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Ptolemy (125 A.D.)• Designed a complete geometrical

model of the universe that accuratelypredicted planetary motions with errorswithin 50

• Most of the geometric devices andbasic foundations of his model did notoriginate with him but were based onthe models of the early Greeks suchas Aristotle & Hipparchus

• Wrote the Almagest (Greatest)– included the original works & models of Ptolemy– included a compilation of past works of Greeks,

especially Hipparchus– 13 volumes

Ptolemy’s Model• Earth was spherical & at center of

cosmos - GEOCENTRIC• Cosmos is finite in size• Earth has no motions• Sun, Moon Planets exhibit uniform,

circular motions– natural motions - no forces

• Used devices of eccentrics, epicycles& deferents to explain the observednon-uniform motions of the Sun andplanets along the ecliptic & retrogrademotion.

• Introduced equant to explain thevariations in retrograde motions

Ptolemy’s Equant• 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 beuniform around the centers of circles

Ptolemy’s Model

Copernicus (1473-1543)

• Developed a Heliocentric (Sun centered) model ofthe cosmos

• Why? Ptolemy’s geocentric model lasted forcenturies mainly because it accurately predictedcelestial motions so there was little reason to discardit

• Copernicus studied the works of Aristotle,Pythagoras & Plato

• an offshoot of Plato’s philosophy asserted that Sunwas godhead of all knowledge

• Copernicus objected to equant based on aesthetics -equant not faithful to ideal of uniform motion - makesmodels to complex

Copernicus’ Model• 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 & rotatedon axis once a day

• His work was published in Derevolutionibus in the year of hisdeath

• De revolutionibus took after theAlmagest in outline and basic intention- to explain planetary motions

• Even though it took 20 years todevelop this model did predict celestialmotions any better than Ptolemy’sgeocentric model

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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 thecosmos– chosen based on aesthetics and

simplicity• The distance from the earth to the

sphere of stars is much greater than thedistance from the earth to the sun– accounts for lack of observed

stellar parallax

Heliocentric Model of Copernicus

• The daily motion of the heavens relative to thehorizon results from the earth’s motion on itsaxis– 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 flungfrom the surface

• The apparent motion of the sun relative to thestars results from the annual revolution of theearth around the sun

• The planets’ retrograde motion occur from themotion of the earth relative to the other planets– retrograde explained as a natural result of the

planet’s revolutions about sun - what weobserve 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

• Copernicus eliminated epicycles toexplain retrograde motion

• Eliminated equant - kept uniform circularmotion

• Needed to account for variations inplanetary motion so he was forced toadd many smaller circles

• Violated Aristotelian physics & did notoffer new physical ideas to support hismodel

• Didn’t predict motions any better thanPtolemy’s model

Copernican Model Ptolemy

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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 mortalscould predict astronomical events

– Very arrogant and hard to get along with andplayed the role of a courtly prince with a goldnose

• Supported by the King Frederick IIof Denmark, he built andobservatory on the island of Hvenwith 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

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 Supernova• The supernova of 1572

– Noticed a brilliant star hadappeared in Cassiopeia

– Observed that it faded slowlyover the following 2 years

– Demonstrated that the universewas changing and imperfect

– Showed that it had the sameposition no matter where on earthit was observed - thus it was at agreat distance

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

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 wentto teach math at a Protestant School in Graz

– While in Graz he felt that the 5 perfect Platonicsolids could explain the motion of the 6 knownplanets

– Attempted to get observations of the planets -but was incompetent

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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– 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

• Kepler got Tycho’s data from the family follow a death bedrequest by Tycho “Let me not seem to have died in vain”

Kepler Solves The Mystery of PlanetaryMotions

• It took Kepler 8 years tosolve the motion of Mars– He tried various combinations of

circular motions– He finally abandoned circular motion

and tried ovals and other geometricshapes

• He found that an ellipsecould produce the motions ofthe planets

Properties of Ellipses• Ellipses belong to the family of

conic sections (the intersection of aplane and a cone)– The shape is defined by the

separation between the two fociof the ellipse

• When the foci coincide theshape is a circle

• As the foci separate from eachother the ellipse gets flatter

• The amount by which an ellipsediffers from a circle is definedas an ellipse’s eccentricity

– Each point on an ellipse has theproperty that the sum of itsdistances to the two foci is thesame

– The line through the foci to bothsides of the ellipse is called themajor axis

Kepler’s 1st Law of Ellipses 1609• Planets move in elliptical orbits with

the sun at one focus• The other focus is empty & located

in space

⇒ Distance between the planetand the sun changes as theplanet moves along its ellipticalorbit⇒Planet move along orbits whichhave a special and regulargeometrical shape⇒Gets rid of idea of uniformcircular motions

Kepler’s 1st Law Kepler’s 2nd Law of Equal Areas 1609

• Kepler noted that Mars slowed down when itapproached the sun

• He found that a line drawn between the sunand a planet sweeps out an equal area in anequal 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 arenear the sun

– The increase in speed as a planet movestoward the sun makes it appear that thesun is “pulling” on the planet

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Kepler’s 2nd Law• 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 planetssuggests a single physical cause for theirorbits

Kepler’s 3nd Harmonic Law 1618

Kepler’s Laws• Kepler’s scheme correctly described

planetary motions when compared to theaccurate observations of Tycho

• Together laws 2 and 3 imply that thereexists a force, between the sun andplanets, that weakens with increasingdistance– Invested Sun with physical properties

necessary for the uniform description ofplanetary motions

• Kepler recognized there must be a force atwork but could not figure out what it was.He speculated it may be magnetism

Astronomy Enters a New Era

• Astronomical objects are viewed as actualbodies not distant ethereal or mythicalentities

• Realization that the laws of nature are thesame for both the earthly and heavenlydomains