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Ptolemy's Model Ptolemy (125 A.D.) Ptolemy’s Model • Designed a complete geometrical • Earth was spherical & at center of model of the universe that accurately cosmos - GEOCENTRIC predicted planetary motions with errors • Cosmos is finite in size 0 within 5 • Earth has no motions • Most of the geometric devices and • Sun, Moon Planets exhibit uniform, basic foundations of his model did not circular motions originate with him but were based on – natural motions - no forces the models of the early Greeks such • Used devices of eccentrics, epicycles as Aristotle & Hipparchus & deferents to explain the observed • Wrote the Almagest (Greatest) non-uniform motions of the Sun and – included the original works & models of Ptolemy planets along the ecliptic & retrograde – included a compilation of past works of Greeks, motion. especially Hipparchus • Introduced equant to explain the – 13 volumes variations in retrograde motions Ptolemy’s Equant 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) • Copernicus worked on his new • Developed a Heliocentric (Sun centered) model of Heliocentric model for 20 years the cosmos – Sun was placed at center of • Why? Ptolemy’s geocentric model lasted for cosmos centuries mainly because it accurately predicted – Earth no longer static, but revolved celestial motions so there was little reason to discard around Sun once a year & rotated it on axis once a day • Copernicus studied the works of Aristotle, • His work was published in De Pythagoras & Plato revolutionibus in the year of his • an offshoot of Plato’s philosophy asserted that Sun death was godhead of all knowledge • De revolutionibus took after the • Copernicus objected to equant based on aesthetics - Almagest in outline and basic intention equant not faithful to ideal of uniform motion - makes - to explain planetary motions models to complex • 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 • The daily motion of the heavens relative to the motions horizon results from the earth’s motion on its – Physics of Aristotle axis • Assumed uniform, circular motions – aesthetic appeal since only 1 sphere is rotating – done for aesthetics - followed not many Aristotle – however, he did not account for the objection • All heavenly spheres revolve around that if the earth rotated, objects should be flung the sun & the sun is at the center of the from the surface cosmos • The apparent motion of the sun relative to the – chosen based on aesthetics and stars results from the annual revolution of the simplicity earth around the sun • The distance from the earth to the • The planets’ retrograde motion occur from the sphere of stars is much greater than the motion of the earth relative to the other planets distance from the earth to the sun – retrograde explained as a natural result of the – accounts for lack of observed planet’s revolutions about sun - what we stellar parallax observe is an illusion Retrograde Motion Explained Retrograde Motion Explained • 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 Copernican Model Ptolemy • 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 2 Copernicus Tycho Brahe (1546-1601) • 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 • He made a long series of very precise observations Tycho’s Supernova of the motions of the planets – Very accurate since they were made with a very • The supernova of 1572 large quadrant circle (<0.5 arc minute) – Made repeated observations and took the – Noticed a brilliant star had appeared in Cassiopeia average to reduce the errors – Observed that it faded slowly – Made observations over many years 1576-1591 over the following 2 years – Realized the motions of planets did not match – Demonstrated that the universe the predictions of Ptolemy 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 Tycho’s Cosmology Kepler (1571-1630) • Refuted Copernican system • Was born a commoner in Germany • Placed the earth at the center – His father was a mercenary – Due to his inability to measure parallax – His mother was raised by an aunt who was – believed in Aristotlean physics burnt as a witch • Realized that the Ptolemaic system failed – Went to college at Tubingen were he was • Compromise recognized as a mathematical genius and went – Stationary earth at center to teach math at a Protestant School in Graz – Sun moves around the earth – While in Graz he felt that the 5 perfect Platonic – All other celestial bodies move around solids could explain the motion of the 6 known the sun planets – His worked no better than Ptolemy - – Attempted to get observations of the planets - He needed better mathematics but was incompetent 3 Kepler Solves The Mystery of Planetary Kepler and Brahe Meet Motions • Brahe had to leave Hven for Prague because Fredrick II drank himself to death • It took Kepler 8 years to • Kepler was driven out because the Catholics took over Graz and drove out the Protestants solve the motion of Mars • Tycho knew he needed a mathematician to help him model his – He tried various combinations of observations circular motions – They meet Feb 4, 1600 and fought continually – He finally abandoned circular motion – Tycho feared that Kepler would eclipse him so he would only show and tried ovals and other geometric him data on Mars (his most challenging object) shapes – Tycho died October 21, 1601 from over partying • Kepler got Tycho’s data from the family follow a death bed • He found that an ellipse request by Tycho “Let me not seem to have died in vain” could produce the motions of the planets Properties of Ellipses Kepler’s 1st Law of Ellipses 1609 • Planets move in elliptical orbits with • Ellipses belong to the family of the sun at one focus conic sections (the intersection of a plane and a cone) • The other focus is empty & located – The shape is defined by the in space 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 ⇒ Distance between the planet • The amount by which an ellipse and the sun changes as the differs from a circle is defined as an ellipse’s eccentricity planet moves along its elliptical – Each point on an ellipse has the orbit property that the sum of its ⇒Planet move along orbits which distances to the two foci is the have a special and regular same geometrical shape – The line through the foci to both sides of the ellipse is called the ⇒Gets rid of idea of uniform major axis circular motions 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 • 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 Kepler’s Laws Astronomy Enters a New Era • Kepler’s scheme correctly described planetary motions when compared to the • Astronomical objects are viewed as actual accurate observations of Tycho bodies not distant ethereal or mythical • Together laws 2 and 3 imply that there exists a force, between the sun and entities planets, that weakens with increasing distance • Realization that the laws of nature are the – Invested Sun with physical properties necessary for the uniform description of same for both the earthly and heavenly planetary motions • Kepler recognized there must be a force at domains work but could not figure out what it was.
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