The Copernican Revolution

Scotland: 4,000-year-old stone circle; the Moon rises as shown here every 18.6 years.

What did ancient civilizations achieve in astronomy?

• Daily timekeeping • Tracking the

seasons and calendar

• Monitoring lunar cycles

• Monitoring planets and stars

• Predicting eclipses• And more…

Mexico: model of the Templo MayorEngland: 1550 BC

Our mathematical and scientific heritage originated with the civilizations of the Middle East

• Greeks were the first people known to make models of nature.

• They tried to explain patterns in nature without resorting to myth or the supernatural.

• This is a very important step in understanding the natural world.

Greek geocentric model (c. 400 B.C.)

Why does modern science trace its roots to the Greeks?

What shape is the Earth?

Prove it!

Eratosthenes measures the Earth (c. 240 BC)

Calculate circumference of Earth:7/360 (circum. Earth) = 5000 stadia circum. Earth = 5000 360/7 stadia ≈ 250,000 stadia

Measurements:Syene to Alexandria distance ≈ 5000 stadia angle = 7°

Compare to modern value (≈ 40,100 km): Greek stadium ≈ 1/6 km 250,000 stadia ≈ 42,000 km

APPARENT RETROGRADE MOTION OF THE PLANETS

We see apparent retrograde motion when we pass by a planet in its orbit.Easy for us to explain today: occurs when we “lap” another planet (or when Mercury or Venus laps us).But very difficult to explain if you think that Earth is the center of the universe!In fact, ancients considered but rejected the correct explanation …

Why did the ancient Greeks reject the real explanation for planetary motion?

Their inability to observe stellar parallax was a major factor.

Not to scale1 AU

p

d

tan p = 1AU/d (AU)

For small angles:

p = 1/d

If the angle p = 1 second of arc then d =206265 AU. We define this distance to be 1 parsec (pc). Can you show that 1 pc = 3.26 light-years?

Here p is in radians, but there are 206265 arcseconds in 1 radian.

The Greeks knew that the lack of observable parallax could mean one of two things:

1. Stars are so far away that stellar parallax is too small to notice with the naked eye

1. The Moon is 30 minutes of arc and the eye can separate objects about 1 minute of arc apart

2. Earth does not orbit the Sun; the Earth is the center of the universe

With rare exceptions such as Aristarchus, the Greeks rejected the correct explanation (1) because they did not think the stars could be that far away

Thus setting the stage for the long, historical showdown between Earth-centered and Sun-centered systems.

Ptolemy

Most sophisticated geocentric model was that of Ptolemy (A.D. 100-170).Sufficiently accurate to remain in use for 1,500 years.

Arabic translation of Ptolemy’s work named Almagest (“the greatest compilation”)

Explaining retrograde motion with the Geocentric model

The Copernican Revolution

• How did Copernicus, Tycho, and Kepler challenge the Earth-centered (geocentric) idea?

• What are Kepler’s three laws of planetary motion?

• How did Galileo solidify the Copernican revolution?

Our goals for learning:

How did Copernicus, Tycho, and Kepler challenge the Earth-centered idea?

Copernicus (1473-1543): • Proposed Sun-centered model (published 1543)

• Used model to determine layout of solar system (planetary distances in AU)

But . . .

• He had the right idea, yet Copernicus still assumed that the orbits of the planets were perfect circles and so his heliocentric model was no more accurate than Ptolemy’s geocentric model in predicting planetary positions!

Danish astronomer Tycho Brahe (1546-1601)

• Compiled the most accurate (one arcminute) naked eye measurements ever made of planetary positions.

• He still could not detect stellar parallax, and thus still thought Earth must be at center of solar system (but recognized that other planets must go around the Sun)

• Hired Johannes Kepler, who later used these detailed observations to discover the truth about planetary motion.

Brahe’s observatory was visual – before the telescope – and really a giant protractor for measuring angles.

Johannes Kepler (1571-1630)

• Kepler first tried to match Tycho’s observations with circular orbits

• But an 8-arcminute discrepancy led him eventually to ellipses…

“If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.”

An ellipse looks like an elongated circle

What is an ellipse?

The semi-major axis is also the “average” distance from the focus.

Eccentricity and Planetary Orbits

• Define c to be the distance from the center to either focus

• Define a to be the semi-major axis length• The eccentricity e of an ellipse is then

e = c/a [ For a circle e = 0 because c = 0]

Perihelion distance = a(1 – e)

Aphelion distance = a(1 + e)

Earth: e = 0.017; a = 149.6 million km; closest = 147.1 million km and farthest = 152.1 million km. Note that a is the average.

ca

Eccentricity of an Ellipse

Kepler’s First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus.

Kepler’s Three Laws of Planetary Motion?

NOTE: None of the planets have orbits as elliptical as this, only some comets.

Kepler’s Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal times.

Þ this law means that a planet travels faster when it is nearer to the Sun and slower when it is farther from the Sun.

More distant planets orbit the Sun at slower average speeds, obeying the relationship

p2 = a3

p = orbital period in years a = avg. distance from Sun in AU

Kepler’s Third Law

If a = 1 AU then 13=1 and p2=1 therefore p=1 (year).

