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the same (constant). This is illustrated in Fig. 2.10a, where the sum of the lengths of the pairs of lines A and B, or C and D, or E and F, is always the same.

II. The line joining the Sun and the planet (called the radius vector) sweeps over equal areas in equal times as the planet travels around the orbit.

In Fig. 2.10b, the planet takes the same length of time to travel along the elliptical arcs bounding the two shaded areas because the areas are the same size.

III. The square of the period of revolution (time for one complete orbit) of a planet about the Sun is proportional to the cube of the average distance of the planet from the Sun.

Mathematically, this means that the ratio T2 to Z>\ where T stands for the period and D the average distance, is the same for all planets.

The third law is illustrated in Table 2.1 above, which gives the period T (in years) of the planets; their average distance, D (in astronomical units, with one astronomical unit equal to 93 million miles); the period squared, T2; and the average distance cubed, D3. With these units, the ratio is equal to one, that is T2 = D\

With these three laws, Kepler completed the program set forth by Copernicus. Copernicus made the Sun the center of the planetary system; Kepler discarded the idea that circular motion was required for heavenly bodies. By using elliptical orbits, the true beauty and simplicity of the heliocentric theory was preserved, in excellent agreement with the data, and with no need for such devices as the eccentric, epicycle, or equant. The fact that the focus of an ellipse is not in the center supplies eccentricity. Kepler's second law performs the same function as an equant, in that it accounts for the varying speed of the planet as it travels around the orbit.

Kepler went further and pointed to the Sun as being the prime mover or causative agent, which he called the anima motrix. He put the cause of the motion at the center of the action, rather than at the periphery as Aristotle had done. Kepler had a primitive idea about gravitational forces exerted by the Sun but failed to connect gravity to planetary orbits. Rather he asserted that the Sun exerted its influence through a combination of rays emitted by the Sun and the natural magnetism of the planet. He was influenced in this belief by some clever demonstrations of magnetic effects by William Gilbert, an English physician and physicist.

Interestingly enough, Kepler discovered his second law, dealing with the speed of the planet as it travels around the orbit, before he discovered the exact shape of the orbit, as given by the first law. The complete translated title of the book he published in 1609 on the first two laws is A New Astronomy Based on Causation, or a Physics of the Sky Derived from Investigations of the Motions of the Star Mars, Founded on Observations of the Noble Tycho Brahe. The book is usually referred to as the New Astronomy. Kepler had actually completed an outline of the book in 1605, but it took four years to get it printed because of a dispute with Tycho Brahe's heirs over the ownership of the data.

Kepler's third law, called the Harmonic Law, was described in a book called Harmony of the World, published while he held a new and lesser position as Provincial Mathematician in the town of Linz, Austria (his former patron, the Emperor Rudolph II, had been forced to abdicate his throne, and so Kepler lost his job). In this book Kepler returned to the Pythagorean mysticism of his early days, searching for harmonic relationships among the distances of the planets from the Sun (Section Fl). Instead of finding a relationship between the "musical notes" of the heavenly spheres and their size, he found the relationship between the speed of the planets (i.e., their periods), which he somehow associated with musical notes and harmony, and the size of their orbits. He supported the connections with harmony by some surprisingly accurate calculations.

Optional Section 2.2 Kepler's Laws and Satellite Orbits

As will be discussed in Chapter 3, Kepler's Laws of Planetary Motion apply equally well to satellites orbiting planets. For example, the time it takes an artificial satellite to make a complete orbit about the Earth depends upon its altitude above the Earth, and can be easily calculated from a knowledge of the distance to the Moon and the period of the Moon. Suppose it were desired to place a satellite in a 3-hour circumpolar orbit, that is, passing over the North and South Poles. Kepler's Third Law says that T\ = k D\ for the satellite and T\ — kD\ for the Moon, with k the constant of proportionality. If the first relation is divided by the second, k cancels out and the result is (Tj/^)2 = (DJ D2y or D] = D\ (7yr2)2 where Tx and T2 are the times for a complete orbit of the satellite and the Moon, respectively, and D, and D2 are the distances of the satellite and the Moon from the center of the Earth. We want Tx to equal 3 hours, and we know that T2 — 281/4 days = 678 hours. We also know the Earth-Moon distance is 250 thousand miles. Putting these numbers into the formula above, D\ = (250)3 (3/678)2. At this point, it is best to take the cube root and therefore Dl = 250 (3/678)2/3 = 6.7 thousand miles from the center of the Earth after rounding off the answer. Since the radius of the Earth is about 4 thousand miles, the satellite should be placed in an orbit which is 6.7 - 4 = 2.7 thousand miles above the surface of the Earth.

