Astronomy · Round I · Foundations · Models of the heavens

Compare Ptolemy, Copernicus, Tycho and Kepler

I can state what Ptolemy's, Copernicus's and Tycho's arrangements each claim, explain the retrograde motion of Mars in the first two, say what the phases of Venus refute and what they leave open, and state Kepler's three laws with their sources.

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Round ICompare Ptolemy, Copernicus, Tycho and Kepler (this page)
Round IIThe moons of Jupiter and the parallax debateComing
Round IIIFrom Kepler's laws to Newton's moon testComing

1 Learn

You will be able to

Describe the three arrangements: what is at rest, what circles what, and what an epicycle, a deferent and an equant are in Ptolemy's models.
Explain retrograde motion as the backward carriage of the epicycle in Ptolemy's model and as the Earth overtaking Mars at opposition in Copernicus's.
Say that a round Venus refutes Ptolemy's arrangement but not Tycho's, and that Copernicus's gain was an explanation of order and scale, not the removal of epicycles or much better prediction.
State Kepler's three laws with the work where each was published (laws 1 and 2 in Astronomia Nova, 1609; law 3 in Harmonices Mundi V, 1619), and use T squared = a cubed, in years and AU, to find a planet's period.

2 Practise

Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.

recall
In Ptolemy's planetary models, what are the epicycle, the deferent and the equant?

Show answer
The epicycle is a small circle that carries the planet, and its centre rides on a larger circle, the deferent. The equant is a third point, off the deferent's centre, about which the epicycle's centre sweeps equal angles in equal times. It broke the old rule of uniform motion about a circle's centre, and Copernicus dropped it.
recall
State Kepler's three laws and say where each was published.

Show answer
Law 1: a planet moves on an ellipse with the Sun at one focus. Law 2: the line from the Sun to a planet sweeps out equal areas in equal times. Both are in Astronomia Nova (1609), which states them for Mars; Kepler later found they held for the other planets. Law 3: the square of a planet's period is proportional to the cube of its semi-major axis, T^2 = a^3 in years and astronomical units, in Harmonices Mundi V (1619).
explain
How does Copernicus explain the retrograde motion of Mars, and why does the loop come at opposition?

Show answer
He makes it an effect of the Earth's own motion: stations and retrogradations are "not movements of the planets, but a motion of the earth". The Earth moves faster than Mars (a mean 29.8 against 24.1 km/s) and overtakes it as it passes between Mars and the Sun, so Mars seems to slide backwards against the stars. With the Earth between Mars and the Sun, Mars stands opposite the Sun in our sky, so the middle of the backward arc falls at opposition.
apply
Ptolemy gives Jupiter an epicycle of 11;30 on a deferent of 60. Reading the ratio as Copernicus would, how far is Jupiter from the Sun in astronomical units?

Show answer
11;30 is 11 and 30/60, or 11.5, so the ratio is 11.5/60, about 0.19. Like those of Mars and Saturn, Jupiter's epicycle stands for the Earth's orbit, so the ratio is the Earth's distance over Jupiter's. Jupiter is therefore about 60/11.5 = 5.2 AU from the Sun.
connect
"Hypotheses, Predictions and Tests" separates strong tests from weak ones. How do the phases of Venus illustrate the difference?

Show answer
A strong test is one on which rivals predict different results. Venus's phases were strong against Ptolemy, whose arrangement forbids a round Venus. They were silent between Copernicus and Tycho, whose arrangements both allow it, so the test removed one rival and left two.
recall
When does Mars appear to move backwards among the stars, and for roughly how long?

Show answer
Around opposition, when the Earth lies between the Sun and Mars and overtakes it. In reality it averages about 72 days and varies from about 60 to 81 days, because Mars's orbit is an ellipse. In the circular model it lasts about 73 days, from 36.4 days before opposition to 36.4 days after.
explain
How does the Sun-centred view explain a retrograde loop without Mars ever reversing?

Show answer
The Earth orbits faster (1 year against 1.881 years), so it overtakes Mars. As it passes, the line of sight from the Earth to Mars swings backwards against the stars, although Mars keeps moving the same way round the Sun.
apply
The Earth's period is 1 year and Mars's is 1.881 years. How long is it between successive oppositions of Mars?

Show answer
Each year the Earth gains 1 − 1/1.881 of a lap on Mars, so it laps Mars every 1 ÷ (1 − 1/1.881) = 2.135 years, about 780 days. NASA gives 779.94 days.
explain
Did Copernicus's Sun-centred system do away with epicycles or predict the planets much better than Ptolemy's?

