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Full text of "Motion in the Heavens, Project Physics Text and Handbook Volume 2"

The Project Physics Course 



Text and Handbook 



2 



Motion in the Heavens 




The Project Physics Course 



Text and Handbook 



UNIT 



2 



Motion in the Heavens 



A Component of the 
Project Physics Course 




Published by 

HOLT, RINEHART and WINSTON, Inc. 

New York, Toronto 



Directors of Harvard Project Physics 



Acknowledgments, Text Section 



Gerald Holton, Department of Physics, Harvard 

University 
F. James Rutherford, Capuchino High School, 

San Bruno, California, and Harvard University 
Fletcher G. Watson, Harvard Graduate School 

of Education 



Special Consultant 
to Project Physics 

Andrew Ahlgren, Harvard Graduate School of 
Education 

A partial bst of staff 

and consultants to Harvard Project 

Physics appears on page iv 



This Text and Handbook, Unit 2 is one of the 
many instructional materials developed for the 
Project Physics Course. These materials include 
Texts, Handbooks, Teacher Resource Books, 
Readers, Programmed Instruction booklets. Film 
Loops, Transparencies, 16mm films, and labo- 
ratory equipment. 



Copyright © 1970, Project Physics 

All Rights Reserved 

2nd Printing April, 1971 

3rd Printing May, 1972 

ISBN 0-03-084498-3 

234567890 039 9876543 

Project Physics is a registered trademark 



The authors and publisher have made every effort 
to trace the ownership of all selections found 
in this book and to make full acknowledgment for 
their use. Many of the selections are in public 
domain. 

Grateful acknowledgment is hereby made to the 
following authors, pubhshers, agents, and in- 
dividuals for use of their copyrighted material. 

Pp. 21, 22 Ptolemy, "The Almagest," Trans, by 
Taliaferro, R. Catesby, Great Books of the 
Western World, Vol. 16, pp. 5-12 not inclusive, 
copyright 1952 by Encyclopaedia Britannica, Inc. 
P. 29 Rosen, Edward, Trans. "The Commentariolus 
of Copernicus," Three Copernican Treatises, 
Columbia University Press, pp. 57-58. 
Pp. 31, 32 Copernicus, "De Revolutionibus," 
("On the Revolutions of Heavenly Spheres"), 
Great Books of the Western World. Vol. 16. 
pp. 514 and 508, copyright 1939 by Encyclopaedia 
Britannica, Inc. 

Pp. 31, 32, 33 Rosen, op. cit., pp. 58-59. 
P. 38 Knedler, John W.. Jr., Masterworks of 
Science, Doubleday and Company, Inc., pp. 67-69. 
Pp. 39, 40 Copernicus, Great Books of the 
Western World, op. cit., pp 514, 515, copyright 
1939 by Encyclopaedia Britannica, Inc. 
P. 43 Buttei-field, H., Origins of Modern Science, 
1300-1800, p. 41, copyright © 1957 by The 
Macmillan Company. 

P. 66 Shapley, Harlow, and Howarth, Helen E., 
A Source Book in Astronomy, copyright © 1956 
by Harvard University Press. 
Pp. 70, 71 Drake, Stillman, Discoveries and 
Opinions of Galileo, pp. 29, 31, 51, copyright © 
1957 by Doubleday & Company. Inc. 
Pp. 73, 74 GaUlei, Galileo. Dialogue Concerning 
the Two Chief World Systems, trans. Stillman 
Drake, pp. 123-124, copyright © 1967 by 
University of California Press. 
Pp. 74, 75 Von Gebler, Karl. Galileo Galilei 
and the Roman Curia, trans. Mrs. George Sturge, 
C. Kegan Paul and Company, p. 26. 
Pp. 85, 86 Needham. Joseph, and Pagel. Walter, 
eds., Background to Modern Science, The Mac- 
millan Company, pp. 73-74. 
P. 86 Stukeley. William. Memoirs of Sir Isaac 
Newton's Life, Taylor and Francis, p. 21. 
written in 1752. 

Pp. 86, 87 Anthony, H. D., Sir'isaac Newton, 
p. 96. copyright © 1960 by Abelard-Schuman Ltd. 
P. 88 Newton. Sir Isaac. The Principia, Vol. I, 
Motte's translation revised by Cajori. pp. xvii- 
xviii, copyright © 1961 by University of 
California Press. 

Pp. 88, 89, 97 Ibid.. Vol. II. pp. 398-408 and 
p. 547. copyright © 1962 by University of 
California Press. 



p. 89 Pope, Alexander, Epitaph Intended for 
Sir Isaac Newton (1732). 
P. 115 Fontenelle, "Preface sur L'UtiUte des 
Mathematlques et de la Physique" from Oeuvres 
Completes (1818), in Randall, John H., Jr., 
The Making of the Modern Mind, p. 254, Revised 
Edition, copyright 1940 by Houghton Mifflin 
Company. 

P 117 Wilson, James, The Works of the Honour- 
able James Wilson, pubUshed under direction of 
Wilson, Bird, Bronson and Chauncey, pp. 411-412, 
not inclusive. 

Pp. 118, 119 Burtt, E. A. The Metaphysical 
Foundations of Modern Science, pp. 238-239, 
Humanities Press, Inc. 
P. 120 Laplace, A Philosophical Essay on 
Probabilities, first published in Paris (1814), 
trans. Truscott, Frederick and Emory, Frederick, 
Dover Pubhcations, Inc., p. 4. 

P 120 Darwan, Erasmus, quoted by Charles Darwin 
in his preface to Krause, Erasmus Darwin, D. 
Appleton and Company, p. 44. 



Picture Credits, Text Section 

Cover photograph : the orrery on the cover 
was made in 1830 by Newton of Chancery Lane, 
London. The earth and moon are geared; the 
rest of the planets have to be set by hand. It is 
from the Collection of Historical Scientific 
Instruments at Harvard University. Photograph 
by Albert Gregory, Jr. 

Opposite p. 1 Aztec Calendar Stone in the 
Museo Nacional, Mexico City. Photo courtesy of 
the American Museum of Natural History, New 
York. 

P. 1 Collection of Historical Scientific Instru- 
ments, Harvard University. 

P. 2 (top) Stephen Perrln. 

P. 3 Courtesy of the Trustees of the British 
Museum, London. 

P. 4 Frontispiece from Recueil de plusieurs 
traitez de Mathematique de I'Academie Royals 
des Sciences, 1676. 

P. 6 Emil Schultess, Black Star Publishing 
Company, Inc. 

P. 9 (top left) John Stofan; (top right) John 
Bufkin, Macon, Missouri, Feb. 1964. 



P. 11 Mount Wilson and Palomar Observa- 
tories 

P. 18 DeGolyer Collection, University of 
Oklahoma Libraries. 

P. 28 Jagiellonian Library, University of 
Krakow, Poland. 

P. 29 Muzeum Okregowe in Torun. Courtesy 
of Dr. Owen Gingerich. 

P. 42 Jagiellonian Library, University of 
Krakow, Poland. 

P. 46 (top left) from Atlas Major, vol. I, Jan 
Blaeu, 1664; (bottom left) The Mansell Collec- 
tion, London; Danish Information Office. 

P. 47 Smithsonian Astrophysical Observatory, 
courtesy of Dr. Owen Gingerich. 

P. 48 (top left) from Brahe, Tycho, Astro- 
nomiae Instauratae Mechanica, Phihp von Ohr, 
Wandsbeck, 1598; (bottom photograph) by John 
Bryson, reprinted with permission from 
HOLIDAY, 1966, the Curtis Publishing Company. 

P. 50 Mount Wilson and Palomar Observa- 
tories. 

P. 54 (portrait) The Bettmann Archive. 

P. 56 Kepler, Johannes, Mysterium cosmo- 
graphicum, Linz, 1596. 

P. 62 Archives, Academy of Sciences, 
Leningrad, U.S.S.R. Photo courtesy of Dr. Owen 
Gingerich. 

P. 69 Instituto e Museo di Storia della Scienza, 
Florence, Italy. 

P. 71 DeGolyer Collection, University of 
Oklahoma Libraries. 

P. 72 (middle photograph) Lowell Observatory 
Photograph 

P. 74 Alinari — Art Reference Bureau. 

P. 77 Bill Bridges. 

P. 81 Courtesy of Biblioteca Nazionale 
Centrale, Florence. 

P. 82 Yerkes Observatory 

P. 85 (drawing) from a manuscript by Newton 
in the University Library, Cambridge; (portrait) 
engraved by Bt. Reading from a painting by Sir 
Peter Lely. Trinity College Library, Cambridge. 

P. 96 Descartes, Rene, Principia Philosophiae, 
1664. 

P. 102 Courtesy of NASA. 

P. 105 Cavendish, Henry, Philosophical Trans- 
actions of the Royal Society, vol. 88, 198. 

P. 1 16 Print Collection of the Federal 
Institute of Technology, Zurich. 

All photographs not credited above were made 
by the staff of Harvard Project Physics. 



Partial List of Staff and Consultants 

The individuals listed below (and on the following pages) have each contributed in some way to the 
development of the course materials. Their periods of participation ranged from brief consultations to 
full-time involvement in the team for several years. The affiliations indicated are those just prior to 
or during the period of participation. 



Advisory Committee 

E. G. Begle, Stanford University, Calif. 

Paul F. Brandwein, Harcourt, Brace & World, 

Inc., San Francisco, Calif. 
Robert Brode, University of California, Berkeley 
Erwin Hiebert, University of Wisconsin, Madison 
Harry Kelly, North Carolina State College, Raleigh 
William C. Kelly, National Research Council, 

Washington, D.C. 
Philippe LeCorbeiller, New School for Social 

Research, New York, N.Y. 
Thomas Miner, Garden City High School, New 

York. 
Philip Morrison, Massachusetts Institute of 

Technology, Cambridge 
Ernest Nagel, Columbia University, New York, 

N.Y. 
Leonard K. Nash, Harvard University 
I. I. Rabi, Columbia University, New York, N.Y. 



Staff and Consultants 

L. K. Akers, Oak Ridge Associated Universities, 

Tenn. 
Roger A. Albrecht, Osage Community Schools, 

Iowa 
David Anderson, Oberlin College, Ohio 
Gary Anderson, Harvard University 
Donald Armstrong, American Science Film 

Association, Washington, D.C. 
Arnold Arons, University of Washington 
Sam Ascher, Henry Ford High School, Detroit, 

Mich. 
Ralph Atherton, Talawanda High School, Oxford, 

Ohio 
Albert V. Baez, UNESCO, Paris 
William G. Banick, Fulton High School, Atlanta, 

Ga. 
Arthur Bardige, Nova High School, Fort 

Lauderdale, Fla. 
RoUand B. Bartholomew, Henry M. Gunn High 

School, Palo Alto, Calif. 
O. Theodor Benfey, Earlham College, Richmond, 

Ind. 
Richard Berendzen, Harvard College Observatory 
Alfred M. Bork, Reed College, Portland, Ore. 
F. David Boulanger, Mercer Island High School, 

Washington 
Alfred Brenner, Harvard University 
Robert Bridgham, Harvard University 
Richard BrinckerhofF, Phillips Exeter Academy, 

Exeter, N.H. 



Donald Brittain, National Film Board of Canada, 

Montreal 
Joan Bromberg, Harvard University 
Vinson Bronson, Newton South High School, 

Newton Centre, Mass. 
Stephen G. Brush, Lawrence Radiation Laboratory. 

University of California, Livermore 
Michael Butler, CIASA Films Mundiales, S. A., 

Mexico 
Leon Callihan, St. Mark's School of Texas, Dallas 
Douglas Campbell, Harvard University 
J. Arthur Campbell, Harvey Mudd College, 

Claremont, Calif. 
Dean R. Casperson, Harvard University 
Bobby Chambers, Oak Ridge Associated 

Universities, Tenn. 
Robert Chesley, Thacher School, Ojai, Calif. 
John Christensen, Oak Ridge Associated 

Universities, Tenn. 
Dora Clark, W. G. Enloe High School, Raleigh, 

N.C. 
David Clarke, Browne and Nichols School, 

Cambridge. Mass. 
Robert S. Cohen, Boston University, Mass. 
Brother Columban Francis. F.S.C., Mater Christi 

Diocesan High School, Long Island City, N.Y. 
Arthur Compton, Phillips Exeter Academy, 

Exeter, N.H. 
David L. Cone, Los Altos High School, Calif. 
Wilham Cooley, University of Pittsburgh, Pa. 
Ann Couch, Harvard University 
Paul Cowan, Hardin-Simmons University, 

Abilene, Tex. 
Charles Davis, Fairfax County School Board. 

Fairfax, Va. 
Michael Dentamaro, Senn High School, Chicago, 

111. 
Raymond Dittman, Newton High School, Mass. 
Elsa Dorfman, Educational Services Inc., 

Watertown, Mass. 
Vadim Drozin, Buc knell University, Lewisburg, 

Pa. 
Neil F. Dunn. Burlington High School. Mass. 
R. T. Ellickson. University of Oregon. Eugene 
Thomas Embry. Nova High School. Fort 

Lauderdale, Fla. 
Walter Eppenstein, Rensselaer Polytechnic 

Institute. Troy, NY. 
Herman Epstein. Brandeis University. Waltham. 

Mass. 
Thomas F. B. Ferguson. National Film Board of 

Canada. Montreal 
Thomas von Foerster. Harvard University 
Kenneth Ford, University of California. Irvine 

(continued on p. 124) 



Science is an adventure of the whole human race to learn to live in and 
perhaps to love the universe in which they are. To he a part of it is to 
understand, to understand oneself, to begin to feel that there is a capacity 
within man far beyond what he felt he had, of an infinite extension of 
human possibilities .... 

I propose that science be taught at whatever level, from the lowest to the 
highest, in the humanistic way. It should be taught with a certain historical 
understanding, with a certain philosophical understanding , with a social 
understanding and a human understanding in the sense of the biography, the 
nature of the people who made this construction, the triumphs, the trials, the 
tribulations. 

I. I. RABI 

Nobel Laureate in Physics 



Preface 



Background The Project Physics Course is based on the ideas and 
research of a national curriculum development project that worked in 
three phases. First, the authors — a high school physics teacher, a 
university physicist, and a professor of science education — collaborated 
to lay out the main goals and topics of a new introductory physics 
course. They worked together from 1962 to 1964 with financial support 
from the Carnegie Corporation of New York, and the first version of 
the text was tried out in two schools with encouraging results. 

These preliminary results led to the second phase of the Project 
when a series of major grants were obtained from the U.S. Office of 
Education and the National Science Foundation, starting in 1964. 
Invaluable additional financial support was also provided by the 
Ford Foundation, the Alfred P. Sloan Foundation, the Carnegie 
Corporation, and Harvard University. A large number of collaborators 
were brought together from all parts of the nation, and the group 
worked together for over four years under the title Harvard Project 
Physics. At the Project's center, located at Harvard University, 
Cambridge, Massachusetts, the staff and consultants included college 
and high school physics teachers, astronomers, chemists, historians 
and philosophers of science, science educators, psychologists, 
evaluation specialists, engineers, film makers, artists and graphic 
designers. The teachers serving as field consultants and the students 
in the trial classes were also of vital importance to the success of 
Harvard Project Physics. As each successive experimental version of 
the course was developed, it was tried out in schools throughout the 
United States and Canada. The teachers and students in those schools 
reported their criticisms and suggestions to the staff in Cambridge, 
and these reports became the basis for the subsequent revisions of 
the course materials. In the Preface to Unit 1 Text you will find a list of the 
major aims of the course. 



We wish it were possible to list in detail the contributions of each 
person who participated in some part of Harvard Project Physics. 
Unhappily it is not feasible, since most staff members worked on a 
variety of materials and had multiple responsibilities. Furthermore, 
every text chapter, experiment, piece of apparatus, film or other item 
in the experimental program benefitted from the contributions of a 
great many people. On the preceding pages is a partial list of 
contributors to Harvard Project Physics. There were, in fact, many 
other contributors too numerous to mention. These include school 
administrators in participating schools, directors and staff members 
of training institutes for teachers, teachers who tried the course after 
the evaluation year, and most of all the thousands of students who 
not only agreed to take the experimental version of the course, but 
who were also willing to appraise it critically and contribute their 
opinions and suggestions. 

The Project Physics Course Today. Using the last of the experimental 
versions of the course developed by Harvard Project Physics in 
1964-68 as a starting point, and taking into account the evaluation 
results from the tryouts, the three original collaborators set out to 
develop the version suitable for large-scale publication. We take 
particular pleasure in acknowledging the assistance of Dr. Andrew 
Ahlgren of Harvard University. Dr. Ahlgren was invaluable because 
of his skill as a physics teacher, his editorial talent, his versatility 
and energy, and above all, his commitment to the goals of Harvard 
Project Physics. 

We would also especially like to thank Miss Joan Laws whose 
administrative skills, dependability, and thoughtfulness contributed so 
much to our work. The publisher, Holt, Rinehart and Winston, Inc. 
of New York, provided the coordination, editorial support, and general 
backing necessary to the large undertaking of preparing the final 
version of all components of the Project Physics Course, including 
texts, laboratory apparatus, films, etc. Damon, a company located in 
Needham, Massachusetts, worked closely with us to improve the 
engineering design of the laboratory apparatus and to see that it was 
properly integrated into the program. 

In the years ahead, the learning materials of the Project Physics 
Course will be revised as often as is necessary to remove remaining 
ambiguities, clarify instructions, and to continue to make the materials 
more interesting and relevant to students. We therefore urge all 
students and teachers who use this course to send to us (in care of 
Holt, Rinehart and Winston, Inc., 383 Madison Avenue, New York, 
New York 10017) any criticism or suggestions they may have. 



F. James Rutherford 
Gerald Holton 
Fletcher G. Watson 



Contents text section 

Prologue 1 

Chapter 5 Where is the Earth? — The Greeks' Answers 

Motions of the sun and stars 7 

Motions of the moon 11 

The "wandering" stars 12 

Plato's problem 15 

The Greek idea of "explanation" 16 

The first earth-centered solution 17 

A sun-centered solution 19 

The geocentric system of Ptolemy 21 

Successes and limitations of the Ptolemaic model 25 

Chapter 6 Does the Earth Move? — The Work of Copernicus and Tycho 

The Copernican system 29 

New conclusions 33 

Arguments for the Copernican system 35 

Arguments against the Copernican system 39 

Historical consequences 44 

Tycho Brahe 45 

Tycho's observations 47 

Tycho's compromise system 49 

Chapter 7 A New Universe Appears — The Work of Kepler and Galileo 

The abandonment of uniform circular motion 55 

Kepler's Law of Areas 57 

Kepler's Law of Elliptical Orbits 60 

Kepler's Law of Periods 66 

The new concept of physical law 68 

Galileo and Kepler 69 

The telescopic evidence 70 

GaHleo focuses the controversy 73 

Science and freedom 75 

Chapter 8 The Unity of Earth and Sky — The Work of Newton 

Newton and seventeenth-century science 83 

Newton's Principia 87 

The inverse-square law of planetary force 90 

Law of Universal Gravitation 91 

Newton and hypotheses 96 

The magnitude of planetary force 98 

Planetary motion and the gravitational constant 102 

The value of G and the actual masses of the planets 104 

Further successes 106 

Some effects and limitation of Newton's work 112 

Epilogue 114 

Contents — Handbook Section 126 

Index/Text Section 193 

Index/ Handbook Section 196 

Answers to End-of-Section Questions 199 

Brief Answers to Study Guide Questions inside back cover 




The Aztec calendar, carved over 100 years before our calendar was adopted, divides 
the year into eighteen months of twenty days each. 



UNIT 



2 



Motion in the Heavens 



CHAPTER 
5 
6 

7 
8 



Where is the Earth? -The Greeks' Answers 
Does the Earth Move? -The Work of Copernicus and Tycho 
A New Universe Appears -The Work of Kepler and Galileo 
The Unity of Earth and Sky -The Work of Newton 



PROLOGUE Astronomy, the oldest science, deals with objects now 
known to be the most distant from us. Yet, to early observers, the sun, 
moon, planets, and stars did not seem to be so far away. Nor were they 
considered of little importance. On the contrary, even as today, the 
majestic display of celestial events powerfully stimulated the imagination 
of curious men. The great variety of objects visible in the sky, the 
regularity of their motions, the strangely slow changes in their position 
and brightness-this whole mysterious pattern of motions required 
some reason, some cause, some explanation. 

The discovery of the causes and the meanings is the subject of this 
unit. It starts with prehistoric attempts to deal with the observations by 
incorporating them, in disguised form, into myths and tales, some of 
the best in world literature. It ends with the Scientific Revolution in the 
seventeenth century, which gave us the explanations that hold to this 
day. These explanations also provided a whole new set of methods for 
solving problems in a scientific manner. 

Astronomical events affected not only the imagination of the 
ancients; they had a practical effect on their everyday life. The working 
day began when the sun rose and ended when the sun set. Before 
electric lighting, human activity was dominated by the presence or 
absence of daylight and by the sun's warmth, which changed season 
by season. 

Of all the time units used in common practice, "one day" is 
probably the most basic and surely the most ancient. For counting 
longer intervals, a "moon" or month was an obvious unit. Over the 
centuries, clocks have been devised to subdivide the days into smaller 
units, and calendars have been devised to record the passage of days 
into years. 

When the early nomadic tribes settled down to live m villages some 
10,000 years ago, and became dependent upon agriculture for their 
food, they needed a calendar for planning their plowing and sowing. 




Even in modern times outdoorsmen 
use the sun by day and the stars by 
night as a clock. Directions are 
indicated by the sun at rising and 
setting time, and true south can be 
determined from the position of the 
sun at noon. The Pole Star gives a 
bearing on true north after dark. The 
sun's position can also be used as 
a crude calendar. Its noontime 
altitude varies with the seasons. 



Motion in the Heavens 



Stonehenge, England, apparently a 
prehistoric observatory. 



Throughout recorded history, most of the world's population has been 
involved in agriculture and so has depended on a calendar. If seeds 
were planted too early, they might rot in the ground, or the young 
shoots might be l<illed by a frost. If they were planted too late, the crops 
would not ripen before winter came. Therefore, a knowledge of the 
times for planting and harvesting was important for survival. Because 
religious festivals were often related to the seasons, the making and 
improving of the calendar by observation of the sun, planets, and stars 
was often the task of priests. The first astronomers were, therefore, 
usually priests. 

Practical needs and imagination acted together to give astronomy 
an early importance. Many of the great buildings of ancient times were 
constructed with careful astronomical orientation. The great pyramids 
of Egypt, tombs of the Pharaohs, have sides that run due north-south 
and east-west. The impressive, almost awesome circles of giant stones 
at Stonehenge in England appear to have been arranged about 2000 
B.C. to permit accurate astronomical observations of the positions of 
the sun and moon. The Mayans and the Incas in America, as well as the 
Chinese, put enormous effort into buildings from which they could 
measure the changes in the position of the sun, moon, and planets. At 
least as early as 1000 B.C. the Babylonians and Egyptians had 
developed considerable ability in timekeeping. Their recorded 
observations are still being unearthed. 

Thus, for thousands of years, the motions of the heavenly bodies 




Prologue 



were carefully observed and recorded. In all sciences, no other field 
has had such a long accumulation of data as astrononny has had. 

But our debt is greatest to the Greeks, who began trying to deal in 
a new way with what they saw. The Greeks recognized the contrast 
between the apparently haphazard and short-lived motions of objects 
on the earth and the unending cycles of motions of the objects in the 
heavens. About 600 B.C. they began to ask a new question: How can 
we explain these cyclic events in the sky in a simple way? What order 
and sense can we make of the heavenly happenings? The Greeks' 
answers, which are discussed in Chapter 5, had an important effect on 
science. For example, as we shall see, the writings of Aristotle (about 
330 B.C.) became widely studied and accepted in western Europe after 
1200 A.D. and were important factors in the scientific revolution that 
followed. 

After the conquests of Alexander the Great, the center of Greek 
thought and science shifted to Egypt at the new city of Alexandria, 
founded in 332 B.C. There a great museum, similar to a modern research 
institute, was created and flourished for many centuries. But as the 
Greek civilization gradually declined, the practical-minded Romans 
captured Egypt, and interest in science died out. In 640 A.D. Alexandria 
was captured by the Muslims as they swept along the southern shore 
of the Mediterranean Sea and moved northward through Spain to the 
Pyrenees. Along the way they seized and preserved many libraries of 
Greek documents, some of which were later translated into Arabic and 
carefully studied. During the following centuries the Muslim scientists 
made new and better observations of the heavens, although they did 
not make major changes in the explanations or theories of the Greeks. 

In western Europe during this period the works of the Greeks were 
largely forgotten. Eventually they were rediscovered by Europeans 
through Arabic translations found in Spain after the Muslims were 
forced out. By 1130 A.D. complete manuscripts of at least one of 
Aristotle's books were known in Italy and France. After the founding of 
the University of Paris around 1200, many other writings of Aristotle 
were acquired and studied both there and at the new English 
universities, Oxford and Cambridge. 

During the next century, the Dominican monk, Thomas Aquinas, 
blended major elements of Greek thought and Christian theology into a 
single philosophy. His work was widely studied and accepted in 
western Europe for several centuries. In achieving this commanding and 
largely successful synthesis. Aquinas accepted the physics and 
astronomy of Aristotle. Because the science was blended with theology, 
any questioning of the science seemed also to be a questioning of the 
theology. Thus for a time there was little effective criticism of the 
Aristotelian science. 

The Renaissance movement, which spread out across Europe from 
Italy, brought new art and music. It also brought new ideas about the 
universe and man's place in it. Curiosity and a questioning attitude 
became acceptable, even prized. Men acquired a new confidence in 
their ability to learn about the world. Among those whose work 
introduced the new age were Columbus and Vasco da Gama, 




The positions of Jupiter from 132 B.C. 
to 60 B.C. are recorded on this section 
of Babylonian clay tablet, now in the 
British Museum. 



In the twelfth century, the Muslim 
scholar Ibn Rashd had attempted a 
similar union of Aristotelianism and 
Islam. 



4 Motion in the Heavens 

Gutenberg and da Vinci, Michelangelo and Raphael, Erasmus and 
Vesalius, Luther, Calvin, and Henry VIM. (The chart in Chapter 6 shows 
their life spans.) Within this emerging Renaissance culture lived Niklas 
Koppernigk, later called Copernicus, whose reexamination of 
astronomical theories is discussed in Chapter 6. 

Further improvements in astronomical theory were made in the 
seventeenth century by Kepler, mainly through mathematical reasoning, 
and by Galileo through his observations and writings; these are 
discussed in Chapter 7. In Chapter 8 you shall see that Newton's work, 
in the second half of the seventeenth century, extended the ideas about 
the motions of objects on earth to explain the motions observed in the 
heavens — a magnificent synthesis of terrestrial and celestial dynamics. 
The results obtained by these men, and by others like them in other 
sciences such as anatomy and physiology, and the ways in which they 
went about their work, were so far-reaching that the resulting changes 
are generally referred to as the Scientific Revolution. 

Great scientific advances can, and often do, affect ideas outside 
science. For example, Newton's impressive work helped to bring a new 
feeling of self-confidence. Man seemed capable of understanding all 
things in the heavens and on the earth. This great change in attitude 



Louis XIV visiting the French Academy 
of Sciences, which he founded in the 
middle of the seventeenth century. 
Seen through the right-hand window 
is the Paris Observatory, then under 
construction. 




Prologue 



was a major characteristic of the eighteenth century, which has been 
called the Age of Reason. To a degree, what we think today and how 
we run our affairs are still based on the effect of scientific discoveries 
made centuries ago. 

Decisive changes in thought developed at the start of the 
Renaissance and grew during a period of about a century, from the 
work of Copernicus to that of Newton. In a sense, this era of invention 
can be compared to the sweeping changes which occurred during the 
past hundred years. This recent period might extend from the 
publication in 1859 of Darwin's Origin of Species to the first large-scale 
release of atomic energy in 1945. Within this interval lived such 
scientists as Mendel and Pasteur, Planck and Einstein, Rutherford and 
Fermi. The ideas they and others introduced into science during the 
last century have become increasingly important. These scientific ideas 
are just as much a part of our time as the ideas and works of people 
such as Roosevelt, Ghandi, and Pope John XXIII, Marx and Lenin, 
Freud and Dewey, Picasso and Stravinsky, Shaw and Joyce. If we 
understand the way in which science influenced the men of past 
centuries, we shall be better prepared to understand how science 
influences our thought and lives today. This is clearly one of the 
essential aims of this course. 

In sum, the material treated in this unit, although historical as well 
as scientific, is still of the first importance today for anyone interested 
in an understanding of science. The reasons for presenting the science 
in its historical context include the following: 



THE ORIGIN OF SPECIES 



BY MRANS OF NATURAL- SELECTION, 



I'UESERVATION OF PAVOUIIED RACES IN TUE STRUGGLE 
FOR LIFE. 



Cv CHARLES PARWTN. M.A., 



LONDON: 
JOHN MIIRnAV, ALBEMARLE STREET 



The results that were finally obtained are still valid and rank 
among the oldest ideas used every day in scientific work. The 
characteristics of all scientific work are clearly visible: the role of 
assumptions, of experiment and observations, of mathematical 
theory; the social mechanisms for collaborating, teaching, and 
disputing; and the possibility of having one's scientific findings 
become part of the established lore of the time. 

There is an interesting conflict between the rival theories used to 
explain the same set of astronomical observations. It illustrates 
what all such disputes have in common down to our day, and 
helps us to see clearly what standards may be used to judge one 
theory against another. 

This subject matter includes the main reasons for the rise of 
science as we understand it now. The story of the revolution in 
science in the seventeenth century and its many effects outside 
science itself is as crucial to the understanding of this current age 
of science as is the story of the American Revolution to an 
understanding of America today. 



5.1 Motions of the sun and stars 7 

5.2 Motions of tlie moon 11 

5.3 Tlie "wandering" stars 12 

5.4 Plato's problem 15 

5.5 The Greek idea of "explanation" 16 

5.6 The first earth-centered solution 17 

5.7 A sun-centered solution 19 

5.8 The geocentric system of Ptolemy 21 

5.9 Successes and limitations of the Ptolemaic model 25 



Midnight sun photographed at five 
minute intervals over the Ross Sea in 
Antarctica. 




CHAPTER FIVE 



Where is the Earth? - 
The Greeks' Answers 



5.1 Motions of the sun and stars 



The facts of everyday astronomy, the heavenly happenings 
themselves, are the same now as in the times of the Greeks. You 
can observe with your unaided eyes most of what these early 
scientists saw and recorded. You can discover some of the long- 
known cycles and rhythms, such as the seasonal changes of the 
sun's height at noon, the monthly phases of the moon, and the 
glorious spectacle of the slowly revolving night sky. If our purpose 
were only to make accurate forecasts of eclipses, planetary positions, 
and the seasons, we could, like the Babylonians and Egyptians, 
focus our attention on recording the details of the cycles and 
rhythms. If. however, like the Greeks, we wish to explain these 
cycles, we must also use these data to construct soine sort of simple 
model or theory with which we can predict the observed variations. 
But before we explore the several theories proposed in the past, let 
us review the major observations which the theories had to explain: 
the motions of the sun. moon, planets, and stars. 

The most basic celestial cycle as seen from our earth is. of 
course, that of day and night. Each day the sun rises above the local 
horizon on the eastern side of the sky and sets on the western side. 
The sun follows an arc across the sky. as is sketched in diagram 
(a) on the top of the next page. At noon, halfway between sunrise 
and sunset, the sun is highest above our horizon. Every day. a 
similar motion can be seen from sunrise to sunset. Indeed all the 
objects in the sky show this pattern of daily motion. They all rise in 
the east, reach a highest point, and drop lower in the west (although 
some stars never actually sink below the horizon). 

As the seasons change, so do the details of the sun's path across 
the sky. In our Northern Hemisphere during winter, the sun rises 
and sets more to the south, its altitude at noon is lower, and hence, 
its run across the sky lasts for a shorter period of time. In summer 
the sun rises and sets more toward the north, its height at noon is 
greater, and its track across the sky lasts a longer time. The whole 
cycle takes a little less than SeSjdays. 



SG5.1 



The motions of these bodies, 
essentially the same as they were 
thousands of years ago, are not 
difficult to observe — you should 
make a point of doing so. 
Handbook 1 has many suggestions 
for observing the sky. both with the 
naked eye and with a small 
telescope. 



This description is for observers in 
the Northern Hemisphere. For 
observers south of the equator, 
exchange "north'" and "south." 



Where is the Earth?— The Greeks' Answers 



sornrner 




tif Son 






N^-o.R.21 JUNE Zl &ePT 2»> Dec 22 '-'AR 



(a) Path of the sun through the sky 
for one day of summer and one day of 
winter. 



(b) Noon altitude of the sun as seen 
from St. Louis, Missouri, throughout 
the year. 



SG 5.2 
SG 5.3 



To reduce the number of leap days 
from 100 to 97 in 400 years, century 
years not divisible by 400 were 
omitted as leap years. Thus the 
year 1900 was not a leap year, but 
the year 2000 will be a leap year. 



SG 5.4 



This year-long cycle north and south is the basis for the 
seasonal or "solar" year. Apparently the ancient Egyptians thought 
that the year had 360 days, but they later added five feast days to 
have a year of 365 days that fitted better with their observations of 
the seasons. Now we know that the solar year is 365.24220 days 
long. The decimal fraction of the day, 0.24220. raises a problem for 
the calendar maker, who works with whole days. If you used a 
calendar of just 365 days, after four years New Year's Day would 
come early by one day. In a century you would be in error by 
almost a month. In a few centuries the date called January 1 would 
come in the summertime! In ancient times extra days or even 
whole months were inserted from time to time to keep a calendar of 
365 days and the seasons in fair agreement. 

Such a makeshift calendar is, however, hardly satisfactory. In 
45 B.C. Julius Caesar decreed a new 365-day calendar (the Julian 
calendar) with one extra whole day (a "leap day") being inserted 
each fourth year. Over many years, the average would therefore be 
3657 days per year. This calendar was used for centuries, during 
which the small difference between jand 0.24220 accumulated to 
a number of days. Finally, in 1582 A.D., under Pope Gregory, a new 
calendar (the Gregorian calendar) was announced. This had only 97 
leap days in 400 years, and the new approximation was close 
enough that it has lasted satisfactorily to this day without revision. 

You have noticed that a few stars are bright and many are 
faint. The brighter stars may seem to be larger, but if you look at 
them through binoculars, they still appear as points of light. Some 
bright stars show colors, but most appear whitish. People have 
grouped many of the brighter stars into patterns, called constellations. 
Examples include the familiar Big Dipper and Orion. 

