Astronomy 171
Solar System Astronomy
Prof. Paul Martini

Lecture 12: Greek Astronomy

Key Ideas

Celestial Motions
Aristarchus of Samos: Distance to the Moon; Distance to the Sun; Early Heliocentric Model
Geocentric Model: Equants, Deferents, and Epicycles; Ptolemy

Summary of Celestial Motions

Fixed Stars: Uniform daily motion about the celestial poles
The Sun: Daily motion around the celestial poles.; Annual motion eastward along the Ecliptic - slower in summer, faster in winter
The Moon: Daily motion around the celestial poles; Eastward motion near the ecliptic over a month
The Planets: Daily motions about the celestial poles; Generally eastward motion near the Ecliptic at different speeds for each planet; Westward "retrograde" motions at opposition (Superior Planets) or inferior conjunction (Inferior Planets); Superior planets are brighter at opposition, fainter at conjunction
Any successful description of the Solar System must explain all of these facts.

A Question of Approach

How to explain these motions?
Phenomenologically: Find a way to compute the motions without worrying about "why" they work this way.; Preserve appearances (make good predictions)
Physically: Discover the underlying physical principles behind the motions - ask "why?"; Predictions then follow from first principles

From Myth to Science

Any satisfactory theory of planetary motions has to have the following characteristics:
Internal Consistency:: They must follow the same basic rules - no special cases
Predictive Power: Must provide measurably accurate predictions of future behavior
This effort marks the true birth of science

The Geocentric System

Geocentric = Earth Centered
Anaximander of Miletus (610-546 BC)
The first Greek philosopher to suggest a geocentric system: Earth was a flat disk (cylinder) fixed and unmoving at the center; Sun, Moon, and Stars were affixed to rotating crystalline spheres centered on the Earth
Sun, Moon, and Stars were physical objects

Pythagoras (d. 497 BC)

Philosopher and Mathematician: Founded the Pythagorean School; Spheres are the perfect shapes
Pythagorean Model: Spherical Earth fixed at the center; Planets and Stars on concentric crystalline spheres; Sizes were ratios of small numbers (2:1, 3:2, etc.)

Aristarchus of Samos (310 - 230 BC)

Distance to the Moon
Distance to the Sun: Sun is much further; Therefore the Sun is much larger; Logical that the Sun is at the center
Earliest known Heliocentric Model

Aristarchus and the Moon

Aristarchus observed that the center of the Moon takes 3 hours to cross the Earth's shadow during a Lunar Eclipse, compared to 27.3 days to orbit the Earth
If this distance corresponds to the diameter of the Earth, he can express the distance of the Moon in terms of the size of the Earth: He estimated that the distance to the Moon was 70 times the radius of the Earth; He also observed that the Moon was half as large as the Earth's shadow and therefore the Earth should be twice as large as the Moon.

Distance to the Moon

Aristarchus of Samos determined that the distance to the Moon was 70 times the radius of the Earth
Modern measurements show that the Moon is actually 60 Earth radii away
Aristarchus assumed that the size of the Earth's shadow was equal to the size of the Earth
Because the Sun is not infinitely far away, the Earth's shadow is actually smaller at the orbit of the Moon than the Earth's diameter
Aristarchus was slightly off because the Sun is not a point light source at an infinite distance - the Earth's shadow is a cone, not a cylinder

Aristarchus and the Sun

Aristarchus reasoned that when the Moon appeared to be exactly 1/2 illuminated (Quarter Moon) the Earth-Moon-Sun angle must be 90 degrees.
If the Sun is close to the Earth, the Moon will not be far from the Sun in the sky at First Quarter
The farther the Sun, the larger the angle between the Moon and Sun at Quarter Moon

Distance to the Sun

Aristarchus measured an angle of 87 degrees between the Moon and Sun.: He determined that the Sun was 20 times further away than the Moon; He did this without trigonometry, which had not been invented yet!
Aristarchus actually measured too small an angle: Modern measurements show that the angle is 89 degrees, 50 arcminutes, and the Sun is therefore 400 times further away than the Moon.

Implications of Aristarchus

Aristarchus demonstrated: The Earth is twice the size of the Moon; The Moon is 70 Earth radii away; The Sun is 20 times further away than the Moon
This implies:: The Sun is 20 times larger than the Moon; The Sun is 10 times larger than the Earth
He accomplished this without knowing the size of the Earth

Heliocentrism in Ancient Greece

Aristarchus inferred that the Sun was much larger than the Earth: It was absurd for the larger Sun to go around the smaller Earth
He instead proposed that the Earth goes around the Sun - a Heliocentric Model
His contemporaries preferred a Geocentric Model
Aristarchus was 1800 years ahead of his time!

Enter Epicycles

Hipparchus of Nicaea (165 - 127 BC): Greatest astronomer of the classical period; Discovered the Precession of the Equinoxes; Developed the system of stellar magnitudes
Elaborated a New Geocentric System: Introduced epicycles, building on ideas of Apollonius of Perga; Located the Earth slightly off-center on an Eccentric

Successes of Epicycles and Eccentrics

Epicyclic models have a number of successes: Combined motion of deferent and spicycle reproduces the retrograde motion of the planets; Superior planets are closer and brighter at opposition when moving retrograde; Placing the Earth at an eccentric away from the deferent center explains the non-uniform motion of the Sun, Moon, and Planets
Can fine-tune the models by adding more epicycles

Ptolemy (c. 150 AD)

Great Astronomer and Geographer of the late classical age
Wrote the Mathematical Syntaxis: Compilation of all Mathematica and Astronomical knowledge of the time; The Arabs referred to this manuscript as "Al Magest," literally "The Greatest"
Today it retains this name as "The Almagest"

The Equant

Ptolemy introduced the Equant to account for observed changes in a planet's speed as it moved around the Earth: Epicycle still moves about the center of the Deferent; Uniform circular motion about the center of the deferent is replaced by uniform angular motion about an off-center equant point.

The Ultimate Geocentric System

Ptolemy's final system was quite complex: 40 epicycles and deferents were required; Equants and eccentrics for all planets, the Moon, and the Sun; It provided accurate predictions of the motions of the planets, Sun, and Moon.
It was to prevail virtually unchallenged for nearly 1500 years

See A Note about Graphics to learn why some of the graphics shown in the lectures are not reproduced with these notes.