Show answer
No. Copernicus still used circles on circles, and the tables computed from his models were little different in accuracy from the older ones. What he changed was the explanation of retrograde motion: it is apparent, produced by the observer's own motion.
apply
A body orbits the Sun with a semi-major axis of 9 AU. What is its period?

Show answer
T = 9^(3/2) = 27 years.

3 Prove it: the mastery check

5 questions drawn from a pool of 12. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.

4 Prove it: the performance task

Draw Venus under two arrangements and read the test

Part A (draw). Draw two diagrams, one for Ptolemy's arrangement and one for Tycho's. In each, show the Earth, the Sun and Venus, put Venus in at least four positions, draw the direction of sunlight, and shade the half of Venus that the Sun lights. In Ptolemy's, keep Venus nearer the Earth than the Sun and keep its epicycle's centre on the line from the Earth to the Sun. In Tycho's, the Sun circles the Earth and Venus circles the Sun. Part B (read). Venus's phase depends on the angle at Venus between the Sun and the Earth, not, as the Moon's nearly does, on its elongation: shade the half facing the Sun and see how much of it faces the Earth. For each position, say what an observer on the Earth would see (new, crescent, half, gibbous or full), and mark any position where Venus would look round. Part C (judge). Write four to six sentences saying which arrangement a round Venus refutes and which it does not, why the test cannot separate Tycho's arrangement from Copernicus's, and whether the test was strong or weak against each (a strong test is one on which rivals predict different results). Finish with one sentence on what Copernicus's arrangement gained and what it did not.

What to hand in: Two labelled diagrams (or a written description of each, listing the positions and what is seen) and a written part of five to seven sentences. Save it on this page or download it.

Check your work against the rubric Each diagram shows the Earth, the Sun and Venus in at least four positions, with the direction of sunlight and the lit half of Venus shaded. The Ptolemaic diagram keeps Venus nearer the Earth than the Sun, with its epicycle's centre on the Earth-Sun line, and the reading finds no position that shows a full disc (at most about half lit). The Tychonic diagram has Venus circling the Sun while the Sun circles the Earth, and the reading finds a position beyond the Sun where Venus is seen round. The judgement says that a round Venus refutes Ptolemy's arrangement but not Tycho's, that the test cannot separate Tycho's from Copernicus's because only what is at rest differs, and that it was strong against Ptolemy's and silent between the other two. The closing sentence says that Copernicus's arrangement gained an explanation of order and scale, and that it did not remove epicycles or predict much better.

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5 Discuss: the seminar

Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.

Astronomy · Round 1

Aristotle, On the Heavens (De Caelo), II.14 (Bekker 296a24-298a20; the arguments for a spherical Earth run from 297a8)J. L. Stocks's Oxford translation on the MIT Internet Classics Archive. The page carries Book II in full without Bekker numbers; chapter 14 is 'Part 14', the last on the page. Aristotle argues for a spherical Earth from the curved shadow in lunar eclipses and from the stars that change as one travels north or south, and cites the mathematicians' circumference of 400,000 stades.
Cleomedes, On the Circular Motion of the Celestial Bodies (Caelestia), Book I, chapter 10 in Ziegler's numbering (chapter 7 in Todd's numbering)T. L. Heath's English translation of the chapter, reproduced on a freely readable page; it also gives Posidonius's method. Cleomedes reports the measurement of Eratosthenes (about 276 to 194 BC). The length of the stade is uncertain, so the result in kilometres is too. Bowen and Todd (2004) is the standard translation but is not free.

Aristotle argues that the Earth is a sphere from the curved shadow in an eclipse and from the stars that change as you travel, and Cleomedes reports that Eratosthenes found its size from one shadow angle and one distance. Which of these arguments depends most on what we already assume, and which could you check yourself this week?

Eratosthenes needs Syene and Alexandria on one meridian and the two lines to the Sun to be parallel. Cleomedes states both. Does his reason for the parallel lines convince you, and what would go wrong if they were not parallel?
Cleomedes also reports Posidonius, who used the star Canopus, on the horizon at Rhodes and one forty-eighth of a circle above it at Alexandria. Compare the two methods: which hypotheses do they share, and which does each add?
Aristotle gives 400,000 stades as the figure of unnamed mathematicians. How much weight can a figure carry when neither its method nor its unit is given?

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Self-administered checks and self-assessed tasks: no credential is awarded. The design borrows from Khan Academy (mastery levels), WGU (competencies proved by assessment) and St. John's College (seminars on primary texts); this site is not affiliated with any of them.