You may have noticed a particular pattern of stars overhead, 
and several hours later, noticed it low in the west. What was 
happening? More detailed observation, say by taking a time-exposure 
photograph, would show that the entire bowl of stars had moved 
from east to west -new stars rising in the east and others setting 
in the west. During the night, as seen from a point on the Northern 



Section 5.1 




A combination trail and star photo- 
graph of the constellation Orion. The 
camera shutter was opened for several 
hours while the stars moved across 
the sky (leaving trails on the photo- 
graphic plate). Then the camera was 
closed for a few minutes and reopened 
while the camera was driven to follow 
the stars. 



Time exposure showing star trails 
around the north celestial pole. The 
diagonal line was caused by the rapid 
passage of an artificial earth satelite. 
You can use a protractor to deter- 
mine the duration of the exposure; the 
stars move about 15° per hour. 



Hemisphere of our earth, the stars appear to move counter-clockwise 
around a point in the sky called the north celestial pole. This 
stationary point is near the fairly bright star Polaris, as can be 
seen in the photograph at the top right of the page. 

Some of the star patterns, such as Orion (the Hunter) and 
Cygnus (the Swan — also called the Northern Cross), were described 
and named thousands of years ago. Since the star patterns 
described by the ancients still fit, we can conclude that star 
positions change very little, if at all, over the centuries. This 
constancy of relative positions has led to the term "fixed stars." 

Thus, we observe in the heavens both stability over the 
centuries and smooth, orderly, motions. But, although the daily 
rising and setting cycles of the sun and stars are similar, they are 



SG 5.5 

See "The Garden of Epicurus' 
in Reader 2. 






^ 



-O d 





URSA MAJOR 



URSA MINOR CASSIOPEIA 



CYGNUS 



LYRA ^••^ c 



^•\ ORION 



10 



Where is the Earth?-The Greeks' Answers 



A very easy but precise way to time 
the motions of the stars is explained 
in Handbook 2. 



The differences between the two 
frames of reference -the horizon 
and the fixed stars -are the basis 
for establishing a position on the 
earth, as in navigation. 



not identical. Unlike the sun's path, the paths of the stars do not 
vary in altitude from season to season. Also, stars do not have quite 
the same rhythm of rising and setting as the sun. but go a little 
faster. Some constellations seen high in the sky soon after sunset 
will, at the same time several weeks later, appear to be catching up 
with the sun. As measured by sun-time, the stars set about four 
minutes earlier each day. 

Thus far, we have described the positions and motions of the 
sun and stars in relation to the observer's horizon. But. because 
different observers have different horizons, the horizon is not an 
unchanging frame of reference from which all observers will see 
the same positions and motions in the sky. However, a frame of 
reference which is the same for all observers is provided by the 
fixed stars. The relative positions of these stars do not change as 
the observer moves over the earth. Also, their daily motions are 
simple circles with virtually no changes during a year or through 
the years. For this reason, positions in the heavens are usually 
described in terms of a frame of reference defined by the stars. 

A description of the sun's motion, using the fixed stars as a 
reference, must include the daily crossing of the sky, the daily 
difference in rising and setting times, and the seasonal change in 
noon altitude. We have already seen that, as measured by sun-time, 
the stars set about four minutes earlier each day. We can just as 
well say that, measured by star-time, the sun sets about four 
minutes later each day. That is, the sun appears to be gradually 
slipping behind the daily east-to-west motion of the stars. 

The difference in noon altitude of the sun during the year 
corresponds to a drift of the sun's path north and south on the back- 
ground of stars. In the first diagram below, the appearance of the 
middle portion of the sky is represented by a band around the earth. 
The sun's yearly path against this background of stars is represented 
by the dark line. If we cut and flatten out this band, as shown in 
the second and third diagrams, we get a chart of the sun's path 
during the year. (The 0° line is the celestial equator, the imaginary 
line in the sky that is directly above the earth's equator.) The sun's 
path against the background of the stars is called the ecliptic — its 
drift north and south of the celestial equator is about 23^°. We also 
need to define one point on the ecliptic so we can locate the sun, or 
other objects along it. For centuries this point has been the place 
where the sun moving eastward on the ecliptic, crosses the 
equator from south to north -about March 21. This point is called 
the "vernal (spring) equinox." It is the zero point from which 
positions among the stars are usually measured. 





March Zl 





f~~-^^i^^^i^ 


oelestial 


equator 


'^\/&yna\ ejs{\j\f 


ox 




^______.^^^ 







June 2.1 



Se|dt..22) 



Deo. 22. 



\Aard^ 21 



Section 5.2 



11 



Thus, there are three apparent motions of the sun: (1) its daily 
westward motion across the sky, (2) its yearly drift eastward among 
the stars, (3) its yearly cycle of north-south drift in noon altitude. 

These phenomena are all the more intriguing because they repeat 
unfailingly and precisely. We must try to explain these phenomena 
by devising a simple model to represent them. 

Q1 If you told time by the stars, would the sun set earlier or 
later each day? 

Q2 For what practical purposes were calendars needed? 

Q3 What are the observed motions of the sun during one year? 



These end-of-section questions are 
intended to help you checit your 
understanding before going on to 
the next section. 



5.2 Motions of the moon 



The moon shares the general east-to-west daily motion of the 
sun and stars. But the moon slips eastward against the background 
of the stars faster than the sun does. Each night the moon rises 
nearly an hour later. When the moon rises in the east at sunset 
(opposite the sun in the sky) it is bright and shows a full disk (full 
moon). Each day thereafter, it rises later and appears less round, 
waning finally to a thin crescent low in the dawn sky. After about 
fourteen days, when the moon is passing near the sun in the sky 
and rises with it, we cannot see the moon at all (new moon). After 
the new moon, we first see the moon as a thin crescent low in the 
western sky at sunset. As the moon rapidly moves further eastward 
from the sun. the moon's crescent fattens to a half disk and then 
within another week goes on to full moon again. After each full 
moon the cycle repeats. 



A "half moon" occurs one-quarter 
of the way through the monthly 
cycle, and is therefore called "first 
quarter" by astronomers. The full 
moon occurs half way through the 
cycle, and another "half moon" 
occurs at "third quarter." 




^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ '■^^^ 


■■ 


^^^^H 


^^^^^^^^^^ t-'^r^ 


jffS^ 


P^^^^^^^^^H 


^^^W^ ^MOL-Mi 


IL^l' 


-S^^^^^l 


^^m ^im;^ 


^ 


I^H 


^^p ' vh*^ 




l^l^^H 


^^H 




nl^^^^H 


^^^^^ w 


J 






26 days after new moon. 



17 days after new moon. 



3 days after new moon. 



12 



Where is the Earth? -The Greeks' Answers 



SG 5.6 



As early as 380 B.C., the Greek philosopher, Plato recognized 
that the phases of the moon could be explained by thinking of the 
moon as a globe reflecting sunlight and moving around the earth in 
about 29 days. Because the moon appears so big and moves so 
rapidly compared to the stars, people in early times assumed the 
moon to be quite close to the earth. 

The moon's path around the sky is close to the yearly path of 
the sun; that is, the moon is always near the ecliptic. But the 
moon's path is tipped a bit with respect to the sun's path; if it were 
not, the moon would come exactly in front of the sun at every new 
moon (causing an eclipse of the sun) and be exactly opposite the 
sun at every full moon, and move into the earth's shadow 
(causing an eclipse of the moon). 

The motions of the moon have been studied with great care for 
centuries, partly because of interest in predicting eclipses, and 
have been found to be very complicated. The precise prediction of 
the moon's position is an exacting test for any theory of motion in 
the heavens. 

Q4 Draw a rough diagram to show the relative positions of the 
sun, earth, and moon during each of the moon's four phases. 
Q5 Why don't eclipses occur each month? 



5.3 The "wandering stars" 



When a planet is observed directly 
opposite from the sun, the planet is 
said to be In opposition. Retrograde 
motions of Mars, Jupiter, and Saturn 
are observed about the time they are 
in opposition. 



Without a telescope we can see, in addition to the sun and 
moon, five rather bright objects which move among the stars. These 
are the "wanderers," or planets: Mercury, Venus, Mars, Jupiter, and 
Saturn. With the aid of telescopes, three more planets have been 
discovered: Uranus, Neptune, and Pluto; but none of these were 
known for nearly a century after the time of Isaac Newton. Like the 
sun and moon, all the planets rise daily in the east and set in the 
west. Also like the sun and moon, the planets generally move slowly 
eastward among the stars. But they have another remarkable and 
puzzling motion of their own: at certain times each planet stops 
moving eastward among the stars and for some months loops back 
westward. This westward or "wrong-way" motion is called 
retrograde motion. The retrograde loops made by Mercury. Mars, 
and Saturn during 1963 are plotted on the next page. Saturn. 
Jupiter, and Mars can at one time or another be anywhere in the 
sky, although always very near the ecliptic. The retrograde motion 
of one of these planets occurs when it is nearly opposite to the sun 
(that is, halfway across the sky at midnight). Mercury and Venus, 
however, have limits to how far away from the sun they can be; 



PLanft-t 



E-arUn 



Section 5.3 



13 





rtxr 




Saturn • . 






i^' 



.•t 



T ^ • 



> 



-20 

To" 



The retrograde motions of Mercury 
(marked at 5-day intervals), Mars (at 
10-day intervals), and Saturn (at 20- 
day intervals) in 1963, plotted on a 
star chart. The dotted line is the an- 
nual path of the sun, called the 
ecliptic. 



14 



Where is the Earth? — The Greeks' Answers 



Sun 



The maximum angles from the sun at 
which we observe Mercury and Venus. 
Both planets can, at times, be ob- 
served at sunset or at sunrise. Mer- 
cury is never observed to be more than 
28° from the sun, and Venus is never 
more than 48° from the sun. 



Merour 





Ea.rt-h 



Earth 



SG 5.7 

Typical Retrograde 
Motions of the Planets 

WESTWARD 



PLANET 


DAYS 


DISPLACEMENT 


Mercury 


34 


15' 


Venus 


43 


19° 


Mars 


83 


22° 


Jupiter 


118 


10° 


Saturn 


139 


7° 


Uranus 


152 


4° 


Neptune 


160 


3° 


Pluto 


156 


2" 



as the above figures show, the greatest angular distance in either 
direction from the sun is 28° for Mercury and 48° for Venus. The 
retrograde motion of Venus or Mercury begins after the planet is 
furthest east of the sun and visible in the evening sky. 

The planets change considerably in brightness. When Venus is 
first seen in the evening sky as the "evening star," the planet is 
only fairly bright. But during the following four to five months, as it 
moves farther eastward from the sun, Venus gradually becomes so 
bright that it often can be seen in the daytime if the air is clear. 
A few weeks later, when Venus scoots westward toward the sun, it 
fades rapidly, passes the sun, and soon reappears in the morning 
sky before sunrise as the "morning star." Then it goes through the 
same pattern of brightness changes, but in the opposite order: 
bright, then gradually fading. The variations of Mercury follow 
much the same pattern. But because Mercury is always seen near 
the sun (that is. only during twilight). Mercury's changes are 
difficult to observe. 

Mars, Jupiter, and Saturn are brightest about the time that they 
are in retrograde motion and opposite to the sun. Yet over many 
years their inaximum brightness differs. The change is most 
noticeable for Mars; the planet is brightest when it is opposite the 
sun during August or September. 

Not only do the sun, moon, and planets generally move 
eastward among the stars, but the moon and planets (except 
Pluto) are always found within a band only 8° wide on either side 
of the sun's path. 

These, then, are some of the main observations of celestial 
phenomena. All of them were known to the ancients, and in their 
day as in ours, the puzzling regularities and variations seemed to 
cry out for some explanation. 

Q6 In what part of the sky must you look to see the planets 
Mercury and Venus? 

Q7 In what part of the sky would you look to see the planet 
which is in opposition? 

Q8 When do Mercury and Venus show retrograde motion? 



Section 5.4 



15 



Q9 When do Mars, Jupiter, and Saturn show retrograde 
motion? 

Q10 Can Mars, Jupiter, and Saturn appear any place in the 
sky? 



5.4 Plato's problem 

In the fourth century B.C., Greek philosophers asked a new 
question: How can we explain the cyclical changes observed in the 
sky? That is, what model can consistently and accurately account 
for all celestial motions? Plato sought a theory to account for what 
was seen, or, as he phrased it, "to save the appearances." The 
Greeks were among the first people to desire explanations for 
natural phenomena that did not require the intervention of gods 
and other supernatural beings. Their attitude was an important step 
toward science as we know it today. 

How did the Greeks begin their explanation of the motions 
observed in the heavens? What were their assumptions? 

Any answers to these questions must be tentative. Although 
many scholars over the centuries have devoted themselves to the 
study of Greek thought, the documents, on which our knowledge of 
the Greeks is based, are mostly copies of copies and translations of 
translations, in which errors and omissions occur. In some cases all 
we have are reports from later writers on what certain philosophers 
did or said, and these accounts may be distorted or incomplete. The 
historians' task is difficult. Most of the original Greek writings were 
on papyrus or cloth scrolls which have decayed through the ages. 
Many wars and much plundering and burning have also destroyed 
many important documents. Especially tragic was the burning of 
the famous library of Alexandria in Egypt, which contained several 
hundred thousand documents. (It was burned three times: in part 
by Caesar's troops in 47 B.C.; then in the fourth century A.D. by 
Christians; and the third time about 640 A.D. by early Muslims 
when they overran the country.) Thus, while the general picture of 
Greek culture seems to be rather well established, many interesting 
details are not known. 

The approach taken by the Greeks and their intellectual 
followers for many centuries was already implied in a statement 
by Plato in the fourth century B.C. He defined the problem to his 
students in this way: the stars — representing eternal, divine, 
unchanging beings — move at a uniform speed around the earth, as 
we observe, in that most regular and perfect of all paths, the 
endless circle. But a few celestial objects, namely the sun, moon, 
and planets, wander across the sky and trace out complex paths, 
including even retrograde motions. Yet, being heavenly bodies, 
surely they too must really move in a way that suits their exalted 
status. Their motions, if not in a perfect circle, must therefore be in 
some combination of perfect circles. What combinations of circular 
motions at uniform speed can account for the peculiar variations 
in the overall regular motions In the sky? 



Several centuries later, a more 
mature Islamic culture led to 
extensive study and scholarly 
commentary on the remains of 
Greek thought. Several centuries 
later still, a more mature Christian 
culture used the ideas preserved by 
the Muslims to evolve early parts of 
modern science. 



16 Where is the Earth? -The Greeks' Answers 

Notice that the problem is concerned only with the changing 
apparent positions of the sun, moon, and planets. The planets 
appear to be only points of light moving against the background of 
stars. From two observations at different times we obtain a rate of 
motion: a value of so many degrees per day. The problem then is to 
find a "mechanism," some combination of motions, that will 
reproduce the observed angular motions and lead to predictions 
which agree with observations. The ancient astronomers had no 
observational evidence about the distances of the planets from the 
earth; all they had were directions, dates, and rates of angular 
motion. Although changes in brightness of the planets were known 
to be related to their positions relative to the sun, these changes in 
brightness were not included in Plato's problem. 

Plato and many other Greek philosophers assumed that there 
were a few basic "elements" that mixed together to cause the 
apparent variety of materials observed in the world. Although not 
everyone agreed as to what these elements were, gradually four 
were accepted as the explanation of phenomena taking place on 
earth. These elements were Fire, Air, Water, and Earth. Because 
substances found on earth were supposed to contain various 
mixtures of these elements, these compound substances would have 
a wide range of properties. (See Unit 1, Chapter 2.) Only perfection 
could exist in the heavens, which were separate from the earth and 
were the abode of the gods. Just as motions in tTie heavens must be 
eternal and perfect, so also the unchanging heavenly objects could 
not be composed of elements normally found on or near the earth. 
Hence, they were supposed to be composed of a changeless fifth 

In Latin the ether became quinta element of their own — the ether. 

essentia (fifth element), whence our Plato's problem in explaining the motion of planets remained 

qum essence. ^.j^^ most significant problem for theoretical astronomers for nearly 

two thousand years. To appreciate the later efforts and consequences 
of the different interpretations developed by Kepler, Galileo, and 
Newton, we will first examine the solutions to Plato's problems as 
they were developed by the Greeks. Let us confess right away that 
for their time these solutions were useful, ingenious, and indeed 
beautiful. 



Q11 What was Plato's problem of planetary motion? 
Q12 Why is our knowledge of Greek science incomplete? 
Q13 Why did the Greeks feel that they should use only 
uniform circular motion to explain celestial phenomena? 



5.5 The Greek idea of "explanation" 

Plato's statement of this historic problem of planetary motion 
illustrates three contributions of Greek philosophers which, with 
modifications, are still basic to our understanding of the nature of 
physical theories: 

1. A theory should be based on simple ideas. Plato regarded it 



Section 5.6 



17 



not merely as simple, but also as self-evident, that heavenly bodies 
must move uniformly along circular paths. Only in recent centuries 
have we come to understand that such common-sense beliefs may 
be misleading. More than that — we have learned that unproved 
assumptions are often necessary, but must be critically examined 
and should at first be accepted only tentatively. As we shall often 
see in this course, the identification of hidden assumptions in 
science has been extremely difficult. Yet in many cases, when the 
assumptions have been identified and questioned, entirely new 
theories have followed. 

2. Physical theory must incorporate the measured results of 
observation of phenomena, such as the motions of the planets. 
Furthermore, our purpose in making a theory is to discover the 
uniformity of behavior, the hidden simplicities underlying apparent 
irregularities. For organizing our observations, the language of 
number and geometry is useful. This use of mathematics, widely 
accepted today, was derived in part from the Pythagoreans, a group 
of mathematicians who lived in southern Italy about 500 B.C. and 
believed that "all things are numbers." Actually the use of 
mathematics and measurement became important only in the later 
development of science. Plato stressed the fundamental role of 
numerical data only in his astronomy, while Aristotle largely 
avoided detailed measurements. This was unfortunate because, as 
the Prologue reported, the arguments of Aristotle, which did not 
include the idea of measurement of change as a tool of knowledge, 
were adopted by such influential philosophers as Thomas Aquinas. 

3. To "explain" complex phenomena means to develop or invent 
a scheme (a physical model, or a geometrical or other mathematical 
construction) which shows the same features as the phenomena to 
be explained. Thus, for example, if one actually constructs a model 
of interlocking spheres, as has been often done, a point on one of 
the spheres has the same motions as the planet which the point 
represents. 

5.6 The first earth-centered solution 

The Greeks observed that the earth was large, solid, and 
permanent, while the heavens seemed to be populated by small, 
remote, ethereal objects that were continually in motion. What was 
more natural than to conclude that our big, heavy earth was the 
steady, unmoving center of the universe? Such an earth-centered 
viewpoint is called geocentric. From this viewpoint the daily motion 
of the stars could easily be explained: they were attached to, or 
were holes in, a large, dark, spherical shell surrounding the earth 
and were all at the same distance from us. Daily, this celestial 
sphere turned once around on an axis through the earth. As a 
result, all the stars fixed on it would move in circular paths around 
the pole of rotation. Thus, a simple model of a rotating celestial 
sphere and a stationary earth could account for the daily motions 
of the stars. 




The annual north-south (seasonal) 
motion of the sun was explained by 
having the sun on a sphere whose axis 
was tilted 23j° from the axis of the 
eternal sphere of the stars. 



18 



Where is the Earth? — The Greeks' Answers 



La Figure & nombrc 



iksSphero. 




DcsCerclesdc la Sphere. Chap. III. 
Quelle choft eft laSphere. 

339 ^SphmrJ}yn corpt cortnu itttne/Kptr/ke Wfw/c,** milljfH 
fe^ rfw tjutU^linftcin^: toutrt t» ttrnts y<« en fiHt fnttr^tCiti 
sSj iw/^Mfj 4 U amtnfcrtTKt font tgtiUx. 

Quelle chofc eft 1 cxicu dt la Sphere. 

Exin it U SfliercCcSmt diCl Ditdtchu) ijl U dmman }«' fdf- 
^P » rr4«m U Sfhm.fur IojkI lUtfi Isunu. 



A geocentric cosmological sciieme. 
The earth is fixed at the center of 
concentric rotating spheres. The 
sphere of the moon (lune) separates 
the terrestrial region (composed of 
concentric shells of the four elements 
Earth, Water, Air, and Fire) from the 
celestial region. In the latter are the 
concentric spheres carrying Mercury, 
Venus, Sun, Mars, Jupiter, Saturn, and 
the stars. To simplify the diagram, only 
one sphere is shown for each planet. 
(From the DeGolyer copy of Petrus 
Apianus' Cosmographia, 1551.) 



The three observed motions of the sun require a somewhat 
more complex model. To explain the sun's motion with respect to 
the stars, a separate invisible shell was imagined that carried the 
sun around the earth. This shell was fixed to the celestial sphere 
and shared its daily motion but had also a slow, contrary motion of 
its own, namely one cycle of 360° per year. 

The yearly north-south motion of the sun was accounted for 
by tipping the axis of this sphere, for the sun was tipped from the 
axis of the dome of the stars. 

The motions of the visible planets — Mercury, Venus. Mars, 
Jupiter, and Saturn — were more difficult to explain. They share 
generally the daily motion of the stars, but they also have peculiar 
motions of their own. Because Saturn moves most slowly among 
the stars (it revolves once in 30 years), its sphere was assumed to 
be largest and closest to the stars. Inside the sphere for Saturn 
would be spheres that carried the faster-moving Jupiter (12 years) 
and Mars (687 days). Since they all require more than a year for a 
complete trip among the stars, these three planets were believed to 
be beyond the sphere of the sun. Venus, Mercury, and the moon 
were placed between the sun and the earth. The fast-moving moon 
was assumed to reflect sunlight and to be closest to the earth. 

Such an imaginary system of transparent shells or spheres can 
provide a rough "machine" to account for the general motions of 
heavenly objects. By choosing the sizes of the spheres and their 
rates and directions of motions, a rough match could be made 
between the model and the observations. If additional observations 
revealed other cyclic variations, more spheres could be added to 
make the necessary adjustment in the model. 

Plato's friend Eudoxus concluded that 26 spheres would account 
for the general pattern of motions. Later Aristotle added 29 more. 
(An interesting description of this general system or cosmological 
scheme is given by the poet Dante in the Divine Comedy, written 
about 1300 A.D., shortly after Aristotle's writings became known 
in Europe.) Yet even Aristotle knew that this system did not get the 
heavenly bodies to their observed positions at quite the right time. 
Moreover, it did not account at all for the observed variations in 
brightness of the planets. 

You may feel that Greek science was bad science because it 
was different from our own or less accurate. But you should 
understand from your study of this chapter that such a conclusion 
is not justified. The Greeks were just beginning the development of 
scientific theories and inevitably made assumptions that we now 
consider invalid. Their science was not "bad science," but in many 
ways it was a different kind of science from ours. And ours is not the 
last word, either. We must realize that to scientists 2000 years from 
now our efforts may seem strange and inept. 

Even today's scientific theory does not and cannot account for 
every detail of each specific situation. Scientific concepts are 
idealizations which treat only selected aspects of observations 
rather than the totality of the raw data. Also, each period in 



Section 5.7 



19 



history has its own Hmits on the range of human imagination. As 
you aheady have seen in Unit 1, important general concepts, such 
as force or acceleration, are specifically invented to help organize 
observations. They are not given to us in final form by some 
supernatural genius. 

As you might expect, the history of science contains many 
examples in which certain factors overlooked by one researcher 
turn out later to be very important. But how would better systems 
for making predictions be developed without first trials? Theories 
are improved through tests and revisions, and sometimes are 
completely replaced by better ones. 

Q14 What is a geocentric system? How does it account for the 
motions of the sun? 

Q15 Describe the first solution to Plato's problem. 



5.7 A sun-centered solution 

For nearly two thousand years after Plato and Aristotle, the 
geocentric model was generally accepted. However, a radically 
different model, based on different assumptions, had been proposed 
in the third century B.C. The astronomer Aristarchus, (perhaps 
influenced by the writings of Heracleides, who lived a century 
earlier) suggested that a simpler explanation of heavenly motion 
would result if the sun were considered to be at the center, with the 
earth, planets, and stars all revolving around it. A sun-centered 
system is called heliocentric. 

Because the major writings of Aristarchus have been lost, our 
knowledge of his work is based mainly on comments made by other 
writers. According to Archimedes, Aristarchus taught that the sun 
must be at least eighteen times farther away than the moon, and 
that the larger body, which was also the source of light', should be 
at the center of the universe. 

Aristarchus proposed that all the daily motions observed in the 
sky could be explained by assuming that the celestial sphere is 
motionless and that the earth rotates once daily on an axis of its 
own. The apparent tilt of the paths of the sun, moon, and all the 
planets is accounted for simply by the tilt of the earth's own axis. 
Furthermore, the yearly changes in the sky, including the retrograde 
motions of the planets, could be explained by assuming that the 
earth and the five visible planets revolve around the sun. In this 
model, the motion previously assigned to the sun around the earth 
was assigned to the earth moving around the sun. Also, the earth 
became just one among several planets. 

How such a system can account for the rectrograde motions of 
Mars, Jupiter, and Saturn can be seen from the diagram in the 
margin in which an outer planet and the earth are assumed to be 
moving around the sun in circular orbits. The outer planet moves 



Sun 




As the earth passes a planet in its 
orbit around the sun, the planet ap- 
pears to move backwards in the sky. 
The arrows show the sight lines to- 
ward the planet for the different 
numbered positions of the earth. The 
lower numbered circles indicate the 
resulting apparent positions of the 
planet against the background of 
distant stars. 



20 



Where is the Earth? — The Greeks' Answers 




B*i-th 



If the earth goes around the sun, then 
the direction in which we have to look 
for a star should change during the 
year. A shift in the relative observed 
positions of objects that is caused by 
a displacement of the observer is 
called a parallax. The greatest ob- 
served parallax of a star caused by the 
earth's annual motion around the sun 
is about 1/2400". This is explained by 
the fact that the distance to this 
nearest star is not just hundreds of 
millions of miles but 25 million million 
miles. 



more slowly than the earth. As a result, when the earth is directly 
between the sun and the planet, the earth moves rapidly past the 
planet. To us the planet appears for a time to be moving backward 
or in retrograde motion across the sky. 

Gone are all the interlocking concentric spheres. The heliocentric 
(sun-centered) hypothesis, which also uses only uniform circular 
motions, has one further advantage. It explains the bothersome 
observation that the planets are brighter during their retrograde 
motion, since at that time the planets are nearer to the earth. Even 
so, the proposal by Aristarchus was essentially neglected in 
antiquity. It was severely criticized for three basic reasons. One 
reason was that the idea of making a moving earth was unaccept- 
able. It contradicted the philosophical doctrines that the earth is 
different from the celestial bodies and that the natural place of the 
earth, both physically and theologically, is the center of the 
universe. In fact, his contemporaries considered Aristarchus 
impious for even suggesting that the earth moved. Also, this new 
picture of the solar system contradicted common sense and everyday 
observations: the earth certainly seemed to be at rest rather than 
rushing through space. 

Another criticism was that certain observational evidence 
seemed to refute Aristarchus. If the earth were moving in an orbit 
around the sun, it would also be moving back and forth under the 
fixed stars. As shown in the sketch in the margin, the angle from 
the vertical at which we have to look for any star would be different 
when seen from the various points in the earth's annual path. This 
annual shift of the fixed stars should occur if the earth moves 
around the sun. But it was not observed by the Greek astronomers. 
This awkward fact could be explained in two ways either (1) the 
earth does not go around the sun and so there is no shift, or (2) the 
earth does go around the sun but the stars are so far away that the 
shift is too small to observe. But as the Greeks realized, for the 
shift to be undetectably small, the stars would have to be enormously 
far away — perhaps hundreds of millions of miles. 

Today we can observe the annual shift of the stars with 
telescopes, so we know that Aristarchus' model is in fact useful. 
The shift is too small to be seen with the naked eye using the best 
sighting instruments — and indeed so small that even with 
telescopes it was not measured until 1838. The largest annual shift 
is an angle of only about 1/2400 of a degree of arc. The smallest 
angle observable by the human eye under ideal conditions is about 
1/25^, so the actual angular shift is about 100 times smaller than 
could possibly have been observed. The shift exists, but we can 
sympathize with the Greeks, who rejected the heliocentric theory 
partly because the shift required by the theory could not be 
observed. Only Aristarchus imagined that the stars might be as 
immensely distant as we now know them to be. 

Finally, Aristarchus was criticized because he did not develop 
his system in detail or use it for making predictions of planetary 
positions. His work seems to have been purely qualitative, a general 
scheme of how things might be. 



Section 5.8 21 

The geocentric and heliocentric systems were two different 
ways to account for the same observations. But the hehocentric 
proposal required such a drastic change in man's image of the 
universe that Aristarchus' heliocentric hypothesis had little influence 
on Greek thought. Fortunately his arguments were recorded and 
reported and eighteen centuries later stimulated the thoughts of 
Copernicus. Ideas are not bound by space or time. 

Q16 What two radically new assumptions were made by 
Aristarchus? What simplification resulted? 

Q17 How can the heliocentric model proposed by Aristarchus 
explain retrograde motion? 

Q18 What change predicted by Aristarchus' theory was not 
observed by the Greeks? 

Q19 Why was Aristarchus considered impious? Why was his 
system neglected? 



5.8 The geocentric system of Ptolemy 

Disregarding the heliocentric model suggested by Aristarchus, 
the Greeks continued to develop their planetary theory as a 
geocentric system. As we noted, the first solution in terms of 
concentric spheres lacked accuracy. During the 500 years after 
Plato and Aristotle, astronomers began to sense the need for a more 
accurate theory for the heavenly timetables. To fit the observations, 
a complex mathematical theory was required for each planet. 

Several Greek astronomers made important contributions 
which culminated about 150 A.D. in the geocentric theory of 
Claudius Ptolemy of Alexandria. Ptolemy's book on the motions of 
the heavenly objects is a masterpiece of analysis. 

Ptolemy wanted a system that would predict accurately the 
positions of each planet. The type of system and the motions he 

accepted were based on the assumptions of Aristotle. In the preface The Arabic title given to Ptolemy's 

to his Almagest, Ptolemy defines the problem and states his book, the Almagest, means "the 

assumptions: 

... we wish to find the evident and certain appearances 

from the observations of the ancients and our own, and 

applying the consequences of these conceptions by means 

of geometrical demonstrations. 

And so, in general, we have to state, that the heavens 

are spherical and move spherically; that the earth, in 

figure, is sensibly spherical . . .; in position, lies right in 

the middle of the heavens, like a geometrical center; in 

magnitude and distance, [the earth] has the ratio of a 

point with respect to the sphere of the fixed stars, having 

itself no local motion at all. 
Ptolemy then argues that each of these assumptions is 
necessary and fits with all our observations. The strength of his 
belief is illustrated by his statement ". . . it is once for all clear 



greatest." 



22 



Where is the Earth? -The Greeks' Answers 



from the very appearances that the earth is in the middle of the 
world and all weights move towards it." Notice that he has 
supported his interpretation of the astronomical observations with 
the physics of falling bodies. This mixture of astronomy and 
physics, when applied to the earth itself and its place in the 
scheme, became highly important when he referred to the proposal 
of Aristarchus that the earth might rotate and revolve: 

Now some people, although they have nothing to 
oppose to these arguments, agree on something, as they 
think, more plausible. And it seems to them there is 
nothing against their supposing, for instance, the heavens 
immobile and the earth as turning on the same axis [as 
the stars] from west to east very nearly one revolution a 
day. . . . 

But it has escaped their notice that, indeed, as far as 
the appearances of the stars are concerned, nothing 
would perhaps keep things from being in accordance with 
this simpler conjecture, but that in the light of what 
happens around us in the air such a notion would seem 
altogether absurd. 



SG 5.8 



The annual path of the sun against 
the celestial sphere. 



Ptolemy believed that if the earth rotated it would not take its 
blanket of air around with it, with the result that all clouds would 
fly past toward the west, and all birds or other things in the air 
would be carried away to the west. If, however, the air was carried 
with the earth, the objects in the air would still be left behind by 
the earth and air together. 

The paragraphs quoted above contain a main theme in Unit 2. 
Ptolemy recognized that the two systems were equally successful in 
describing motion — in the kinematics; but the one was to be 
preferred over the others because it better fit the causes of 
motion — the dynamics — as understood at the time. Much later, when 
Newton had developed a completely different dynamics, the choice 
would fall the other way. 

Ptolemy developed very clever and rather accurate procedures 
by which the positions of each planet could be derived on a 
geocentric model. In the solutions he went far beyond the scheme 
of concentric spheres of the earlier Greeks, constructing a model 
out of circles and three other geometrical devices. Each device 
provided for variations in the rate of angular motion as seen from 
the earth. To appreciate Ptolemy's ingenious solution, let us examine 
one of the many small variations he was attempting to explain. 

We can divide the sun's yearly 360° path across the background 
of stars into four 90° parts. If the sun is at the point on March 21, 
it will be 90° farther east on June 21, 90° further still on September 

30* 

20* 




so* 



March Zl 



June 2.1 



Se|[^2& 



Deo. 27 



March 21 



Section 5.8 



23 



23, 90° farther on December 22, and back at the starting point on 
March 21, a year later. If the sun moves uniformly on a circle 
around the earth, the times between these dates ought to be equal. 
But, as you will find by consulting a calendar, they aren't equal. 
The sun takes longer to move 90° in spring and summer than it 
does in fall and winter. So any simple circular system based on 
motion with constant speed will not work for the sun. 

The three devices that Ptolemy used in proposing an improved 
geocentric theory were the eccentric, the epicycle and equant. 

In agreement with Plato, astronomers had held previously that 
the motion of a celestial object must be at a uniform angular rate 
and at a constant distance from the center of the earth. Although 
Ptolemy beheved that the earth was at the center of the universe, 
he did not insist that it be at the geometrical centers of all the 
perfect circles. He proposed that the center C of the circle could be 
off-center from the earth, in an eccentric position. Thus, motion 
that was really uniform around the center C would not appear to be 
uniform motion when observed from the earth. An eccentric orbit 
of the sun will therefore account for the type of seasonal irregularity 
observed in the sun's rate of motion. 

While the eccentric can also account for small variations in the 
rate of motion of planets, it cannot describe any such radical 
change as retrograde motion of the planets. To account for 
retrograde motion, Ptolemy used another device, the epicycle (see 
the figure at the right). The planet is considered to be moving at a 
uniform rate on the circumference of a small circle, called the 
epicycle. The center of the epicycle moves at a uniform rate on the 
large circle, called the deferent, around the earth. 

If a planet's speed on the epicycle is greater than its speed on 
the large circle, the planet as seen from above the planetary system 
would appear to move through loops. When observed from a 
location near the center, these loops would look like the retrograde 
motions actually observed for planets. The photographs below show 
two views of the motions produced by a simple mechanical model, 
an "epicycle machine" with a small lamp in place of the planet. 
The photo on the left was taken from "above," like the diagram in 
the margin; the photograph on the right was taken "on edge," 



PUai-»«t. 




