CE – c. 168 CE) z z z Another academic at the Library of Alexandria Wrote the Almagest (as it was later called in Arabic), the only surviving major astronomical treatise from antiquity. In Planetary Hypotheses, he laid out a physical structure of nested spheres to explain planetary motions. This introduced the epicycle-deferent-equant concept. z z z Concept of “saving the appearances”. His Geographia was the great summation of all that was known about the world at the time. His maps were scientific, in that they attempted to handle the projection of the sphere onto a flat map correctly. His Harmonics revived the Pythagorean ratio concept, and emphasized how such ratios related to mathematical equations. Claudius Ptolemy (c. 90 CE – c. 168 CE) Ptolemy's particular contribution Skipping over much history z z z z z z Ptolemy's system was picked up by Islamic scholars who, in the 8th to 10th centuries were avid translators of Greek texts and aggressive pursuers of the ideas therein. There was a significant argument between those who felt the Greek knowledge was irrelevant to Islam and those who did not. Islamic follow-through on Greek science declined from the 11th century on, as one side took ascendance. At about that time, though, a number of Western European scholars became interested in both the Greek and Arabic work, and began an active translation movement, with a special interest in the works of Aristotle. Again, there were pro- and anti-Aristotle factions, as well as pro- and anti-Averroes (Ibn Rushd) factions. (Ibn Rushd had promoted a reconciliation of faith and reason in order to pursue science,) From the 13th century on, much work on natural philosophy was carried out in the universities, which spread rapidly from Bologna, Paris, and Oxford to the rest of Europe. Mikołaj Kopernik (Nicolaus Copernicus; 1473 – 1543) z z z Born in Torun (Thorn), part of Poland under the Kingdom of Prussia. Studied from 1491 to 1495 at the University of Krakow (now Jagiellonian University), leaving without a degree. Studied canon law and humanities from 1496 to 1501 at the University of Bologna. z z Studied medicine from 1501 to 1503 at the University of Padua. z z Became more focused on astronomy, reading complaints by Puerbach and Regiomontanus about the Ptolomaic system, and making observations of Aldeberan that seemed to confirm their concerns. Began studying Greek to be able to work with the original texts that related both to astronomy and to general humanist ideas. Eventually settled as an administrator (and eventually chancellor) in the city of Frauenberg, in the region of Warmia, in the NE part of Poland (but full of Prussians; the original Warmians were a Prussian tribe – and Warmia was eventually annexed by Prussia in the First Partition of Poland, 1772). The Political Situation z z z z z z Martin Luther posted his Disputation of Martin Luther on the Power and Efficacy of Indulgences (known later as the Ninety-Five Theses) in 1517. He was excommunicated, and declared outlaw in the Holy Roman Empire, in 1521. The various parts of the Holy Roman Empire began to identify themselves either with the Roman Catholic church or the Lutheran church. Conflict among them triggered the Peasant Revolts of 1524-1526. Political and social tensions continued to increase as what we now call the Reformation began to work itself out. Eventually, Pope Julius III convened the Council of Trent in 1551-1552, condemning Protestantism and initiating what we now call the Catholic Counter-reformation The Emperor, Charles V, convened a Diet in Augsberg in 1555, and established the principle of Cuius regio, eius religio, religious uniformity in each state according to the faith of the ruler, so long as it was either Catholic or Lutheran. Mikołaj Kopernik (Nicolaus Copernicus; 1473 – 1543) • Having thought through the matter, and having made relevant observations over the course of his travels for his work, Copernicus wrote up a short treatise on his version of the heliocentric theory: Nicolai Copernici de hypothesibus motuum coelestium a se constitutis commentariolus, now known as the Commentariolus. z z z z z z It was just argument without mathematical or detailed observational support, intended only for distribution to his colleagues, in order to get feedback. In his role as chancellor (economist), he wrote Monetae cudendae ratio, which expressed the first version of Gresham's law (bad money drives out good money). In 1532, he completed De revolutionibus orbium coelestium, but withheld it from publication. People were already aware of what he was doing, and not liking it. In 1539, Georg Joachim Rheticus came to be his student. In the course of that work, he published a short description of the theory, the Narratio prima. Rheticus pressed Copernicus to publish De revolutionibus, and was eventually given the manuscript to get it published in Nurenberg. Rheticus