lecture notes 6

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Published on January 19, 2008

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Slide1:  11-hour star trails http://apod.nasa.gov/apod/astropix.html Slide2:  Western scientific history begins with the ancient Greek civilization about 600 BCE Ancient Greek Astronomers (1):  Ancient Greek Astronomers (1) Unfortunately, there are no written documents about the significance of stone and bronze age monuments. First preserved written documents about ancient astronomy are from ancient Greek philosophy. Greeks tried to understand the motions of the sky and describe them in terms of mathematical (not physical!) models. (Is it indeed so??) Ancient Greek Astronomers (2):  Ancient Greek Astronomers (2) Models were generally wrong because they were based on wrong “first principles”, believed to be “obvious” and not questioned: Geocentric Universe: Earth at the Center of the Universe. “Perfect Heavens”: Motions of all celestial bodies described by motions involving objects of “perfect” shape, i.e., spheres or circles (Plato). Ancient Greek Astronomers (3):  Ancient Greek Astronomers (3) Eudoxus (409 – 356 B.C.): Model of 27 nested spheres Aristotle (384 – 322 B.C.), major authority of philosophy until the late middle ages: Universe can be divided in 2 parts: 1. Imperfect, changeable Earth, He expanded Eudoxus’ Model to use 55 spheres. 2. Perfect Heavens (described by spheres) Clockwork universe:  Clockwork universe Slide9:  Pythagoras of Samos ~580-500 BC Aristotle 384-322 BC Aristarchus of Samos ~310-230 BC Eratosthenes of Cyrene 276-194 BC Claudius Ptolemy 85-165 AD Slide10:  Aristarchus of Samos (~310-230 BCE) Scientific method, not just pure philosophy! Proposed a theory that Earth rotated on its axis and revolved around the sun! Slide12:  Using the distance between the Earth and the Moon as a baseline Slide14:  Eratosthenes of Cyrene Born: 276 BC in Cyrene, North Africa (now Shahhat, Libya) Died: 194 BC in Alexandria, Egypt  Measured circumference of the Earth He got R = 6739 km Actual radius 6378 km Eratosthenes’s Experiment:  Eratosthenes’s Experiment (SLIDESHOW MODE ONLY) Eratosthenes (~ 200 B.C.): Calculation of the Earth’s radius:  Eratosthenes (~ 200 B.C.): Calculation of the Earth’s radius Angular distance between Syene and Alexandria: ~ 70 Linear distance between Syene and Alexandria: ~ 5,000 stadia  Earth Radius ~ 40,000 stadia (probably ~ 14 % too large) – better than any previous radius estimate. Slide17:  Eratosthenes Sieve Slide18:  The problem of retrograde motion Later refinements (2nd century B.C.) :  Later refinements (2nd century B.C.) Hipparchus: Placing the Earth away from the centers of the “perfect spheres” Ptolemy: Further refinements, including epicycles Slide20:  Claudius Ptolemy 85-165 AD Mathematical Syntaxis (Almagest) Epicycles:  Epicycles The Ptolemaic system was considered the “standard model” of the Universe until the Copernican Revolution. Introduced to explain retrograde (westward) motion of planets Epicycles:  Epicycles (SLIDESHOW MODE ONLY) Slide23:  Aryabhata the Elder Born: 476 in Kusumapura (now Patna), India Died: 550 in India Most accurate value of Pi among ancients! Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given. This gives π = 62832/20000 = 3.1416 which is a surprisingly accurate value: π = 3.14159265 correct to 8 places! Aryabhata gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent approximation to the currently accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors! Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. Slide24:  Aryabhata calculated the Sidereal day (the rotation of the earth against the fixed stars) as 23 hours 56 minutes and 4.1seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the true value (over 365 days). The very notion of sidereal time was very advanced for the time, so this kind of accurate computation speaks of a very sophisticated understanding of the universe. The 8th century Arabic edition of the Āryabhatīya was translated into Latin in the 13th century, well before Copernicus. Through this translation, European mathematicians may have learned methods for calculating sines and cosines (exact to 4 significant digits!), as well as square and cube roots, and it is likely that some of Aryabhata's results also influenced European astronomy. Slide25:  Solar and Sidereal Day Slide26:  Solar and Sidereal Day The fact that our clocks are based on the solar day (24 hours) and the Sun appears to drift eastward with respect to the stars (or lag behind the stars) by about 1 degree per day means that if you look closely at the positions of the stars over a period of several days, you will notice that according to our clocks, the stars rise and set 4 minutes earlier