An eccentric 




An epicycle 



SG 5.9 





Retrograde motion created by a sim- 
ple epicycle machine. 

(a) Stroboscopic photograph of epi- 
cyclic motion. The flashes where made 
at equal time intervals. Note that the 
motion is slowest in the loop. 

(b) Loop seen from near its plane. 



24 



Where is the Earth? -The Greeks' Answers 




Ptolemy did not picture the 
planetary motions as those in an 
interlocking machine where each 
planet determined the motion of the 
next. Because there was no infor- 
mation about the distances of the 
planets, Ptolemy adopted the old 
order of distances from the earth: 
stars being the most remote, then 
Saturn, Jupiter, Mars, the sun, 
Venus, Mercury, and the moon. The 
orbits were usually shown nested 
inside one another so that their 
epicycles did not overlap. 




Simplified representation of the Ptolemaic system. The scale of the 
upper drawing, which shows the planets between the earth and the 
sun, is eight times that of the lower drawing, which shows the planets 
that are farther than the sun. The planets' epicycles are shown along 
one straight line to emphasize the relative sizes of the epicycles. 



Section 5.9 



25 






almost in the plane of the motion; thus, the appearance of the loop 
is very much like it would be if viewed from near the center. 

Epicycles can be used to describe many kinds of motion. So it 
was not too difficult to produce a system that had all the main 
features of observed planetary motion. One particularly interesting 
feature of Ptolemy's system was the epicycles for the outer planets 
all had the same period: exactly one year! Moreover, as the sketches 
on the opposite page indicate, the positions of the outer planets on 
their epicycles always matched the position of the sun relative to 
the earth. This correspondence of epicycles to the relative motion of 
sun and earth was, fourteen centuries later, to be a key point of 
concern to Copernicus. 

So far, the system of epicycles and deferents "works" well 
enough. It explains not only retrograde motion, but also the greater 
brightness of the planets when they are in retrograde motion. Since 
the planet is on the inside of its epicycle during retrograde motion, 
it is closest to the earth, and so appears brightest. This is an 
unexpected bonus, since the model was not designed to explain this 
feature. 

But even with combinations of eccentrics and epicycles, 
Ptolemy was not able to fit the motions of the five planets precisely. 
For example, as we see in the three figures above, the retrograde 
motion of Mars is not always of the same angular size or duration. 
To allow for these variations, Ptolemy used a third geometrical 
device, called the equant, which is a modification of an eccentric. 
As shown in the margin, the earth is again off-center from the 
geometric center C of the circle, but the motion along the circle is 
not uniform around C ; instead, it is uniform as seen from another 
point C, which is as far off-center as the earth is, but on the other 
side of the center. 

5.9 Successes and limitations of the Ptolemaic model 



Mars plotted at four-day intervals on 
three consecutive oppositions. Note 
the different sizes and shapes of the 
retrograde curves. 



PUanet 




An equant. C is the center of the circle. 
The planet P moves at a uniform rate 
around the off-center point C 



Ptolemy's model always used a uniform rate of angular 
motion around some center, and to that extent stayed close to the 
assumptions of Plato. But Ptolemy was willing to displace the 
centers of motion from the center of the earth, as much as was 
necessary to fit the observations. By a combination of eccentrics, 
epicycles, and equants he described the positions of each planet 
separately. For each planet, Ptolemy had found a combination of 
motions that predicted its observed positions over long periods of 
time to within about two degrees (roughly four diameters of the 
moon) — a considerable improvement over earlier systems. 



SG 5.10 

Astronomical observations were all 
observations of angles — a small 
loop in the sky could be a small 
loop fairly near, or a larger loop 
much farther away. 



26 Where is the Earth?-The Greeks' Answers 

The success of Ptolemy's model, especially the unexpected 
explanation of variation in brightness, might be taken as proof that 
objects in the sky actually moved on epicycles and deferents around 
off-center points. It seems, however, that Ptolemy himself did not 
believe he was providing an actual physical model of the universe. 
He was content to give a mathematical model for the computation 
of positions. 

Of course, some difficulties remained. For example, to explain 
the motions of the moon, Ptolemy had to use such large epicycles 
that during a month the moon would appear to grow and shrink in 
size appearing to have at some times twice the diameter than at 
other times! Ptolemy surely knew that this was predicted by his 
model — and that it does not happen in actual observation. But, his 
model was not intended to be "real," it was only a basis for 
computing positions. 

The Ptolemaic description was a series of mathematical devices 
to match and predict the motion of each planet separately. His 
geometrical analyses were equivalent to finding a complicated 
equation of motion for each individual planet. Nevertheless, in the 
following centuries most scholars, including the poet Dante, 
SG 5.11 believed that the planets really moved on some sort of crystalline 
spheres as Eudoxus had suggested earlier. 

Although now discarded, the Ptolemaic form of the geometric 
model of the planetary system, proposed in 150 A.D., was used for 
about 1500 years. There were good reasons for this long acceptance. 

It predicted fairly accurately the positions of the sun, moon. 

and planets. 

It explained why the fixed stars do not show an annual shift 

when observed with the naked eye. 

It agreed in most details with the philosophical doctrines 

developed by the earlier Greeks, including the idea of "natural 

motion" and "natural place." 

It had common-sense appeal to all who saw the sun. moon, 

planets, and stars moving around them. 

It agreed with the comforting assumption that we live on an 

immovable earth at the center of the universe. 

Also, later, it fitted into Thomas Aquinas' widely accepted 

synthesis of Christian belief and Aristotelian physics. 
Yet, Ptolemy's system was eventually displaced by a heliocentric 
one. Why did this occur? What advantages did the new theory have 
SG 5.12 over the old? From this epic argument about competing theories 

what can we learn about the relative value of rival theories in 
SG 5.13 science today? These are some of the questions to consider in the 
next chapter. 



STUDY GUIDE 



5.1 The Project Physics learning materials 
particularly appropriate for Chapter 5 include the 
following: 

Experiments 

Naked-Eye Astronomy (cont.) 

Size of the Earth 

Height of Piton, A Mountain on the Moon 
Activities 

Making Angular Measurements 

Celestial Sphere Model 

How Long is a Sidereal Day? 

Scale Model of the Solar System 

Build a Sundial 

Plot an Analemma 

Stonehenge 

Moon Crater Names 

Literature 

Size of the Earth — Simplified Version 
Reader Articles 

The Boy Who Redeemed His Father's Name 

Four Poetic Fragments About Astronomy 
Film Strip 

Retrograde Motion of Mars 
Film Loops 

Retrograde Motion of Mars and Mercury 

Retrograde Motion — Geocentric Model 
Transparencies 

Stellar Motion 

Celestial Sphere 

Retrograde Motion 

Eccentrics and Equants 

In addition, the following Reader articles are of 
general interest for Unit 2: 
The Black Cloud 
Roll Call 

A Night at the Observatory 
The Garden of Epicurus 
The Stars Within 22 Light-years That Could 

Have Habitable Planets 
Scientific Study of UFO's 

5.2 How could you use the shadow cast by a 
vertical stick on horizontal ground to find 

(a) the local noon? 

(b) which day was June 21st? 

(c) the length of a solar year? 

5.3 What is the difference between 365.24220 
days and 3657 days (a) in seconds (b) in percent? 

5.4 (a) List the observations of the motions of 

heavenly bodies that you might make 
which would also have been possible in 
ancient Greek times, 
(b) For each observation, list some reasons 
why the Greeks thought these motions 
were important. 

5.5 Which of the apparent motions of the stars 
could be explained by a flat earth and stars fixed 
to a bowl that rotated around it? 

5.6 Describe the motion of the moon during one 
month. (Use your own observations if possible.) 



5.7 Mercury and Venus show retrograde motion 
after they have been farthest east of the sun and 
visible in the evening sky. Then they quickly move 
ahead westward toward the sun. pass it, and 
reappear in the morning sky. During this 
motion they are moving westward relative to the 
stars, as is shown by the plot of Mercury on page 
13. Describe the rest of the cyclic motion of 
Mercury and Venus. 

5.8 Center a protractor on point C in the top 
diagram on page 23 and measure the number of 
degrees in the four quadrants. Consider each 1° 
around C as one day. Make a table of the days 
needed for the planet to move through the four 
arcs as seen from the earth. 

5.9 (a) How many degrees of longitude does 

the sun move each hour? 
(b) Tell how you could roughly obtain the 
diameter of the earth from the following 
information: 

i. Washington. D.C. and San Francisco 
have about the same latitude. How can 
one easily test this? 

ii. A non-stop jet plane, going upwind at 
a ground speed of 500 mph from Washing- 
ton, D.C. to San Francisco, takes 5 hours 
to get there. 

iii. When it is just sunset in Washington, 
D.C, a man there turns on his TV set to 
watch a baseball game that is just be- 
ginning in San Francisco. The game goes 
into extra innings. After three hours the 
announcer notes that the last out occurred 
just as the sun set. 

5.10 In Ptolemy's theory of the planetary motions 
there were, as in all theories, a number of 
assumptions. Which of the following did Ptolemy 
assume? 

(a) the vault of stars is spherical in form 

(b) the earth has no motions 

(c) the earth is spherical 

(d) the earth is at the center of the sphere of 
stars 

(e) the size of the earth is extremely small 
compared to the distance to the stars 

(f ) uniform angular motion along circles (even 
if measured from an off-center point) is the 
only proper behavior for celestial objects 

5.11 As far as the Greeks were concerned, and 
indeed as far as we are concerned, a reasonable 
argument can be made for either the geocentric 
or the heliocentric theory of the universe. 

(a) In what ways were both ideas successful? 

(b) In terms of Greek science, what are some 
advantages and disadvantages of each 
system? 

(c) What were the major contributions of 
Ptolemy? 

5.12 Why was astronomy the first successful 
science, rather than, for example, meteorology or 
zoology? 



27 



6.1 The Copernican system 29 

6.2 New conclusions 33 

6.3 Arguments for the Copernican system 35 

6.4 Arguments against the Copernican system 39 

6.5 Historical consequences 44 

6.6 Tycho Brahe 45 

6.7 Tycho's observations 47 

6.8 Tycho's compromise system 49 













■^ .f.fM •fit -lUr 

r--^ Char -••""■ »;>y*''*^ 



rvT ^ r-v^* ^ vuT ».._ t** / "-*-».< 



a $Ur rrro -» _-, /P- ^~*' Kj^ &— ..j 






Copernicus' diagram of his heliocentric system (from his manuscript, 
of De Revolutionibus, 1543). This simplified representation omits the 
many small epicycles actually used in the system. 



CHAPTER SIX 



Does the Earth Move? - 
The Work of 
Copernicus and Tycho 



6.1 The Copernican system 

Nicolaus Copernicus (1473-1543) was a young student in 
Poland when America was discovered. An outstanding astronomer 
and mathematician, Copernicus was also a talented and respected 
churchman, jurist, administrator, diplomat, physician, and 
economist. During his studies in Italy he read the writings of Greek 
and other earlier philosophers and astronomers. As Canon of the 
Cathedral of Frauenberg he was busy with civic and church affairs 
and also worked on calendar reform. It is said that on the day of 
his death in 1543, he saw the first copy of his great book, on which 
he had worked most of his life and which opened a whole new 
vision of the universe. 

Copernicus titled his book De Revolutionibus Orbium 
Coelestium, or On the Revolutions of the Heavenly Spheres, which 
suggests the early Greek notion of concentric spheres. Copernicus 
was indeed concerned with the old problem of Plato: the construction 
of a planetary system by combinations of the fewest possible 
uniform circular motions. He began his study to rid the Ptolemaic 
system of the equants, which were contrary to Plato's assumptions. 
In his words, taken from a short summary written about 1530, 

. . . the planetary theories of Ptolemy and most other 
astronomers, although consistent with the numerical data, 
seemed likewise to present no small difficulty. For these 
theories were not adequate unless certain equants were 
also conceived; it then appeared that a planet moved with 
uniform velocity neither on its deferent nor about the 
center of its epicycle. Hence a system of this sort seemed 
neither sufficiently absolute nor sufficiently pleasing to 
the mind. 

Having become aware of these defects, I often 
considered whether there could perhaps be found a more 
reasonable arrangement of circles, from which every 
apparent inequality would be derived and in which every- 
thing would move uniformly about its proper center. 



SG 6.1 




Nicolas Copernicus (1473-1543). (in 
Polish his name was Koppernigk, but, 
in keeping with the scholarly tradition 
of the age, he gave it the Latin form 
Copernicus.) 



29 



1350 



CQ 



1400 



1450 



1500 



1550 



1600 



o 






T3 




<8 


c 


E 


etc 


^ 


> 


^ 






c 


c 






>> 


a 



s ^ 



o 



Bi Q 



RICHARD II 



LORENZO DE MEDICI 

ISABELLA OF CASTILE 

I 

FERDINAND of Aragon 

I 

RICHARD III HENRY VIII of England 

I 

MONTEZUMAof Mexico 



ELIZABETH I of England 

I 

IVAN THE TERRIBLE of Russia 



HENRY IV of France 



PRINCE HENRY the Navigator 



GUTENBERG 



CHRISTOPHER COLUMBUS 



TYCHO BRAHE 
GIORDANO BRUNO 

JOHANNES KEPLER 



JEAN FERNEL 



ANDREAS VESALIUS 
i I, 
AMBROISE PARE 

I 

VMLLIAM GILBERT 



THOMAS A KEMPIS 

I 
JOHN HUSS 



JOAN of ARC 



MARTIN LUTHER 
SAVONAROLA JOHN CAL\1\ 

ERASMUS 

I 
MACHIAVELLl 
I 
THOMAS MORE ^ 

I m 

IGNATIUS LOYOLA 



PETRARCH 



RABELAIS 



GEOFFREY CHAUCER 

I 

JEAN FROISSART 



FRANCOIS VILLON 



SHAKESPEARE 

I 

MONTAIGNE 

I 

CERVANTES 

EDMUND SPENSER 



DONATELLO 



SANDRO BOTTICELLI 

I 

LEONARDO DA \ 1N( I 

I 

MlCllKl. ANGELO 



FRA FILLIPO LIPPI 
I 
ROGIER VAN DER WEYDEN 



HAl'llAKL 



TINTORETTO 

I 
PIETER BRUEGHEL 
I 



EL GRECO 



• ] 



1 

WILLIAM BYRD 

I 

PALESTRINA 



GUILLAUME DUFAY 
JOHN DUNSTABLE 



JOSQUIN DEPHKS 
I 



ANDKKA GABRIELLI 




Section 6.1 31 

In De Revolutionibus he wrote: 

We must however confess that these movements [of 
the sun, moon, and planets] are circular or are composed 
of many circular movements, in that they maintain these 
irregularities in accordance with a constant law and with 
fixed periodic returns, and that could not take place, if 
they were not circular. For it is only the circle which can 
bring back what is past and over with. . . . 

I found first in Cicero that Nicetas thought that the 
Earth moved. And afterwards I found in Plutarch that 
there were some others of the same opinion. . . . Therefore 
I also . . . began to meditate upon the mobility of the 
Earth. And although the opinion seemed absurd, never- 
theless, because I knew that others before me had been 
granted the liberty of constructing whatever circles they 
pleased in order to demonstrate astral phenomena, I 
thought that I too would be readily permitted to test 
whether or not. by the laying down that the Earth had 
some movements, demonstrations less shaky than those 
of my predecessors could be found for the revolutions of 
the celestial spheres. ... I finally discovered by the help 
of long and numerous observations that if the movements 
of the other wandering stars are correlated with the 
circular movement of the Earth, and if the movements 
are computed in accordance with the revolution of each 
planet, not only do all their phenomena follow from that 
but also this correlation binds together so closely the 
order and magnitudes of all the planets and of their 
spheres or orbital circles and the heavens themselves 
that nothing can be shifted around in any part of them 
without disrupting the remaining parts and the universe 
as a whole. 

After nearly forty years of study, Copernicus had proposed a See the preface to Copernicus' 

system of more than thirty eccentrics and epicycles which would ^® Revolutionibus in Reader 2. 

"suffice to explain the entire structure of the universe and the entire 
ballet of the planets." Like Ptolemy's Almagest, De Revolutionibus 
uses long geometrical analyses and is difficult to read. Comparison 
of the two books strongly suggests that Copernicus thought he was 
producing an improved version of the Almagest. He used many of 
Ptolemy's observations plus some more recent ones. Yet his system 
differed from that of Ptolemy in several fundamental ways. Above 
all. he adopted a sun-centered system which in general outline was 
the same as that of Aristarchus (who was still discredited, and 
whom Copernicus did not think it wise to mention in print). 

Like all scientists. Copernicus made a number of assumptions 
in his system. In his own words (rendered in modern equivalent in 
several places), his assumptions were: 

1. There is no one precise, geometrical center of all 
the celestial circles or spheres. 

2. The center of the earth is not the center of the 



32 



Does the Earth Move?— The Work of Copernicus and Tycho 




universe, but only of gravitation and of the lunar sphere. 

3. All the spheres revolve about the sun . . . and 
therefore the sun has a central location in the universe. 

4. The distance from the earth to the sun is 
inperceptible in comparison with the distance to the 
stars. 

5. Whatever motion appears in the sky arises not 
from any motion of the sky, but from the earth's motion. 
The earth together with its water and air performs a 
complete rotation on its fixed poles in a daily motion, 
while the sky remains unchanged. 

6. What appear to us as motions of the sun arise not 
from its motion but from the motion of the earth and . . . 
we revolve about the sun like any other planet. The 
earth has, then, more than one motion. 

7. The apparent retrograde motion of the planets 
SG 6.2 arises not from their motion but from the earth's. The 

motions of the earth alone, therefore, are sufficient to 
explain so many apparent motions in the sky. 

Comparison of this list with the assumptions of Ptolemy, given 
in Chapter 5, will show some close similarities and important 
differences. 

Notice that Copernicus proposed that the earth rotates daily. 
As Aristarchus and others had realized, this rotation would account 
for all the daily risings and settings observed in the sky. Copernicus 
also proposed, as Aristarchus had done, that the sun was stationary 
and occupied the central position of the universe. The earth and 
each of the other planets moved about different central points near 
the sun. 

The figure at the left shows the main concentric spheres 
carrying the planets around the sun (sol). His text explains the 
basic features of his system: 

The ideas here stated are difficult, even almost 
impossible, to accept; they are quite contrary to popular 
notions. Yet with the help of God, we will make everything 
as clear as day in what follows, at least for those who 
are not ignorant of mathematics. . . . 

The first and highest of all the spheres is the sphere 
of the fixed stars. It encloses all the other spheres and is 
itself self-contained; it is immobile; it is certainly the 
portion of the universe with reference to which the 
movement and positions of all the other heavenly bodies 
must be considered. If some people are yet of the 
opinion that this sphere moves, we are of contrary mind; 
and after deducing the motion of the earth, we shall 
show why we so conclude. Saturn, first of the planets, 
which accomplishes its revolution in thirty years, is 
nearest to the first sphere. Jupiter, making its revolution 
in twelve years, is next. Then comes Mars, revolving 
once in two years. The fourth place in the series is 
occupied by the sphere which contains the earth 
and the sphere of the moon, and which performs an 



Section 6.2 33 

annual revolution. The fifth place is that of Venus, 
revolving in nine months. Finally, the sixth place is 
occupied by Mercury, revolving in eighty days. ... In the 
midst of all, the sun reposes, unmoving. 

Already we see an advantage in Copernicus' system that makes it SG 6.3 

"pleasing to the mind." The rates of rotation for the heavenly 
spheres increase in order from the motionless sphere of stars to 
speedy Mercury. 

Q1 What reasons did Copernicus give for rejecting the use of 
equants? 

Q2 In the following list of propositions, mark with a P those 
made by Ptolemy and with a C those made by Copernicus. 

(a) The earth is spherical. 

(b) The earth is only a point compared to the distance to the 
stars. 

(c) The heavens rotate daily around the earth. 

(d) The earth has one or more motions. 

(e) Heavenly motions are circular. 

(f ) The observed retrograde motion of the planets results from 
the earth's motion around the sun. 



6.2 New conclusions 

A new way of looking at old observations — a new theory — can 
suggest quite new kinds of observations to make, or new uses of old 
data. Copernicus used his moving-earth model to obtain two 
important results which were not possible with the Ptolemaic 
theory. He was able to calculate (a) the period of motion of each 
planet around the sun, and (b) the sizes of each planet's orbit 
compared to the size of the earth's orbit. This, for the first time, 
gave a scale for the dimensions of the universe. 

To calculate the periods of the planets around the sun, 
Copernicus used observations that had been recorded over many 
centuries. The method of calculation is similar to the "chase 

problem" of how often the hands on a clock pass one another. The SG 6.4, 6.5 

details of the calculation are shown on page 34. In Table 6.1 below, 
Copernicus' results are compared with the currently accepted 
values. 





Table 6.1 




PLANET 


COPERNICUS' VALUE 


MODERN VALUE 


Mercury 


0.241 y (88 d) 


87.97 d 


Venus 


0.614 y (224 d) 


224.70 d 


Mars 


1.88 y (687 d) 


686.98 d 


Jupiter 


11.8 y 


11.86 y 


Saturn 


29.5 y 


29.46 y 



Copernicus was also able, for the first time in history, to derive 
relative distances between the planets and the sun. Remember that 
the Ptolemaic system had no distance scale; it provided only a way 



The Periods of Revolution of the Planets 



The problem is to find the rate at which 
a planet moves around the sun by using 
observations made from the earth — which is 
itself moving around the sun. Say, for 
example, that a planet closer to the sun 
than the earth is, goes around the sun at the 
frequency (rate) of I7 cycles per year. The 
earth moves around the sun also, in the same 
direction, at the rate of 1 cycle per year. 
Because the earth follows along behind the 
planet, the planet's motion around the sun 
would, as seen from the earth, appear to be at a 
rate less than I7 cycles per year. In fact, as 
the diagrams below suggest, the planet's 




A planet that is inside the Earth's 
orbit and moves 1:7 revolutions around 
the sun in a year would, as seen from 
the earth, appear to have made only 
7 cycle. 




A planet that is outside the Earth's 
orbit and moves only 7 revolution 
around the sun in a year would, as 
seen from the earth, appear to make 
about I7 revolution. 



apparent rate of motion around the sun would 
be the difference between the planet's rate and 
the earth's rate around the sUn: I7 cycle per 
year - 1 cycle per year = 7 cycle per year. In 
general, if an inner planet moves around the 
sun at frequency f„ and the earth moves 
around the sun with frequency f^., then the 
planet's apparent rate of motion, fp^., as seen 
from the earth, will be fpe = fp — f^. 

A similar argument holds for planets 
farther from the sun than the earth is. (See 
diagram B.) Since these outer planets revolve 
about the sun more slowly than the earth, the 
earth repeatedly leaves the planets behind. Con 
sequently, for the outer planets the sign in 
the equation for f,„. is reversed: fp,. = f„ - f^,. 

The apparent frequency fp,. is what is 
actually observed and ^e is by definition 1 
cycle per year, so either equation is easily 
solved for the unknown actual rate f„ of the 
planet around the sun: 

For inner planets: fp = 1 cycle/yr -r f^,^ 

For outer planets: f^ = 1 cycle/yr - ^p,. 

Copernicus used some observations by 
Ptolemy and some of his own. A typical data 
statement in De Revolutionibus is "Jupiter 
is outrun by the earth 65 times in 71 solar 
years minus 5 days 45 minutes 27 sec- 
onds . . . ." In the table below, Copernicus' 
data have been rounded off to the nearest 
year (but they were very near to whole years 
to begin with). The cycle used for the inner 
planets is from one position of greatest 
eastern displacement from the sun to the 
next. The cycle used for the outer planets is 
from one opposition to the next. 



TABLE 6.2 





NUMBER 


APPARENT 


APPARENT 


FREQUENCY f. 


PERIOD 




OF YEARS 


NUMBER OF 


FREQUENCY 


AROUND SUN 


AROUND 




OF OB- 


CYCLES WITH 


f,.r IN 


IN CYCLES 


SUN 




SERVA- 


RESPECT TO 


CYCLES PER 


PER YEAR 


(1/^,) 




TION 


SUN 


YEAR 




IN YEARS 




(t) 


(n) 


(n/t) 






Mercury 


46 


145 


3.15 


4.15 


.241 


Venus 


8 


5 


.625 


1.625 


.614 


Mars 


79 


37 


.468 


.532 


1.88 


Jupiter 


71 


65 


.915 


.085 


11.8 


Saturn 


59 


57 


.966 


.034 


29.4 



Section 6.3 



35 



of deriving the directions to the planets or the angle through which 
they move. 

So in Ptolemy's system, only the relative sizes of epicycle and 
deferent circle were known, separately for each planet. In 
Copernicus' system, the motions of the sun and five planets that 
had previously been attributed to one-year epicycles or deferent 
circles were all replaced by the single motion of the earth's yearly 
revolution around the sun. The details of how this can be done are 
given on pages 36 and 37. Thus, it became possible to compare the 
radii of the planet's orbit with that of the earth. Because the 
distances were all compared to the distances between the sun and 
the earth, the sun-earth distance is conveniently called 1 
astronomical unit, abbreviated 1 AU. 

Table 6.3 below compares Copernicus" values of the orbit radii 
with the currently accepted values. 



MODERN VALUE 



0.39 AU 

0.72 

1.00 

1.52 

5.20 

9.54 









TABLE 6.3 


PLANET 


RADII 


OF 


PLANETARY ORBITS 




COPERNICUS' VALUES 


Mercury 






0.38 


Venus 






0.72 


Earth 






1.00 


Mars 






1.52 


Jupiter 






5.2 


Saturn 






9.2 




In the Ptolemaic system, only the 
relative size of epicycle and deferent 
was specified. Then size could be 
changed at will, so long as they kept 
the same proportions. 



Notice that Copernicus now had one system in which the size of 
each planet's orbit was related to the sizes of all the other planets' 
orbits. Contrast this to Ptolemy's solutions which were completely 
independent for each planet. No wonder that Copernicus said, in 
the quote on p. 31 that: "nothing can be shifted around in any part 
of them without disrupting the remaining parts and the universe 
as a whole." 

Q3 What new kinds of results did Copernicus obtain with a 
moving-earth model which were not possible with a geocentric 
model for the planetary system? 



SG 6.6 



6.3 Arguments for the Copernican system 



Since Copernicus knew that to many his work would seem 
absurd, "nay. almost contrary to ordinary human understanding," 
he tried in several ways to meet the old arguments against a 
moving earth. 

1. Copernicus argued that his assumptions agreed with 
theological dogma at least as well as Ptolemy's. Copernicus' book 
has many sections on the limitations of the Ptolemaic system (most 
of which had been known for centuries). Other sections pointed out 
the harmony of his own system and how well it reflects the thought 




Earth 



Changing Frame of Reference from 
the Earth to the Sun 

The change of viewpoint from 
Ptolemy's system to Copernicus' 
involved what today would be called 
a shift in frame of reference. The 
apparent motion previously attributed 
to the deferent circles and epicycles 
was attributed by Copernicus to the 
earth's orbit and the planet's orbits 
around the sun. 

For example, consider the motion 
of Venus. In Ptolemy's earth-centered 
system the center of Venus' epicycle 
was locked to the motion of the sun, 
as shown in the top diagram at the 
left. The size of Venus' deferent 
circle was thought to be smaller than 
the sun's, and the epicycle was 
thought to be entirely between the 
earth and sun. However, the observed 
motions to be explained by the 
system required only a certain relative 
size of epicycle and deferent. The 
deferent could be changed to any 
size, as long as the epicycle was 
changed proportionally. 

The first step toward a sun- 
centered system is taken by moving 
the center of Venus' 1-year deferent 
out to the sun and enlarging Venus' 
epicycle proportionally, as shown in 
the middle diagram at the left. Now 
the planet moves about the sun, while 
the sun moves about the earth. Tycho 
actually proposed such a system with 
all the observed planets moving about 
the moving sun. 

Copernicus went further and 
accounted for the relative motion of 
the earth and sun by considering the 
earth to be moving around the sun, 
instead of the sun moving about the 
earth. In the Copernican system, 
Venus' epicycle becomes its orbit 
around the sun and Venus' deferent 
is replaced by the earth's orbit around 
the sun, as shown in the bottom 



diagram at the left. All three systems, 
Ptolemy's, Tycho's and Copernicus', 
explain the same observations. 
For the outer planets the argument 
is similar, but the roles of epicycle 
and deferent circle are reversed. 

For the outer planets, it was the 
epicycles instead of the deferent 
circles which had a 1-year period 
and which were synchronized with 
the sun's orbit. The sizes of the 
deferents were chosen so that the 
epicycle of each planet would just 
miss the epicycles of the planets next 
nearest and next furthest from the 
sun. (This was a beautiful example 
of a simplifying assumption — it 
filled the space with no overlap and 
no gaps.) This system is represented 
in the top diagram at the right (in 
which the planets are shown in the 
unlikely condition of having their 
epicycle centers along a single 
line.) 

The first step in shifting to an 
earth-centered view was to adjust 
the sizes of the deferent circles, 
keeping the epicycles in proportion, 
until the 1-year epicycles were the 
same size as the sun's 1-year orbit. 
This adjustment is shown in the 
middle diagram at the right. Next, 
the sun's apparent yearly motion 
around the earth is accounted for 
just as well by having the earth 
revolve around the sun. Also, the 
same earth orbit would account for 
the retrograde loops associated with 
all the outer planets' matched 1-year 
epicycles. So all the synchronized 
epicycles of the outer planets and the 
sun's orbit are replaced by the single 
device of the earth's orbit around 
the sun. This shift is shown in the 
bottom diagram at the right. The 
deferent circles of the outer planets 
became their orbits around the sun. 




SAt-urn 



Satur-n 



38 Does the Earth Move?-The Work of Copernicus and Tycho 

of the Divine Architect. To Copernicus, as to many scientists, the 
complex events he saw were merely symbols of God's thinking. To 
find order and symmetry in the observed changes was to Copernicus 
an act of piety. To him the symmetry and order were another 
proof of the existence of the Deity. As an important church 
dignitary, he would have been appalled if he had been able to 
foresee that his theory would contribute to the conflict, in Galileo's 
time, between religious dogma and the interpretations that 
scientists gave to their experiments. 

2. Copernicus' analysis was as thorough as that of Ptolemy. The 
relative radii and speeds of the circular motions in his system were 
calculated so that tables of planetary motion could be made. 
Actually the theories of Ptolemy and Copernicus were about equally 
accurate in predicting planetary positions, which for both theories 
often differed from the observed positions by as much as 2° (about 
four diameters of the moon). 

3. Copernicus tried to answer several other objections that were 
certain to be raised, as they had been against Aristarchus' 
heliocentric system nearly nineteen centuries earlier. In reply to the 
argument that the rapidly rotating earth would surely fly apart, he 
asked, "Why does the defender of the geocentric theory not fear the 
same fate for his rotating celestial sphere — so much faster because 
so much larger?" It was argued that birds and clouds in the sky 
would be left behind by the earth's rotation and revolution. He 
answered this objection by indicating that the atmosphere is 
dragged along with the earth. To the lack of observable annual shift 
for the fixed stars, he could only give the same answer that 
Aristarchus had proposed, namely: 

. . . though the distance from the sun to the earth appears 
very large as compared with the size of the spheres of 
some planets, yet compared with the dimensions of the 
sphere of the fixed stars, it is as nothing. 

4. Copernicus claimed that the greatest advantage of his 
scheme was its simple description of the general motions of the 
planets. There certainly is a basic overall simplicity to his system, 
as is shown in his own diagram on p. 28. (Yet for precise computa- 
tions, because Copernicus would not use equants, he needed more 
small motions than did Ptolemy to account for the observations. A 
diagram from Copernicus' manuscript that shows more detail is 
reproduced on page 42.) 

5. Last of all, Copernicus pointed out that the simplicity of his 
system was not merely convenient, but also beautiful and "pleasing 
to the mind. " It is not often stressed in textbooks that this sort of 
esthetic pleasure which a scientist finds in his models" simplicity is 
one of the most powerful experiences in the actual practice of 
science. Far from being a "cold," merely logical exercise, scientific 
work is full of such recognitions of harmony and therefore of 
beauty. One beauty that Copernicus saw in his system was the 
central place given to the sun, the biggest, brightest object — the 
giver of light and warmth and life. As Copernicus himself put it: 



Section 6.4 39 

In the midst of all, the sun reposes, unmoving. Who, Look again at SG 6.2 

indeed, in this most beautiful temple would place the 
light-giver in any other part than whence it can illumine 
all other parts? So we find underlying this ordination an 
admirable symmetry in the Universe and a clear bond of 
the harmony in the motion and magnitude of the spheres, 
such as can be discovered in no other wise. 



Q4 Which of these arguments did Copernicus use in favor of 
his system? 

(a) it was obvious to ordinary common sense 

(b) it was consistent with Christian theology 

(c) it was much more accurate in predicting planet positions 

(d) its simplicity made it beautiful 

(e) the stars showed an annual shift in position due to the 
earth's motion around the sun 

Q5 What were the largest differences between observed 
planetary positions and those predicted by Ptolemy? by Copernicus? 

Q6 Did the Copernican system allow simple calculations of 
where the planets should be seen? 



6.4 Arguments against the Copernican system 

Copernicus' hopes for acceptance of his theory were not quickly 
fulfilled. More than a hundred years passed before the heliocentric 
system was generally accepted even by astronomers — and then, as 
we shall see, the acceptance came on the basis of arguments quite 
different from those of Copernicus. In the meantime the theory and 
its few champions met powerful opposition. Most of the arguments 
were the same as those used by Ptolemy against the heliocentric 
system of Aristarchus. 

1. Apart from its apparent simplicity, the Copernican system 
had no clear scientific advantages over the geocentric theory. There 
was no known observation that was explained by one system and 
not by the other. Copernicus had a different viewpoint but no new 
types of observations, no experimental data that could not be 
explained by the old theory. Furthermore, the accuracy of his 
predictions of planetary positions was little better than that by 
Ptolemy. As Francis Bacon wrote in the early seventeenth century: 
"Now it is easy to see that both they who think the earth revolves 
and they who hold the old construction are about equally and 
indifferently supported by the phenomena." 