Note how close the satellite is to the surface of the Earth as compared to the Moon. This illustrates the general rule of Kepler's Laws, the closer the satellite to the Earth (or planet to the Sun), the faster it travels.

It is clear that Kepler was the right person at the right time to discover the laws of planetary motion. First and perhaps most significantly, Kepler was an intense Pythagorean mystic. He was an extremely capable mathematician who believed that the universe was full of mathematical harmonies. Because he was so interested in figures and shapes, it was relatively easy for him to consider elliptical orbits—once he finally decided that circular constructions must be wrong. Finally, Kepler was Brahe's assistant, so that he had access to Brahe's measurements and knew how accurate they were. No one else would have had the ability or the interest to perform all the mathematical calculations required to discover the laws of planetary motion, and no one else would have regarded an 8-minute angular discrepancy as significant.

Optional Section 2.3. Accepted Astronomical Dimensions Prior to the Telescope

At the time that Copernicus revived the heliocentric theory, there was a generally accepted set of data on the sizes of the various planets and their distances from the Earth. Although individual investigators used somewhat different values for these quantities, with very few exceptions the differences were not very large. This was true of the supporters of both the heliocentric and the geocentric systems.

In the geocentric systems these distances were expressed in terms of the Earth's radius. In the Ptolemaic system, the distance from the Earth to the Moon varied from about 33 to about 64 times the Earth's radius, with an average of about 48. This was determined by the use of "eclipse" diagrams, which noted that a total eclipse of the Sun by the Moon, as seen from one location on the surface of the Earth would be seen as a partial eclipse from some other location on the Earth's surface. Knowing the distance between the two locations on the Earth, the distance from the Earth to the Moon can be calculated from geometry. Using this distance, it is possible to return to the eclipse diagram and calculate the distance to the Sun as ranging between about 1100 and 1200 times the Earth's radius, corresponding to 4.4 to 4.8 million miles. (The correct value is about 20 times larger. The method requires accurate measurements of the apparent size of the Sun as seen in different positions. The calculations are of such a nature that the results are very sensitive to small errors in these measurements and to correction factors that are necessary because of different thicknesses of the atmosphere traversed by the line of sight to the Sun.)

Copernicus chose the distance from the Earth to the Sun to be about the same as that chosen by Ptolemy, that is, about 1200 Earth radii. After Copernicus adapted eccentrics and epicycles to his system, there was a substantial range of distances of each planet from the Sun, but the resulting shell was still thin enough as to leave large volumes of' 'empty space'' that were not traversed by a planet. The range of distances for Saturn, the most distant planet from the Sun, was 9881 to 11,073 earth radii. If the sphere of the fixed stars had a radius of 11,073 earth radii, then the Copernican universe would have been even smaller than the Ptolemaic universe, and star parallax as a result of the Earth's movement along its orbit should have been very easily observed. It was for this reason that Copernicus suggested that the sphere of the fixed stars should be very large, because no parallax was observed. Tycho Brahe calculated that in the Copernican universe the fixed stars should be at a distance of at least 8 million earth radii, that is, 32,000 million miles; otherwise he would have detected star parallax. This meant even more unused or empty space, which was simply unacceptable to him and most other learned people, who were thoroughly imbued with the Aristotelian concepts of nature. (Remarkably enough, there was one medieval astronomer, Levi ben Gerson, who set the distance to the fixed stars as being even 5000 times larger than the value that Tycho had rejected.)

One objection to a universe containing empty space was that there seemed to be no way to explain the sizes and positions of the void regions and the planetary shells. Thus one of the goals of Kepler's Mysterium Cosmographicum was to give a reason for the sizes of the shells for each planet.

Within the 30 or 40 years after Galileo first began using the telescope to study the heavens, the measurements of the apparent size of the Sun and of the planets became sufficiently refined that it was recognized that the previously accepted values for the distance from the Earth to the Sun were far too small. By the year 1675 a new type of telescope had come into widespread use and new measuring techniques were developed. As a result the accuracy of the measurements was significantly improved. By the year 1700 it was generally accepted that the distance from the Earth to the Sun was over 90 million miles. This compares well with the modern value of 93 million miles. Once this distance is known, it is possible to scale the distances to all the other planets by use of Kepler's third law of planetary motion.

4. Slow Acceptance of Kepler's Theory

Kepler sent copies of his work to many of the best known astronomers of his day and corresponded copiously with them. In general, although he was greatly respected, the significance and implications of his results remained unrecognized for a long time, except by a few English astronomers and philosophers. Most astronomers had become so accustomed to the relatively poor agreement between theory and data, and so imbued with the idea of circular motion, that they were not impressed with Kepler's results.