each day. Our clocks say that the day is 24 hours long, so the stars move around the Earth in 23 hours 56 minutes. This time period is called the sidereal day because it is measured with respect to the stars. This is the true rotation rate of the Earth and stays the same no matter where the Earth is in its orbit---the sidereal day = 23 hours 56 minutes on every day of the year. One month later (30 days) a given star will rise 2 hours earlier than it did before (30 days × 4 minutes/day = 120 minutes). A year later that star will rise at the same time as it did today. Sidereal and tropical year:  Sidereal and tropical year The precession of the Earth's rotation axis introduces another difference between sidereal time and solar time. This is seen in how the year is measured. A year is defined as the orbital period of the Earth. However, if you use the Sun's position as a guide, you come up with a time interval about 20 minutes shorter than if you use the stars as a guide. The time required for the constellations to complete one 360° cycle around the sky and to return to their original point on our sky is called a sidereal year. This is the time it takes the Earth to complete exactly one orbit around the Sun and equals 365.2564 solar days. The slow shift of the star coordinates from precession means that the Sun will not be at exactly the same position with respect to the celestial equator after one sidereal year. The tropical year is the time interval between two successive vernal equinoxes. It equals 365.2422 solar days and is the year our calendars are based on. After several thousand years the 20 minute difference between sidereal and tropical years would have made our summers occur several months earlier if we used a calendar based on the sidereal year. Arab Astronomy:  Arab Astronomy Science in Islamic world was built on the sciences of two great cultures, the Greek and the Indian. Blending and expanding these often different ideas led to a new science which later profoundly influenced Western scientific exploration beginning in the Renaissance. Al-Khwarizmi (9th century), the inventor of algebra. He based the system on the Indian numbers borrowed by the Arabs (what we today call Arabic numerals). Detailed positions of the sun and planets, detailed map of the world. Omar Khayyam was a great Persian scientist, philosopher, and poet who lived from 1048-1131. He compiled many astronomical tables and performed a reformation of the calendar which was more accurate than the Julian and came close to the Gregorian. An amazing feat was his calculation of the year to be 365.24219858156 days long, which is accurate to the 6th decimal place! The Copernican Revolution:  The Copernican Revolution Nicolaus Copernicus (1473 – 1543): Heliocentric Universe (Sun in the Center) Slide30:  1. There is no one centre in the universe. 2. The Earth's centre is not the centre of the universe. 3. The centre of the universe is near the sun. 4. The distance from the Earth to the sun is imperceptible compared with the distance to the stars. 5. The rotation of the Earth accounts for the apparent daily rotation of the stars. 6. The apparent annual cycle of movements of the sun is caused by the Earth revolving round it. 7. The apparent retrograde motion of the planets is caused by the motion of the Earth from which one observes. Church cleric, but rejected a 2000-yr old paradigm Seven axioms written in a pamphlet “Little Commentary” (1514) Born: 19 Feb 1473 in Torun, Poland Died: 24 May 1543 in Frombork, Poland Copernicus’ new (and correct) explanation for retrograde motion of the planets:  Copernicus’ new (and correct) explanation for retrograde motion of the planets This made Ptolemy’s epicycles unnecessary. Retrograde (westward) motion of a planet occurs when the Earth passes the planet. Slide34:  Tycho Brahe (1546-1601) Contributions to Astronomy: Tycho was the first to suggest a non-circular orbit for a celestial body (a comet). Used calibrated and bigger instruments, new techniques to measure angles (similar to a sextant). Built an observatory (remember - no telescopes yet) and made accurate and continuous measurements for 20 years. His measurements helped to prove that planets orbited the sun. Accurate map of the stars with 777 stars. Observed “a new star” – supernova 1572 (Tycho’s Supernova). Measured length of the year to within 1 second. Was still unable to choose between the geocentric and heliocentric model. He had his own model with the Earth at the center, orbited by the sun and the moon, with planets orbiting the sun. Never worked out the mathematical details, and his model was never accepted. Using Tycho's data, a German astronomer (Kepler) was able to refute the geocentric model Slide37:  Galileo Galilei (1564-1642) Was the first