Basically, the rival systems differed in their choice of reference 
systems used to describe the observed motions. Copernicus himself 
stated the problem clearly: 

Although there are so many authorities for saying 
that the Earth rests in the centre of the world that people 
think the contrary supposition . . . ridiculous; ... if, 
however, we consider the thing attentively, we will see 



40 



Does the Earth Move? — The Work of Copernicus and Tycho 



Ptolemy too had recognized the 
possibility of alternative frames of 
reference. (Reread the quotation on 
p. 22 in Ch. 5.) Most of Ptolemy's 
followers did not share this insight. 



that the question has not yet been decided and accordingly 
is by no means to be scorned. For every apparent change 
in place occurs on account of the movement either of the 
thing seen or of the spectator, or on account of the 
necessarily unequal movement of both. For no movement 
is perceptible relatively to things moved equally in the 
same directions — I mean relatively to the thing seen and 
the spectator. Now it is from the Earth that the celestial 
circuit is beheld and presented to our sight. Therefore, if 
some movement should belong to the Earth ... it will 
appear, in the parts of the universe which are outside, as 
the same movement but in the opposite direction, as 
though the things outside were passing over. And the 
daily revolution ... is such a movement. 



SG 6.8 



SG 6.7 



In this statement Copernicus invites the reader to shift the frame 
of reference from that of an observer on the earth to one at a 
remote position looking upon the whole system with the sun at the 
center. As you may know from personal experience, such a shift is 
not easy, and we can sympthize wdth those who preferred to hold to 
an earth-centered system for describing what they saw. 

Physicists now generally agree that all systems of reference are 
in principle equally valid for describing phenomena, although some 
will be easier and others more complex to use or think about. 
Copernicus and those who followed him felt that the heliocentric 
system was right in some absolute sense — that the sun was really 
fixed in space, and the same claim was made for the earth by his 
opponents. But the modern attitude is that the choice of a frame of 
reference depends mainly on which frame will allow the simplest 
discussion of the problem being studied. We should not speak of 
reference systems as being right or wrong, but rather as being 
convenient or inconvenient. (To this day, navigators use a geocentric 
model for their calculations; see the page of a navigation book 
shown on p. 41.) Yet even if it is recognized that different frames 
of reference are possible mathematically, a reference system that 
is acceptable to one person may involve philosophical assumptions 
that are unacceptable to another. 

2. The lack of an observable annual shift for the fixed stars 
spoke against Copernicus' model. His only possible reply was 
unacceptable because it meant that the stars were at an enormous 
distance away from the earth. The naked-eye instruments allowed 
positions in the sky to be measured to a precision of about 1/10°; 
for an annual shift to be less than 1/10°, the stars would have to be 
more than 1000 times further from the sun than the earth is! To us 
this is no shock, because we have been raised in a society that 
accepts the idea of enormous extensions in space and in time. 
Even so, such distances do strain our imagination. To the opponents 
of Copernicus such distances were absurd. Indeed we may well 
speculate that even if an annual shift in star position had then been 
observable, it would not have been taken as unmistakable evidence 
against one and for the other theory. One can usually modify a 



Section 6.4 



41 



theory -more or less pleasingly — to accomodate a bothersome 
finding. 

The Copernican system led to other conclusions that were also 
puzzling and threatening. Copernicus determined the actual 
distances between the sun and the planetary orbits. Perhaps, then, 
the Copernican system was not just a mathematical procedure for 
predicting the observable positions of the planets! Perhaps 
Copernicus was describing a real system of planetary orbits in 
space (as he thought he was). But this would be difficult to accept, 
for the orbits were far apart. Even the small epicycles which 
Copernicus still needed to account for the variations in the motions 
did not fill up the spaces between the planets. Then what did fill up 
these spaces? Because Aristotle had stated that "nature abhors a 
vacuum," it was agreed there had to be something in all that space. 
As you might expect, even many of those who believed Copernicus' 
system felt that space should be full of something and invented 
various sorts of invisible fluids and ethers to fill up the emptiness. 
More recently, similar fluids were used in theories of chemistry and 
of heat, light, and electricity, as you will see in later units. 

3. Since no definite decision between the Ptolemaic and the 
Copernican theories could be made on the astronomical evidence, 
attention focused on the argument concerning the central, 
immovable position of the earth. Despite his efforts, Copernicus was 
unable to persuade most of his readers that his heliocentric system 
reflected the mind of God at least as closely as did the geocentric 
system. All religious faiths in Europe, including the new Protestants, 
found enough biblical quotations (for example, Joshua 10:12-13) to 
assert that the Divine Architect must have worked from a 
Ptolemaic blueprint. Indeed, Martin Luther called Copernicus "the 
fool who would overturn the whole science of astronomy." 

Eventually, in 1616, when storm clouds were raised by the case 
of Galileo, the Papacy put De Revolutionibus on the Index of 
forbidden books as "false and altogether opposed to Holy 
Scriptures." Some Jewish communities also prohibited the teaching 
of his theory. It seems that man, believing himself central to God's 
plan, had to insist that his earth be the center of the physical 
universe. 

The assumption that the earth was not the center of the 
universe was offensive enough. But worse, the Copernican system 
suggested that the other planets were similar to the earth. Thus, the 
concept of the distinctly different heavenly matter was threatened; 
who knew but what some rash person might next suggest that the 
sun and possibly even the stars were made of earthly materials? If 
the other celestial bodies, either in our solar system or beyond, were 
similar to the earth, they might even be inhabited. And the 
inhabitants might be heathens, or beings as well-beloved by God as 
man is, or possibly even more beloved! Thus, the whole Copernican 
scheme led to profound philosophical questions which the Ptolemaic 
scheme avoided. 

4. The Copernican theory conflicted with the basic propositions 



CELESTIAL 
OBSERVATIONS 

15)- Philip Kissam, C. E. 

Professor of Civil Engineering, 
Princeton University 

I. The Principles upon >vhich 
Celestial Observations are 
Based. 

A. CONCEPTS. 

1. The Celestial Sphere. To simplify the 
computations necessary for the determinations 
of the direction of the meridian, of latitude, and 
of longitude or time, certain concepts of the 
heavens have been generally adopted. They are 
the following: 

a. The earth is stationary. 

b. The heavenly bodies have been projected 
outward, along lines which extend from 
the center of the earth, to a sphere of 
infinite radius called the celestial sphere. 

The celestial sphere has the following char- 
acteristics: 

a. Its center is at the center of the earth. 

b. Its equator is on the projection of the 
earth's equator. 

c. With respect to the earth, the celestial 
sphere rotates from east to west about 
a line which coincides with the earth's 
axis. Accordingly, the poles of the celes- 
tial sphere are at the prolongations of the 
earth's poles. 

d. The speed of rotation of the celestial 
sphere is 360° 59.15' per 24 hours. 

e. With the important exception of bodie? 
in the solar system, which change position 
slowly, all heavenly bodies remain prac- 
tically fixed in their positions on the 
celestial sphere, never changing more than 
negligible amounts in 24 hours, and ac- 
cordingly are often called fixed stars. 



Celestial navigation involves compar- 
ing the apparent position of the sun 
(or star) with the "actual" position as 
given in a table called an "ephe- 
meris." Above is an excerpt from the 
introduction to the tables in the Solar 
Ephemeris for 1950. (Keuffel and Es- 
ser Co.) 



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Section 6.4 43 

of Aristotelian physics. This conflict is well described by H. 
Butterfield in Origins of Modern Science: 

... at least some of the economy of the Copernican 
system is rather an optical illusion of more recent 
centuries. We nowadays may say that it requires smaller 
effort to move the earth round upon its axis than to swing 
the whole universe in a twenty- four hour revolution 
about the earth; but in the Aristotelian physics it required 
something colossal to shift the heavy and sluggish earth, 
while all the skies were made of a subtle substance that 
was supposed to have no weight, and they were 
comparatively easy to turn, since turning was concordant 
with their nature. Above all, if you grant Copernicus a 
certain advantage in respect of geometrical simplicity, 
the sacrifice that had to be made for the sake of this was 
tremendous. You lost the whole cosmology associated 
with Aristotelianism — the whole intricately dovetailed 
system in which the nobility of the various elements and 
the hierarchical arrangement of these had been so 
beautifully interlocked. In fact, you had to throw 
overboard the very framework of existing science, and it 
was here that Copernicus clearly failed to discover a 
satisfactory alternative. He provided a neater geometry 
of the heavens, but it was one which made nonsense of 
the reasons and explanations that had previously been 
given to account for the movements in the sky. 

In short, although the sun-centered Copernican scheme was 
scientifically equivalent to the Ptolemaic in explaining the 
astronomical observations, to abandon the geocentric hypothesis 
seemed philosophically false and absurd, dangerous, and fantastic. 
Most learned Europeans at that time recognized the Bible and the 
writings of Aristotle as their two supreme sources of authority. Both 
appeared to be challenged by the Copernican system. Although the 
freedom of thought that marked the Renaissance was beginning, 
the old image of the universe provided security and stability to 
many. So, to believe in a sun-centered rather than an earth- 
centered universe in Copernicus' time required that a partial gain 
in simplicity be considered more important than common sense and 
observation, the teaching of philosophy and religion, and physical 
science. No wonder Copernicus had so few believers! 

Similar conflicts between the assumptions underlying accepted 
beliefs and the philosophical content of new scientific theories have 
occurred many times, and are bound to occur again. During the last 
century there were at least two such conflicts. Neither is completely 
resolved today. In biology, the evolutionary theory based on 
Darwin's work has caused major philosophical and religious 
reactions. In physics, as Units 4, 5. and 6 indicate, evolving theories 
of atoms, relativity, and quantum mechanics have challenged other 
long-held assumptions about the nature of the world and our 
knowledge of reality. As the dispute between Copernicans and 
Ptolemaists illustrates, the assumptions which common sense holds. 

Opposite: A page from Copernicus' manuscript of De Revolutionibus, 
showing detail of some epicycles in his model. 



44 Does the Earth Move?-The Work of Copernicus and Tycho 

SG 6 9 ^° dearly and defends so fiercely are often only the remnants of an 
earlier, less complete scientific theory. 

Q7 Why were many people, such as Francis Bacon, undecided 
about the correctness of the Ptolemaic and Copemican systems? 

Q8 How did the astronomical argument become involved with 
religious beliefs? 

Q9 From a modem viewpoint, was the Ptolemaic or the 
Copemican system of reference more valid? 



6.5 Historical consequences 

Eventually the moving-earth model of Copemicus was accepted. 
The slowness with which that acceptance came is illustrated by a 
passage in the published diary of John Adams (who later became 
the second president of the United States). He wrote that he 
attended a lecture at Harvard College in which the correctness of 
the Copemican viewpoint was argued -on June 19, 1753. 

Soon we shall follow the work which gradually led to the 
general acceptance of the heliocentric viewpoint -a heliocentric 
viewpoint without, however, the detailed Copemican system of 
uniform circular motions with eccentrics and epicycles. We shall 
see that the real scientific significance of Copemicus' work lies in 
the fact that a heliocentric view opened a new approach to 
understanding planetary motion. This new way became dynamic, 
rather than just kinematic -it involved the laws relating force and 
motion, developed in the 150 years after Copernicus, and the 
application of these laws to motions in the heavens. 

The Copemican model with moving earth and fixed sun opened 
a floodgate of new possibilities for analysis and description. 
According to this model the planets could be considered as real 
bodies moving along actual orbits. Now Kepler and others could 
consider these planetary paths in quite new ways. In science, the 
sweep of possibilities usually cannot be foreseen by those who begin 
the revolution — or by their critics. 

Today the memory of Copernicus is honored not so much 
because of the details of his theory, but because he challenged the 
prevailing world-picture, and because his theory became a principal 
force in the intellectual revolution which shook man out of his 
self-centered view of the universe. As men gradually accepted the 
Copemican system, they necessarily found themselves accepting 
the view that the earth was only one among several planets circling 
the sun. Thus, it became increasingly difficult to assume that all 
creation centered on mankind. At the same time, the new system 
SG 6.10 stimulated a new self-reliance and curiosity about the world. 

Acceptance of a revolutionary idea based on quite new 
assumptions, such as Copemicus' shift of the frame of reference, 
is always slow. Sometimes compromise theories are proposed as 
attempts to unite two conflicting alternatives, to "split the 
difl'erence." As you will see in later units, such compromises are 



Section 6.6 



45 



rarely successful. Often the new ideas do stimulate new observations 
and concepts that in turn may lead to a very useful development or 
restatement of the original revolutionary theory. 

Such a restatement of the heliocentric theory came during the 
150 years after Copernicus. Many men provided observations and 
ideas, and in Chapters 7 and 8 we will follow the major contribu- 
tions made by Kepler, Galileo, and Isaac Newton. But first, we will 
consider here the work of Tycho Brahe, who devoted his life to 
improvements in the pi'ecision with which planetary positions were 
observed and to the proposal of a compromise theory of planetary 
motion. 

Q10 In terms of our historical perspective, what were the 
greatest contributions of Copernicus to modem planetary theory? 



6.6 Tycho Brahe 

Tycho Brahe was born in 1546 of a noble, but not particularly 
rich. Danish family. By the time Tycho was thirteen or fourteen, he 
had become intensely interested in astronomy. Although he was 
studying law, he secretly spent his allowance money on astronomical 
tables and books such as the Almagest and De Revolutionibus. 
Soon he discovered that both Ptolemy and Copernicus had relied 
upon tables of planetary positions that were inaccurate. He 
concluded that before a satisfactory theory of planetary motion 
could be created, new astronomical observations of the highest 
possible accuracy, gathered over many years, would be necessary. 

Tycho's interest in studying the heavens was increased by an 
exciting observation in 1572. Although the ancients had taught that 
the stars were unchanging. Tycho observed a "new star" in the 
constellation Cassiopeia. It soon became as bright as Venus and 
could be seen even during the daytime. Then over several years it 
faded until it was no longer visible. To Tycho these events were 
astonishing — changes in the starry sky! Evidently at least one 
assumption of the ancients was wrong. Perhaps other assumptions 
were wrong, too. 

After observing and writing about the new star, Tycho traveled 
through northern Europe where he met many other astronomers 
and collected books. Apparently he was considering moving to 
Germany or Switzerland where he could easily meet other 
astronomers. To keep the young scientist in Denmark, King 
Frederick II made Tycho an offer that was too attractive to turn 
down. Tycho was given an entire small island and also the income 
derived from various farms to allow him to build an observatory on 
the island and to staff and maintain it. The offer was accepted, and 
in a few years Uraniborg ("Castle of the Heavens") was built. It 
was a large structure, having four large observatories, a library, a 
laboratory, shops, and living quarters for staff, students, and 
observers. There was even a complete printing plant. Tycho 
estimated that the observatory cost Frederick II more than a ton of 
gold. For that time in history this magnificent laboratory was at 



Although there were precision sight- 
ing instruments, all observations 
were with the naked eye — the 
telescope was not to be invented 
for another 50 years. 





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At the top left is a plan of the observatory and gardens 
built for Tycho Brahe at Uraniborg, Denmark. 

The cross section of the observatory, above center, 
shows where most of the important instruments including 
large models of the celestial spheres were housed. 

The picture at the left shows the room containing 
Tycho's great quadrant. On the walls are pictures of some 
of his instruments. He is making an observation, aided 
by assistants. 

Above is a portrait of Tycho. painted about 1597. 



Section 6.7 



47 




least as significant, complex, and expensive as some of the great 
research establishments of our own time. Primarily a research 
center, Uraniborg was a place where scientists, technicians, and 
students from many lands could gather to study astronomy. Here 
was a unity of action, a group effort under the leadership of an 
imaginative scientist to advance the boundaries of knowledge in 
one science. 

In 1577 Tycho observed a bright comet, a fuzzy object whose 
motion across the sky seemed to be erratic, unlike the orderly 
motions of the planets. To find the distance to the comet, Tycho 
compared its position as observed from Denmark with its positions 
as observed from elsewhere in Europe. At any given time, the comet 
was found to have the same position with respect to the stars, even 
though the observing places were many hundreds of miles apart. 
By contrast, the moon's position in the sky was measurably 
different when observed from places so far apart. Therefore, Tycho 
concluded, the comet must be at least several times farther away 
than the moon. 

This was an important conclusion. Up to that time comets had 
been believed to be some sort of local event, like clouds or hghtning, 
rather than something in the realm of eternal things beyond the 
moon. Now comets had to be considered distant astronomical 
objects which seemed to move right through the crystalline spheres 
that were still generally beheved to carry the planets. Tycho's book 
on this comet was widely read and helped to undermine belief in 
the old assumptions about the nature of the heavens. 



Q11 What event stimulated Tycho's interest in astronomy? 

Q1 2 In what ways was Tycho's observatory like a modern 
research institute? 

Q13 Why were Tycho's conclusions about the comet of 1577 
important? 



The bright comet of 1965. 



Two articles on comets appear in 
Reader 2: "The Great Comet of 
1965 ' and "The Boy Who Redeemed 
His Father's Name." 



SG6.11 



6.7 Tycho's observations 

Tycho's fame results from his lifelong devotion to making 
unusually accurate observations of the positions of the stars, sun, 
moon, and planets. He did this before the telescope was invented. 
Over the centuries many talented observers had been recording the 
positions of the celestial objects, but the accuracy of Tycho's work 
was much greater than that of the best astronomers before him. 
How was Tvcho Brahe able to do what no others had done before? 



For a more modern example of this 
same problem of instrumentation, 
you may wish to read about the 
development and construction of the 
200-inch Hale telescope on Mt. 
Palomar. Also see "A Night at the 
Observatory" in Reader 2. 



48 



Does the Earth Move?-The Work of Copernicus and Tycho 




One of Tycho's sighting devices. 
Unfortunately Tycho's instruments 
were destroyed in 1619 during the 
Thirty Years War. 



Singleness of purpose was certainly one of Tycho's assets. He 
knew that highly precise observations must be made during many 
years. For this he needed improved instruments that would give 
consistent readings. Fortunately he possessed both the mechanical 
ingenuity to devise such instruments and the funds to pay for their 
construction and use. 

Tycho's first improvement on the astronomical instruments of 
the day was to make them larger. Most of the earlier instruments 
had been rather small, of a size that could be moved by one person. 
In comparison, Tycho's instruments were gigantic. For instance, 
one of his early devices for measuring the angular altitude of 
planets (shown in the etching in the margin) had a radius of about 
six feet. This wooden instrument was so large that it took several 
men to set it into position. Tycho also put his instruments on heavy, 
firm foundations or else attached them to a wall that ran exactly 
north-south. By increasing the stability of the instruments. Tycho 
increased the reliability of the readings over long periods of time. 
Thoughout his career Tycho also created better sighting devices, 
more precise scales, and stronger support systems and made dozens 
of other changes in design which increased the precision of the 
observations. 

Apparent distortion of the setting sun. The light's path through the 
earth's atmosphere is bent, making the sun appear flattened and rough- 
edged. 




Section 6.8 



49 



Not only did Tycho devise better instruments for making his 
observations, but he also determined and specified the actual limits 
of precision of each instrument. He realized that merely making 
larger and larger instruments does not always result in greater 
precision; ultimately, the very size of the instrument introduces 
errors, since the parts will bend under their own weight. Tycho 
therefore tried to make his instruments as large and strong as he 
could without at the same time introducing errors due to bending. 
Furthermore, in modern style, Tycho calibrated each instrument 
and determined its range of error. (Nowadays many commercially 
available scientific instruments designed for precision work are 
accompanied by a measurement report, usually in the form of a 
table, of corrections to be applied to the readings.) 

Like Ptolemy and the Muslim astronomical observers, Tycho 
knew that the light coming from each celestial body was bent 
downward by the earth's atmosphere and increasingly so as the 
object neared the horizon. To increase the precision of his 
observations, Tycho carefully determined the amount of refraction 
so that each observation could be corrected for refraction effects. 
Such careful work was essential if improved records were to be 
made. 

Tycho worked at Uraniborg from 1576 to 1597. After the death 
of King Frederick II, the Danish government became less interested 
in helping to pay the cost of Tycho's observatory. Yet Tycho was 
unwilling to consider any reduction in the costs of his activities. 
Because he was promised support by Emperor Rudolph of Bohemia, 
Tycho moved his records and several instruments to Prague. There, 
fortunately, he hired as an assistant an able, imaginative young 
man named Johannes Kepler. When Tycho died in 1601, Kepler 
obtained all his records of observations of the motion of Mars. As 
Chapter 7 reports, Kepler's analysis of Tycho's observations solved 
many of the ancient problems of planetary motion. 






Q14 What improvements did Tycho make in astronomical 
instruments? 

Q15 In what way did Tycho correct his observations to provide 
records of higher accuracy? 



6.8 Tycho's compromise system 

Tycho's observations were intended to provide a basis for a new 
theory of planetary motion which he had outlined in an early 
publication. Tycho saw the simplicity of the Copemican system by 
which the planets moved around the sun, but he could not accept 
the idea that the earth had any motion. In Tycho's system, all the 
planets except the earth moved around the sun, but the sun moved 
around the stationary earth, as shown in the sketch in the margin. 
Thus he devised a compromise model which, as he said, included 
the best features of both the Ptolemaic and the Copemican systems. 




Refraction, or bending, of light from a 
star by the earth's atmosphere. The 
amount of refraction shown in the 
figure is greatly exaggerated over 
what actually occurs. 




Main spheres in Tycho Brahe's system 
of the universe. The earth was fixed 
and was at the center of the universe. 
The planets revolved around the sun, 
while the sun, in turn, revolved around 
the fixed earth. 




Buildings and instruments of modern observatories. Top: the 200-inch telescope 
and its dome on Mt. Palomar. Bottom: the complex of buildings on Mt. Wilson. 



Section 6.8 51 

However, he did not live to publish quantitative details of his theory. 

The compromise Tychonic system was accepted by only a few 
people. Those who accepted the Ptolemaic model objected to 
Tycho's proposal because he had the planets move around the sun. 
Those who accepted the Copernican model objected to having the 
earth held stationary. So the argument continued between those 
holding the seemingly self-evident position that the earth was 
stationary and those who accepted, at least tentatively, the strange, 
exciting proposals of Copernicus that the earth might rotate and 
revolve around the sun. The choice for one or the other was based 
on philosophical or esthetic preferences, for each of the three sys- 
tems could account about equally well for the observational evidence. 

All planetary theories up to that time had been developed only 
to provide some system by which the positions of the planets could 
be predicted fairly precisely. In the terms used in Unit 1, these 
would be called kinematic descriptions. The causes of the motions — 
what we now call dynamics — had not been considered in any detail. 
The angular motions of objects in the heavens were, as Aristotle 
said and everyone (including Ptolemy, Copernicus, and Tycho) 
agreed, "natural"; the heavens were still considered to be completely 
different from earthly materials and to behave in quite different 
ways. That a single theory of dynamics could describe both earthly 
and heavenly motions was a revolutionary idea yet to be proposed. 

As long as there was no explanation of the causes of motion, 
there remained a question of whether the orbits proposed for the 
planets in the various systems were actual paths of real objects in 
space or only convenient imaginary devices for making computa- 
tions. The status of the problem in the early part of the seventeenth 
century was later described well by the English poet John Milton 
in Paradise Lost: 

. . . He his fabric of the Heavens 
Hath left to their disputes, perhaps to move 
His laughter at their quaint opinions wide 
Hereafter, when they come to model Heaven 
And calculate the stars, how they will wield 
The mighty frame, how build, unbuild, contrive 
To save appearances, how gird the sphere 
With centric and eccentric scribbled o'er 
Cycle and epicycle, orb in orb. 

You will see that the eventual success of Newton's universal 

dynamics led to the belief, one which was held confidently for about 

two centuries, that scientists were describing the "real world." 

Later chapters of this text that deal with recent discoveries and 

theories will indicate that today scientists and philosophers are SG 6.12 

much less certain that the common-sense notion of "reality" is so 

useful in science. 

Q16 In what ways did Tycho's system for planetary motions 
resemble the Ptolemaic and the Copernican systems? 



STUDY GUI 



6.1 The Project Physics learning materials 
particularly appropriate for Chapter 6 include the 
following : 

Experiments 

The Shape of the Earth's Orbit 
Using Lenses to Make a Telescope 

Activities 

Two Activities on Frames of Reference 

Reader Articles 

The Boy Who Redeemed His Father's Name 
The Great Comet of 1965 
A Night at the Observatory 



Film Loop 

Retrograde Motion - 



Heliocentric Model 



6.2 The first diagram on the next page shows 
numbered positions of the sun and Mars (on its 
epicycle) at equal time intervals in their motion 
around the earth, as described in the Ptolemaic 
system. You can easily redraw the relative 
positions to change from the earth's frame of 
reference to the sun's. Mark a sun-sized circle in 
the middle of a thin piece of paper; this will be a 
frame of reference centered on the sun. Place the 
circle over each successive position of the sun, 
and trace the corresponding numbered position of 
Mars and the position of the earth. (Be sure to 
keep the piece of paper straight.) When you have 
done this for all 15 positions, you will have a 
diagram of the motions of Mars and the earth as 
seen in the sun's frame of reference. 

6.3 What reasons did Copernicus give for 
believing that the sun is fixed at or near the 
center of the planetary system? 

6.4 Consider the short and long hands of a clock 
or watch. If, starting from 12:00 o'clock, you rode 
on the slow short hand, how many times in 12 
hours would the long hand pass you? If you are 
not certain, slowly turn the hands of a clock or 
watch, and keep count. From this information, 
can you derive a relation by which you could 
conclude that the period of the long hand around 
the center was one hour? 

6.5 The diagram at the upper right section of the 



next page shows the motions of Mercury and 
Venus east and west of the sun as seen from the 
earth during 1966-1967. The time scale is 
indicated at 10-day intervals along the central 
line of the sun's position. 

(a) Can you explain why Mercury and Venus 
appear to move from farthest east to farthest 
west more quickly than from farthest west 
to farthest east? 

(b) From this diagram can you find a period for 
Mercury's apparent position in the sky 
relative to the sun? 

(c) Can you derive a period for Mercury's actual 
orbital motion around the sun? 

(d) What are the major sources of uncertainty 
in the results you derived? 

(e) Similarly, can you estimate the orbital 
period of Venus? 

6.6 From the sequence of orbital radii from 
Mercury to Saturn, guess what the orbital radius 
would be for a new planet if one were discovered. 
What is the basis for your guess? 

6.7 If you have had some trigonometry, try this 
problem: the largest observed annual shift in star 
position is about 1/2400 of a degree. What is 

the distance (in AU's) to this closest star? 

6.8 How might a Ptolemaic astronomer have 
modified the geocentric system to account for 
observed stellar parallax? 

6.9 Do you know of any conflicts between 
scientific theories and common sense today? 

6.10 How did the Copernican system encourage 
the suspicion that there might be life on objects 
other than the earth? Is such a possibility 
seriously considered today? What important kinds 
of questions would such a possibility raise? 

6.11 How can you explain the observed motion of 
Halley's comet during 1909-1910, as shown on the 
star chart on the next page? 

6.12 To what extent do you feel that the 
Copernican system, with its many motions in 
eccentrics and epicycles, reveals real paths in 
space, rather than being only another way of 
computing planetary positions? 



52 



East 
60° 



W 20' 



H 1 ^ -i ^ 1 h 



Wast 
60° 




aing sky 



Positions of Veaus and Mercury 

Relative to Sun, 1966-67 

(5-day intervals) 



Apparent motion of Mars and Sun 
around the earth. 




Sun's position at 
10-day intervals 



Positions of Venus and Mercury 
relative to Sun. 



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Observed motion of Halley's comet 
during 1909-1910. 



53 



7.1 The abandonment of uniform circular motion 55 

7.2 Kepler's law of areas 57 

7.3 Kepler's law of elliptical orbits 59 

7.4 Kepler's law of periods 66 

7.5 The new concept of physical law 68 

7.6 Galileo and Kepler 69 

7.7 The telescopic evidence 70 

7.8 Galileo focuses the controversy 73 

7.9 Science and freedom 75 



Galileo's notes on his observations 
beginning "on day 7 of January 1610," 



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CHAPTER SEVEN 



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A New Universe Appears — 
The Work of Kepler and Galileo 



7.1 The abandonment of uniform circular motion 

Kepler's lifelong desire was to perfect the heliocentric theory. 
He viewed the harmony and simplicity of that theory with 
"incredible and ravishing delight." To Kepler, such patterns of 
geometric order and numerical relation were clues to Gods mind. 
To unfold these patterns further through the heliocentric theory. 
Kepler attempted in his first major work to explain the spacing of 
the planetary orbits, as calculated by Copernicus (p. 35 in 
Chapter 6). 

Kepler was searching for the reasons just six planets (including 
the earth) were visible and were spaced as they are. These are 
excellent scientific questions, though even today too difficult to 
answer. Kepler felt the key lay in geometry, and he began to wonder 
whether there was any relation between the six known planets and 
the five "regular solids." A regular solid is a polyhedron whose faces 
all have equal sides and angles. From the time of the Greeks, it was 
known that there were just five regular geometrical solids. Kepler 
imagined a model in which these five regular solids could be nested, 
one inside the other, somewhat like a set of mixing bowls. Between 
the five solids would be spaces for four planetary spheres. A fifth 
sphere could be inside the whole nest and a sixth sphere could be 
around the outside. Kepler then sought some sequence of the five 
solids; just touching the spheres, that would space the spheres at 
the same relative distances from the center as were the planetary 
orbits. Kepler said: 

I took the dimensions of the planetary orbits according 
to the astronomy of Copernicus, who makes the sun 
immobile in the center, and the earth movable both 
around the sun and upon its own axis; and I showed that 
the differences of their orbits corresponded to the five 
regular Pythagorean figures. . . . 




SG 7.1 




The five "perfect solids" taken from 
Kepler's Harmonices Mundi (Har- 
mony of the World). The cube is a 
regular solid with six square faces. 
The dodecahedron has twelve five- 
sided faces. The other three regular 
solids have faces which are equilateral 
triangles: the tetrahedron has four 
triangular faces, the octahedron has 
eight triangular faces, and the ico- 
sahedron has twenty triangular faces. 



For Kepler, this geometric view was 
related to ideas of harmony. (See 
"Kepler's Celestial Music" in 
Reader 2. ) 



55 



56 



A New Universe Appears-The Work of Kepler and Galileo 




A model of Kepler's explanation of the 
spacing of the planetary orbits by 
means of the regular geometrical 
solids. Notice that the planetary 
spheres were thick enough to include 
the small epicycle used by Copernicus. 



In keeping with Aristotelian physics, 
Kepler believed that force was 
necessary to drive the planets along 
their circles, not to hold them in 
circles. 



SG 7.2 



By trial and error Kepler found a way to arrange the solids so 
that the spheres fit within about five percent of the actual 
planetary distances. To Kepler this remarkable (but as we now 
know, entirely accidental) arrangement explained both the spacings 
of the planets and the fact that there were just six. Also it indicated 
the unity he expected between geometry and scientific observations. 
Kepler's results, published in 1597, demonstrated his imagination 
and computational ability. Furthermore, it brought him to the 
attention of major scientists such as Galileo and Tycho. As a result, 
Kepler was invited to become one of Tycho's assistants at his new 
observatory in Prague in 1600. 

There Kepler was given the task of determining in precise detail 
the orbit of Mars. This unusually difficult problem had not been 
solved by Tycho and his other assistants. As it turned out, the in- 
vestigation of the motion of Mars was the start from which Kepler 
could redirect the study of celestial motion, just as Galileo used the 
motion of falling bodies to redirect the study of terrestrial motion. 

Kepler began his study of Mars by trying to fit the observations 
with motions on an eccentric circle and an equant. Like Copernicus. 
Kepler eliminated the need for the large epicycle by putting the 
sun motionless at the center and having the earth move around it 
(see p. 31). But Kepler made an assumption which differed from that 
of Copernicus. Recall that Copernicus had rejected the equant as an 
improper type of motion, but he used small epicycles. Kepler 
used an equant, but refused to use even a single small epicycle. To 
Kepler the epicycle seemed "unphysical" because the center of the 
epicycle was empty, and empty space could not exert any force on a 
planet. Thus, from the start of his study on Mars, Kepler was 
assuming that the orbits were real and that the motion had some 
physical causes. Even though Kepler's teacher advised him to make 
only "astronomical" (observational) and not physical assumptions, 
Kepler stubbornly stuck to his idea that the motions must be 
produced and explained by forces. When finally he published his 
results on Mars in his book Astronomia Nova, the New Astronomy, 
it was subtitled Celestial Physics. 

For a year and a half Kepler struggled to fit Tycho's observations 
of Mars by various arrangements of an eccentric and an equant. 
When after 70 trials success finally seemed near, he made a 
discouraging discovery. Although he could represent fairly well the 
motion of Mars in longitude (east and west along the ecliptic), he 
failed markedly with the latitude (north and south of the ecliptic). 
However, even in longitude his very best fit still had differences of 
eight minutes of arc between Tycho's observed positions and the 
positions predicted by the model. 

Eight minutes of arc, about a fourth of the moon's diameter, 
may not seem like much of a difference. Others might have been 
tempted to explain it away, perhaps charging it to observational 
error. But Kepler knew from his own studies that Tycho's 
instruinents and observations were rarely in error by even as much 
as two minutes of arc. Those eight minutes of arc meant to Kepler 



Section 7.2 



57 



that his best system, using the old, accepted devices of eccentric 
and equant, would never be adequate to match the observations. 
In his New Astronomy, Kepler wrote: 



Since divine kindness granted us Tycho Brahe, the 
most diligent observer, by whose observations an error of 
eight minutes in the case of Mars is brought to light in 
this Ptolemaic calculation, it is fitting that we recognize 
and honor this favor of God with gratitude of mind. Let 
us certainly work it out, so that we finally show the true 
form of the celestial motions (by supporting ourselves 
with these proofs of the fallacy of the suppositions 
assumed). I myself shall prepare this way for others in 
the following chapters according to my small abilities. 
For if I thought that the eight minutes of longitude were 
to be ignored, I would already have corrected the 
hypothesis which he had made earlier in the book and 
which worked moderately well. But as it is, because they 
could not be ignored, these eight minutes alone have 
prepared the way for reshaping the whole of astronomy, 
and they are the material which is made into a great part 
of this work. 



Fortunately Kepler had made a 
major discovery earlier which was 
crucial to his later work. He found 
that the orbits of the earth and other 
planets were in planes which 
passed through the sun. Ptolemy 
and Copernicus required special 
explanations for the motion of 
planets north and south of the 
ecliptic, but Kepler found that these 
motions were simply the result of 
the orbits lying in planes tilted to 
the plane of the earth's orbit. 



Kepler concluded that the orbit was not a circle and there was no 
point around which the motion was uniform. So Plato's aim of 
fitting perfect circles to the heavens, which had for twenty 
centuries engaged the minds of brilliant men, had to be abandoned. 
Kepler had in his hands the finest observations ever made, but now 
he had no theory by which they could be explained. He would have 
to start over to account for two altogether new questions: what is 
the shape of the orbit followed by Mars, and how does the speed of 
the planet change as it moves along the orbit? 

Q1 When Kepler joined Tycho Brahe what task was he 
assigned? 