had to leave Nurenberg, so Andreas Osiander oversaw completion of the printing, adding an infamous preface. The legend is that Copernicus was given a galley copy on his death bed in 1543. What Copernicus Didn't Like About the Ptolemaic Theory z z z z z Why do Mercury and Venus only appear close to the Sun? Why do the retrograde motions of the outer planets only appear at opposition? There is no particular reason for ordering the planets the way we do. Why should all the various epicycles be such as to exactly correct for the Earth-Sun motion? A coincidence? Why should the vast shell of stars be moving, rather than the little Earth? What Copernicus Liked About His Theory z z z z z z z He explained why all the various epicycles were such as to 'exactly' correct for the Earth-Sun motion. The reflection of the Earth-Sun motion explains all the retrograde motions with one concept; all Ptolemy's epicycles are ad hoc. The order of the planets is fixed by the theory, and explains the maximum elongations of Mercury and Venus, and the opposition retrograde motions of the superior planets. The relative sizes of the planetary orbits can be determined, rather than just posited. He got rid of the equant (which violated the spirit of uniform circular motion) by adding additional epicycles. Didn't look so complicated (from the above-ecliptic view). It used a physical theory (the 'true causes') to explain observations, rather than just an explanatory structure ( a basic philosophical disagreement with his contemporaries). When Copernicus Says That His System is 'Simpler'... PTOLEMY COPERNICUS He Means Particularly That the Motions of the Planets, as Seen from Above the Plane, Don't Look Like This. Complaints to Copernicus (well, not directly to him, because he was already dead) z Why don't we have any sensation of the diurnal motion of the Earth? Why is it that, when you throw something over your head, it doesn't fall behind you (because the Earth has moved under it)? z z z z z And by the way, if the other planets aren't naturally falling toward the Earth, then why does stuff on the Earth still fall toward it? And by the way, why is the Moon still orbiting the Earth, not the Sun? Copernicus says that this means the stars are quite far away. All these circles on circles... you have more epicycles than Ptolemy! z z The counter argument to that is: where does this 'natural' motion come from. You need a new physics to explain this. If the Earth is going around the Sun, we should see a parallax shift of the nearby stars. But we don't. z z Copernicus argued that the rotation of the Earth is 'natural', not 'violent', so all objects on it naturally partake in its rotation. Copernicus had to do that to get rid of the equants. Speaking of the principle of economy, why is there so much space between the orbit of Saturn and the fixed stars? It seems a waste... Bottom Line z z z z Simplicity really isn't there in the details, to a large part because Copernicus was still wedded to the idea of circular motion. If you only think of a theory as a method for calculating the appearances of the phenomena (which was the approach put forward by Osiander, totally contrary to what Copernicus would have wanted), then either Ptolemy's theory or Copernicus' theory was good enough for most predictions. The big difference was that Copernicus had created a physical theory, which claimed that it wasn't just useful, but actually represented what the real universe looked like, and how the real universe acted. And as a physical theory, it could give answers for questions that hadn't been asked, like “What is the order of the planets?”, and “How far away are the stars?” Clicker Question What part of Ptolemy's theory of the heavens did Copernicus agree with? A – The planets revolve about the Earth. B – The orbits of the planets are circular. C – The fixed stars are close to the most distant planet (Saturn). D – The most important criterion for a theory is that it fits the data, “saves the appearances”. Clicker Answer: B What part of Ptolemy's theory of the heavens did Copernicus agree with? A – The planets revolve about the Earth. (Copernicus said they revolve about the Sun.) B – The orbits of the planets are circular. (Yes, but it meant he needed more epicycles.) C – The fixed stars are close to the most distant planet (Saturn). (Copernicus put the stars far away.) D – The most important criterion for a theory is that it fits the data, “saves the appearances”. (Copernicus wanted his theory to be physical, to correspond to the physical truth.) Tyge Ottesen Brahe (Tycho Brahe; 1546 – 1601) z z An exceptional astronomer because he was good at getting funds, from the Kings of Denmark and Bohemia. Established observatories in Hven (Uraniborg, in Denmark) and Benátky nad Jizerou (in the part of Bohemia that is now the Czech Republic) z z z z These were true research institutes