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Transcript

Ptolemy (125 A.D.) Ptolemy’s Model • Designed a complete geometrical model of the universe that accurately predicted planetary motions with errors within 50 • Most of the geometric devices and basic foundations of his model did not originate with him but were based on the models of the early Greeks such as Aristotle & Hipparchus • Wrote the Almagest (Greatest) • • • • – natural motions - no forces • – included the original works & models of Ptolemy – included a compilation of past works of Greeks, especially Hipparchus – 13 volumes • Ptolemy’s Equant • • • Earth was spherical & at center of cosmos - GEOCENTRIC Cosmos is finite in size Earth has no motions Sun, Moon Planets exhibit uniform, circular motions Used devices of eccentrics, epicycles & deferents to explain the observed non-uniform motions of the Sun and planets along the ecliptic & retrograde motion. Introduced equant to explain the variations in retrograde motions Ptolemy’s Model point inside a circle, not at the center, from which motion along the circumference of the circle would appear to be uniform opposite the circle’s center from the eccentric (the Earth) nonphysical geometrical device that broke fundamental assumption of uniform circular motion Celestial motions no longer had to be uniform around the centers of circles Copernicus’ Model Copernicus (1473-1543) • • • • • • Developed a Heliocentric (Sun centered) model of the cosmos Why? Ptolemy’s geocentric model lasted for centuries mainly because it accurately predicted celestial motions so there was little reason to discard it Copernicus studied the works of Aristotle, Pythagoras & Plato an offshoot of Plato’s philosophy asserted that Sun was godhead of all knowledge Copernicus objected to equant based on aesthetics equant not faithful to ideal of uniform motion - makes models to complex • • • Copernicus worked on his new Heliocentric model for 20 years – Sun was placed at center of cosmos – Earth no longer static, but revolved around Sun once a year & rotated on axis once a day His work was published in De revolutionibus in the year of his death De revolutionibus took after the Almagest in outline and basic intention - to explain planetary motions Even though it took 20 years to develop this model did predict celestial motions any better than Ptolemy’s geocentric model 1 Heliocentric Model of Copernicus Heliocentric Model of Copernicus • • • • • Cosmos finite in size Assumed no forces for heavenly motions – Physics of Aristotle Assumed uniform, circular motions – done for aesthetics - followed Aristotle All heavenly spheres revolve around the sun & the sun is at the center of the cosmos – chosen based on aesthetics and simplicity The distance from the earth to the sphere of stars is much greater than the distance from the earth to the sun – accounts for lack of observed stellar parallax Retrograde Motion Explained • The daily motion of the heavens relative to the horizon results from the earth’s motion on its axis – aesthetic appeal since only 1 sphere is rotating not many – however, he did not account for the objection that if the earth rotated, objects should be flung from the surface • • The apparent motion of the sun relative to the stars results from the annual revolution of the earth around the sun The planets’ retrograde motion occur from the motion of the earth relative to the other planets – retrograde explained as a natural result of the planet’s revolutions about sun - what we observe is an illusion Retrograde Motion Explained • • Copernican Model • Copernicus eliminated epicycles to explain retrograde motion • Eliminated equant - kept uniform circular motion • Needed to account for variations in planetary motion so he was forced to add many smaller circles • Violated Aristotelian physics & did not offer new physical ideas to support his model • Didn’t predict motions any better than Ptolemy’s model When the earth passes any of the outer planets or when the earth is passed by the inner ones, retrograde motion occurs Passing is key to retrograde motion in the heliocentric model Ptolemy 2 Tycho Brahe (1546-1601) Copernicus • Danish born member of nobility – Stolen by his uncle and adopted as his own son – Trained as a lawyer but secretly studied astronomy after being impressed that mortals could predict astronomical events – Very arrogant and hard to get along with and played the role of a courtly prince with a gold nose • Supported by the King Frederick II of Denmark, he built and observatory on the island of Hven with one ton of gold – His adopted father saved Frederick from drowning but then died of pneumonia Tycho’s Observations • Tycho’s Supernova He made a long series of very precise observations of the motions of the planets – Very accurate since they were made with a very