Q2 Why did Kepler conclude that Plato's problem, to describe 
the motions of the planets by combinations of circular motions, 
could not be solved? 




The diagram depicts a nearly edge- 
on view of orbital planes of earth and 
another planet, both intersecting at 
the sun. 



7.2 Kepler's law of areas 



Kepler's problem was immense. To solve it would demand the 
utmost of his imagination and computational skills. 

As the basis for his study, Kepler had Tycho's observed 
directions to Mars and to the sun on certain dates. But these 
observations were made from a moving earth whose orbit was not 
well known. Kepler realized that he must first determine more 
accurately the shape of the earth's orbit so that he would know 
where it was on the dates that the various observations of Mars had 
been made. Then he might be able to use the observations to 



58 



A New Universe Appears — The Work of Kepler and Galileo 





determine the shape and size of the orbit of Mars. Finally, to 
predict positions for Mars he would need to discover some rule or 
regularity that described how fast Mars moved along different 
parts of its orbit. 

As we follow his brilliant analysis here, and particularly if we 
repeat some of this work in the laboratory, we will see the series of 
problems that he solved. 

To derive the earth's orbit he began by considering the 
moments when the sun, earth, and Mars are essentially in a straight 
line (Fig. A). After 687 days, as Copernicus had found. Mars would 
return to the same place in its orbit (Fig. B). Of course, the earth 
at that time would not be at the same place in its own orbit as 
when the first observation was made. But as Figs. B and C indicate, 
the directions to the sun and Mars as they might be seen from the 
earth against the fixed stars would be known. The crossing point of 
the sight-lines to the sun and to Mars must be a point on the earth's 
orbit. By working with several groups of observations made 687 
days apart (one Mars "year"), Kepler was able to determine fairly 
accurately the shape of the earth's orbit. 




MAr» 





MAfS 



Q BArt-h 

The orbit Kepler found for the earth appeared to be almost a 
circle, with the sun a bit off center. From his plotted shape and 
the record of the apparent position of the sun for each date of the 
year, he could locate the position of the earth on its orbit, and its 
speed along the orbit. Now he had an orbit and a timetable for 
the earth's motion. You made a similar plot in the experiment The 
Shape of the Earth's Orbit. 

In Kepler's plot of the earth's motion around the sun, it was 
evident that the earth moves fastest when nearest the sun. (Kepler 
wondered why this occurred and speculated that the sun might 
exert some force that drove the planets along their orbits; his 
concern with the physical cause of planetary motion marked a 
change in attitude toward motion in the heavens.) The drawings 
at the left represent (with great exaggeration) the earth's motion 
for two parts of its orbit. The different positions on the orbit are 
separated by equal time intervals. Between points A and B there 
is a relatively large distance, so the planet is moving rapidly; 
between points C and D it moves more slowly. Kepler noticed, 
however, the two areas swept over by a line from the sun to the 
planet are equal. Kepler, it is believed, had actually calculated such 
areas only for the nearest and farthest positions of two planets. 
Earth and Mars, yet the beautiful simplicity of the relation led him 
to conclude that it was generally true, for all parts of orbits. In its 



Section 7.2 



59 



general form the Law of Areas states: The line from the sun to the 
moving planet sweeps over areas that are proportional to the time 
intervals. Later, when Kepler found the exact shape of orbits, his 
law of areas became a powerful tool for predicting positions along 
the orbit. In the next section we shall use both laws and see in 
detail how they work. 

You may be surprised that the first rule we have encountered 
about the motions of the planets is concerned with the areas swept 
over by the line from the sun to the planet. After we considered 
circles, eccentric circles, epicycle, and equants, we come upon a 
quite unexpected property: the area swept over per unit time is 
the first property of the orbital motion to remain constant. (As we 
shall see in Chapter 8, this major law of nature applies to all orbits 
in the solar system and also to double stars.) Here was something 
that, besides being new and different, also drew attention to the 
central role of the sun, and so bolstered Kepler's faith in the still 
widely neglected Copernican idea of a heliocentric system. 

As you will see, Kepler's other labors would have been of little 
use without this basic discovery, although the rule does not give 
any hint why this regularity should exist. The law of areas 
describes the relative rate at which the earth and. Mars (and, 
Kepler thought, any other planet) move at any point of their orbits. 
Kepler could not fit the rule to Mars by assuming a circular orbit, 
and so he set out to find what shape Mars' orbit was. 

Q3 What observations did Kepler use to plot the earth's orbit? 

Q4 State Kepler's law of areas. 

Q5 Where in its orbit does a planet move the fastest? 



7.3 Kepler's law of elliptical orbits 

With the orbit and timetable of the earth known, Kepler could 
reverse the analysis and find the shape of Mars' orbit. For this 
purpose he again used observations separated by one Martian year. 
Because this interval is somewhat less than two earth years, the 
earth is at different positions in its orbit at the two times, so the 
two directions from the earth toward Mars differ. Where they cross 
is a point on the orbit of Mars. From such pairs of observations 
Kepler fixed many points on the orbit of Mars. The diagrams below 
illustrate how two such points might be plotted. From a curve 
drawn through such points, he obtained fairly accurate values 



'^ h^^r» 





SG 7.4 

Another way to express this 
relationship for the nearest and 
farthest positions would be to say 
the speeds were inversely propor- 
tional to the distance; but this rule 
does not generalize to any other 
points on the orbit. (A modification 
of the rule that does hold is 
explained on pages 64 and 65.) 




Kepler's Law of Areas. A planet moves 
along its orbit at a rate such that the 
line from the sun to the planet sweeps 
over areas which are proportional to 
the time intervals. The time taken to 
cover AB is the same as that for BC, 
CD, etc. 



60 



A New Universe Appears-The Work of Kepler and Galileo 



In this experiment the orbit of Mars 
is plotted from measurements made 
on pairs of sky photographs taken 
one Martian year apart. 



SG7.5 




r— H 



An ellipse showing the major axis a 
the minor axis b, and the two foci F, 
and F.2. The shape of an ellipse is 
described by its eccentricity e, where 
e = c/a. 



SG7.6 
SG 7.7 

In the Orbit of Mercury Experiment, 
you can plot the shape of Mercury's 
very eccentric orbit from observa- 
tional data. See also SG 7.8. 

SG 7.9 




for the size and shape of Mars' orbit. Kepler saw at once that the 
orbit of Mars was not a circle around the sun. You will find the 
same result from the experiment, The Orbit of Mars. But what sort 
of path was this? How could it be described most simply? As Kepler 
said, "The conclusion is quite simply that the planet's path is not a 
circle — it curves inward on both sides and outward again at opposite 
ends. Such a curve is called an oval." But what kind of oval? 

Many different closed curves can be called ovals. Kepler 
thought for a time that the orbit was egg-shaped. Because such a 
shape did not agree with Kepler's ideas of physical interaction 
between the sun and the planet, he rejected that possibility. Kepler 
concluded that there must be some better way to describe the orbit, 
and that he could find it. For many months, Kepler struggled with 
the question. Finally he was able to show that the orbit was a 
simple curve which had been studied in detail by the Greeks two 
thousand years before. The curve is called an ellipse. It is the shape 
you see when you view a circle at a slant. 

Ellipses can differ greatly in shape. They have many interesting 
properties. For example, you can draw an ellipse by looping a piece 
of string around two thumb tacks pinned to a drawing board at 
points F, and Fa as shown at the left. Pull the loop taut with a 
pencil point (P) and run the pencil once around the loop. You will 
have drawn an ellipse. (If the two thumb tacks had been together, 
what curve would you have drawn? What results do you get as you 
separate the two tacks more?) 

Each of the points F, and F., is called a focus of the ellipse. The 
greater the distance between F, and F2. the flatter, or more 
"eccentric" the ellipse becomes. As the distance between F, and F, 
shrinks to zero, the ellipse becomes more nearly circular. A 
measure of the eccentricity of the ellipse is the ratio of the distance 
F1F2 to the long axis. If the distance between F, and F2 is c and 
length of the long axis is a, then the eccentricity e is defined as 
e = da. 

The eccentricities are given for each of the ellipses shown in 
the series of photographs in the margin of the next page. You can 
see that a circle is the special case of an ellipse with e = 0, and that 
the greatest possible eccentricity for an ellipse is e = 1.0. 



Section 7.3 



61 



What Kepler discovered was not merely that the orbit of Mars 
is an ellipse -a remarkable enough discovery in itself- but also that 
the sun is at one focus. (The other focus is empty.) Kepler stated 
these results in his Law of Elliptical Orbits: The planets move in 
orbits which are ellipses and have the sun at one focus. 

As Table 7.1 shows, the orbit of Mars has the largest eccentricity 
of all the orbits that Kepler could have studied, namely those of 
Venus, Earth, Mars. Jupiter, and Saturn. Had he studied any planet 
other than Mars, he might never have noticed that the orbit was an 
ellipse! Even for the orbit of Mars, the difference between the 
elliptical orbit and an off-center circle is quite small. No wonder 
Kepler later wrote that "Mars alone enables us to penetrate the 
secrets of astronomy which otherwise would remain forever hidden 
from us." 

Table 7.1 The Eccentricities of Planetary Orbits 





ORBITAL 




PLANET 


ECCENTRICITY 


NOTES 


Mercury 


0.206 


Too few observations for Kepler to 
study 


Venus 


0.007 


Nearly circular orbit 


Earth 


0.017 


Small eccentricity 


Mars 


0.093 


Largest eccentricity among planets 
Kepler could study 


Jupiter 


0.048 


Slow moving in the sky 


Saturn 


0.056 


Slow moving in the sky 


Uranus 


0.047 


Not discovered until 1781 


Neptune 


0.009 


Not discovered until 1846 


Pluto 


0.249 


Not discovered until 1930 



The work of Kepler illustrates the enormous change in outlook 
in Europe that had begun well over two centuries earlier. Kepler 
still shared the ancient idea that each planet had a "soul," but he 
refused to rest his explanation of planetary motion on this idea. 
Instead, he began to search for physical causes. When Copernicus 
and Tycho were willing to settle for geometrical models by which 
planetary positions could be predicted, Kepler was one of the first 
to seek dynamic causes for the motions. This new desire for 
physical explanations marks the beginning of one of the chief 
characteristics of modern physical science. 

Like Kepler, we believe that our observations represent some 
aspects of a reality that is more stable than the changing emotions 
and wishes of human beings. Like Plato and all subsequent 
scientists, we assume that nature is basically orderly and consistent 
and, therefore, understandable in a simple way. This faith has led 
to great theoretical and technical gains. Kepler's work illustrates 
one of the scientific attitudes — to regard a wide variety of 
phenomena as better understood when they can be summarized by 
simple law, preferably one expressed in mathematical form. 

After Kepler's initial joy over the discovery of the law of 
elliptical paths, he may have asked himself the question: why are 
the planetary orbits elliptical rather than some other geometrical 



e = 0.3 




= 0.5 




e = 0.7 




e = 0.8 




e = 0.94 




e = 0.98 




Ellipses of different eccentricities. 
(The pictures were made by photo- 
graphing a saucer at different angles.) 














/ 

















^- 



tr. rr. 



. -yr -x^.H' CCA /V» #•!/ 



T^r^c^^^/PC . II. u. ir 



»t€r-^^. 













/* 






•^Z •^'^c t*t> f" 



-V^/'- 



Section 7.3 



63 



shape? While we might understand Plato's desire for uniform 
circular motions, nature's insistence on the ellipse is a surprise. 

In fact, there was no satisfactory answer to this question until 
Newton showed, almost eighty years later, that these elliptical 
orbits were necessary results of a more general law of nature. Let 
us accept Kepler's laws as rules that contain the observed facts 
about the motions of the planets. As empirical laws, they each 
summarize the data obtained by observation of the motion of any 
planet. The law of orbits, which describes the paths of planets as 
elliptical around the sun, gives us all the possible positions each 
planet can have if we know the size and eccentricity. That law, 
however, does not tell us when the planet will be at any one 
particular position on its ellipse or how rapidly it will be moving 
then. The law of areas does not specify the shape of the orbit, but 
does describe how the angular speed changes as the distance from 
the sun changes. Clearly these two laws complement each other. 
With these two general laws, and given the values for the size and 
eccentricity of the orbit (and a starting point), we can determine 
both the position and angular speed of a given planet at any time, 
past or future. Since we can also find where the earth is at the 
same instant, we can calculate the position of the planet as it 
would have been or will be seen from the earth. 

The elegance and simplicity of Kepler's two laws are impressive. 
Surely Ptolemy and Copernicus would have been amazed that the 
solution to the problem of planetary motions could be given by such 
short statements. But we must not forget that these laws were 
distilled from Copernicus' idea of a moving earth, the great labors 
and expense that went into Tycho's fine observations, and the 
imagination and devotion, agony and ecstasy of Kepler. 

Q6 What was special about Mars' orbit that made Kepler's 
study of it so fortunate? 

Q7 If the average distance and eccentricity of a planet's orbit 
are known, which of the following can be predicted from the law of 
areas alone? From the law of Elliptical orbits alone? Which require 
both? (Mark A, E, or A + E). 

(a) All possible positions in the orbit, 

(b) speed at any point in an orbit, 

(c) position at any given time. 



Empirical means based on 
observation, but not on theory. 



Conic Sections are figures produced 
by cutting a cone with a plane — the 
eccentricity of a figure is related to 
the angle of the cut. In addition to 
circles and ellipses, parabolas and 
hyperbolas are conic sections, with 
eccentricities greater than ellipses. 
Newton eventually showed that all of 
these shapes are possible paths for a 
body moving under the influence of 
the sun. 





Ci'rcLe 





E-LL\toe>c 






PA.ra.boLa. 



H v_j per boLcL 




Opposite; A page from Kepler's notebooks. 




(A) 




ViAt- 



(B) 




(C) 




A General Equation for Orbital Speed 

Figure A represents the elliptical orbit of a planet, with 
the sun at one focus. By a short analysis we can find 
the ratio of the speeds at the position nearest to the 
sun (perihelion) and farthest from the sun (aphelion). 

Figure B shows a small part of the planet's path 
around perihelion, during a time interval At. If Af is very 
short, then the average speed along the path will be 
virtually equal to the instantaneous speed at perihelion, 
i^p, and the path length will be v,, x Af. 

Also, if At is very short, the section of orbit is 
almost straight, so it can be considered the base of a 
long, thin triangle of altitude R,„ shaded in Figure C. 
The area of any triangle is 7 base x altitude, so 
the area A^ of this triangle is T(Vp x Af)f?„. 

Similarly, the area A^ of a triangle swept out 
during At at aphelion is j(v.^ x At)R^. By Kepler's 
law of areas, equal areas are swept out in equal 
times, so A^ = Ap. Then 

jv^ X Atx R^= ^Vp X Atx Rp 

and, dividing both sides by jAt. 

We can rearrange the equation to the form 



R, 



which shows that the speeds at aphelion and 
perihelion are inversely proportional to the distances 
from the sun: at a larger distance the speed is smaller. 

The derivation for these two points was easy, 
because at these points the velocity is perpendicular to 
the line drawn to the sun. When the planet is at some 



(D) 



position other than perihelion or aphelion, the velocity 
vector Is not perpendicular, as shown in Figure D. 
However, we can approximate the area swept out, 
shaded in Figure E, by a triangle of altitude R, as 
shown In Figure F. Notice that it Includes a tiny 
corner of extra area, but also leaves out a tiny 
corner. For a very short time Interval Af, the 
triangle will be very thin and the difference 
between the two tiny corners will virtually 
vanish. As shown In figure G, the base of 
the triangle is not v x At, but i/^ x At, where 
v'^ Is the portion or component of v 
perpendicular to the sun-planet line. Thus the 
area swept out during At can be expressed as 

jv^ xAtxR 

This same derivation for area swept out will hold 
for any part of the orbit over a short time interval At. By 
Kepler's law of areas, the areas swept out during 
equal time intervals would be equal, so we can write 

TV, xAtxR^^v, xAtxR'^ Yv" xAtx R" etc. 

or, dividing through by \ At, 

v^R = v[R' = v'[R" etc. 

Therefore, we can express Kepler's law of areas as 
v^R = constant. 
If the shape (eccentricity) of the orbit Is known, 
together with the speed and distance at any one 
point, we can use this equation to calculate the speed 
at any other point In the orbit. (See SG 7.10.) Moreover, 
the law of areas, from which this relation is derived. Is 
true for the motion of any body that experiences a 
force directed toward one of the foci of the ellipse — 
a so-called "central force." So the relation Vj^R = 
constant applies to double stars and to atoms as well 
as to the solar system. 




(E) 




(F) 




(G) 




(H) 



66 A New Universe Appears— The Work of Kepler and Galileo 

7.4 Kepler's Law of Periods 



As Einstein later put it: "The Lord is 
subtle, but He is not malicious." 



For the earth, T is one year. The 
average distance R.,, of the earth 
from the sun is one astronomical 
unit, 1AU. So one way to express 
the value of the constant ic is fr = 1 
year'/AU'. 



SG 7.11-7.14 



Kepler's first two laws were published in 1609 in his book 
Astronomia Nova, but he was still dissatisfied because he had not 
yet found any relation among the motions of the different planets. 
Each planet seemed to have its own elliptical orbit and speeds, but 
there appeared to be no overall pattern relating all planets to one 
another. Kepler had begun his career by trying to explain the 
number of planets and their spacing. He was convinced that the 
observed orbits and speeds could not be accidental, but that there 
must be some regularity linking all the motions in the solar 
system. His conviction was so strong, that he spent years examining 
many possible combinations of factors to find, by trial and error, a 
third law that would relate all the planetary orbits. His long search, 
almost an obsession, illustrates a belief that has run through the 
whole history of science: that despite apparent difficulties in 
getting a quick solution, underneath it all, nature's laws are 
understandable. This belief is to this day a chief source of inspiration 
in science, often sustaining one's spirit in periods of seemingly 
fruitless labor. For Kepler it made endurable a life of poverty, 
illness, and other personal misfortunes, so that in 1619 he could 
write triumphantly in his Harmony of the World: 

. . . after I had by unceasing toil through a long period of 
time, using the observations of Brahe, discovered the 
true relation . . . overcame by storm the shadows of my 
mind, with such fullness of agreement between my 
seventeen years' labor on the observations of Brahe and 
this present study of mine that I at first believed that I 
was dreaming .... 

Kepler's law of periods, also called the "harmonic law," relates 
the periods of the planets to their average distances from the sun. 
The period is the time taken to go once completely around the orbit. 
The law states that the squares of the periods of the planets are 
proportional to the cubes of their average distances from the sun. 
In the short form of algebra, calling the period T and the average 
distance Rav. this law can be expressed as 

y2 



T2 ex i?^^.3 or T' = feRav' 



or 



^av 



= fe 



where fe is a constant. Because this relation applies to all the 
planets and even to comets in orbit around the sun, we can use it 
to find the period of any planet once we know its average distance 
froin the sun, and vice versa. 

Kepler's three laws are so simple that their great power may be 
overlooked. When they are combined with his discovery that each 
planet moves in a plane passing through the sun, they let us 
derive the past and future history of each planet from only six 
quantities. Two of these quantities are the size and eccentricity of 
the orbit, three others are angles that relate the plane of the orbit 
to that of the earth's orbit, while the sixth tells where in its orbit 



Section 7.4 



67 



the planet was on any one certain date. These quantities are 
explained more fully in the Activities section of Handbook 2 for 
Chapters 7 and 8. 





Copernicus' Values 




Modern Values 




PLANET 


PERIOD T, 


AVERAGE DIS- 


7^ 


PERIOD 7, 


AVERAGE DIS- 


r 




(YEARS) 


TANCE Rav(AU) 


«av-^ 


(YEARS) 


TANCE flav(AU) 


«av^ 


Mercury 


0.241 


0.38 


1.06 


0.241 


0.387 


1.00 


Venus 


0.614 


0.72 


1.01 


0.615 


0.723 


1.00 


Mars 


1.881 


1.52 


1.01 


1.881 


1.523 


1.00 


Jupiter 


11.8 


5.2 


0.99 


1 1 .862 


5.20 


1.00 


Saturn 


29.5 


9.2 


1.12 


29.458 


9.54 


1.00 




The value of R.„ for an ellipse is just 
half the major axis. 



It is astonishing that in this manner the past and future 
positions of each planet (and, as we now know, also each comet) 
can be derived in a simpler and more precise way than through the 
multitude of geometrical devices on which all other planetary 
theories depended, whether those of Ptolemy, Copernicus, or Tycho. 
With different assumptions and procedures Kepler had at last solved 
the astronomical problem on which so many great men had worked 
over the centuries. Although he had to abandon the geometrical 
devices of the Copernican system. Kepler did depend on the 
Copernican viewpoint of a sun-centered universe. None of the 
earth-centered models could have led to Kepler's three laws. 

In 1627, after many troubles with his publishers and Tycho's 
heirs, Kepler finally published a set of astronomical tables. In these 
tables Kepler combined Tycho's observations and the three laws in 
a way that permitted accurate calculations of planetary positions 
for any time, whether in the past or future. These tables remained 
useful for a century, until telescopic observations of greater 
precision replaced Tycho's observations. 

Kepler's scientific interest was not confined to the planetary 
problem alone. Like Tycho, who was much impressed by the new 
star of 1572, Kepler observed and wrote about new stars that 
appeared in 1600 and 1604. His observations and interpretations 
added to the impact of Tycho's earlier observations that changes 
did occur in the starry sky. As soon as Kepler learned of the 
development of the telescope, he spent most of a year making 
careful studies of how the images were formed. These he 
published in a book titled Dioptrice (1611), which became the 
standard work on optics for many years. In addition to a number of 
important books on mathematical and astronomical problems, 
Kepler wrote a popular and widely read description of the Copernican 
system as modified by his own discoveries. This added to the 
growing interest in and acceptance of the sun-centered model of 
the planetary system. 



The tables, named for Tycho's and 
Kepler's patron. Emperor Rudolph II, 
were called the Rudolphine Tables. 
They were also important for a 
quite different reason. In them 
Kepler pioneered in the use of 
logarithms for making rapid 
calculations and included a long 
section, practically a textbook, on 
the nature and use of logarithms 
(first described in 1614 by Napier in 
Scotland). His tables spread the use 
of this computational aid, widely 
needed for nearly three centuries, 
until modern computing machines 
came into use. 



Q8 State Kepler's law of periods. 



68 A New Universe Appears-The Work of Kepler and Galileo 

7.5 The new concept of physical law 

One general feature of Kepler's life-long work has had a 
far-reaching effect on the way in which all the physical sciences 
developed. When Kepler began his studies, he still accepted Plato's 
assumptions about the importance of geometric models and 
Aristotle's emphasis on natural place to explain motion. But later 
he came to concentrate on algebraic laws describing how planets 
moved. His successful statement of empirical laws in mathematical 
form helped to establish the use of the equation as the normal 
form of stating laws in physical science. 

More than anyone before him, Kepler expected an acceptable 
theory to agree with precise and quantitative observation. From 
Tycho's observations he learned to respect the power of precision 
measurement. Models and theories can be modified by human 
ingenuity, but good data endure regardless of changes in assump- 
tions or viewpoints. 

Kepler went beyond observation and mathematical description, 
and attempted to explain motion in the heavens by the action of 
physical forces. In Kepler's system the planets no longer were 
thought to revolve in their orbits because they had some divine 
nature or influence, or because this motion was "natural." or 
because their spherical shapes were self-evident explanation for 
circular motion. Rather, Kepler was the first to look for a physical 
law based on observed phenomena to describe the whole universe 
in a detailed quantitative manner. In an early letter he expressed 
his guiding thought: 

I am much occupied with the investigation of the 
physical causes. My aim in this is to show that the 
celestial machine is to be likened not to a divine 
organism but rather to a clockwork . . . insofar as nearly 
all the manifold movements are carried out by means of 
a single; quite simple magnetic force, as in the case of a 
clockwork, all motions are caused by a simple weight. 
Moreover, I show how this physical conception is to be 
presented through calculation and geometry. [Letter to 
Herwart, 1605] 

To show the celestial machine to be like a clockwork propelled 
by a single force — this was a prophetic goal indeed. Stimulated by 
William Gilbert's work on magnetism published a few years earlier, 
Kepler could imagine magnetic forces from the sun driving the 
planets along their orbits. This was a reasonable and promising 
hypothesis. As it developed, the basic idea that a single kind of 
force controls the motions of all the planets was correct; but the 
force is not magnetism, and it is needed not to keep the planets 
moving forward, but to deflect their paths to form closed orbits. 

Kepler's statement of empirical laws reminds us of Galileo's 
suggestion, made at about the same time, that we deal first with 
the how of motion in free fall and then with the why. A half 
century later Newton used the concept of gravitational force to tie 



Section 7.6 69 

together Kepler's three planetary laws with laws of terrestrial 
mechanics to provide a magnificent synthesis. (See Chapter 8.) 



Q9 In what ways did Kepler's work exemplify a "new" concept 
of physical law? 



7.6 Galileo and Kepler 

One of the scientists with whom Kepler corresponded about 
scientific developments was Galileo. While Kepler's contributions to 
planetary theory were mainly his empirical laws based on the 
observations of Tycho, Galileo contributed to both theory and 
observation. As was reported in Chapters 2 and 3. Galileo's theory of 
motion was based on observations of bodies moving on the earth's 
surface. His development of the new science of mechanics 
contradicted the assumptions on which Aristotle's physics and 
interpretation of the heavens had been based. Through his books 
and speeches Galileo triggered wide discussion about the differences 
or similarities of earth and heaven. Outside of scientific circles, as 
far away as England, the poet John Milton wrote, some years after 
his visit to Galileo in 1638: 

. . . What if earth {Paradise Lost, Book V, line 574, 

Be but the shadow of Heaven, and things therein published 1667.) 

Each to the other like, more than on earth is thought? 

Galileo challenged the ancient interpretations of experience. As 
we saw earlier, he focused attention on new concepts: time and 
distance, velocity and acceleration, forces and matter, in contrast 
to the Aristotelian qualities of essences, final causes, and fixed 
geometric models. In Galileo's study of falling bodies he insisted on 
fitting the concepts to the observed facts. By seeking results that 
could be expressed in concise algebraic form, Galileo paralleled 
the new style being used by Kepler. 

The sharp break between Galileo and most other scientists of 
the time arose from the kind of questions he asked. To his 
opponents, many of Galileo's problems seemed trivial. What was 
important about watching pendulums swing or rolling balls down 
inclines, when philosophical problems needed clarification? His 
procedures for studying the world seemed peculiar, even fantastic. 

Although Kepler and Galileo lived at the same time, their lives 
were quite different. Kepler lived a hand-to-mouth existence under 
stingy patrons and was driven from city to city by the religious wars 
of the time. Few people, other than a handful of friends and 
correspondents, knew of or cared about his studies and results. He 
wrote lengthy, tortuous books which demanded expert knowledge 
to read. 

Galileo, on the other hand, wrote his numerous essays and 
books in Italian, in a language and a style which could be 
understood by his contemporaries who did not read scholarly Latin. 



70 



A New Universe Appears— The Work of Kepler and Galileo 



In recent times, similar receptions 
were initially given to such artists 
as the painter Picasso, and the 
sculptor Giacometti, and the 
composers Stravinski and Schon- 
berg. The same has often been true 
in most fields, whether literature or 
mathematics, economics or politics. 
But while great creative novelty is 
often attacked at the start, it does 
not follow that, conversely, every- 
thing that is attacked must be 
creative. 



Galileo was a master at publicizing his work. He wanted as many 
as possible among the reading public to know of his studies and to 
accept the Copernican theory. He took the argument far beyond a 
small group of scholars out to the nobles, civic leaders, and religious 
dignitaries. His arguments included satire on individuals or ideas. 
In return, his efforts to inform and persuade on a topic as 
"dangerous" as cosmological theory stirred up the ridicule and even 
violence often poured upon those who have a truly new point of 
view. 

Q10 Which of the following would you associate more with 
Galileo's work than with that of his predecessors: qualities and 
essences, popular language, concise mathematical expression, 
final causes? 



7.7 The telescopic evidence 




Two of Galileo's telescopes, dis- 
played at the Museum of Science in 
Florence. 



Like Kepler, Galileo was surrounded by colleagues who were 
convinced the heavens were eternal and could not change. Hence. 
Galileo was especially interested in the sudden appearance in 1604 
of a new star, one of those observed by Kepler. Where there had 
been nothing visible in the sky, there was now a brilliant star. 
Galileo, like Tycho and Kepler, realized that such changes in the 
starry sky conflicted with the old idea that the stars could not 
change. Furthermore, this new star awakened in Galileo an interest 
in astronomy which lasted throughout his life. 

Consequently, Galileo was ready to react to the news he 
received four or five years later: that a Dutchman "had constructed 
a spy glass by means of which visible objects, though very distant 
from the eye of the observer, were distinctly seen as if nearby." 
Galileo (as he tells it) quickly worked out some of the optical 
principles involved and set to work to grind the lenses and build 
such an instrument himself. His first telescope made objects appear 
three times closer than when seen with the naked eye. Reporting 
on his third telescope in his book The Starry Messenger: 



Galileo meant that the area of the 
object was nearly 1000 times 
greater. The area is proportional to 
the square of the magnification (or 
"power") as we define it now. 



Finally, sparing neither labor nor expense, I succeeded in 
constructing for myself so excellent an instrument that 
objects seen by means of it appeared nearly one thousand 
times larger and over thirty times closer than when 
regarded with our natural vision. 

What would you do if you were handed "so excellent an 
instrument"? Like the men of Galileo's time, you probably would 
put it to practical uses. "It would be superfluous," Galileo agreed. 



to enumerate the number and importance of the 
advantages of such an instrument at sea as well as on 
land. But forsaking terrestrial observations, I turned to 
celestial ones, and first I saw the moon from as near at 



Section 7.7 



71 



hand as if it were scarcely two terrestrial radii away. 
After that I observed often with wondering delight both 
the planets and the fixed stars .... 

In the period of a few short weeks in 1609 and 1610, Galileo 
used his telescope to make several discoveries, each of which is of 
first rank. 

First, Galileo pointed his telescope at the moon. What he saw 
led him to the conviction that 

. . . the surface of the moon is not smooth, uniform, and 
precisely spherical as a great number of philosophers 
believe it (and other heavenly bodies) to be, but is 
uneven, rough, and full of cavities and prominences, 
being not unlike the face of the earth, relieved by chains 
of mountains, and deep valleys. 

Galileo did not stop with that simple observation, so contrary to 
the Aristotelian idea of heavenly perfection. He supported his 
conclusions with several kinds of evidence, including careful 
measurement. For instance, he worked out a method for determining 
the height of a mountain on the moon from the shadow it is seen to 
cast there. (His value of about four miles for the height of some 
lunar mountains is not far from modern results such as those 
obtainable in the experiment, The Height of Piton: A Mountain on 
the Moon. 

Next he looked at the stars. To the naked eye the Milky Way 
had seemed to be a continuous blotchy band of light; through the 
telescope it was seen to consist of thousands of faint stars. 
Wherever Galileo pointed his telescope in the sky, he saw many 
more stars than could be seen with the unaided eye. This observa- 
tion was contrary to the old argument that the stars were created 
to provide light so men could see at night. If that were the 
explanation, there should not be stars invisible to the naked eye — 
but Galileo found thousands. 

After his observations of the moon and the fixed stars, Galileo 
turned his attention to the discovery which in his opinion ". . . de- 
serves to be considered the most important of all — the disclosure 
of four Planets never seen from the creation of the world up to our 
own time." He is here referring to his discovery of four of the 
satellites which orbit about Jupiter. Here before his eyes was a 
miniature solar system with its own center of revolution. Today, as 
to Galileo so long ago, it is a sharp thrill to see the moons of Jupiter 
through a telescope for the first time. One is immediately struck by 
this evidence so directly opposed to the Aristotelian notion that the 
earth was at the center of the universe and the chief center of 
revolution. 

The manner in which Galileo discovered Jupiter's "planets" is a 
tribute to his ability as an observer. Each clear evening during this 
period he was discovering dozens if not hundreds of new stars 
never before seen by man. When looking in the vicinity of Jupiter 




Two of Galileo's early drawings of the 
moon (from Galileo's Siderius Nun- 
cius). 



72 



A New Universe Appears-The Work of Kepler and Galileo 




Telescopic photograph of Jupiter and 
its four bright satellites. This is ap- 
proximately what Galileo saw and 
what you see through the simple 
telescope described in the Handbook. 

As of 1970, 12 satellites of Jupiter 
have been observed. 






3-'~i^ 



O * ^ 



3HoA 



*o 



»o 



r «— «' 



lD__jl 



^*-^//i?. -* * . o 



• o 



* o 



14 "^ 



o 



I^C^r^H^ 






J 7. evi:^ 



* o » 



♦ O ' 



l-l . -tMrAt' ♦ 



it. 



* •• o 



« ><• o 






o 



^ 



►>y ir-^rj.' 



• D 



♦^•^ 



<r 



; I ^^w^ Cm*. 



These sketches of Galileos are from 
the first edition of The Starry Mes- 
senger. 



on the evening of January 7, 1610, he noticed ". . . that beside the 
planet there were three starlets, small indeed, but very bright. 
Though I believe them to be among the host of fixed stars, they 
aroused my curiosity somewhat by appearing to lie in an exact 
straight line . . . ." (The first page of the notebook in which he 
recorded his observations is reproduced on p. 81 at the end of this 
chapter.) When he saw them again the following night, he saw that 
they had changed position with reference to Jupiter. Each clear 
evening for weeks he observed that planet and its roving "starlets" 
and recorded their positions in drawings. Within days he had 
concluded that there were four "starlets" and that they were indeed 
satellites of Jupiter. He continued his observations until he was 
able to estimate the periods of their revolutions around Jupiter. 

Of all of Galileo's discoveries, that of the satellites of Jupiter 
caused the most stir. His book. The Starry Messenger, was an 
immediate success, and copies were sold as fast as they could be 
printed. For GalOeo the result was a great demand for telescopes 
and great public fame. 

Galileo continued to use his telescope with remarkable results. 
By projecting an image of the sun on a screen, he observed 
sunspots. This was additional evidence that the sun, like the moon, 
was not perfect in the Aristotelian sense: it was disfigured rather 
than even and smooth. From his observation that the sunspots 
moved across the face of the sun in a regular pattern, he concluded 
that the sun rotated with a period of about 27 days. 




Photographs of Venus at various 
phases with a constant magnification. 

He also found that Venus showed all phases, just as the moon 
does (see photos above). Therefore. Venus must move completely 
around the sun as Copernicus and Tycho had believed, rather than 
be always between the earth and sun as the Ptolemaic astronomers 
assumed. Saturn seemed to carry bulges around its equator, as 
indicated in the drawings on the next page, but Galileo's telescopes 
were not strong enough to show that they were rings. With his 
telescopes he collected an impressive array of new information 
about the heavens — all of it seemed to contradict the basic 
assumptions of the Ptolemaic world scheme. 