Galileo, for example, totally ignored Kepler's work even though they had corresponded and Galileo was quick to seek Kepler's approval of his own telescopic observations. It is not clear whether Galileo was put off by Kepler's Pythagorean mysticism, or whether he was so convinced of circular motion as the natural motion for the Earth and other planets that he simply could not accept any other explanation. The fact is, however, that Galileo did not embark on his telescope work until after Kepler published the New Astronomy, nor did he publish his Dialogues until 1632, two years after Kepler's death.

Only after Kepler calculated a new set of astronomical tables, called the Rudolphine Tables (which were published in 1627), was it possible to use his ideas and the data of Tycho Brahe to calculate calendars that were much more accurate than any previously calculated. For example, discrepancies in tables based on the Copernican theory had been found fairly quickly by Tycho Brahe within about 10 years of their publication; the Rudolphine Tables were used for about 100 years. During that time Kepler's work gradually became more and more accepted. It was not until some 70 years later, however, when Isaac Newton published his laws of motion and his law of universal gravitation in order to explain how Kepler's laws had come about, that they became completely accepted. The work of Newton is discussed in the next chapter.

### 5. The Course of Scientific Revolutions

The replacement of the geocentric theory of the universe by the heliocentric theory is often referred to as the Copernican revolution. The Copernican ' 'revolution" is the archetype of scientific revolution, against which other scientific revolutions are compared. In fact, no other scientific revolution can match it in terms of its effect on modes of thought, at least in Western civilization. The only one that comes close to it in that respect is the Darwinian revolution, which is associated with the development of biological evolutionary theory.

The Copernican revolution is often pointed to as illustrating the "conflict between Science and Religion"; yet its leaders, Copernicus, Galileo, and Kepler, were devoutly religious. On the other side, within the religious establishment, there were many who did not follow the narrowly literal interpretation of religious dogma. Initially, at least, the biggest fear of the revolutionaries was not censure by religious authorities, but rather ridicule by their scientific peers. How could they challenge the established body of scientific knowledge and belief?

The revolution, moreover, did not take place overnight. The time span from the serious studies of Copernicus to Newton's publication of his gravitational studies was about 150 years. The seeds of the revolution went back a long time, about 2000 years before Copernicus, to Philolaus, a Pythagorean. As the older theories began to accumulate more and more anomalous and unexplained facts, the burden of necessary modifications simply became overwhelming, and the older theories broke down. Nevertheless, the older theories had been phenomenally successful. In fact, there were only a few heavenly bodies out of the whole host of the heavens (i.e., the planets) that caused major difficulties for the established theory. The revolution came about because of quantitative improvements in the data and the recognition of subtle, rather than gross, discrepancies, not only in the data, but in the way in which the data were assimilated. The development of new technologies and instruments that yielded entire new classes and qualities of data also played a large role. At the same time, the new concepts themselves initially could not account for all the data, and their advocates were required to be almost unreasonably persistent in the face of severe criticism. A new theory is expected, quite properly, to overcome a large number of objections, and even then some of the advocates of the old theory are never really convinced.

Not all revolutionary ideas in science are successful, or even deserve to be. If they fail to explain the existing facts or future facts to be developed, or if they are incapable of modification, development, and evolution in the face of more facts, they are of little value. The heliocentric theory was successful because it was adaptable. Kepler, Copernicus, and others were able to incorporate into the scheme such things as the eccentrics and epicycles, and finally a different shape orbital figure, the ellipse, without destroying the essential idea, so that it could be stretched to accommodate a wide variety of experimental facts.

Study Questions

1. What is meant by irrational numbers? Give some examples.

2. What were the connections perceived by the Pythagoreans between irrational numbers, geometry, and the concept of atoms?

3. Why did the ancient Greeks consider a circle to be a perfect figure?

4. What does Plato's Allegory of the Cave have to do with science?

5. Why did the ancient Greeks consider that heavenly objects had to move in circular paths?

6. What is the stellar or celestial sphere? Describe its appearance.

7. What is the celestial equator, celestial pole, celestial horizon, polar axis?

8. What is the ecliptic?

9. What is diurnal rotation?

10. Describe the annual motion of the Sun.

11. What is retrograde motion? alternate motion?

12. What is meant by a geocentric model of the universe?

13. What is meant by a homocentric model of the universe?

14. Name the prime substances described by Aristotle.

15. Name and describe the devices used by Ptolemy in his calculations.

16. What particular aspects of Ptolemy's approach did Copernicus not like?

17. How does the heliocentric theory account for retrograde motion?

18. Which of Ptolemy's devices did Copernicus have to use and why?

19. List and describe the scientific objections to the Copernican theory.

20. What were the philosophical and religious objections to the Copernican theory?

21. What are the common characteristics of Tychonic theories of the universe?

22. What was Tycho Brahe's real contribution to the science of astronomy?

23. What were the new discoveries and arguments that tipped the balance toward the heliocentric theory?

24. State Kepler's laws of planetary motion.

25. Why were Kepler's laws of planetary motion not immediately grasped by Galileo? (Part of the answer to this question is in the next chapter.)