Reported the November 11, 1572 “new” star (now known as SN 1572) in Cassiopeia, in De nova stella, coining the word 'nova'. Its lack of parallax indicated it must be much farther away than the moon. At the time, it was thought that changes only occurred in the sublunar regions; the idea that there could be a change among the fixed stars was revolutionary. His observations of the Great Comet of 1577 showed that it was farther away than the orbit of the Moon. At the time, comets were thought to be atmospheric phenomena. z Kepler later showed from these data that the comet was crossing the “spheres” of the planets, which meant that they could not be physical. SN 1572 Great Comet of 1577 Tycho's Instrumentation enabled the best measurements visual astronomy could make, with accuracies ~ one arc minute. Equatorial armillary Sextant Tycho's Theory The superior planets orbit the Earth. Venus and Mercury orbit the Sun, as it orbits the Earth The Earth is still in the center Johannes Kepler (1571 – 1630) z z In 1594, at 23, began teaching mathematics and astronomy at the Protestant School of Graz, Austria (which became the University of Graz). While talking about the occurrences of the conjunctions of Jupiter and Saturn, he realized that their orbits defined circles with ratios that match those of the circles circumscribed and inscribed by a triangle. z z He soon extended this to having the spheres of the orbits be spaced by regular polygons, and published the result in the Mysterium cosmographicum. This began a long fascination with searching out geometric and harmonic structure in the universe. 1570-1770 1300-1490 Johannes Kepler (1571 – 1630) z z In 1600, Kepler began working with Tycho, and continued to analyze Tycho's data after his death in 1601. He was expelled from Graz because he refused to become a Catholic. Fortunately, he was hired by Emperor Rudolf II to be the imperial mathematician. z z In 1603, Kepler completed Astronomiae Pars Optica (The Optical Part of Astronomy), which described the inverse square law for brightness versus distance, reflections from mirrors (both flat and curved), the red color of the Moon when it is eclipsed, and even the optical performance of the human eye. * z z Imperial mathematician meant court astrologer. Kepler was the first to show that the image produced by the lens on the retina was upside-down. He also worked out significant elements of projective geometry, including the idea that, as the foci of a conic section shift, the figure changes continuously from one form to another. *The full title: Ad Vitellionem paralipomena, quibus Astronomiae pars optica traditur Conic Sections and the Ellipse in Particular Are Very Important to Kepler's Work The ellipse is defined as the set of points with the same sum of distances from the two foci. FOCI Kepler showed how the conics smoothly change from one to another. Astronomia Nova z z z Kepler put most of his effort at calculations in understanding the orbit of Mars, which was the most difficult to fit to existing theories. Part of what helped was thinking that there might be some motive force directing the planets towards the Sun, similar to the magnetic force he had read about in William Gilbert's De Magnete (1600). In 1604, he was able to establish two laws of planetary motion: z z All planets move on ellipses, not circles, with the Sun at one focus. As they move along their orbits, they sweep out equal areas in equal times. Area of section a = Area of section b = Area of section c Galileo Galilei (1564 – 1642) z z z z z z Born in Pisa, Galileo trained in medicine at the University of Pisa, but was distracted by doing experiments in physics. Became chair of mathematics first at Pisa, and then at the University of Padua. In 1608, various Dutch spectacle makers, including Hans Lipperhey and Zacharias Janssen, in Middelburg, and Jacob Metius in Alkmaar, began to produce refracting telescopes. Galileo acquired one, and made some modifications of his own, and turned it to the heavens. He published the results in 1610, in the Sidereus Nuncius (The Starry Messenger). This began a series of important celestial observations of surprising accuracy. Galileo's Observations of Jupiter If there are bodies that go around Jupiter, then there can be other centers of motion In the universe than just the Earth or the Sun! JAN 7 JAN 8 JAN 10 JAN 11 JAN 13 JAN 15 JAN 17 JAN 17 4h later JAN 18 JAN 19 Clear evidence that the Moon has mountains and other surface structure, and so is like the Earth. Thomas Hariot (1610) Moon as seen through telescope Motion of Jupiter's moons The Telescope Shows Many Faint Stars Not Visible to the Naked Eye Neptune Clicker Question Which of these was not one of Galileo's discoveries? A – Sunspots B – All revolutions don't have to be around the same center. C – Annual parallax of the nearest stars. D – Venus has phases like the