Section 7.8 



73 



Q11 Could Galileo's observations of all phases of Venus 
support the heliocentric theory, the Tychonic system, or Ptolemy's 
system? 

Q12 In what way did telescopic observation of the moon and 
sun weaken the earth-centered view of the universe? 

Q13 What significance did observations of Jupiter have in 
weakening the Ptolemaic view of the world? 



7.8 Galileo focuses the controversy 

Galileo's observations supported his belief in the heliocentric 
Copernican system, but they were not the cause of his belief. In 
his great work, Dialogue Concerning the Two Chief World Systems 
(1632), his arguments were based more on assumptions that 
seemed self-evident to him than on observations. Galileo recognized, 
as Ptolemy and Copernicus had, that the observed motions of 
planets alone do not decide uniquely between a heliocentric and a 
geocentric hypothesis. With proper modifications of the systems, 
says Galileo, "The same phenomena would result from either 
hypothesis." But Galileo accepted the earth's motion as real 
because the heliocentric system seemed to him simpler and more 
pleasing. Elsewhere in this course you wUl find other cases where 
a scientist accepted or rejected an idea for reasons arising from a 
strong belief or feeling that frankly could not, at the time, be 
verified by experiment. 

In the Dialogue Concerning the Two Chief World Systems, 
Galileo presents his arguments in a systematic and lively way. Like 
his later book, Discourses Concerning Two New Sciences, men- 
tioned in Chapter 2, it is in the form of a discussion between three 
learned men. Salviati, the voice of Galileo, wins most of the 
arguments. His antagonist is Simplicio, an Aristotelian who speaks 
for and defends the Ptolemaic system. The third member, Sagredo, 
represents the objective and intelligent citizen not yet committed 
to either system. However, Sagredo's role is written so that he 
usually accepts most of Galileo's arguments in the end. 

Galileo's arguments in favor of the Copernican system as set 
forth in Two Chief World Systems were mostly those given by 
Copernicus. Oddly enough. Galileo made no use of Kepler's laws. 
However, Galileo's observations did provide new evidence for 
Kepler's laws. After determining the periods of Jupiter's four 
moons, Galileo found that the larger the orbit of the satellite, the 
longer was its period of revolution. Copernicus had already found 
that the periods of the planets increased with their average 
distances from the sun. (Kepler's law of periods stated the relation 
for the planets in detailed quantitative form.) Now Jupiter's 
satellite system showed a similar pattern. These new patterns of 
regularities would soon replace the old assumptions of Plato. 
Aristotle, and Ptolemy. 

Two Chief World Systems relies upon Copernican arguments, 









cm) 



Drawings of Saturn 
made in the 
seventeenth century. 



74 A New Universe Appears-The Work of Kepler and Galileo 

Galilean observations, and arguments of plausibility to attack the 
basic assumptions of the geocentric model. In response, Simplicio, 
seemingly in desperation, tries to dismiss all of Galileo's arguments 
with a characteristic counter argument: 

. . . with respect to the power of the Mover, which is 
infinite, it is just as easy to move the universe as the 
earth, or for that matter a straw. 

But to this Galileo makes a very interesting reply; notice how he 
quotes Aristotle against the Aristotelians: 

. . . what I have been saying was with regard not to the 
Mover, but only the movables . . . Giving our attention, 
then, to the movable bodies, and not questioning that it 
is a shorter and readier operation to move the earth than 
the universe, and paying attention to the many other 
simplifications and conveniences that follow from merely 
this one, it is much more probable that the diurnal 
motion belongs to the earth alone than to the rest of the 
universe excepting the earth. This is supported by a very 
true maxim of Aristotle's which teaches that ... 'it is 
pointless to use many to accomplish what may be done 
with fewer.' 

With characteristic enthusiasm, Galileo thought his telescopic 
discoveries would soon cause everyone to realize how absurd the 
assumptions were that prevented wide acceptance of the Copemican 
theory. But men cannot believe what they are not ready to believe. 
In their fight against the new Copernicans, the followers of 
Aristotle were convinced that they were adhering to the facts, that 
the heliocentric theory was obviously false and in contradiction to 
observation and also to common sense. The evidences of the 
telescope could be due to distortions; after all, glass lenses change 
the path of light rays. And even if telescopes seemed to work for 
terrestrial observation, nobody could be sure they worked equally 
well when pointed at these vastly more distant celestial objects. 

Furthermore, the Aristotelians could not even consider the 
Copernican system as a possible theory without giving up many of 
their basic assumptions, as we saw in Chapter 6. This would have 
required them to do what is nearly humanly impossible: give up 
many of their common-sense ideas and find new bases for their 
theological and moral doctrines. They would have to admit that 
the earth is not at the center of creation. Then perhaps the universe 
was not created especially .for mankind. Is it any wonder that 
Galileo's arguments stirred up a storm of opposition? 

Galileo's observations intrigued many, but they were unaccept- 
able to Aristotelian scholars. Most of these had reasons one can 
respect. But a few were driven to positions that must have seemed 
silly at that time, too. For example, the Florentine astronomer 
Francesco Sizzi argued in 1611 why there could not, indeed must 
not, be any satellites around Jupiter: 



Section 7.9 



75 



There are seven windows in the head, two nostrils, 
two ears, two eyes and a mouth; so in the heavens there 
are two favorable stars, two unpropitious, two luminaries, 
and Mercury alone undecided and indifferent. From 
which and many other similar phenomena of nature 
such as the seven metals, etc., which it were tedious to 
enumerate, we gather that the number of planets is 
necessarily seven [including the sun and moon]. . . . 
Besides, the Jews and other ancient nations, as well as 
modern Europeans, have adopted the division of the 
week into seven days, and have named them from the 
seven planets; now if we increase the number of 
planets, this whole system falls to the ground. . . . 
Moreover, the satellites are invisible to the naked eye 
and therefore can have no influence on the earth, and 
therefore would be useless, and therefore do not exist. 

A year after his discoveries, Galileo wrote to Kepler: 

You are the first and almost the only person who, 
even after a but cursory investigation, has . . . given 
entire credit to my statements. . . . What do you say of the 
leading philosophers here to whom I have offered a 
thousand times of my own accord to show my studies, 
but who with the lazy obstinacy of a serpent who has 
eaten his fill have never consented to look at the planets, 
or moon, or telescope? 

Q14 Did Galileo's telescopic observations cause him to believe 
in the Copernican viewpoint? 

Q15 What reasons did Galileo's opponents give for ignoring 
telescopic observations? 



Some of the arguments that were 
brought forward against the new 
discoveries were so silly that it is 
hard for the modern mind to take 
them seriously. . . . One of his 
[Galileo's] opponents, who admitted 
that the surface of the moon looked 
rugged, maintained that it was 
actually quite smooth and spherical 
as Aristotle had said, reconciling 
the two ideas by saying that the 
moon was covered with a smooth 
transparent material through which 
mountains and craters inside it 
could be discerned. Galileo, sar- 
castically applauding the ingenuity 
of this contribution, offered to accept 
it gladly — provided that his opponent 
would do him the equal courtesy of 
allowiny him then to assert that the 
moon was even more rugged than 
he had thought before, its surface 
being covered with mountains and 
craters of this invisible substance 
ten times as high as any he had 
seen. [Quoted from Discoveries and 
Opinions of Galileo, translated by 
Stillman Drake.] 

SG 7.16 



7.9 Science and freedom 



The political and personal tragedy that befell Galileo is 
described at length in many books. Here we shall only mention 
briefly some of the major events. GalHeo was warned in 1616 by the 
Inquisition to cease teaching the Copernican theory as true (rather 
than as just one of several possible methods to compute the 
planetary motions) for that theory was held contrary to Holy 
Scripture. At the same time Copernicus' book was placed on the 
Index of Forbidden Books and suspended "untO corrected." As we 
saw before, Copernicus had, whenever possible, used Aristotelian 
doctrine to make his theory plausible. But Galileo had reached a 
new point of view: he urged that the heliocentric system be 
accepted on its merits alone. While he was himself a devoutly 
religious man. he deliberately ruled out questions of religious faith 
from scientific discussions. This was a fundamental break with 
the past. 

When Cardinal Barberini, formerly a close friend of Galileo, was 



76 



A New Universe Appears-The Work of Kepler and Galileo 



Pantheism refers to the idea that 
God is no more (and no less) than 
the forces and laws of nature. 



According to a well-known, but 
probably aprocryphal story, at the 
end of these proceedings Galileo 
muttered, "E pur se muove — but 
it does move." 



elected in 1623 to be Pope Urban VIII, Galileo talked with him 
regarding the decree against the Copernican ideas. As a result of 
the discussion, Galileo considered it safe enough to write again on 
the controversial topic. In 1632, having made some required 
changes, Galileo obtained the necessary papal consent to publish 
Two Chief World Systems. This book presented very persuasively 
the Ptolemaic and Copernican viewpoints and their relative merits. 
After the book's publication, his opponents argued that Galileo 
seemed to have tried to get around the warning of 1616. Further- 
more, Galileo's forthright and sometimes tactless behavior and the 
Inquisition's need to demonstrate its power over suspected heretics 
combined to mark him for punishment. 

Among the many factors in this complex story, we must 
remember that Galileo, though a suspect of the Inquisition, 
considered himself religiously faithful. In letters of 1613 and 1615 
Galileo wrote that God's mind contains all the natural laws; 
consequently he held that the occasional glimpses of these laws 
which the human investigator may gain were direct revelations of 
God, just as valid and grand as those in the Bible: "From the 
Divine Word, the Sacred Scripture and Nature did both alike 
proceed. . . . Nor does God less admirably discover himself to us in 
Nature's action than in the Scripture's sacred dictions." These 
opinions are held by many today whether they are scientists or not. 
and are no longer regarded as being in conflict with theological 
doctrines. But in Galileo's time they could be regarded as symptoms 
of pantheism. This was one of the heresies for which Galileo's 
contemporary, Giordano Bruno, was burned at the stake. The 
Inquisition was alarmed by Galileo's contention that the Bible was 
not a certain source of knowledge for the teaching of natural 
science. In reply, arrogant as Galileo often was. he quoted Cardinal 
Baronius: "The Holy Spirit intended to teach us how to go to 
heaven, not how the heavens go." 

Though he was old and ailing, Galileo was called to Rome and 
confined for a few months. From the proceedings of Galileo's trial, 
of which parts are still secret, we learn that he was tried, threatened 
with torture, forced to make a formal confession of holding and 
teaching forbidden ideas and to make a denial of the Copernican 
theory. In return for his confessions and denial, he was sentenced 
only to perpetual house arrest. Galileo's friends in Italy did not dare 
to defend him publicly. His book was placed on the Index where it 
remained, along with that of Copernicus and one of Kepler's, until 
1835. Thus, he was used as a warning to all men that the demand 
for spiritual conformity also required intellectual conformity. 

But without intellectual freedom, science cannot flourish for 
long. Perhaps it is not a coincidence that for two centuries after 
Galileo, Italy, which had been the mother of many outstanding 
men, produced hardly a single great scientist, while elsewhere in 
Europe they appeared in great numbers. Today scientists are 
acutely aware of this famous part of the story of the development 
of planetary theories. Teachers and scientists in our time have had 



Section 7.9 



77 



to face strong enemies of open-minded inquii-y and of unrestricted 
teaching. Today, as in Galileo's time, men and women who create 
or publicize new thoughts must be ready to stand up before those 
who fear and wish to suppress the open discussion of new ideas and 
new evidence. 

Plato knew that an authoritarian state is threatened by 
intellectual nonconformists and recommended for them the now 
well-known treatment: re-education, prison, or death. Not long ago, 
Soviet geneticists were required to discard well-established theories, 
not on the basis of compelling new scientific evidence, but because 
of conflicts with political doctrines. Similarly, discussion of the 
theory of relativity was banned from textbooks in Nazi Germany 
because Einstein's Jewish parentage was said to invalidate his 
work. Another example of intolerance was the condition that led 
to the "Monkey Trial" held during 1925 in Tennessee, where the 
teaching of Darwin's theory of biological evolution was attacked 
because it conflicted with certain types of biblical interpretation. 

On two points, one must be cautious not to romanticize the 
lesson of this episode. While a Galileo sometimes still may be 
persecuted or ridiculed, not everyone who is persecuted is, therefore, 
a Galileo. He may in fact be just wrong, or a crank. Secondly, it has 
turned out that, at least for a time, science in some form can 
continue to live in the most hostile surroundings. When political 
philosophers decide what may be thought and what may not, 
science will suffer (like everything else), but it will not necessarily 
be extinguished. Scientists can take comfort from the judgment of 
history. Less than 50 years after Galileo's trial, Newton's great 
book, the Principia. brilliantly united the work of Copernicus, 
Kepler, and Galileo with Newton's new statement of the principles 
of mechanics. Without Kepler and Galileo, there probably could 
have been no Newton. As it was, the work of these three, together 
with the work of many contemporaries working in the same 
spirit, marked the triumphant beginning of modern science. Thus, 
the hard-won new laws of science and new views of man's place in 
the world were established. What followed has been termed by 
historians The Age of Enlightenment. 

Q16 Which of the following appears to have contributed to 
Galileo's being tried by the Inquisition? 

(a) He did not believe in God. 

(b) He was arrogant. 

(c) He separated religious and scientific questions. 

(d) He wrote in Italian. 



Palomar Observatory houses the 
200-inch Hale reflecting telescope. 
It is located on Palomar Mountain in 
southern California. 




Over 200 years after his confinement 
in Rome, opinions had changed so 
that Galileo was honored as in the 
fresco "Galileo presenting his tele- 
scope to the Venetian Senate" by 
Luigi Sabatelli (1772-1850). 



SG 7.17 




7.1 The Project Physics learning materials 
particularly appropriate for Chapter 7 include the 
following: 
Experiments 

The Orbit of Mars 

The Orbit of Mercury 
Activities 

Three-dimensional Model of Two Orbits 

Inclination of Mars Orbit 

Demonstrating Satelhte Orbits 

Galileo 

Conic-Section Models 

Challenging Problems: Finding Earth-Sun 
Distance 

Measuring Irregular Areas 
Reader Articles 

Kepler 

Kepler on Mars 

Kepler's Celestial Music 

The Starry Messenger 

Galileo 
Film Loop 

Jupiter Satellite Orbit 
Transparency 

Orbit Parameters 

7.2 How large was an error of 8 minutes of arc in 
degrees? How far right or left do you think a dot 
over an i on this page would have to be before 
you would notice it was off-center? What angle 
would this shift be as seen from a reading distance 
of 10 inches? 

7.3 Summarize the steps Kepler used to determine 
the orbit of the earth. 

7.4 For the orbit positions nearest and furthest 
from the sun, a planet's speeds are inversely 
proportional to the distances from the sun. What 
is the percentage change between the earth's 
slowest speed in July when it is 1.02 AU from the 
sun, and its greatest speed in January when it is 
0.98 AU from the sun? 

7.5 Summarize the steps Kepler used to determine 
the orbit of Mars. 



7.6 In any ellipse the sumlor/the distances from 
the two foci to a point on the curve equals the 
length of the major axis, or (F,P + F2P) = a. This 
property of ellipses allows us to draw them by 
using a loop of string around two tacks at the 
foci. What should the length of the string be? 




7.7 In describing orbits around the sun, the point 
nearest the sun is called the perihelion point and 
the point farthest from the sun is called the 
aphelion point. The distances of these two points 
from the sun are called the perihelion distance 
and the aphelion distance respectively. The terms 
perihelion and aphelion come from the Greek, in 
which helios is the sun, peri means near, and apo 
means away from. 

(a) List some other words in which the prefixes 
peri and apo or ap have similar meanings. 

(b) In describing earth satellite orbits, the terms 
apogee and perigee are often used. What do 
they mean? 

(c) What would such points for satellites 
orbiting the moon be called? 

7.8 For the planet Mercury the perihelion distance 
(closest approach to the sun) has been found to be 
about 45.8 x 10'^ kilometers, and the aphelion 
distance (greatest distance from the sun) is about 
70.0 X lO** kilometers. What is the eccentricity of 
the orbit of Mercury? 

7.9 The eccentricity of Pluto's orbit is 0.254. What 
will be the ratio of the minimum orbital speed to 
the maximum orbital speed of Pluto? 

7.10 The rule Vj^R = const, makes it easy to find v^ 
for any point on an orbit if the speed and distance 
at any other point are known. Make a sketch to 
show how you would find v once you know v_^. 

7.11 Halley's comet has a period of 76 years, and 
its orbit has an eccentricity of .97. 

(a) What is its average distance from the sun? 

(b) What is its greatest distance from the sun? 

(c) What is its least distance from the sun? 

(d) How does its greatest speed compare with 
its least speed? 

7.12 The mean distance of the planet Pluto from 
the sun is 39.6 AU. What is the orbital period of 
Pluto? 

7.13 Three new major planets have been 
discovered since Kepler's time. Their orbital 
periods and mean distances from the sun are 
given in the table below. Determine whether 
Kepler's law of periods holds for these planets 
also. 



Discov- Orbital 
ery Period 
Date 



Average Eccentri- 

Distance city 

From of 

Sun Orbit 



Uranus 1781 
Neptune 1846 
Pluto 1930 



84.013yr 
164.783 
248.420 



19.19 AU 0.047 

30.07 0.009 

39.52 0.249 



78 



7.14 Considering the data available to him. do you 
think Kepler was justified in concluding that the 
ratio T^/Rav' is a constant? 

7.15 The chart on p. 79 is reproduced from the 
January, 1969. issue of Sky and Telescope. 

(a) Make a sketch of how Jupiter and its 
satellites appeared at one week intervals, 
beginning with dav "0." 



STUDY GUIDE 



JUPITER'S 
SATELLITES 

The four curving lines 
represent Jupiter's four 
bright (Galilean) satel- 
lites: I, lo; II, Europa; 
III, Ganymede; IV, Cal- 
listo. The location of 
the planet's disk is in- 
dicated by the pairs of 
vertical lines. If a moon 
is invisible because it is 
behind the disk (that 
is, occulted by Jupiter), 
the curve is broken. 

For successive dates, the 
horizontal lines mark 
0^ Universal time, or 7 
p.m. Eastern standard 
time (or 4 p.m. Pacific 
standard time) on the 
preceding date. Along 
the vertical scale, 1/16 
inch is almost seven 
hours. In this chart, 
west is to the left, as in 
an inverting telescope 
for a Northern Hemi- 
sphere observer. At the 
bottom, "d" is the point 
of disappearance of a 
satellite in the shadow 
of Jupiter; "r" is the 
point of reappearance. 
From the American 
Ephemeris and Nauti- 
cal Almanac. 



SATELLITES OF JUPITER, 1969 

CONFIGlfRATIONS OF SATELLITES I-IV FOR JANUARY 

INIVERSAI. TIMK 




\ 

,A 

PHASES OF THE ECLIPSES 



1 /- 


~-\ 


III ^ 




;(; 


3 


w : ;(^ 


') 


II X- 


— ^ 


IV . , ^ 


-^ 


w : ^ 


E3 E 


W a , |e 


S3 E 



(b) Make measurements of the chart to find the 
Rav and T for each satellite. (For this 
problem. Rav can be to any convenient scale, 
such as cm on the diagram.) 
(c) Does Kepler's law of periods T^/Rav* = 
constant hold for Jupiter's satellites? 
7.16 Below are two passages from Galileo's 
"Letters on Sunspots." On the basis of these 
quotations, comment on Galileo's characteristics 
as an observer and as a scientist. 

(May 4th, 1612) 

1 have resolved not to put anything around 
Saturn except what 1 have already observed and 
revealed -that is, two small stars which touch it, 
one to the east and one to the west, in which no 
alteration has ever yet been seen to take place 
and in which none is to be expected in the future, 
barring some very strange event remote from 
every other motion known to or even imagined 
by us. But as to the supposition of Apelles that 
Saturn sometimes is oblong and sometimes 
accompanied by two stars on its flanks. Your 
Excellency may rest assured that this results 



either from the imperfection of the telescope or 
the eye of the observer, for the shape of Saturn is 
thus: oQo ' as shown by perfect vision and perfect 
instruments, but appears thus:<0*. where 
perfection is lacking, the shape and distinction of 
the three stars being imperfectly seen 1, who 
have observed it a thousand times at different 
periods with an excellent instrument, can assure 
you that no change whatever is to be seen in it. 
And reason, based upon our experiences of all 
other stellar motions, renders us certain that none 
ever wUl be seen, for if these stars had any 
motion similar to those of other stars, they would 
long since have been separated from or conjoined 
with the body of Saturn, even if that movement 
were a thousand times slower than that of any 
other star which goes wandering through the 
heavens. 

(December 1, 1612) 

About three years ago 1 wrote that to my great 
surprise 1 had discovered Saturn to be three- 
bodied; that is, it was an aggregate of three stars 
arranged in a straight line parallel to the 

79 



STUDY GUIDE 



ecliptic, the central star being much larger than 
the others. I believed them to be mutually 
motionless, for when I first saw them they 
seemed almost to touch, and they remained so for 
almost two years without the least change. It was 
reasonable to believe them to be fixed with 
respect to each other, since a single second of arc 
(a movement incomparably smaller than any 
other in even the largest orbs) would have 
become sensible in that time, either by separating 
or by completely uniting these stars. Hence I 
stopped observing Saturn for more than two 
years. But in the past few days I returned to it 
and found it to be solitary, without its customary 
supporting stars, and as perfectly round and 
sharply bounded as Jupiter. Now what can be 
said of this strange metamorphosis? That the two 
lesser stars have been consumed in the manner 
of the sunspots? Has Saturn devoured his 
children? Or was it indeed an illusion and a fraud 
with which the lenses of my telescope deceived 
me for so long — and not only me, but many 
others who have observed it with me? Perhaps 
the day has arrived when languishing hope may 
be revived in those who, led by the most profound 
reflections, once plumbed the fallacies of all my 
new observations and found them to be incapable 
of existing! 

I need not say anything definite upon so strange 
an event; it is too recent, too unparalleled, and I 
am restrained by my own inadequacy and the 
fear of error. But for once I shall risk a little 
temerity; may this be pardoned by Your 
Excellency since I confess it to be rash, and 



protest that I mean not to register here as a 
prediction, but only as a probable conclusion. I 
say, then, that I believe that after the winter 
solstice of 1614 they may once more be observed. 

(Discoveries and Opinions of Galileo, translated 
by Stillman Drake, Doubleday, 1957, pp. 101-102, 
143-144.) 

7.17 What are the current procedures by which 
the public is informed of new scientific theories? 
Do you think they are adequate? To what extent 
do news media emphasize clashes of points of 
view? Bring in some examples from news 
magazines. 

7.18 Recently the Roman Catholic Church 
decided to reconsider its condemnation of Galileo. 
The article reproduced opposite, which appeared 
in The New York Times, July 1968, quotes 
passages from an Austrian Cardinal's view of 
the question. 

(a) In the quoted remarks Cardinal Konig lists 
three forms of knowledge: "divine 
revelations," "philosophical constructions," 
and "spontaneously naive views of reality." 
Under which of these do you think he would 
classify Galileo's claims? Would Galileo 
agree? 

(b) What seems to be the basis for the 
reconsideration? Is it doubt about the 
conclusions of the trial, or about the 
appropriateness of trying scientific ideas at 
all? Is it being reconsidered because of a 
change in Church philosophy, or because 
Galileo turned out to be right? 



80 



To Rehabilitate Galileo 



The following are excerpts 
from a speech entitled "ReUgion 
and Natural Sciences" by Franz 
Cardinal Konig of Vienna at a 
meeting of Nobel Prize winners 
in Germany last week. 

Neither the Christian churches 
nor modern science have man- 
aged to date to control that com- 
ponent of human nature which 
mirrors visibly a like phenome- 
non in the animal kingdom; ag- 
gressiveness. I hold that the 
neutralization of this instinct, 
which now is creating more 
dangers than ever before, ought 
to be a prime goal of objective 
cooperation between theologians 
and scientists. This work should 
try to bridge the incongruity be- 
tween man's complete and per- 
fected power of destruction and 
his psychic condition which re- 
mains unbridled and prey to 
atavism. 

Removing Barriers 

To enable such cooperation 
it is first of all necessary to 
remove the barriers of the past. 
Perhaps the biggest obstacle, 
blocking for centuries coopera- 
tion between religion and sci- 
ence, was the trial of Galileo. 

For the church after the sec- 
ond Vatican Council, turning as 
it is to the world as an advocate 
of legitimate rights and the 
freedom of the human mind, the 
time appears to have come to 
terminate as thoroughly as pos- 
sible the era of unpleasantness 
and distrust which began with 
Galileo's censure in 1G33. For 
over 300 years the scientific 
world has rightly regarded as a 
painful, unhealing wound the 
church's unjust verdict on one 
of those men who prepared the 
path for modern science. Gali- 



leo's judgment Is felt all the 
more painful today since all in- 
telligent people inside and out- 
side the church have come to the 
( onclusion that the scientist 
Galileo was right and that 
his work particularly gave mod- 
ern mechanics and physics a 
first, firm basis. His Insights er • 
abled the human mind to de- 
velop a new understanding of 
nature and universe, thus re- 
placing concepts and notions in- 
herited from antiquity. 

An open and honest clari- 
fication of the Galileo case ap- 
pears all the more necessary to- 
day if the church's claim to 
speak for truth, justice and 
freedom is not to suffer in cred- 
ibility and if those people are 
not to lose faith in the church 
who in past and picsent have 
defended freedom and the right 
to independent thought against 
various forms of totalitariansim 
and the so-called raison d'etat. 

I am in a position to announce 
before this meeting that com- 
petent authorities have already 
initiated steps to bring the 
Galileo case a clear and open 
solution. 

The Catholic Church is un- 
doubtedly ready today to sub- 
ject the judgment in the Galileo 
trial to a revision. Clarification 
of the questions which at Gal- 
ileo's time were still clouded 
allow the church today to re- 
sume the case with full confi- 
dence in itself and without 
prejudice. Faithful minds have 
struggled for truth under pain 
and gradually found the right 
way through experience and dis- 
cussions conducted with pas- 
sion. 
The church has learned to 



treat science with frankness and 
respect. It now knows that har- 
mony is possible between mod- 
ern man's scientific thinking 
and religion. The seeming con- 
tradiction between the Coperni- 
can system or, more precisely, 
the initial mechanics of modern 
physics and the Biblical story 
of creation has gradually disap- 
peared. Theology now differen- 
tiates more sharply between es- 
sentially divine revelations, phil- 
osophical constructions and 
spontaneously naive views of 
reality. 

What used to be insurmount- 
able obstacles for Galileo's con- 
temporaries have stopped long 
ago to irritate today's educated 
faithful. From their perspective 
Galileo no longer appears as a 
mere founder of a new science 
but also as a prominent propo- 
nent of religious thinking. In this 
field, too, Galileo was in many 
respects a model pioneer. 

Trial and Error 

In Gahleo's wake and in the 
spirit of his endeavors the Cath- 
olic church has through trial 
and error come to recognize the 
possibility of harmonious coop- 
eration between free research 
and free thinking on the one 
hand and absolute loyalty to 
God's word on the other. To- 
day's task is to draw the con- 
sequences from this recognition. 
Without fixing borders, God has 
opened his creation — the uni- 
verse — to man's inquiring mind. 

The church has no reason 
whatsoever to shun a revision 
of the disputed Galileo verdict. 
To the contrary, the case pro- 
vides the church with an op- 
portunity to explain its claim 
to infallibility in its realm and 
to define its limits. However, it 
will also be a chance to prove 
that the church values justice 
higher than prestige. 



Excerpt from The New York Times, July 1968. 



81 



8.1 Newton and seventeenth-century science 83 

8.2 Newton's Principia 87 

8.3 The inverse-square law of planetary force 90 

8.4 Law of universal gravitation 91 

8.5 Newton and hypotheses 96 

8.6 The magnitude of planetary force 98 

8.7 Planetary motion and the gravitational constant 102 

8.8 The value of G and the actual masses of the planets 104 

8.9 Further successes 106 

8.10 Some effects and limitations of Newton's work 111 



Isaac Newton (1642-1727) 



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CHAPTER EIGHT 



The Unity of Earth and Sky 
—The Work of Newton 



8.1 Newton and seventeenth-century science 



In the forty-five years between the death of Galileo in 1642 and 
the publication of Newton's Principia in 1687, major changes 
occurred in the social organization of scientific studies. The new 
philosophy of experimental science, applied by enthusiastic and 
imaginative men, was giving a wealth of new results. These men 
were beginning to work together and organize scientific societies in 
Italy, France, and England. One of the most famous is the Royal 
Society of London for Improving Natural Knowledge, which was 
founded in 1662. Through these societies the scientific experi- 
menters exchanged information, debated new ideas, argued against 
the opponents of the new experimental activities, published 
technical papers, and sometimes quarreled heatedly. Each society 
sought public support for its work and published studies in widely 
read scientific journals. Through the societies, scientific activities 
were becoming well-defined, strong, and international. 

This development of scientific activities was part of the general 
cultural, political, and economic changes occurring in the 1500's 
and 1600's (see the time chart on p. 84). Craftsmen and men of 
wealth and leisure became involved in scientific studies. Some 
sought the improvement of technological methods and products. 
Others found the study of nature through experiment a new and 
exciting hobby. But the availability of money and time, the growing 
interest in science, and the creation of organizations are not 
enough to explain the growing success of scientific studies. 
Historians agree that this rapid growth of science depended upon 
able men, well-formulated problems, and good experimental and 
mathematical tools. 

Some of the important scientists who lived between 1600 and 
1750 are shown in the time chart for the Age of Newton. The list 
includes amateurs as well as university professors. 

83 



SG8.1 



The forms "1500's" and "16th 
century" are used interchangeably 
in referring to the time period 
roughly between 1500 and 1600. 



1600 



1650 



1700 



3 
O 

S 




3 

.2 
o 
O 



CARDINAL RICHELIEU 

I 

OLIVER CROMWELL 

JEAN COLBERT 



WILLIAM III of England 




Oh 



PETER THE GREAT of Russia 



I 

LOUIS XIV of France 



RENE DESCARTES 

I 

WILLIAM HARVEY 

I 

JOHANNES KEPLER 




I 

ROBERT HOOKE 

1 

BLAISE PASCAL 



EDMUND HALLEY 

JEAN BERNOULLI 



ROBERT BOYLE 




CHRISTIAAN HUYGENS 
I 
GOTTFRIED LEIBNITZ 




Philosophy 
and Theolog 


THOMAS HOBBES 




SPINOZA 
JOHN LOCKE 



;:___ ,_ 

GEORGE BERKELEY 

I 

MONTESQUIEU 



BEN JONSON 

JOHN MILTON 



PETER PAUL RUBENS 



VELAZQUEZ 



GIOVANNI BERNINI 

REMBRANDT VAN RUN 




Section 8.1 



85 



Well-formulated problems were numerous in the writings of 
Galileo and Kepler. Their studies showed how useful mathematics 
could be when used together with experimental observation. 
Furthermore, their works raised exciting new questions. For 
example, what forces act on the planets to explain the paths 
actually observed? And why do objects fall as they do at the earth's 
surface? 

Good experimental and mathematical tools were being created. 
With mathematics being applied to physics, studies in each field 
stimulated developments in the other. Similarly, the instrument- 
maker and the scientist aided each other. 

Another factor of great importance was the rapid accumulation 
of scientific knowledge itself. From the time of Galileo, repeatable 
experiments reported in books and journals were woven into 
testable theories and were available for study, modification, and 
application. Each study could build on those done previously. 

Newton, who lived in this new scientific age, is the central person 
in this chapter. However, before we follow Newton's work, we must 
recall that in science, as in any other field, many men made useful 
contributions. The whole structure of science depends not only 
upon those whom we recognize as geniuses, but also upon many 
lesser-known men. As Lord Rutherford, one of the founders of 
modern atomic theory, said: 



It is not in the nature of things for any one man to make 
a sudden violent discovery; science goes step by step, and 





Newton entered Trinity College. Cam- 
bridge University, in 1661 at the age 
of eighteen. He was doing experi- 
ments and teaching while still a stu- 
dent. This early engraving shows the 
quiet student wearing a wig and heavy 
academic robes. 



This drawing of the reflecting tele- 
scope he invented was done by 
Newton while he was still a student. 



86 The Unity of Earth and Sky-The Work of Newton 

every man depends upon the work of his predecessors. . . . 
Scientists are not dependent on the ideas of a single man, 
but on the combined wisdom of thousands of men. 

To tell the story properly, we should trace fully each man's 
dependence upon those who worked before him, the influences of 
his contemporaries, and his influence upon his successors. While 
this would be interesting and rewarding, within the space 
available to us we can only briefly hint at these relationships. 

Isaac Newton was bom on Christmas Day, 1642, in the small 
English village of Woolsthorpe in Lincolnshire. He was a quiet 
farm boy, who, like young Galileo, loved to build mechanical 
gadgets and seemed to have a liking for mathematics. With 
financial help from an uncle he went to Trinity College of 
Cambridge University in 1661. There he enrolled in the study of 
mathematics (perhaps as applied to astrology) and was an 
enthusiastic and successful student. In 1665 the Black Plague, 
which swept through England, caused the college to be closed, and 
Newton went home to Woolsthorpe. There, by the time he was 
twenty-four, he had made spectacular discoveries in mathematics 
(the binomial theorem and diff'erential calculus), in optics (theory 
of colors), and in mechanics. During his isolation. Newton had 
formulated a clear concept of the first two laws of motion, the law 
of gravitational attraction, and the equation for centripetal 
acceleration. However, he did not announce the centripetal 
acceleration equation until many years after Huygens' equivalent 
statement. 

This must have been the time of the famous and disputed fall of 
the apple. One of the records of the apple story is in a biography of 
Newton written in 1752 by his friend, William Stukeley. In it we 
read that on a particular occasion Stukeley was having tea with 
Newton; they were sitting under some apple trees in a garden, and 
Newton recalled that: 

he was just in the same situation, as when formerly, the 
notion of gravitation came into his mind. It was occasion'd 
by the fall of an apple, as he sat in a contemplative mood. 
Why should that apple always descend perpendicularly 
to the ground, thought he to himself. Why should it not go 
sideways or upwards, but constantly to the earth's centre? 