Graphical version of Kepler’s Third Law

Planet’s average speed is v = 2πa/p (circumference of orbit divided by period), so v2 = (2π)2 a2/p2 but p2 = a3 so, v2 = (2π)2 a2/a3 = (2π)2/a, thus v = 2π/√a. Using v in km/s then v = 30/√a.

p2 plotted against a3

v plotted against a

Thought Question: An asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun?

A. 4 years

B. 8 years

C. 16 years

D. 64 years

Hint: Remember that p2 = a3

An asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun?

A. 4 yearsB. 8 yearsC. 16 yearsD. 64 years

We need to find p so that p2 = a3 Since a = 4, a3 = 43 = 64Therefore p = 8, p2 = 82 = 64

How did Galileo Galilei solidify the Copernican revolution?

Galileo (1564-1642) overcame major objections to Copernican view. Threekey objections rooted in Aristotelian view were:

1. Earth could not be moving because objects in air would be left behind.

2. Non-circular orbits are not “perfect” as heavens should be.

3. If Earth were really orbiting Sun,we’d detect stellar parallax.

Galileo’s experiments showed that objects in air would stay with a moving Earth.

Overcoming the first objection (nature of motion):

• Aristotle thought that all objects naturally come to rest.

• Galileo showed by experiment that an object will stay in motion unless a force acts to slow it down.

• Galileo proved this with countless experiments involving falling and rolling objects.

Overcoming the second objection (heavenly perfection):

• Tycho’s observations of a comet and a supernova (new star) already challenged this idea.

• Using his telescope (1609), Galileo saw:

• Sunspots or “imperfections” on the Sun

• Mountains and valleys on the Moon (proving it is not a perfect sphere)

• Tycho thought he had measured stellar distances, so lack of parallax seemed to rule out an orbiting Earth.

• Galileo showed stars must be much farther than Tycho thought — in part by using his telescope to resolve the Milky Way into countless individual stars.

If stars were much farther away, then lack of detectable parallax was no longer so troubling.

The eventual discovery of the tiny angular shifts in a star’s position due to parallax as the Earth orbits the Sun required more than another century of technology development to make much bigger telescopes.

Overcoming the third objection (parallax):

Galileo saw four moons orbiting Jupiter, proving that not all objects orbit the Earth

Galileo’s observations of phases of Venus proved that it orbits the Sun and not Earth.

The Nature of Science

• How can we distinguish science from nonscience?

• What is a scientific theory?

Our goals for learning:

The previous example of the evolution of ideas from the false geocentric model to the true heliocentric model leads us nicely to a discussion of the Scientific Method.

The idealized Scientific Method

• Based on proposing and testing hypotheses through careful and repeated experiments

• hypothesis = educated guess

• Around the time of Galileo there was a fundamental shift away from mere speculation and towards tangible experimental evidence.

Hallmarks of Science: #1

Modern science seeks explanations for observed phenomena that rely solely on natural causes.

(A scientific model cannot include divine intervention)

Example: Kepler sought a natural explanation for observations made by Tycho.

Hallmarks of Science: #2

Science progresses through the creation and testing of models of nature that explain the observations as simply as possible.

(Simplicity = “Occam’s razor”)

Occam’s principle states that “all things being equal, the simplest explanation tends to be the right one.” The heliocentric model is simpler than the geocentric one.

Hallmarks of Science: #3

A scientific model must make testable predictions about natural phenomena that would force us to revise or abandon the model if the predictions do not agree with observations.

Example: each of the competing models offered predictions that were tested. Kepler’s model worked best. Kepler’s model can still be tested. In fact, slight discrepancies found at much later dates led to new discoveries, such as Einstein’s theories…

What is a scientific theory?

• The word theory has a different meaning in science than in everyday life.

• In science, a theory is NOT the same as a hypothesis, it is NOT a guess, rather:

• A scientific theory must:— Explain a wide variety of observations with a few

simple principles, AND— Must be supported by a large, compelling body of

evidence. — Must NOT have failed any crucial test of its validity.

Scientific Theories

• The theory of gravity• The theory of electromagnetism and light• The theory of atoms• The theory of evolution by natural selection• The theory of relativity• The theory of plate tectonics• The quantum theory

Other modes of reasoning• Appeal to authority – weak because does not rely on experimental

evidence and advance predictions, even if the authority figure is a famous scientist.

• Opposing advocates (the legal system) – good in principle if both sides present all the evidence and argue rationally for the truth. Fails in practice because each advocate presents only those facts that support their position.

• Myths, superstitions and divine intervention – weakest of all because there is no rational way to seek truth when explanations are attributed to unseen and unknowable forces. Today, after millennia of study, no belief systems based on superstitions stand the test of experiment.

• The Scientific Method, especially in modern times where many scientists of equal skill compete to verify facts by experiment, is the most powerful method of obtaining new knowledge.

What have we learned?

• How can we distinguish science from non-science?– Science: seeks explanations that rely solely on

natural causes; progresses through the creation and testing of models of nature; models must make testable predictions

• What is a scientific theory?– A model that explains a wide variety of

observations in terms of a few general principles and that has survived repeated and varied testing