to report using the telescope to view the heavens. Telescope invented in 1604 by Hans Lippershay. Galileo used the telescope in 1609. Built his own. Two lenses in a metal tube about 4 feet long, diameter = 4 cm (1.6 inches). Magnification 3X to 33X. His observations between 1609 and 1612 changed our ideas about the universe. What did he see? New stars (Milky Way made up of stars) Mountains and valleys on the moon Four moons orbiting Jupiter (now called Galilean moons) Phases of Venus Sunspots (rotating around the sun about once a month) The rings of Saturn (sketches. was puzzling; not identified as rings until about 50 years later.) Planets are disks, not pinpoints of light like the stars Slide38:  32X power Major Discoveries of Galileo:  Major Discoveries of Galileo Moons of Jupiter (4 Galilean moons) Rings of Saturn (What he really saw) 0 Major Discoveries of Galileo (2):  Major Discoveries of Galileo (2) Surface structures on the moon; first estimates of the height of mountains on the moon Major Discoveries of Galileo (3):  Major Discoveries of Galileo (3) Sun spots (proving that the sun is not perfect!) Major Discoveries of Galileo (4):  Major Discoveries of Galileo (4) Phases of Venus (including “full Venus”), proving that Venus orbits the sun, not the Earth! Slide43:  The significance of what he saw: Cast doubt on the view of the "perfection of the heavens" (of Aristotle and Plato) Showed deficiencies of the geocentric (Ptolemaic) model Rotation of sunspots around sun suggested that if the sun could rotate, perhaps the Earth could too. Phases of Venus would be a natural consequence of the heliocentric model. Jupiter's moons showed that centers of motion other than Earth existed. Galileo published in Italian, not Latin. Widely read. Language of the people, rather than language of the scholars. Arguments against the geocentric model were so forceful that he came under fire from the Catholic Church and was forced to give a public denial of the heliocentric/Copernican system, and was placed under house arrest for the last 10 years of his life. Was not pardoned by the Church until 1992. Science in Italy was dealt a severe blow. The center of scientific investigation shifted to northern Europe. Many scholars refused to believe his ideas and a few even refused to look through the telescope. Many clung to old ideas. Slide44:  Sentence of the Tribunal of the Supreme Inquisition against Galileo Galilei, given the 22nd day of June of the year 1633 "It being the case that thou, Galileo, son of the late Vincenzio Galilei, a Florentine, now aged 70, wast denounced in this Holy Office in 1615: "That thou heldest as true the false doctrine taught by many, that the Sun was the centre of the universe and immoveable, and that the Earth moved, and had also a diurnal motion: That on this same matter thou didst hold a correspondence with certain German mathematicians.... "That the Sun is the centre of the universe and doth not move from his place is a proposition absurd and false in philosophy, and formerly heretical; being expressly contrary to Holy Writ: That the Earth is not the centre of the universe nor immoveable, but that it moves, even with a diurnal motion, is likewise a proposition absurd and false in philosophy, and considered in theology ad minus erroneous in faith..... "We say, pronounce, sentence, and declare, that thou, the said Galileo, by the things deduced during this trial, and by thee confessed as above, hast rendered thyself vehemently suspected of heresy by this Holy Office, that is, of having believed and held a doctrine which is false, and contrary to the Holy Scriptures, to wit: that the Sun is the centre of the universe, and that it does not move from east to west, and that the Earth moves and is not the centre of the universe: and that an opinion may be held and defended as probable after having been declared and defined as contrary to Holy Scripture; and in consequence thou hast incurred all the censures and penalties of the Sacred Canons, and other Decrees both general and particular, against such offenders imposed and promulgated. From the which We are content that thou shouldst be absolved, if, first of all, with a sincere heart and unfeigned faith, thou dost before Us abjure, curse, and detest the above-mentioned errors and heresies and any other error and heresy contrary to the Catholic and Apostolic Roman Church, after the manner that We shall require of thee. … “ http://www.fordham.edu/halsall/mod/1630galileo.html Slide45:  GALILEO'S ABJURATION I, Galileo Galilei, son of the late Vincenzio Galilei of Florence, aged 70 years, tried personally by this court, and kneeling before You, the most Eminent and Reverend Lord Cardinals, Inquisitors-General throughout the Christian Republic against heretical