The main emphasis in this story should probably be placed on the 
contemplative mood and not on the apple. Moreover, it fits again 
the pattern we have seen before: a great puzzle (here, that of the 
forces acting on planets) begins to be solved when a clear-thinking 
person contemplates a long-known phenomenon (such as the fall 
of objects on earth). Where others had seen no relationship, 
Newton did. Refering to the plague years Newton once wrote, 

I began to think of gravity extending to the orb of the 
moon, and . . . from Kepler's rule [third law, law of 



Section 8.2 



87 



periods] ... I deduced that the forces which keep the 
Planets in their orbs must be reciprocally as the squares 
of their distances from the centers about which they 
revolve: and thereby compared the force requisite to keep 
the moon in her orb with the force of gravity at the 
surface of the earth, and found them to answer pretty 
nearly. All this was in the two plague years of 1665 and 
1666, for in those days I was in the prime of my age for 
invention, and minded mathematics and philosophy more 
than at anv time since. 



Soon after Newton's return to Cambridge, he was chosen to 
follow his former teacher as professor of mathematics. He taught at 
the university and contributed papers to the Royal Society. At first, 
his contributions were mainly on optics. His Theory of Light and 
Colors, finally published in 1672, was the occasion of so long and 
bitter a controversy with certain other scientists that the 
introspective and complex man resolved never to publish anything 
more. 

In 1684 Newton's devoted friend Halley, a noted astronomer, 
came to ask his advice in a controversy with Wren and Hooke about 
the force that would have to act on a body to cause it to move along 
an ellipse in accord with Kepler's laws. Halley was pleasantly 
surprised to learn that Newton had already derived the exact 
solution to this problem ("and much other matter"). Halley then 
persuaded his friend to publish these studies that solved one of the 
most debated and interesting scientific problems of the time. To 
encourage Newton, Halley became responsible for all the costs of 
publication. Less than two years later, after incredible labors. 
Newton had the Principia ready for the printer. Publication of the 
Principia in 1687 quickly estabhshed Newton as one of the greatest 
thinkers in history. 

Several years afterward. Newton had a nervous breakdown. He 
recovered, but from then until his death, thirty-five years later, he 
made no major scientific discoveries. He rounded out earlier studies 
on heat and optics and turned more and more to writing on 
theology. During those years he received many honors. In 1699 he 
was appointed Warden of the Mint and subsequently its Master, 
partly because of his great interest in and knowledge about the 
chemistry of metals. In that office he helped to re-establish the 
value of British coins, in which lead and copper were being 
included in place of silver and gold. In 1689 and 1701 he represented 
Cambridge University in Parhament, and he was knighted in 1705 
by Queen Anne. He was president of the Royal Society from 1703 
to his death in 1727. He was buried in Westminster Abbey. 



PHILOSOPHIZE 

NATURALIS 

PRINCI PI A 

MATHEMATICA 



Autorc J S. NEWrOiV, Tnn. CM. Ciivjh. Sx. Mathrftos ' 
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luff. .<^,ci^„ R^t.x ic Typis Jrfcph, S,r,jirr. Proftjt apud 
f pluri. BibliopoUs. Am MDCLXXXV II. 



Title page of Newton's Principia 
mathematica. Because the Royal So- 
ciety sponsored the book, the title 
page includes the name of the So- 
ciety's president, Samuel Pepys, fa- 
mous for his diary, which describes 
life during the seventeenth century. 



8.2 Newton's Principia 



In the original preface to Newton's Principia we find a clear 
outline of the book: 



88 



The Unity of Earth and Sky -The Work of Newton 



These rules are stated by Newton 
at the beginning of Book ill of the 
Principia. 



Since the ancients (as we are told by Pappus) 
esteemed the science of mechanics of greatest importance 
in the investigation of natural things, and the modems, 
rejecting substantial forms and occult qualities, have 
endeavored to subject the phenomena of nature to the 
laws of mathematics, I have in this treatise cultivated 
mathematics as far as it relates to philosophy [we would 
say 'physical science'] ... for the whole burden of 
philosophy seems to consist in this— from the phenomena 
of motions to investigate [induce] the forces of nature, 
and then from these forces to demonstrate [deduce] the 
other phenomena, and to this end the general propositions 
in the first and second Books are directed. In the third 
Book I give an example of this in the explication of the 
system of the World; for by the propositions mathe- 
matically demonstrated in the former Books, in the third 
I derive from the celestial phenomena the forces of 
gravity with which bodies tend to the sun and the several 
planets. Then from these forces, by other propositions 
which are also mathematical, I deduce the motions of 
the planets, the comets, the moon, and the sea [tides] .... 

The work begins with definitions— mass, momentum, inertia, 
force. Next come the three laws of motion and the principles of 
addition for forces and velocities (discussed in Unit 1). Newton 
also Included an equally important and remarkable passage 
on "Rules of Reasoning in Philosophy." The four rules, or 
assumptions, reflect his profound faith in the uniformity of all 
nature. They were intended to guide scientists in making hypotheses, 
and also, we might say, to lay his philosophical cards on the table. 
These rules which had their roots in ancient Greece, are still useful. 
The first has been called a Principle of Parsimony, the second and 
third. Principles of Unity. The fourth expresses a faith needed for 
us to use the process of logic. 

In a brief form, and using some modern language, Newton's 
rules are: 

1. "Nature does nothing ... in vain, and more is in vain 
when less will serve." Nature is essentially simple; 
therefore we ought not to introduce more hypotheses than 
are sufficient and necessary for the explanation of 
observed facts. This fundamental faith of all scientists is 
nearly a paraphrase of Galileo's "Nature . . . does not that 
by many things, which may be done by few." Galileo in 
turn was reflecting an opinion of Aristotle. Thus, the 
belief in simplicity has a long history. 

2. "Therefore to the same natural eff"ects we must, as far 
as possible, assign the same causes. As to respiration in a 
man and in a beast; the descent of stones in Europe and 
in America; . . . the reflection of light in the earth, and 

in the planets." 

3. Properties common to all those bodies within reach of 
our experiments are to be assumed (even if only 
tentatively) to apply to all bodies in general. For example, 



Section 8.2 



89 



since all physical objects known to experimenters had 
always been found to have mass, this rule would guide 
Newton to propose that every object has mass (even 
those beyond our reach, in the celestial region). 
4. In "experimental philosophy," hypotheses or 
generalizations which are based on experience are to be 
accepted as "accurately or very nearly true, notwith- 
standing any contrary hypotheses that may be imagined" 
until we have additional evidence by which our 
hypotheses may be made more accurate or revised. 

The Principia was an extraordinary document. Its three main 
sections contained a wealth of mathematical and physical 
discoveries. But overshadowing everything else in the book is the 
theory of universal gravitation, with the proofs and arguments 
leading to it. Newton uses a form of argument patterned after that 
of Euclid — the type of proofs you encountered in studying geometry. 
Because the style of detailed mathematical steps used in the 
Principia is no longer so familiar, many of the steps given above 
have been restated in modern terms. 

The central idea of universal gravitation can be simply stated: 
every object in the universe attracts every other object. 
Moreover, the amount of attraction depends in a simple way on the 
masses of the objects and the distance between them. 

This was Newton's great synthesis, boldly bringing together the 
terrestrial laws of force and motion and the astronomical laws of 
motion. Gravitation is a universal force that applies to the earth 
and apples, to the sun and the planets, and to all other 
bodies (such as comets, moving in the solar system). Heaven and 
Earth were united in one grand system dominated by the Law of 
Universal Gravitation. The general astonishment and awe were 
reflected in the words of the English poet Alexander Pope: 



Notice that Newton's assumption 
denies the distinction between 
terrestrial and celestial matter. 



You should restate these rules in 
your own words before going on to 
the next section. (A good topic for 
an essay would be whether 
Newton's rules of reasoning are 
applicable outside of science.) 



Nature and Nature's laws lay hid in night: 
God said. Let Newton be! and all was light. 



As you will find by inspection, the Principia, written in Latin, 
was filled with long, geometrical arguments and was difficult to 
read. Happily, gifted popularizers wrote summaries that allowed a 
wide circle of readers to learn of Newton's arguments and 
conclusions. One of the most widely read of these popular books was 
published in 1736 by the French philosopher and reformer Voltaire. 

Readers of these books must have been excited and perhaps 
puzzled by the new approach and assumptions. From ancient 
Greece until well after Copernicus, the ideas of natural place and 
natural motion had been used to explain the general position and 
movements of the planets. From the time of the Greeks it was 
widely believed that the planets moved in their orbits because that 
was their "natural motion. " However, to Newton the natural motion 
of a body was at a uniform rate along a straight line. Motion 
in a curve was evidence that a net force was continuously 



90 The Unity of Earth and Sky-The Work of Newton 

accelerating the planets away from their natural motion along 
straight lines. Yet the force acting on the planets was entirely 
natural, and acted between all bodies in heaven and on the earth. 
Furthermore, it was the same force that caused bodies on the earth 
to fall. What a reversal of the assumptions about what was 
"natural"! 

8.3 The inverse-square law of planetary force 

Newton believed that the natural path of a planet was a straight 
line and that it was forced into a curved path by the influence of 
the sun. He was able to show that Kepler's law of areas could be 
true if. and only if, forces exerted on the planets were always 
directed toward a single point. (Details of his argument for this 
"central" force are given on the special pages. Motion under a 
central force.) He showed also that the single point was the 
location of the sun. The law of areas will be satisfied no matter 
what the magnitude of the force is, as long as it is always directed 
to the same point. So it was still necessary to show that a central 
gravitational force would cause the precise relationship observed 
between orbital radius and period. But how great was the 
gravitational force, and how did it differ for different planets? 

Newton proved that the centripetal accelerations of the six 
known planets toward the sun decreased inversely as the square 
of the planets' average distances from the sun. The proof for 
circular orbits is very short. The expression for centripetal 
acceleration a^. of a body moving uniformly in a circular path, in 
terms of the radius R and the period T. is 

47r-R 
flc — jr~ 

(We derived this expression in Chapter 4.) As Kepler claimed in his 
law of periods, there is a definite relation between the orbital 
periods of the planets and their average distances from the sun: 

T'^/Rav'' = constant 

If we use the symbol k for the constant, we can write 

T' = kR.,J 

For circular orbits, i?,, is just R. Substituting kR' for T- in the 
centripetal force equation gives 

_ 4tt-R _ 4tt'- 
^'■~ kR' ~kR- 

Since 4Tr^lk is a constant, we can write simply 

1 

a '^ — 

This conclusion follows necessarily from Kepler's law of periods 
and the definition of acceleration. If Newton's second law F ^ a 



Section 8.4 



91 



holds for planets as well as for bodies on earth, then there must be 
a centripetal force F^ acting on a planet, and it must decrease in 
proportion to the square of the distance of the planet from the sun: 
F ^ a holds for planets as well as for bodies on earth, then there 
must be a centripetal force F^. acting on a planet, and it must 
decrease in proportion to the square of the distance of the planet 
from the sun: 



Newton showed that the same result holds for ellipses— indeed 
that any object moving in an orbit, that is a conic section (circle, 
ellipse, parabola, or hyperbola), around a center of force is being 
acted upon by a centripetal force that varies inversely with the 
square of the distance from the center of force. 

Newton had still more evidence from the telescopic observations 
of Jupiter's satellites and Saturn's satellites. The satellites of 
Jupiter obeyed Kepler's law of areas around Jupiter as a center, and 
the satellites of Saturn obeyed it around Saturn as a center. For 
Jupiter's satellites. Kepler's law of periods T- R^ = constant held, 
but the value of the constant was different from that for the 
planets around the sun. It held also for Saturn's satellites, but with 
still a different constant. Therefore. Jupiter's satellites were acted 
on by a central force directed toward Jupiter, and the force decreased 
with the square of the distance from Jupiter and similarly for 
Saturn's satellites and Saturn. So Newton was able to show that the 
observed interactions of astronomical bodies could be accounted 
for by a "1/R-" attractive central force. 

Q1 What can be proved from the fact that the planets sweep 
out equal areas with respect to the sun in equal times? 

Q2 With what relationship can T^lRay^ ^constant be combined 
to prove that the gravitational attraction varies as 1/R-? 

Q3 What simplifying assumption was made in the derivation 
given in this section? 

Q4 Did Newton limit his own derivation by the same 
assumption? 



In Newton's time, four of Jupiter's 
satellites and four of Saturn's 
satellites had been observed. 



SG 8.2 



8.4 Law of universal gravitation 



Subject to further evidences, we shall now accept that a central 
force is holding the planets in their orbits. Furthermore, the 
strength of this central force changes inversely with the square of 
the distance from the sun. This strongly suggests that the sun is 
the source of the force— but it does not necessarily require this 
conclusion. Newton's results so far include no physical mechanism. 

The French philosopher Descartes (1596-1650) had proposed a 
theory in which all space was filled with a subtle, invisible fluid 



Motion under a central force 



How will a moving body respond to a 
central force? In order to follow Newton's 
analysis, we shall need to remennber that 
the area of a triangle equals t base x 
altitude. Any of the three sides can be chosen 
as the base, and the altitude is the 
perpendicular distance to the opposite 
corner. 

Suppose that a body was initially 
passing some point P, already moving at 
uniform speed v along the straight line 
through PQ. (See Fig. A below.) If it goes on 



(A^ 



with no force acting, then in equal intervals 
of time Af it will continue to move equal 
distances, PQ, QR, RS, etc. 

How will its motion appear to an 
observer at some point O? Consider the 
triangles OPQ and OQR in Fig. B below. 




The triangles have equal bases, PQ = QR = RS, 
and also equal altitudes, ON for all three. 
Therefore the triangles OPQ and OQR have 
equal areas. And therefore the line drawn from 
an observer at point O to the body moving 
at a uniform speed in a straight line PQR will 
sweep over equal areas in equal times. 




So, strange as it may seem at first, 
Kepler's law of areas applies even to a body 
on which there is no net force, and which 
therefore is moving uniformly along a straight 
line. 

How will the motion of the object we 
discussed in Fig. A be changed if, while 
passing through point 0, it is exposed to a 
brief force, such as a blow, directed toward 
point O? (Refer to Fig. D below.) 




(D) 



First consider what happens if a body 
initially at rest at point Q were exposed to 
the same blow. The body would be accelerated 
during the blow toward O. It would then 
continue to move toward O at constant speed, 
and after some definite time interval At, it will 
have moved a definite distance to a new point 
Q'. (See Fig. E on the next page.) 




Now consider the effect of the blow on 
the object that was initially moving toward 
point R. The resultant motion is the combination 
of these two components— to point R'. (See 
Fig. F below.) 




Earlier we found that the areas of the 
triangles OPQ and OQR were equal. Is the 
area of the triangle OQR' the same? Both 
triangles OQR and OQR' have a common base, 
00. Also, the altitudes of both triangles are 
the perpendicular distance from line 00 to 
line RR'. (See Fig. G.) Therefore, the areas of 
triangles OQR and OQR' are equal. 

If now another blow directed toward 
were given at point R', the body would move 




o (G) 

to some point S", as indicated in Fig. H below. 
By a similar analysis you can find that the 
areas of triangles OR'S" and OR'S' are equal. 
Their areas also equal the area of triangle 
OPQ. 




In this geometrical argument we have 
always applied the force toward the same 
point, O. A force always directed toward a 
single point is called a central force. (Notice 
that the proof has nothing to do with the 
magnitude of the force, or how it changes 
with distance from O.) Also, we have applied 
the force at equal intervals Af. If each time 
interval Af were made vanishingly small, so 
that the force would appear to be applied 
continuously, the argument would still hold. 
We then have an important conclusion: If a 
body Is acted upon by any central force, it will 
move in accordance with Kepler's law of areas. 



94 



The Unity of Earth and Sky-The Work of Newton 



SG 8.3 



To us, who have heard about 
gravity from our early school years, 
this may not seem to have been a 
particularly clever idea. But in 
Newton's time, after centuries of 
believing celestial events to be 
completely different from earthly 
events, it was the mental leap of a 
genius. Newton had already 
assumed the planets to be subject 
to the earth's laws of motion when 
he derived a 1/R- force law using 
the formula for a,.. But it was a still 
greater step to guess that the force 
on planets was not some special 
celestial force, but nothing other 
than familiar old weight of everyday 
objects. 



which carried the planets around the sun in a huge whirlpool-hke 
motion. This was a useful idea, and at the time it was widely 
accepted. However, Newton was able to prove by an elaborate and 
precise argument that this mechanism could not account for the 
details of planetary motion summarized in Kepler's laws. 

Kepler had made a different suggestion some years earlier. He 
proposed that some magnetic force reached out from the sun to 
keep the planets moving. His model was inadequate, but at least 
he was the first to regard the sun as the controlling mechanical 
agent behind planetary motion. And so the problem remained: was 
the sun actually the source of the force? If so. on what characteristics 
of the sun or the planets did the amount of the force depend? 

As you read in Sec. 8.1, Newton had begun to think about the 
planetary force during the years at home in the time of the Black 
Plague. The idea came to him — perhaps while watching an apple 
fall and perhaps not — that the planetary force was the same kind of 
force that pulled objects down to the earth near its surface. He first 
tried this idea on the earth's attraction for the moon. From the data 
available to him, Newton knew that the distance between the 
center of the earth and the center of the moon was nearly sixty 
times the radius of the earth. If the attractive force varied as 1/R-, 
the gravitational acceleration the earth exerts on matter at the 
distance of the moon should be only 1/60^ (or 1/3600) of that exerted 
upon matter (say, an apple) at the surface of the earth. From 
observations of falling bodies it was long known that the 
gravitational acceleration at the earth's surface was about 9.80 
meters per second per second. Therefore, the moon should fall at 
1/3600 of that acceleration value: (9.80/3600) meters per second per 
second, or 2.72 x 10~^ m/sec^ Does it? 

Newton started from the knowledge that the orbital period of the 
moon was very nearly 277 days. The centripetal acceleration a^ 
of a body moving uniformly with period T in a circle or radius R 
(developed in Sec. 4.6 of Unit 1) is a^ = 47r^RIT'-. When we put in 
values for the known quantities R and T (in meters and seconds) 
for the moon, and do the arithmetic, we find that the observed 
acceleration is: 



a, = 2.74 X 10-'' m/sec- 

This is a very good agreement. From the values available to 
Newton, which were close to these, he concluded that he had 

. . . compared the force requisite to keep the moon in her 
orbit with the force of gravity at the surface of the earth, 
and found them to answer pretty nearly. 



Therefore, the force by which the moon is retained in 
its orbit becomes, at the very surface of the earth, equal 
to the force of gravity which we observe in heavy bodies 
there. And, therefore, (by Rules of Reasoning 1 and 2) the 
force by which the moon is retained in its orbit is that 
very same force which we commonly call gravity .... 



Section 8.4 



95 



This was really a triumph: the same gravity that brings little 
apples down from the tree also holds the moon in its orbit. This 
assertion is the first portion of what is known as the Law of 
Universal Gravitation and says: every object in the universe attracts 
every other object with a gravitational force. If this is so, there 
must be gravitational forces not only between a rock and the earth, 
but also between the earth and the moon, between Jupiter and its 
satellites — and between the sun and each of the planets. 

But Newton did not stop by saying only that there is a 
gravitational force between the planets and the sun. He further 
claimed that the force is just exactly the right size to account 
completely for the motion of every planet. No other mechanism is 
needed — no whirlpools in invisible fluids, no magnetic forces. 
Gravitation, and gravitation alone, underlies the dynamics of the 
heavens. 

Because this concept is so commonplace to us. we are in 
danger of passing it by without really understanding what it was 
that Newton was claiming. First, he proposed a truly universal 
physical law. Guided by his Rules of Reasoning, which allowed him 
to extend to the whole universe what he found true for its observable 



Sun 




\ 



Earth 



The sun, moon, and earth each 
pull on the other. The forces are in 
matched pairs, in agreement with 
Newton's third law of motion. As the 
moon moves through space, the gravi- 
tational attraction of the earth causes 
the moon to ■fall " toward the earth. 
The continuous combination of its 
straight line inertial motion and its 
"fall " produce the curved orbit. 



A drawing by which Descartes (1596- 
1650) illustrated his theory of space 
being filled with whirlpools of matter 
that drive the planets along their 
orbits. 



96 The Unity of Earth and Sky-The Work of Newton 

parts, he excluded no object in the universe from the effect of 
gravity. 

Less than a century before, it would have been fooHsh and even 
dangerous to suggest that terrestrial laws and forces were the same 
as those that regulated the whole universe. But Kepler and Galileo 
had begun the unification of the physics of the heavens and earth 
which Newton was able to carry to its conclusion. This extension of 
the mechanics of terrestrial objects to explain also the motion of 
celestial bodies is called the Newtonian synthesis. 

A second feature of Newton's claim, that the orbit of a planet 
is determined by the gravitational attraction between it and the 
sun, was to move physics away from geometrical explanations and 
toward physical ones. Most philosophers and scientists before 
Newton had been occupied mainly with the question "What are the 
motions?" Newton shifted this to ask, "What force explains the 
motions?" In both the Ptolemaic system and Copernicus" system 
the planets moved about points in space rather than about objects. 
and they moved as they did owing to their "nature " or geometrical 
shape, not because forces acted on them. Newton, on the other 
hand, spoke not of points, but of things, of objects, of physical 
bodies. Without the gravitational attraction to the sun to deflect 
them continuously from straight-line paths, the planets would fly 
out into the darkness of space. Thus, it was the physical sun which 
was important, rather than the point at which the sun happened to 
be located. 

Newton's synthesis centered on the idea of gravitational force. 
In calling it a force of gravity. Newton knew, however, that he was 
not explaining why it should exist. When you hold a stone above the 
surface of the earth and release it, it will accelerate to the ground. 
Our laws of motion tell us that there must be a force acting on the 
stone accelerating it toward the earth. We know the direction of 
the force, and we can find the magnitude of the force by multiplying 
the mass of the stone by the acceleration. We can give it a name: 
weight, or gravitational attraction to the earth. But why there is 
such an interaction between bodies remains a puzzle. It is still 
an important problem in physics today. 

Q5 What idea came to Newton while he was thinking about 
falling objects and the moon's acceleration? 

Q6 Kepler, too, believed that the sun exerted forces on the 
planets. How did his view differ from Newton's? 

Q7 The central idea of Chapter 8 is the "Newtonian 
synthesis." What did Newton bring together? 



8.5 Newton and hypotheses 

Newton's claim that there is a mutual force (gravitational 
interaction) between a planet and the sun raised a new question. 



Section 8.5 97 

How can a planet and the sun act upon each other at enormous 
distances without any visible connections between them? On earth 
you can exert a force on an object by pushing it or pulling it. We are 
not troubled when we see a cloud or a balloon drifting across the 
sky, even though nothing seems to be touching it; although air is 
invisible, we know that it is actually a material substance which 
we can feel when it blows against us. Objects falling to the earth 
and iron objects being attracted to a magnet are more troublesome 
examples, but the distances are small. However, the earth is over 
90 million miles, and Saturn more than 2 billion miles, from the 
sun. How could there possibly be any physical contact between such 
distant objects? How can we account for such "action at a 
distance"? 

There were in Newton's time, and for a long time afterward, a 
series of suggestions to explain how mechanical forces could be 
exerted at such distances. Most of these involved imagining space 
to be filled with some invisible substance that would transmit the 
force. But, at least in public. Newton refused to speculate on 
possible mechanisms, because he could find no way to devise an 
experiment to test his private guess that an ether was involved. As 
he said in a famous passage in the General Scholium which he 
added in his second edition of the Principia (1713): 

. . . Hitherto I have not been able to discover the cause of 
those properties of gravity from phenomena, and I frame 
no hypotheses; for whatever is not deduced from the 
phenomena is to be called an hypothesis; and hypotheses, 
whether metaphysical or physical, whether of occult 
qualities or mechanical, have no place in experimental 
philosophy .... And to us it is enough that gravity does 
really exist, and act according to the laws which we have 
explained, and abundantly serves to account for all the 
motions of the celestial bodies, and of our sea. 

We quoted Newton at length because one particular phrase is 
frequently misquoted and misinterpreted. The original Latin reads: 
hypotheses nonfingo. This means "I frame no hypotheses" or "I do 
not feign hypotheses," in the sense of "I do not make false 
hypotheses." We know that Newton did make numerous hypotheses 
in his many publications, and his letters to friends contain many 
other speculations which he did not publish. So his stern disavowal 
of hypotheses in the General Scholium must be properly interpreted. 
The lesson to be drawn (and it is equally useful today) is that there 
are two main kinds of hypotheses or assumptions: 
(1) The most frequently encountered kind is a proposal of some 
hidden mechanism to explain observations. For example, after 
looking at the moving hands of a watch, we can quickly invent or 
imagine some arrangement of gears and springs that causes the 
motion. This would be a hypothesis that is directly or indirectly 
testable, at least in principle, by reference to phenomena. Our 
hypothesis about the watch, for example, can be tested by opening 
the watch or by x-raying it. Newton felt that the hypothesis of an 



98 The Unity of Earth and Sky -The Work of Newton 

invisible fluid that transmitted gravitational force, the so-called 
"ether," has directly testable consequences quite apart from what 
it was first invented to account for. Many experiments had been 
tried to "catch" the ether; to see, for example, if any wind, or 
pressure, or friction due to the ether remained in a bottle from 
which air had been evacuated. Nothing of this sort worked (nor has 
it since). So Newton wisely refrained from making a hypothesis that 
he felt should be testable, but that was not at that time. 
(2) A quite different type of assumption, often made in published 
scientific work, both of Newton and of scientists to this day, is a 
hypothesis of the sort which everyone knows is not directly testable, 
but which is necessary nevertheless jwst to get started on one's 
work. An example is such a statement as "nature is simple" or any 
other of Newton's Four Rules of Reasoning. The commitment to 
either the heliocentric system or the geocentric system was of the 
same kind, since all "the phenomena" could be equally accommo- 
dated in either. In choosing the heliocentric system over its rival, in 
making the hypothesis that the sun is at the center of the universe, 
Copernicus, Kepler, and Galileo were not proposing a directly 
testable hypothesis; rather, they were adopting a point of view 
which seemed to them more convincing, more simple, and as 
Copernicus put it, more "pleasing to the mind." It was this kind of 
hypothesis that Newton used without apology in his published work. 

Every scientist's work involves both kinds of hypothesis. In 
addition — and quite contrary to the commonly held stereotype of a 
scientist — as a person who uses only deliberate, logical, objective 
thoughts — the scientist feels quite free to consider any guess, 
speculation, or hunch, whether it is yet provable or not, in the hope 
it might be fruitful. (Sometimes these are dignified by the phrase 
"working hypotheses.") But. like Newton, most scientists today do 
not like to publish something which is still only an unproven hunch. 

Q8 Did Newton explain the gravitational attraction of all 
bodies? 

Q9 What was the popular type of explanation for "action at a 
distance"? 

Q10 Why didn't Newton use this type of explanation? 

Q11 What are two main types of hypotheses used in science? 



8.6 The magnitude of planetary force 

The general statement that gravitational forces exist universally 
must now be turned into a quantitative law that gives an 
expression for both the magnitude and direction of the forces any 
two objects exert on each other. It was not enough to assert that 
a mutual gravitational attraction exists between the sun and say, 
Jupiter. To be convincing. Newton had to specify what quantitative 
factors determine the magnitudes of those mutual forces, and how 
they could be measured, either directly or indirectly. 



Section 8.6 



99 



The first problem was defining precisely the distance R- should 
it, for example, be taken as the distance between the surface of 
the earth and the surface of the moon? For many astronomical 
problems, the sizes of the interacting bodies are extremely small 
compared to the distances between them; for these cases the 
distance between the surfaces is practically the same as the 
distance between the centers, or any other part of the bodies. (For 
Newton's original case of the earth and the moon, the distance 
between centers was about 2% greater than the distance between 
surfaces.) Some historians believe that Newton's uncertainty about 
a rigorous answer to this problem led him to drop the study for 
many years. 

Eventually Newton was able to prove that the gravitational 
force exerted by a spherical body was the same as if all its mass 
were concentrated at its center. Conversely, the gravitational force 
exerted on a spherical body by another body is the same as would 
be exerted on it if all its mass were concentrated at its center. 
Therefore, the distance R in the Law of Gravitation is the distance 
between centers. 

This was a very critical discovery. It allows us to consider the 
gravitational attraction between spherical bodies as though their 
masses were concentrated at single points; in thought we can 
replace the objects by mass-points. 

If we believe that Newton's third law (action equals reaction) is 
applicable universally, we must conclude that the amount of force 
the sun exerts on the planet is exactly equal to the amount of force 
the planet exerts on the sun. The claim that the forces are equal 
and opposite, even between a very large mass and a small mass, 
may seem contrary to common sense. But the equality is easy to 
prove if we assume only that Newton's third law holds between 
small chunks of matter: for example, that a 1-kg chunk of Jupiter 
pulls on a 1-kg chunk of the sun as much as it is pulled by it. 
Consider for example, the attraction between Jupiter and the sun, 
whose mass is about 1000 times greater than Jupiter's. As the 
figure in the right margin indicates, we could consider the sun as a 
globe containing 1000 Jupiters. Let us call one unit of force the 
force that two Jupiter-sized masses exert on each other when 
separated by the radius of Jupiter's orbit. Then Jupiter pulls on the 
sun (a globe of 1000 Jupiters) with a total force of 1000 units. 
Because each of the 1000 parts of the sun pulls on the planet 
Jupiter with one unit, the total pull of the sun on Jupiter is also 
1000 units. Each part of the massive sun not only pulls on the 
planet, but is also pulled upon by the planet. The more mass there 
is to attract, the more there is to be attracted. (But although the 
mutual attractive forces are equal in magnitude, the resulting 
accelerations are not. Jupiter pulls on the sun as hard as the sun 
pulls on Jupiter, but the sun responds to the pull with only 1/1000 
of the acceleration -its inertia, remember, is 1000 times Jupiter's.) 

In Sec. 3.8 of Unit 1, we developed an explanation for why 
bodies of different mass fall with the same acceleration near the 



PLa-ne-t- 




The gravitational force on a planet 
owing to the sun's pull is equal and 
opposite to the gravitational force on 
the sun owing to the planet. 



Sun - lOOO Jopit-ers 




UlilllUI 

Jujoite-r joulLs or\ lOOO 
pa-cts o-f t-hc. sun 

\000 parts of tA^e. ^>on 
pull on jL-ipit-cr 



100 The Unity of Earth and Sky-The Work of Newton 

earth's surface: the greater the inertia of a body, the more strongly 
it is acted upon by gravity. Or, more precisely: near the earth's 
surface, the gravitational force on a body is directly proportional to 
its mass. Like Newton, let us generalize this earthly effect to all 
gravitation and assume that the gravitational force exerted on a 
planet by the sun is proportional to the mass of the planet. 
Similarly, the gravitational force exerted on the sun by the planet 
is proportional to the mass of the sun. Since the forces the sun and 
a planet exert on each other are equal in magnitude, it follows that 
the magnitude of the gravitational force is proportional to the mass 
of the sun and to the mass of the planet. That is, the gravitational 
attraction between two bodies is proportional to the product of their 

SG 8.4 masses. If the mass of either body is tiipled. the force is tripled. If 
the masses of both bodies are tripled, the force is increased by a 
factor of 9. If we use the symbol F^^^v for the magnitude of the 
forces we can write F„av oc Wpianet^sun- 

Thus far we have concluded that the amount of attraction 
between the sun and a planet will be proportional to the product of 

SG 8.5 the masses. Earlier we concluded that the attraction also depends 
universally on the square of the distance between the bodies. Once 
again we combine the two proportionalities to find one force law 
that now includes masses as well as distance: 

J-, "^planet'^sun 

i' grav °^ JJ2 

Such an expression of proportionality can be written as an 
equation by introducing a constant to allow for the units of 
measurement used. Using G for the proportionality constant, we can 
write the law of planetary forces as: 

F-G - 

This equation is a bold assertion that the force between the sun 
and any planet depends only upon the masses of the sun and planet 
SG 8.6 and the distance between them. This equation seems unbelievably 
simple when we remember the observed complexity of the planetary 
motions. Yet every one of Kepler's empirical Laws of Planetary 
Motion, as we shall see, is consistent with this relation. More than 
that, Kepler's empirical laws can be derived from this force law 
together with Newton's second law of motion. But more important 
still, the force law allowed the calculation of details of planetary 
motion that were not possible using only Kepler's laws. 

Newton's proposal that such a siinple equation describes 
completely the forces between the sun and planets was not the final 
step. He believed that there was nothing unique or special about 
the mutual force between the sun and planets, or the earth and 
apples: so that an identical relation should apply universally to any 
two bodies separated by a distance that is large compared to the 
dimensions of the two bodies — even to two atoms or two stars. That 
is. he proposed that we can write a geiieral Law of Universal 
Gravitation: 



Section 8.6 



101 



where m, and m.^ are the masses of the bodies and R is the distance 
between their centers. The numerical constant G, called the 
constant of universal gravitation, Newton assumed to be the 
same for all gravitational interaction, whether between two grains 
of sand, two members of a solar system, or two stars in different 
parts of the sky. As we shall see. the successes made possible by this 
simple relationship have been so great that we have come to 
assume that this equation applies everywhere and at all times, past, 
present, and future. 

Even before we gather more supporting evidence, the sweeping 
majesty of Newton's theory of universal gravitation commands our 
wonder and admiration. It also leads to the question of how such 
a bold universal theory can be proved. There is no complete proof, 
of course, for that would mean examining every interaction 
between all bodies in the universe! But the greater the variety of 
single tests we make, the greater will be our belief in the 
correctness of the theory. 



SG 8.7 
SG 8.8 



Q12 According to Newton's law of action and reaction, the 
earth should experience a force and accelerate toward a falling 
stone. 

(a) How does the force on the earth compare with the force on 
the stone? 

(b) How does the earth's acceleration compare with the stone's 
acceleration? 



Q13 The top diagram at the right 
represents two bodies of equal mass which 
exert gravitational forces of magnitude F 
on one another. What will be the magnitude 
of the gravitational attractions in each 
case? 



•? 



• -? 



?• 



^ 



Q14 A, B, C, and D are bodies with 
equal masses. How do the forces of 
attraction that A and B exert on each other 
compare with the force that C and D exert 
on each other? 

(a) F^B = 3 X FcD 

(b) Fab = 4 X Fc .„ 

(c) Fab = 9 X Fen 

(d) F.,B = 16 X F<.o 



V 



U- 



••• 



© ® 



102 



The Unity of Earth and Sky -The Work of Newton 



This photograph, taken from an un- 
manned capsule orbiting the moon, 
shows some latter-day evidence that 
the laws of mechanics for heavenly 
bodies are the same as for the earth: 
the trails of two huge boulders that 
rolled 1000 ft down a lunar slope. 