large quadrant circle (<0.5 arc minute) – Made repeated observations and took the average to reduce the errors – Made observations over many years 1576-1591 – Realized the motions of planets did not match the predictions of Ptolemy • Tycho’s Cosmology • • • • Refuted Copernican system Placed the earth at the center – Due to his inability to measure parallax – believed in Aristotlean physics Realized that the Ptolemaic system failed Compromise – Stationary earth at center – Sun moves around the earth – All other celestial bodies move around the sun – His worked no better than Ptolemy He needed better mathematics The supernova of 1572 – Noticed a brilliant star had appeared in Cassiopeia – Observed that it faded slowly over the following 2 years – Demonstrated that the universe was changing and imperfect – Showed that it had the same position no matter where on earth it was observed - thus it was at a great distance Kepler (1571-1630) • Was born a commoner in Germany – His father was a mercenary – His mother was raised by an aunt who was burnt as a witch – Went to college at Tubingen were he was recognized as a mathematical genius and went to teach math at a Protestant School in Graz – While in Graz he felt that the 5 perfect Platonic solids could explain the motion of the 6 known planets – Attempted to get observations of the planets but was incompetent 3 Kepler Solves The Mystery of Planetary Motions Kepler and Brahe Meet • • • Brahe had to leave Hven for Prague because Fredrick II drank himself to death Kepler was driven out because the Catholics took over Graz and drove out the Protestants Tycho knew he needed a mathematician to help him model his observations • It took Kepler 8 years to solve the motion of Mars – He tried various combinations of circular motions – He finally abandoned circular motion and tried ovals and other geometric shapes – They meet Feb 4, 1600 and fought continually – Tycho feared that Kepler would eclipse him so he would only show him data on Mars (his most challenging object) – Tycho died October 21, 1601 from over partying • • He found that an ellipse could produce the motions of the planets Kepler got Tycho’s data from the family follow a death bed request by Tycho “Let me not seem to have died in vain” Properties of Ellipses Kepler’s 1st Law of Ellipses 1609 • • Ellipses belong to the family of conic sections (the intersection of a plane and a cone) – The shape is defined by the separation between the two foci of the ellipse • • When the foci coincide the shape is a circle • As the foci separate from each other the ellipse gets flatter • The amount by which an ellipse differs from a circle is defined as an ellipse’s eccentricity Planets move in elliptical orbits with the sun at one focus The other focus is empty & located in space ⇒ Distance between the planet and the sun changes as the planet moves along its elliptical orbit ⇒Planet move along orbits which have a special and regular geometrical shape ⇒Gets rid of idea of uniform circular motions – Each point on an ellipse has the property that the sum of its distances to the two foci is the same – The line through the foci to both sides of the ellipse is called the major axis Kepler’s 1st Law Kepler’s 2nd Law of Equal Areas 1609 • • Kepler noted that Mars slowed down when it approached the sun He found that a line drawn between the sun and a planet sweeps out an equal area in an equal period of time anytime during its orbit – Planetary motions are non-uniform but vary in a regular way – Planets move more slowly when they are far from the sun and fastest when they are near the sun – The increase in speed as a planet moves toward the sun makes it appear that the sun is “pulling” on the planet 4 Kepler’s 2nd Law Kepler’s 3nd Harmonic Law 1618 • The Harmonic Law - p2 = k a3 – p is the period of the planets orbit – a is the planets average distance to the Sun – k is a constant which is the same for all planets • • Kepler’s Laws • • Kepler’s scheme correctly described planetary motions when compared to the accurate observations of Tycho Together laws 2 and 3 imply that there exists a force, between the sun and planets, that weakens with increasing distance – Invested Sun with physical properties necessary for the uniform description of planetary motions • Kepler recognized there must be a force at work but could not figure out what it was. He speculated it may be magnetism The more distant a planet from the Sun – The slower it moves in its orbit – The longer it takes to complete and orbit Since a single constant works for all planets suggests a single physical cause for their orbits Astronomy Enters a New Era • Astronomical objects are viewed as actual bodies not distant ethereal or mythical entities • Realization that the laws of nature are the same for both the earthly and heavenly domains 5