depravity, having before my eyes the Most Holy Gospels, and laying on them my own hands; I swear that I have always believed, I believe now, and with God's help I will in future believe all which the Holy Catholic and Apostolic Church doth hold, preach, and teach. But since I, after having been admonished by this Holy Office entirely to abandon the false opinion that the Sun was the centre of the universe and immoveable, and that the Earth was not the centre of the same and that it moved, and that I was neither to hold, defend, nor teach in any manner whatever, either orally or in writing, the said false doctrine; and after having received a notification that the said doctrine is contrary to Holy Writ, I did write and cause to be printed a book in which I treat of the said already condemned doctrine, and bring forward arguments of much efficacy in its favour, without arriving at any solution: I have been judged vehemently suspected of heresy, that is, of having held and believed that the Sun is the centre of the universe and immoveable, and that the Earth is not the centre of the same, and that it does move. Nevertheless, wishing to remove from the minds of your Eminences and all faithful Christians this vehement suspicion reasonably conceived against me, I abjure with sincere heart and unfeigned faith, I curse and detest the said errors and heresies, and generally all and every error and sect contrary to the Holy Catholic Church. And I swear that for the future I will neither say nor assert in speaking or writing such things as may bring upon me similar suspicion; and if I know any heretic, or one suspected of heresy, I will denounce him to this Holy Office, or to the Inquisitor and Ordinary of the place in which I may be. I also swear and promise to adopt and observe entirely all the penances which have been or may be by this Holy Office imposed on me. And if I contravene any of these said promises, protests, or oaths, (which God forbid!) I submit myself to all the pains and penalties which by the Sacred Canons and other Decrees general and particular are against such offenders imposed and promulgated. So help me God and the Holy Gospels, which I touch with my own hands. I Galileo Galilei aforesaid have abjured, sworn, and promised, and hold myself bound as above; and in token of the truth, with my own hand have subscribed the present schedule of my abjuration, and have recited it word by word. In Rome, at the Convent della Minerva, this 22nd day of June, 1633. I, GALILEO GALILEI, have abjured as above, with my own hand. Galileo Galilei (1594 – 1642):  Galileo Galilei (1594 – 1642) Invented the modern view of science: Transition from a faith-based “science” to an observation-based science. Rejected the “old” view (still alive now!) that the only path to true understanding is through religious faith Observations are correct even if they contradict the Scripture “The Bible tells us how to go to heaven, not how the heavens go.” Slide47:  Johannes Kepler (1571 – 1630) Slide48:  Kepler hypothesized that a physical force moved the planets, and that the force diminished with distance. Planets closer to the sun feel a stronger force and move faster. Elliptical orbits – key to the problem of the planetary motion Kepler’s Laws of Planetary Motion:  Kepler’s Laws of Planetary Motion The orbits of the planets are ellipses with the sun at one focus. Eccentricity e = c/a c Slide51:  e = Ra - Rp Ra + Rp Remember parameters: perihelion, aphelion, semimajor axis, eccentricity Elliptical orbits Eccentricities of Ellipses:  Eccentricities of Ellipses e = 0.02 e = 0.1 e = 0.2 e = 0.4 e = 0.6 1) 2) 3) 4) 5) Eccentricities of Planetary Orbits:  Eccentricities of Planetary Orbits Orbits of planets are virtually indistinguishable from circles: Earth: e = 0.0167 Most extreme example: Pluto: e = 0.248 Slide54:  LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time The closer to the sun, the larger the orbital velocity Planetary Orbits (2):  Planetary Orbits (2) A planet’s orbital period (P) squared is proportional to its average distance from the sun (a) cubed: Py2 = aAU3 A line from a planet to the sun sweeps over equal areas in equal intervals of time. (Py = period in years; aAU = distance in AU) Slide56:  LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes: For the Earth P2 = 1 yr, a2 = 1 AU Note units!! Johannes Kepler (1571 – 1630):  Johannes Kepler (1571 – 1630) Used the precise observational tables of Tycho Brahe (1546 – 1601) to study planetary motion mathematically. Circular motion and Planets move around the sun on elliptical paths, with non-uniform velocities. Found a consistent description by abandoning both Uniform motion. Historical Overview:  Historical Overview

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