Notes

Outline

Shape of the Earth is a sphere Pythagoreans (550 BC) Original arguments lost From Aristotle Curvature of Earth Shadow of Earth on Moon during an eclipse View of heavens from N and S hemispheres Path of falling object

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Sphere, cont. Different stars seen in north vs. south Objects move toward center of Earth Sphere viewed as a “perfect” shape

Constitution of matter (elements) Earth (solid materials) air (atmosphere) fire (combustion) water (fluids)

Properties of the elements earth - dense (sinks in water) moves toward the center of the Earth air - light, moves away from the Earth fire - light also water - denser that air, lighter than earth

Motions of things all elements move is straight lines motion cause by external force removal of force stops motion

Nature of things Earth is corruptible (storms, earthquakes, decay) Stars are perfect (divine creations) In between are the planets, Sun and Moon These are intermediary in perfection

Fifth element Aether Eternal Main constituent of the heavenly bodies Moves in circular motion Transparent Pure Associated with the stars

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Measuring the whole thing- Questions How big is the Earth? How far away are the Sun and Moon? How big are the Sun and Moon?

Size of the Earth Assumptions Sun is far away Earth is small compared to everything Result is that the Sun’s rays are parallel Based on shadows cast by objects on Earth

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Calculation Angle at Alexandria is 1/50 of a circle Angle at Syene (Aswan) is 0 Distance between Alexandria and Syene is 5,000 stades Therefore the circumference = 50 x 5000 = 250,000 stades roughly 25,000 miles (27,650 real value) Great agreement

More difficult is the distance to the Sun and Moon Based solely on geometry (algebra, trig. missing)

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Calculation Angle formed by Sun-Earth-Moon is measured Value was 87 degrees This gives the ratio Earth-Sun/Earth-Moon as about 18 In other words the Sun is 18 times further away than is the Moon Real value is 390

Problems with the method Very difficult to tell when the Moon is exactly half illuminated No trigonometric table yet Needed to use inexact approximations

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Results Ratio of Sun diameter to Earth diameter Value was 19/3 or 6.3 With diameter of Earth we get Sun’s radius = 41,457 km Real value is 669,000 km Error is largely in the estimate of distances to Sun and Moon from Earth

Second approach based on parallax Parallax is the apparent displacement of an object viewed from two different locations a known distance apart Proposed by Hipparchus in ~200 BC based on an actual solar eclipse of 189 BC

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Objects in universe Sun - large, always a disc, bright and hot Moon - phases, bright but cold Planets - move against the stars Mercury, Venus - variable brightness, close to the Sun Mars, Jupiter, Saturn - retrograde motion Stars -  immovable, bright objects

Stars assumed to be fixed to a sphere Planets of two types - inferior (Mercury and Venus) and superior (Mars, Jupiter, Saturn) Challenge is to arrange these in order of distance from the Earth (at center)

Position of the Earth in the universe Two views At the center of the heavens Not at the center of the heavens Problem – how to test this? Use simple observations views of the stars Use simple arguments

Two proposals Aristarchus - heliocentric system Aristotle (and others) - geocentric system Aristotle won because - idea seemed reasonable simple aim was utility

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Aristotelian assumptions Earth at the center of everything Other objects move in circular orbits The orbits are part of transparent spheres The objects move at constant velocities in their circular orbits

Simple problem - Sun The apparent size of the Sun is roughly constant - roughly fixed distance from Earth Assumed to move in a circular path centered on the Earth Therefore, we would expect a constant length to the four seasons

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Predictions .    Centered     Observed Spring 91 93 Summer 91 93 Autumn 91 90 Winter 91 89 This was observed as early as 330 BC Called the Solar Anomaly

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More difficult problem - the planets Simple motion in a circular orbit centered on the Earth does not work Example - inferior planets appear to oscillate back and forth about the Sun Example - superior planets move about the zodiac but exhibit retrograde motion (the apparent reversal of direction)

More difficult problem - the planets Simple motion in a circular orbit centered on the Earth does not work Example - inferior planets appear to oscillate back and forth about the Sun Example - superior planets move about the zodiac but exhibit retrograde motion (the apparent reversal of direction)

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Solution - Epicycles A break with Aristotle, but there was no choice Earth at center Planet moved at constant velocity on a circular orbit BUT The circular orbit was centered on the circle centered on the Earth Dates from 2 Cent. BC

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Results Model 2 explains retrograde motion Knowing the sizes of the epicycle and deferent and the velocities the positions of the retrograde motions and their dates could be calculated This was a good qualitative model for the superior planets

Three kinds of problems for the Greeks Based on observation alone - size of the Earth Observation plus some assumptions - motion of the planets Assumptions only - order of the planets (e.g., the inferior planets)

What about the inferior planets? Mercury and Venus They move close to the Sun There is an apparent oscillating motion of these planets about the Sun These planets then are either the morning star or evening star An epicycle ties this together

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Problems with epicycles The major difficulty was the accurate prediction of the motions of Mars Mars is a very powerful force in astrology Necessary to improve the understanding of Mars Enter Ptolemy

Claudius Ptolemy He was active between 127-145 AD in Alexandria Almost nothing known about his life He was a mathematician, astronomer, geographer His great astronomical work is the Almagest

What was the problem? The position and width of the retrograde motion can be observed The position and width of the retrograde motion can be predicted from the epicylic model They did not agree!

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The simple epicycle model Simple explanation of retrograde motion Simple explanation of the variation of brightness of the superior planets Predicts that all retrograde events (loops) are equally spaced around the ecliptic AND these will have the same widths

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Two kinds of problem The spacing of the retrograde motions is not equally spaced The width of the retrograde motions is not constant but varies from event to event The closer the spacing - the wider the motion

First attempt at improvement Move the Earth away from the center of the deferent This changes the apparent spacing This changes the apparent width

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Almost there We can change the apparent width and spacing to the retrograde events by moving the Earth away from the center Cannot however match both the width and spacing

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But this does not agree with Aristotle Earth is not at the center of the universe (not a great problem) Aristotle did not mention epicycles The only thing kept is the circular motion There is a non-uniform motion of the epicycle along the deferent (bothered many including Copernicus)

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Success of Ptolemy Explained retrograde motion Ordered the Sun, Moon and planets Estimated size of the solar system Mechanism to calculate positions of planets at any date Lasted roughly 1400 years

Popular story To improve the agreement between predictions and observations - needed multiple epicycles. This became too complicated Search began for a new view of the solar system

Reality There were no new observations of the positions of the planets Not really a problem with predictions Switch to the Copernican (heliocentric) system was equally complex and unfamiliar Copernican system was not accurate - still used circular orbits

What was needed More accurate observations New ways of observing New instruments An organization of observing effort “Big science”