Moon. Clicker Answer: C Which of these was not one of Galileo's discoveries? A – Sunspots (Yes, it's in the book, not in the lecture.) B – All revolutions don't have to be around the same center. (Yes, that's what the moons of Jupiter prove.) C – Annual parallax of the nearest stars. (No, his telescope wasn't strong enough.) D – Venus has phases like the Moon. (Yes, it's in the book, not in the lecture.) More Kepler z z z z z Responding to the Sidereus Nuncius, Kepler began further work on optics, leading to the Dioptrice (1611), on the optics of the telescope. At the time, the primary optical theory presumed that light originated in the eye, and illuminated the objects we see (the emission theory). Although there were a number of contrary works, especially the Kitab alManazir (Book of Optics; 1011-1021) by Abū Alī al-Ḥasan ibn al-Ḥasan ibn al-Haytham (Alhazen; 965 – 1040), it was Kepler's work that finally settled the understanding of light as an external phenomenon. It's a good thing that he did this, because without a theory of how telescopes form images, people were not easily convinced about what Galileo was seeing. Over the course of 1617–1623, Kepler published Epitome astronomiae Copernicanae (Epitome of Copernican Astronomy), which extended his analysis of observations, and introduced his third law, the harmonic law. Kepler's Third Law log (T2a-3) = log k 2logT – 3loga = log k logT = (3/2)loga + log k So, on a log-log plot, we have a straight line. But the constant, log k, is different for Jupiter's moons! The Galileo Affair z z Galileo's Dialogue Concerning the Two Chief World Systems, was published in 1632, as the summation of the arguments for the Copernican theory. There were issues political, personal, and religious in the subsequent controversy. But the end result was that, in 1633, Galileo was tried, found "vehemently suspect of heresy", and sentenced to house arrest; and the Dialogue was put on the Index of Forbidden Books, until 1835. Isaac Newton (1642 – 1727) z z z z In 1661, started as a sizar (a work-study student) at Trinity College, Cambridge. Took degree in 1665, and went home to avoid the plague. Returned as a fellow in 1667. After a conversation with Edmund Halley about the orbits of comets, he worked up a paper for the Royal Society in 1684, De motu corporum in gyrum (On the motion of bodies in an orbit), which showed that Kepler's Laws were a consequence of motion under an inverse square-law force. He then wrote up his years of work on the mathematics of the motions of bodies, Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy; 1687), in which he presents his Three Laws of Motion. Newton's First Two Laws z z The velocity of a body is constant if there are no forces on it. If there is a force (F), then the change of velocity with time (the acceleration, a) is proportional to the force, and inversely proportional to the mass, m. VELOCITY is the PAIR of SPEED and DIRECTION ACCELERATION is the CHANGE IN VELOCITY – either the CHANGE IN SPEED or the CHANGE IN DIRECTION or the CHANGE IN BOTH. a= F m Newton's Third Law Newton's Third Law To hold the rock up, Sisyphus pushes on it, exerting a force. Sisyphus pushing on the rock Newton's Third Law By Newton's third law, the rock exerts an equal and opposite force. Sisyphus pushing on the rock The rock pushing on Sisyphus Newton's Third Law In order to counter that, Sisyphus exerts a force on the ground. Sisyphus pushing on the rock The rock pushing on Sisyphus Sisyphus pushing on the ground Newton's Third Law So that by Newton's third law, the ground pushes back. Sisyphus pushing on the rock The rock pushing on Sisyphus The ground pushing on Sisyphus Sisyphus pushing on the ground Newton's Third Law The forces of the rock material add up to balance this push. Everything balances. It's an equilibrium. Sisyphus pushing on the rock The rock pushing on Sisyphus The ground pushing on Sisyphus These two forces balance each other. Sisyphus pushing on the ground The forces of the ground material add up to balance this push. Newton's Law of Gravitation z z z z z G is the constant of proportionality. Suppose we have two masses, m1 and m2. Each exerts a gravitational force, F, on the other. The force is proportional to m1 and is proportional to m2. The force is inversely proportional to the square of the distance between them, r. The distance, r, is measured between the centers of the two masses. Newton's Law of Gravitation z z z z G is the constant of proportionality. z If you increase m1 to twice its mass, then the force is increased by a factor of 2. If you increase the size of m1 without increasing its mass, then the force stays the same. If you increase the distance, r, between the two masses, the force drops to one-quarter of the original. We adopt for our units: z Kilograms for mass z Meters for distance z Newtons for force. Then G = 6.67300 × 10-11 m3 kg-1 s-2