7. Ptolemy, Copernicus, Tycho Brahe, and Kepler

A. Aristarchus and others measure the distance to the Moon

The ancient Greeks gave several reasons for believing that the Earth was a sphere. One of the most convincing was based on lunar eclipses. They realized that in a lunar eclipse, the Earth's shadow was projected on the Moon. The edge of this shadow was always circular, regardless of the relative positions of the Earth and the Moon. This could only be the case if the Earth were a sphere (a disk would cast a narrow elliptical shadow in some orientations).

Aristarchus (c. 250 B.C.) realized that lunar eclipses could be used for estimating the distance to the Moon in terms of the diameter of the Earth. Although his value was too low by about a factor of 3, later Greek astronomers, using his method, got very nearly the correct answer of about 30 Earth diameters. The steps in this calculation are as follows:

• The umbral shadow of the Earth during an eclipse of the Moon has a diameter of about 3 times the diameter of the Moon.
• From solar eclipses, we know that the Moon's shadow just barely touches the Earth, so shadows from the Moon to the Earth (and therefore from the Earth to the Moon) must converge by about 1 Moon diameter, so the true diameter of the Earth must be 3 + 1 = 4 times that of the Moon.
• The angular size of the Moon is about 1/2 of a degree, or 1/720 of a full circle, so the diameter of the Earth at the same distance would be 1/180 of a full circle.
• A full circle is 2(pi) (about 6.28) times the radius (in this case, the distance to the Moon), so the distance to Moon is about (180/6.28) = 29 times the diameter of the Earth.

B. The motion of the planets on the celestial sphere.

• Motion of planets on the celestial sphere was more complicated than that of the Sun and the Moon (though ancient astronomers regarded these also as "planets", while the Earth was not regarded as a planet).
• Mercury and Venus always stayed close to the Sun (within 28 degrees and 47 degrees, respectively).
• Mars, Jupiter, and Saturn had no such restriction, but went completely around the ecliptic. However, they sometimes reversed their motion for a brief period. This reversal was called retrograde motion.
• The challenge of the various models proposed for the observed sky was to account for the uneven motion of the planets in a satisfactory way, and predict their future positions.

C. The Ptolemaic system

• Claudius Ptolemy (ca. 150 AD) compiled and systematized the observations of his predecessors into a model that survived essentially unchanged for 1300 years.
• The Ptolemaic model is Earth centered: the Moon, the Sun, the planets, and the stars of the celestial sphere all circulate around the Earth.
• All motions are (1) uniform and (2) circular.
• The uneveness and even reversal of the actual motion of planets on the celestial sphere is accomplished by a system of epicycles and deferents.
• Example of actual paths of planets in the Ptolemaic system (you need a QuickTime plug-in for your browser in order to view).
• The culmination of the the Ptolemaic system was the calculation over a period of 10 years of the massive Alphonsine Tables by a team of astronomers financed by King Alfonso X of Castile in the mid-13th century.

D. The Copernican system

• Nicholas Copernicus (1473--1543) was a Polish cleric who revived (after 15 centuries of neglect) the view that the Earth, as well as the other planets (in the modern sense) revolved around the Sun.
• The apparent daily motion of the celestial sphere around the Earth is simply a reflection of the Earth's rotation.
• All motions are still uniform and circular, but they now go around the Sun, rather than the Earth.
• Because Copernicus retained the assumption of uniform circular motion, he also had to retain a few epicycles.
• The Copernican model provided a natural explanation for the restricted movement of Mercury and Venus from the Sun along the ecliptic, as well as a natural explantion for retrograde motion.
• Example of paths of planets in the (modernized) Copernican model.
• Main objection to the Copernican model: Lack of observed parallax meant that the stars had to be much farther away than Saturn--too much wasted space!

E. Tycho Brahe: precision observing without a telescope

• Tycho Brahe (1546--1601) was arrogant, vain, and insulting to nearly everyone he encountered, but he was a meticulous observer who greatly improved the accuracy and completness of observations of stars and planets--all without a telescope.
• He was dismayed at the lack of accuracy of both the Ptolemaic and the Copernican models, and he determined to make observations that would form the basis of a better model.
• With support from King Frederik II of Denmark, he built an observatory on an island in Denmark, furnishing it with precise instruments of his own design.
• For 20 years he made observations nearly every day of the positions of the Sun, Moon, and planets.
• His observations of a "new star" (what we would now call a supernova) shook his faith in the Ptolemaic model, but he was too bothered by the lack of observed parallax to embrace the Copernican model. Instead, he proposed a compromise model, with the Earth at the center, the Moon and Sun circling the Earth, but all of the other planets circling the Sun.

F. Kepler: How the Planets Move (Part 1)

a. Tycho's excellent judgement

• At Tycho's recommendation (from his deathbed), Kepler became his successor as imperial mathematician to the Holy Roman Emperor Rudolph II.
• Kepler was an accomplished mathematician, and he was probably one of the few people in Europe that had the background to make good use of the huge body of observations that Tycho had amassed.

b. Kepler's first law

The orbits of the planets around the Sun are ellipses, with the Sun at one focus.

• What is an ellipse?
• A circle is a special case of an ellipse.
• We will need to know the terms focus and semimajor axis as applied to an ellipse. In this diagram, the distance labeled a is the semimajor axis, and F₁ and F₂ indicate the foci.

G. Overview

• With hindsight, we can see that what had become by now an almost unconcious faith in uniform circular motion prevented the intellectual breakthrough that was needed to make further progress.
• The intellectual breakthrough that Copernicus did make was to see the Earth as a planet, rather than as a totally unique location in the Universe.
• Kepler was one of the first to find a way to apply the power of mathematics to natural phenomena.
• All of these earlier models--Ptolemy's, Copernicus', Tycho's and even Kepler's were purely phenomenological rather than physical. That is, they tried to account for the observations, but without any guiding principle based on an understanding of the underlying physical mechanisms that govern the motions.
• The building of this physical foundation was largely the work of two scientists of the following 100 years: Galileo, and Newton, whose work we will look at in the next lectures. But first, we will have to look at two further findings of Kepler.

End lecture 7 2/4/03

Lecture 8: Kepler: How the planets move (Part 2); Galileo's experiments in dynamics

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Last updated 4 February 2003

Alan Stockton (stockton@ifa.hawaii.edu)