### Questions for Further Thought and Discussion

1. Discuss the similarities and differences between the reception of the heliocentric description of the universe and the reception of Darwin's ideas of evolution.

2. In light of what you understand about the history of scientific revolutions, what should be your attitude toward claims by proponents of extra-sensory perception (ESP), dianetics, precognition?

3. Some people contend that with the aid of modern computers, it is possible to construct a geocentric theory that would agree with astronomical observations—all that is necessary is to introduce enough eccentrics, epicycles, equants, and even elliptical orbits. Would such a theory be acceptable? Why or why not?

### Problems

1. The calculated speed of the Earth in the orbit proposed by Copernicus is about 3500 miles per hour. Show how to calculate how far it will travel in 1 sec. (Hint: Convert miles per hour to miles per second.)

2. Calculate the length of a ' 'pointer'' that would give an accuracy of 1 sec of arc for a ¿-inch error in positioning its far end. {Hint: Tycho Brahe claimed an accuracy of 4 minutes of arc and this required a 3-ft-long pointer.) Does your answer seem to be a reasonable length, i.e., one that could be easily handled?

3. Satellites that are used for worldwide telecommunications are apparently stationary over one spot on the Earth. They are not really stationary but rather orbit the Earth with the same period as the Earth's rotation about the axis. Calculate the altitude of these satellites above the Earth's surface. (Hint: See the discussion of circumpolar orbits at the end of Section F3 above.)

References

E. A. Abbott, Flatland, Dover Publications, New York, 1952.

A very entertaining book, originally written a century ago, about a two-dimensional world.

Cyril Bailey, The Greek Atomists and Epicurus, Russell and Russell, New York, 1964. This was originally written in 1928. Pages 64-65 give a brief discussion of the Indian philosopher Kanada and his atomic theory, and a few references. More recent material is contained in an article by Alok Kumar to be published in the forthcoming Encyclopedia of the History of Science, Technology and Medicine in non-Western Cultures (1995).

Arthur Berry, A Short History of Astronomy: From Earliest Times Through the Nineteenth Century, Dover Publications, New York, 1961. Chapter II discusses the contributions of the Greeks.

J. Bronowski, The Ascent of Man, Little, Brown and Company, Boston, 1973.

Based on a television series of the same name. Chapters 5, 6, and 7 are pertinent to this chapter.

Herbert Butterfield, The Origins of Modern Science, rev. ed., Free Press, New York, 1965. A brief book written by a historian, which discusses scientific developments up to the time of Isaac Newton.

F. M. Cornford, The Republic of Plato, Oxford University Press, London, 1941. An account of Plato's Allegory of the Cave is found here.

J. L. E. Dreyer, A History of Astronomy from Thales to Kepler, 2nd ed., Dover Publications, New York, 1953.

Provides a fairly detailed account of the subject matter of this chapter.

Gerald Holton and Stephen G. Brush, Introduction to Concepts and Theories in Physical Science, 2nd ed., Addison-Wesley, Reading, Mass., 1973.

Chapters 1 through 5 and pp. 154-160 are pertinent, with many references to other sources.

Hugh Kearney, Science and Change, 1500-1700, McGraw-Hill, New York, 1971. A small book that discusses the general scientific ferment in Europe during the period 1500-1700.

Arthur Koestler, The Watershed (Anchor Books), Doubleday, Garden City, New York, 1960.

A thin biographical discussion of Kepler's life and work.

Alexander Koyre, Discovering Plato, Columbia University Press, New York, 1945. This book also discusses Plato's views about science and philosophy.

Thomas S. Kuhn, The Copernican Revolution, Random House, New York, 1957. A detailed discussion of the subject of this chapter.

Thomas S. Kuhn, The Structure of Scientific Revolutions, 2nd ed., University of Chicago

Press, Chicago, 1970.

### See comment in references for Chapter 1.

Cecil J. Schneer, The Evolution of Physical Science, Grove Press, New York, 1964 (a reprint of The Search for Order, Harper & Row, New York, 1960). See comment in references for Chapter 1.

Stephen Toulmin and June Goodfield, The Fabric of the Heavens, Harper & Row, New York, 1961.

Albert Van Helden, Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley, University of Chicago Press, Chicago, 1985.

A thorough discussion by a historian of the various methods used by astronomers to determine distances and apparent sizes of heavenly objects, based on examination and evaluation of original documents.

Isaac Newton. (Photo courtesy Yerkes Observatory, University of Chicago.)