8.7 Planetary motion and the gravitational constant 

According to Newton's mechanics, if a planet of mass m„ is 
moving along an orbit of radius R and period T, there is continually 
a centripetal acceleration Oc = 4rr-RIT-. Therefore, there must 
continually be a central force F,. = m^a^. == ^Tr-RmJT'^. If we identify 
gravity as the central force, then 

•* grav ' " c 



or 



^ R' 






By simplifying this equation and rean-anging some terms, we 
can get an expression for G: 



G = 



Air (R 



m^,„AT- 



We know from Kepler that for the planets' motion around the sun, 
the ratio R VT^ is a constant; 47r'- is a constant also. If we assume 
that the mass of the sun is constant, then all the factors on the 
right of the equation for G are constant. So G must be a constant 
for the gravitational effect of the sun on the planets. By similar 
reasoning the value of G must be a constant for the effect of Jupiter 
on its moons — and for the effect of Saturn on its moons — and for an 
apple and the moon above the earth. But is it the same value of G 
for all these cases? 

It is impossible to prove that G is the same for the gravitational 
interaction of all bodies. If. however, we assume that G is a 
universal constant, we can get some remarkable new information — 
the relative masses of the sun and the planets! 



Section 8.7 



103 



We begin by again equating the centripetal force on the 
planets with the gravitational attraction to the sun, but this time we 
solve the equation for rrisun- 



'■ grav 
G nip TTtsun 



4TT^Rmp 
fi 

477- R-^ 

GT' 



If we write fesun for the constant ratio T'^IR\ we have 

4tt^ 



m. 



GK 



By similar derivation, mjup 



TTts 



TTlp 



477^ 



'-''^Jupiter 

477^ 
^'^Saturn 

477=^ 

vj/tpnrth 



where fejupiter. 'isatum. and feeartn are the known values of the constant 
ratios T^/R^ for the satellites of Jupiter, Saturn, and the earth. 

To compare Jupiter's mass to the mass of the sun, we have 
only to divide the formula for mjupiter by the formula for rrisun : 

477^ 



Gfejuf 



ms. 



4772 



Gfes 



or 



^WjUf 



ms„n 



hiw 



Similarly, the mass of any two planets can be compared if the 
values of T^/R^ are known for them both -that is, if they both have 
satellites whose motion has been carefully observed. 

These comparisons are based on the assumption that G is a 
universal constant. Calculations based on this assumption have led 
to consistent results on a wide variety of astronomical data, 
including the behavior of a space ship orbiting and landing on the 
moon. Results consistent with this assumption were also found 
when more difficult calculations were made for the small disturbing 
effects that the planets have on each other. There is still no way 
of proving G is the same everywhere and always, but it is a 
reasonable working assumption until evidence to the contrary 
appears. 

If the numerical value of G were known, the actual masses of 
the earth, Jupiter, Saturn, and the sun could be calculated. G is 
defined by the equation F^rav = G rn.mJR-. To find the value of G it 
is necessary to know values for all the other variables -that is, to 



SG 8.9 

Masses Compared to Earth 
Earth 1 

Saturn 95 

Jupiter 318 

Sun 333.000 



104 



The Unity of Earth and Sky-The Work of Newton 



measure the force F^rav between two measured masses m, and m-, 
when they are separated by a measured distance R. Newton knew 
this, but in his time there were no instruments sensitive enough to 
measure the very tiny force expected between any masses small 
enough for experimental use. 

Q15 What information can be used to compare the masses of 
two planets? 

Q16 What additional information is necessary for calculation 
of the actual masses? 



8.8 The value of G and the actual masses of the planets 



Calculation of G from approximate 
experimental values: 

_^ Mm 

_ F ,. R- 



Mm 
(10 '■ N) (0.1 m) 
(100 kg) (1 kg) 
10" X 10 - 



10- 



N m/kg^ 



10 '" N m-/kg- 



Nm7kg ' can be expressed as 
m 7kg sec- 



SG 8.10-8.13 



The masses of small solid objects can be found easily enough 
from their weights. Measuring the distance between solid objects 
of spherical shape is also not a problem. But how is one to measure 
the small mutual gravitational force between relatively small 
objects in a laboratory, particularly when each is also experiencing 
separately a huge gravitational force toward the tremendously 
massive earth? 

This serious technical problem of measurement was eventually 
solved by the English scientist, Henry Cavendish (1731-1810). As 
a device for measuring gravitational forces, he employed a torsion 
balance in which the gravitational force of attraction between two 
pairs of lead spheres twisted the wire holding up one of the pairs. 
The twist of the wire could be calibrated by measuring the twist 
produced by small known forces. A typical result for a 100 kg 
sphere and a 1 kg sphere at a center-to-center distance of 0.1m 
would be a force of about one-millionth of a newton! As the 
calculations in the margin show, these data lead to a value for G 
of about 10"'" (Nm^/kg-). This experiment has been progressively 
improved, and the accepted value of G is now: 

G = 6.67 X iO->' Nm-/kg- 

Evidently gravitation is a weak force which becomes important 
only when at least one of the masses is very great. The gravitational 
force on a 1-kg mass at the surface of the earth is 9.8 newtons. 
(which we know because, if released, it will fall with an acceleration 
of 9.8 m/sec.) Substituting 9.8 newtons for F^rav and substituting 
the radius of the earth for R. you can calculate the mass of the 
earth! (See SG 8.11) 

By assuming that the same value for G applies to all 
gravitational interaction, we can calculate values for the masses of 
bodies from the known values of T"-/fi ' for their satellites. Since 
Newton's time, satellites have been discovered around all of the 
outer planets except Pluto. The values of their masses, calculated 
from m = 4tt-iG x R^jT-, are given in the table on the next page. 
Venus and Mercury have no satellites, but values for their masses 



Section 8.8 



105 




T 



Cavendish's original drawing of his apparatus for 
determining the value of G. To prevent disturbance from air 
currents, he inclosed it in a sealed case. He observed the deflection 
of the balance rod from outside w/ith telescopes. 



are found by analyzing the slight disturbing effects each has on 
other planets. Modem values for the actual masses of the planets 
are listed in the margin. Notice that the planets taken together add 
up to not much more than 1/ 1000th part of the mass of the solar 
system. By far, most of the mass is in the sun, and this of course 
accounts for the fact that the sun dominates the motion of the 
planets, acting like an almost infinitely massive and fixed object. 

In all justice to the facts, we should modify this picture a little. 
Newton's third law tells us that for every pull the sun exerts on a 
planet the sun itself experiences an equally strong pull in the 
opposite direction. Of course the very much greater mass of the sun 
keeps its acceleration to a correspondingly smaller value. But this 
acceleration is, after all, not exactly zero. Hence, the sun cannot be 
really fixed in space even in the heliocentric system, if we accept 
Newtonian dynamics, but rather it moves a little about the point 
that forms the common center of mass of the sun and the moving 
planets that pull on it. This is true for every one of the 9 planets; 
and since these generally do not happen to be moving all in one 
line, the sun's motion is actually a complex superposition of 9 small 
ellipses. Still, unless we are thinking of a solar system in which the 
planets are very heavy compared to their sun, such motion of the 
sun is not large enough to be of interest to us for most purposes. 

Q17 Which of the quantities in the equation F„^v = G mim-JR^ 
did Cavendish measure? 

Q18 Knowing a value for G, what other information can be 
used to find the mass of the earth? 




Schematic diagram of the device used 
by Cavendish for determining the 
value of the gravitational constant G. 
Large lead balls of masses /W, and M., 
were brought close to small lead balls 
of masses m, and m-i. The mutual 
gravitational attraction between M, 
and m, and between Mo and m.,, caused 
the vertical wire to be twisted by a 
measurable amount. 



Actual M 


asses 


Sun 


1,980,000 X lO^^kg 


Mercury 




.328 


Venus 




4.83 


Earth 




5.98 


Mars 




.637 


Jupiter 




1,900 


Saturn 




567 


Uranus 




88.0 


Neptune 




103. 


Pluto 




1.1 


SG 8.14- 


8.17 





106 



The Unity of Earth and Sky -The Work of Newton 



Q19 Knowing a value for G, what other information can be 
used to find the mass of Saturn? 

Q20 The mass of the sun is about 1000 times the mass of 
Jupiter. How does the sun's acceleration due to Jupiters attraction 
compare with Jupiter's acceleration due to the sun's attraction? 



6 OR 



1/592 -~_j__ 
l/60'l J 



Tidal Forces. 

The earth-moon distance indicated in 
the figure is greatly reduced because 
of the space limitations. 




8.9 Further successes 

Newton did not stop with the fairly direct demonstrations we 
have described so far. In the Principia he showed that his law of 
universal gravitation could account also for more complicated 
gravitational interactions, such as the tides of the sea and the 
perverse drift of comets across the sky. 

The tides: The flooding and ebbing of the tides, so important to 
navigators, tradesmen, and explorers through the ages, had 
remained a mystery despite the studies of such men as Galileo. 
Newton, however, through the application of the law of gravitation, 
was able to explain the main features of the ocean tides. These he 
found to result from the attraction of the moon and sun upon the 
waters of the earth. Each day two high tides normally occur. Also, 
twice each month, when the moon, the sun. and the earth are in 
line, the tides are significantly higher than average. 

Two questions about tidal phenomena demand special attention. 
First, why do high tides occur on both sides of the earth, including 
the side away from the moon? Second, why does the time of high 
tide occur at a location some hours after the location has passed 
directly under the moon? 

Newton reahzed that the tides result from the difference 
between the acceleration (due to the moon and sun) of the whole 
solid earth and the acceleration of the fluid waters at the earth's 
surface. The moon's distance from the earth's center is 60 earth 
radii. On the side of the earth nearer the moon, the distance of the 
water from the moon is only 59 radii. On the side of the earth, away 
from the moon, the water is 61 earth radii from the moon. The 
accelerations are shown in the figure at the left. On the side of the 
earth nearer the moon, the acceleration of the water toward the 
moon is greater than the acceleration of the earth as a whole -the 
net effect is that the water is accelerated away from the earth. 
On the side of the earth away from the moon, the acceleration of 
the water toward the moon is less than that of the earth as a 
whole -the net result is that the earth is accelerated away from the 
water. 

If you have been to the seashore or examined tide tables, you 
know that high tide does not occur when the moon is highest in the 
sky, but some hours later. To explain this even qualitatively, we 
must remember that on the whole the oceans are not very deep. As 
a result, the waters moving in the oceans in response to the moon's 
attraction are slowed by friction with the ocean floors, especially in 
shallow water, and consequently the time of high tide is delayed. 



Section 8.9 



107 



In any particular place, the amount of the delay and the height of 
the tides depends greatly upon the ease with which the waters can 
flow. No general theory can be expected to account for all the 
particular details of the tides. Most of the local predictions in the 
tide tables are based on empirical rules using the cyclic variations 
recorded in the past. 

Since there are tides in the seas, you may wonder if there are 
tides in the fluid atmosphere and in the earth itself. There are. The 
earth is not completely rigid, but bends somewhat like steel. The 
tide in the earth is about a foot high. The atmospheric tides are 
generally masked by other weather changes. However, at heights of 
about a hundred miles, where satellites have been placed in orbit, 
the thin atmosphere rises and falls considerably. 

Comets: Comets, whose unexpected appearances throughout 
antiquity and the Middle Ages had been interpreted as omens of 
disaster, were shown by Halley and Newton to be nothing more 
than a sort of shiny, cloudy mass that moved around the sun 
according to Kepler's laws, just as planets do. They found that 
most comets were visible only when closer to the sun than the 
distance of Jupiter. Several of the very bright comets were found 
to have orbits that took them well inside the orbit of Mercury, to 
within a few million miles of the sun, as the figure at the right 
indicates. Many of the orbits have eccentricities near 1.0 and are 
almost parabolas; these comets have periods of thousands or even 
millions of years. Some other faint comets have periods of only five 
to ten years. 

Unlike the planets, whose orbits are nearly in a single plane, 
the planes of comet orbits are tilted at all angles. Yet, like all 
members of the solar system, they obey all the laws of dynamics, 
including that of universal gravitation. 

Edmund Halley applied Newton's concepts of celestial motions 
to the motion of bright comets. Among the comets he studied were 
those seen in 1531, 1607. and 1682. whose orbits he found to be 
ver\' nearly the same. Halley suspected that these might be the 
same comet, seen at intervals of about seventy-five years and 
moving in a closed orbit. He predicted that it would return in about 
1757 — which it did, although Halley did not live to see it. Halley's 
comet appeared in 1833 and 1909 and is due to be near the sun 
and bright again in 1985. 

With the period of this bright comet known, its approximate 
dates of appearance could be traced back in history. In the records 
found in ancient Indian, Chinese, and Japanese documents, this 
comet has been identified at all expected appearances except one 
since 240 B.C. That the records of such a celestial event are 
incomplete in Europe is a sad commentary upon the level of 
interests and culture of Europe during the so-called Dark Ages. One 
of the few European records of this comet is the famous Bayeux 
tapestry, embroidered with seventy-two scenes of the Norman 
Conquest of England in 1066; it shows the comet overhead and the 



Comets orbit 




Schematic diagram of the orbit of a 
comet projected onto the ecliptic 
plane; comet orbits are tilted at all 
angles. 



SG 8.18 



108 



The Unity of Earth and Sky-The Work of Newton 




I f.^ i — iU^=4=J^^d^ 



A scene from the Bayeux tapestry, 
which was embroidered about 1070. 
The bright comet of 1066 can be seen 
at the top of the figure. This comet 
was later identified as being Haliey's 
comet. At the right, Harold, pretender 
to the throne of England, is warned 
that the comet is an ill omen. Later 
that year at the Battle of Hastings, 
Harold was defeated by William the 
Conqueror. 



SG 8.19 



frightened ruler and court cowering below. A major triumph of 
Newtonian science was its use to explain that comets, which for 
centuries had been fearful events, were regular members of the 
solar system. 

The scope of the principle of universal gravitation: Newton made 
numerous additional applications of his law of universal 
gravitation which we cannot consider in detail here. He investigated 
the causes of the somewhat irregular motion of the moon and 
showed that these causes are explainable by the gravitational 
forces acting on the moon. As the moon moves around the earth, 
the moon's distance from the sun changes continually. This 
changes the resultant force of the earth and sun on the orbiting 
moon. Newton also showed that other changes in the moon's 
motion occur because the earth is not a perfect sphere, but has an 
equatorial diameter twenty-seven miles greater than the diameter 
through the poles. Newton commented on the problem of the 
moon's motion that "the calculation of this motion is difficult. " Even 
so. he obtained predicted values in reasonable agreement with the 



Section 8.9 



109 



observed values available at that time, and even predicted some 
details of the motion which had not been noticed before. 

Newton investigated the variations of gravity at different 
latitudes on the spinning and bulging earth. From the differences 
in the rates at which pendulums swung at different latitudes, he 
was able to derive an approximate shape for the earth. 

In short, Newton had created a whole new quantitative 
approach to the study of astronomical motion. Because some of 
his predicted variations had not been observed, improved instru- 
ments were built. These were needed anyway to improve the 
observations which could now be fitted together under the grand 
theory. Numerous new theoretical problems also clamored for 
attention. For example, what were the predicted and observed 
influences among the planets themselves upon their motions? 
Although the planets are small compared to the sun and are very 
far apart, their interactions are observable. As precise data have 
accumulated, the Newtonian theory has permitted calculations 
about the past and future states of the planetary system. For past 
or future intervals up to some hundreds of millions of years (beyond 
which the extrapolations become too uncertain) the Newtonian 
theory tells us that the planetary system has been and will be about 
as it is now. 

What astonished Newton's contemporaries and increases our 
own esteem for him was not only the scope and genius of his work 
in mechanics, not only the great originality and the elegance of his 
proofs, but the detail in which he developed the implications of 
each of his ideas. Having satisfied himself of the correctness of his 
principle of universal gravitation, he applied it to a wide range of 
terrestrial and celestial problems, with the result that the theory 
became more and more widely accepted. Nor has it failed us since 
for any of the new problems concerning motion in the solar system; 
for example, the motion of every artificial satellite and space probe 
has been reliably calculated on the assumption that at every instant 
a gravitational force is acting on it according to Newton's law of 
universal gravitation. We can well agree with the reply given to 
ground control as Apollo 8 returned from man's first trip to the 
moon — ground control: "Who's driving up there?" Apollo 8: "I think 
Isaac Newton is doing most of the driving right now." 



Tiny variations from a 1/R^ centrip- 
etal acceleration of satellites in orbit 
around the moon have led to a map- 
ping of "mascons" on the moon — 
usually dense concentrations of mass 
under the surface. 



Beyond the solar system: We have seen how Newton's laws have 
been applied to explain the motions and other observables about 
the earth and the entire solar system. But now we turn to a new 
and even more grandiose question. Do Newton's laws, which are 
so useful within the solar system, also apply at greater distances, 
for example among the stars? 

Over the years following publication of the Principia, several 
sets of observations provided an answer to this important question. 
Toward the end of the 1700's, William Herschel, a British musician 
turned amateur astronomer, was. with the help of his sister 



110 



The Unity of Earth and Sky-The Work of Newton 



Paths of the outer three planets during 
1969 (Diagrams reproduced from Sky 
and Telescope magazine.) 



-3' 



+ 1 



T-T 



-| r 



Jan 1970 



- July 



Vl RGO 




J L 



I2''00" 



URAMUS IN I96Q 



I I \ L 



I i I I I L 



12^20*" 



12^30" 



-17' 



1 1 — I r 



-| — r 



MEPTUME I isl 1969 



I5"35" 



1 — r~ 

47 



Jan 1970 



Jan 1969 




J I L 



Libra 



I I LJJ \ L 



I5"45'' 



I5"50' 



The planet Uranus was discovered in 
1781 with a reflecting telescope. Dis- 
turbances in Uranus' orbit observed 
over many years led astronomers to 
seek another planet beyond Uranus: 
Neptune was observed in 1846 just 
where it was expected to be from anal- 
ysis of Uranus' orbit disturbance (by 
Newtonian mechanics). A detailed 
account of the Neptune story appears 
in the Project Physics Supplementary 
Unit, Discoveries in Physics. Dis- 
turbances observed in Neptune's orbit 
over many years led astronomers to 
seek still another planet. Again the 
predictions from Newtonian me- 
chanics were successful, and Pluto 
(too faint to be seen by eye even in the 
best telescopes) was discovered in 
1930 with a long time-exposure photo- 
graph. 




niGHT ASCENSION II9S0I 



Section 8.10 



111 



Caroline, making a remarkable series of observations of the sky 
through his homemade, but high quality telescopes. While 
planning how to measure the parallax of stars due to the earth's 
motion around the sun, he noted that sometimes one star had 
another star quite close. He suspected that some of these pairs 
might actually be double stars held together by their mutual 
gravitational attractions rather than being just two stars in nearly 
the same line of sight. Continued observations of the directions and 
distances from one star to the other of the pair showed that in some 
cases one star moved during a few years in a small arc of a curved 
path around the other (the figure shows the motion of one of the two 
stars in a system.) When enough observations had been gathered, 
astronomers found that these double stars, far removed from the 
sun and planets, also moved around each other in accordance with 
Kepler's laws, and therefore, in agreement with Newton's law of 
universal gravitation. Using the same equation as we used for 
planets (see p. 103), astronomers have calculated that the masses 
of these stars range from about 0.1 to 50 times the sun's mass. 

A theory can never be completely proven; but it becomes 
increasingly acceptable as it is found useful over a wider and 
wider range of problems. No theory has stood this test better than 
Newton's theory of universal gravitation as applied to the planetary 
system. After Newton, it took nearly a century for physicists and 
astronomers to comprehend, verify, and extend his work as applied 
to the problems of planetary motion. After two centuries (in the late 
1800's), it was still reasonable for leading scientists and philoso- 
phers to claim that most of what had been accomplished in the 
science of mechanics since Newton's day was but a development or 
application of his work. 



Q21 Why does the moon cause the water level to rise on both 
sides of the earth? 

Q22 In which of the following does the moon produce tides? 
(a) the seas (b) the atmosphere (c) the solid earth 

Q23 Why is the precise calculation of the moon's motion so 
difficult? 

Q24 How are the orbits of comets different from the orbits of 
the planets? 

Q25 Do these differences affect the validity of Newton's law of 
universal gravitation for comets? 



8.10 Some effects and limitations of Newton's work 

Today we honor Newton and his system of mechanics for many 
valid reasons. The content of the Principia historically formed the 
basis for the development of much of our physics and technology. 
Also, the success of Newton's approach to his problems became the 
method used in all the physical sciences for the subsequent two 
centuries. 

Throughout Newton's work, we find his basic belief that 




The motion over many years for one 
of the two components of a binary 
star system. Each circle indicates the 
average of observations made over an 
entire year. 



112 The Unity of Earth and Sky-The Work of Newton 

celestial phenomena can be explained by apphing quantitative 
earthly laws. He felt that his laws had physical meaning and were 
not just mathematical conveniences behind which unknow able 
laws were hiding, but rather just the opposite: the natural physical 
laws governing the universe were accessible to man. and the simple 
mathematical forms of the laws were evidence of their reality. 

Newton combined the skills and approaches of both the 
experimental and the theoretical scientist. He made ingenious 
pieces of equipment, such as the first reflecting telescope, and 
performed skillful experiments, especially in optics. Yet he also 
applied his great mathematical and logical powers to the creation of 
explicit, testable predictions. 

Many of the concepts which Newton used came from his 
scientific predecessors and contemporaries. For example. Galileo 
and Descartes had contributed the first steps to a proper idea of 
inertia, which became Newtons First Law of Motion: Keplers 
planetary laws were central in Newtons consideration of planetary 
motions; Huygens. Hooke, and others clarified the concepts of force 
and acceleration, ideas which had been evolving for centuries. 

In addition to his own experiments. Newton selected and used 
data from a large number of sources. Tycho Brahe was only one of 
several astronomers whose observations of the motion of the moon 
he used. When he could not complete his own measurements, he 
knew whom he could ask. 

Lastly, we must recall how exhaustively and how fruitfully he 
used and expanded his own specific contributions. For instance, 
in developing his theor>' of universal gravitation, he used his laws 
of motion and his various mathematical inventions again and 
again. Yet Newton was modest about his achievements, and he once 
said that if he had seen further than others "it was by standing 
upon the shoulders of Giants." 

We recognize today that Newton's mechanics holds only within 
a well-defined region of our science. For example, although the 
forces within each galaxy appear to be New tonian. this may not be 
true for forces acting between one galaxy and another. At the other 
end of the scale, among atoms and subatomic particles, we shall see 
that an entirely non-Newtonian set of concepts had to be developed 
to account for the observations. 

Even within the solar system, there are several small 
discrepancies between the predictions and the observations. The 
most famous is the angular motion of the axis of Mercury's orbit. 
which is greater than the value predicted from Newton's laws by 
about 1/80° per centun,'. For a while, it was thought that the error 
might arise because gravitational force does not vary inversely 
exactly with the square of the distance — perhaps, for example, the 
law was F„Hv= 1,R- ""<""". 

Such difficulties should not be hastily assigned to some minor 
imperfection in the Law of Gravitation, which applies so well with 
unquestionable accuracy to all the other planetary motions. It may 



Section 8.10 



113 



be that the whole idea behind the theory is mistaken, as was the 
idea behind the Ptolemaic system of epicycles. Out of many 
studies has come the conclusion that there is no way that details of 
Newtonian mechanics can be modified to explain certain 
observations. Instead, these observations can be accounted for only 
by constructing new theories based on some very different 
assumptions. The predictions from these theories are almost 
identical to those from Newton's laws for phenomena familiar to us, 
but they are accurate in some extremes where the Newtonian 
predictions begin to show inaccuracies. Newtonian science is 
linked at one end with relativity theory, which is important for 
bodies with ver>' great mass or moving at very high speeds. At the 
other end Newtonian science approaches quantum mechanics. 
which is important for particles of extremely small mass and size — 
atoms, molecules, and nuclear particles. For a vast range of 
problems between these extremes, the Newtonian theory gives 
accurate results and is far simpler to use. Moreover, it was in 
Newtonian mechanics that relativity theory and quantum 
mechanics took root. 



Newtonian mechanics refers to 
the science of the motion of bodies, 
based on Newton's work. It includes 
his laws of motion and of gravitation 
as applied to a range of bodies from 
microscopic size to stars, and 
incorporates developments of 
mechanics for over two centuries 
after Newton's own work. 



SG 8.20-8.22 





114 Motion in tlie Heavens 



EPILOGUE In this unit we started at the beginnings of recorded 
history and followed the attempts of men to explain the cyclic 
motions observed in the heavens. We had several purposes. The first 
was to examine with some care the difficulties of changing from an 
earth-centered view of the heavens to the modern one in which the 
earth came to be seen as just another planet moving around the 
sun. We also wanted to put into perspective Newton's synthesis of 
earthly and heavenly motions. From time to time we have also 
suggested that there was an interaction of these new world views 
with the general culture. We stressed that each contributor was a 
creature of his times, limited in the degree to which he could 
abandon the teachings on which he was reared. Gradually, through 
the successive work of many, a new way of looking at heavenly 
motions arose. This in turn opened new possibilities for even further 
new ideas, and the end is not in sight. 

Still another purpose was to see how theories are made and 
tested, what is the place of assumption and experiment, of 
mechanical models and mathematical description. In later parts 
of the course, we will come back to the same questions in more 
recent context, and we shall find that the attitudes developed 
toward theory-making during the seventeenth-century scientific 
revolution are still immensely fruitful today. 

In our study we have referred to scientists in Greece. Egypt. 
Poland. Denmark, Austria. Italy. England, and other countries. 
Each, as Newton said, stood on the shoulders of others. And for 
each major success there are many lesser advances or. indeed, 
failures. We see science as a cumulative intellectual activity not 
restricted by national boundaries or by time. It is not inevitably and 
relentlessly successful, but it grows more as a forest grows, new 
growth replacing and drawing nourishment from the old. sometimes 
with unexpected changes in its different parts. It is not a cold, 
calculated pursuit, for it may involve passionate controversy, 
religious convictions, esthetic judgments of what beauty is. and 
sometimes wild private speculation. 

It is also clear that the Newtonian synthesis did not put an end 
to the study of science by solving all problems. In many ways it 
opened whole new lines of investigations, both theoretical and 
observational. In fact, much of our present science and also our 
technology had their effective beginnings with the work of Newton. 
New models, new mathematical tools, and new self-confidence 
(sometimes misplaced, as the study of the nature of light will show) 
encouraged those who followed, to attack the new problems. A 
never-ending series of questions, answers, and more questions was 
well launched. The modern view of science is that it is a continuing 
quest into ever more interesting fields. 

Among the many problems remaining after Newton's work was 
the study of objects interacting not by gravitational forces, but by 



Epilogue 



115 



friction and collision. This led. as the next unit shows, to the 
concepts of momentum and energy, and then to a much broader 
view of the connection between different parts of science -physics, 
chemistry, and biology. Eventiially. from this line of study, emerged 
other statements as grand as Newton's law of universal 
gravitation: the conservation laws on which so much of modem 
science - and technology- - is based, especially the part having to 
do with many interacting bodies making up a system. That account 
will be the main subject of Unit 3. 

Newton's influence was not. however, limited to science alone. 
The centur>- following the death of Newton in 1727 was a period of 
consolidation and further application of Newton's discoveries and 
methods, whose eff"ects were felt especially in philosophy and 
literature, but also in may other fields outside science. Let us round 
our view of Newton by considering some of these effects. 

During the 1700's. the so-called Age of Reason or Century of 
Enlightenment, the viewpoint called the Newtonian cosmology 
became firmly entrenched in European science and philosophy. The 
impact of Newton's achievements may be summarized thus: he 
had shown that man. by observing and reasoning, by considering 
mechanical models and deducing mathematical laws, could 
uncover the workings of the physical universe. Therefore, it was 
argued, man should attempt by the same method to understand not 
only nature but also society and the human mind. As the French 
writer Fontenelle (1657-1757) expressed it: 

The geometric spirit is not so bound 
up with geometry that it cannot be 
disentangled and carried into other 
fields. A work of morals, or politics, 
of criticism, perhaps even of eloquence, 
will be the finer, other things being 
equal, if it is written by the hand 
of a geometer. 

The EngUsh philosopher John Locke (1632-1704) was greatly 
influenced by Newton's work and in turn reinforced Newton's 
influence on others: he argued that the goal of philosophy should 
be to solve problems, including those that affect our daily life, by 
observation and reasoning. "Reason must be our best judge and 
guide in all things. " he said. Locke thought that the concept of 
"natural law " could be used in religion as well as in physics: and 
indeed the notion of a religion "based on reason" appealed to many 
Europeans who hoped to avoid a revival of the bitter religious wars 
of the 1600s. 

Locke advanced the theor\' that the mind of the new-bom 
child contains no "innate ideas ": it is like a blank piece of paper on 
which anything may be written. If this were true, it would be futile 
to search within oneself for a God-given sense of what is true or 
morally right. Instead, one must look at nature and society to 
discover whether there are anv "natural laws " that mav exist. 












Mf d . 




116 



Motion in the Heavens 




The engraving of the French Academy 
by Sebastian LeClerc (1698) reflects 
the activity of learned societies at that 
time. The picture does not depict an 
actual scene, of course, but in al- 
legory shows the excitement of com- 
munication that grew in an informal 
atmosphere. The dress is symbolic of 
the Greek heritage of the sciences. 
Although all the sciences are repre- 
sented, the artist has put anatomy, 
botany, and zoology, symbolized by 
skeletons and dried leaves, toward the 
edges, along with alchemy and theol- 
ogy. Mathematics and the physical 
sciences, including astronomy, oc- 
cupy the center stage. 



Conversely, if one wants to improve the quality of man's mind, one 
must improve the society in which he lives. 

Locke's view also implied an "atomistic" structure of society: 
each person is separate from other individuals in the sense that he 
has no "organic" relation to them. Previously, political theories had 
been based on the idea of society as an organism in which each 
person has a perscribed place, function and obligation. Later 
theories, based on Locke's ideas, asserted that government should 
have no function except to protect the freedom and property of the 
individual person. 

Although "reason" was the catchword of the eighteenth-century 
philosophers, we do not have to accept their own judgment that 
their theories about improving religion and society were necessarily 
the most reasonable. Like most others, these men would not give up 
a doctrine such as the equal rights of all men merely because they 
could not find a strictly mathematical or scientific proof for it. 
Newtonian physics, religious toleration, and republican government 
were all advanced by the same movement; but this does not mean 
there was really a logical connection among them. Nor. for that 
matter, did many of the eighteenth-century thinkers in any field or 
nation seem much bothered by another gap in logic and feeling; 
they believed that "all men are created equal," and yet they did 
little to remove the chains of black slaves, the ghetto walls 
imprisoning Jews, or the laws that denied voting rights to women. 

Still, compared with the previous century, the dominant theme 
of the 1700's was moderation — the happy medium, based on 



Epilogue 117 

toleration of different opinions, restraint of excess in any 
direction, and balance of opposing forces. Even reason was not 
allowed to ride roughshod over religious faith; atheism, which 
some philosophers thought to be the logical consequence of 
unlimited rationality, was still regarded with horror by most 
Europeans. 

The Constitution of the United States of America, with its 
ingenious system of "checks and balances" to prevent any faction 
from getting too much power, is one of the most enduring 
achievements of this period. It attempts to establish in politics a 
stable equilibrium of opposing trends similar to the balance 
between the sun's gravitational pull and the tendency of a planet to 
fly off in a straight line. If the gravitational attraction increased 
without a corresponding increase in planetary speed, the planet 
would fall into the sun; if its speed increased without a correspond- 
ing increase in gravitational attraction, the planet would escape 
from the solar system. 

Just as the Newtonian laws of motion kept the earth at its 
proper distance from the sun, so the political philosophers, some of 
whom used Newtonian physics as a model of thought, hoped to 
devise a system of government which would avoid the extremes of 
dictatorship and anarchy. According to James Wilson (1742-1798), 
who played a major role in drafting the American Constitution: 

In government, the perfection of the whole depends on 
the balance of the parts, and the balance of the parts 
consists in the independent exercise of their separate 
powers, and, when their powers are separately exercised, 
then in their mutual influence and operation on one 
another. Each part acts and is acted upon, supports and 
is supported, regulates and is regulated by the rest. 
It might be supposed, that these powers, thus mutually 
checked and controlled, would remain in a state of 
inaction. But there is a necessity for movement in human 
aff"airs, and these powers are forced to move, though still 
to move in concert. They move, indeed, in a line of 
direction somewhat different from that, which each 
acting by itself would have taken; but, at the same time, 
in a line partaking of the natural directions of the 
whole -the true line of public liberty and happiness. 

A related effect of Newton's work in physics on other fields was 
the impetus Newton as a person and Newton's writing gave to the 
idea of political democracy. A former farm boy had penetrated to 
the outermost reaches of the human imagination, and what he 
found there meant, first of all, that God had made only one set of 
laws for heaven and earth. This smashed the old hierarchy and 
raised what was once thought base to the level of the noble. It was 
an extension of a new democracy throughout the universe: Newton 
had shown that all matter, whether of sun or of ordinary stone, was 
created equal, was of the same order in "the Laws of Nature and of 



118 



Motion in the Heavens 



Nature's God," to cite the phrase used at the beginning of the 
Declaration of Independence to justify the elevation of the colonists 
to an independent people. The whole political ideology was heavily 
influenced by Newtonian ideas. The Principia, many thought, gave 
an analogy and extension direct support to the proposition being 
formulated also from other sides that all men, like all natural 
objects, are created equal before nature's creator. Some of these 
important trends are discussed in articles in Reader 2. 

In literature, too, many welcomed the new scientific viewpoint 
as a source of metaphors, allusions, and concepts which they used 
in their poems and essays. Newton's discovery that white light is 
composed of colors was referred to in many poems of the 1700's 
(see Unit 4). Samuel Johnson advocated that words drawn from the 
natural sciences be used in literary works, defining such words in 
his Dictionary and illustrating their application in his "Rambler" 
essays. 

Other writers distrusted the new cosmology and so used it for 
purposes of satire. In his epic poem The Rape of the Lock, 
Alexander Pope exaggerated the new scientific vocabulary for comic 
effect. Jonathan Swift, sending Gulliver on his travels to Laputa, 
described an academy of scientists and mathematicians whose 
experiments and theories were as absurd as those of the Fellows 
of the Royal Society must have seemed to the layman of the 1700's. 

The first really powerful reaction against Newtonian cosmology 
was the Romantic movement, begun in Germany about 1780 by 
young writers inspired by Johann Wolfgang von Goethe. The most 
familiar examples of Romanticism in English literature are the 
poems and novels of Blake, Coleridge, Wordsworth, Shelley. Byron, 
and Scott. 

The Romantics scorned the mathematical view of nature, and 
emphasized the importance of quality rather than quantity. They 
preferred to study the unique element of an individual person or 
experience, rather than make abstractions. They exalted emotion 
and feeling at the expense of reason and calculation. In particular, 
they abhorred the theory that the universe is in any way like a 
clockwork, made of inert matter set into motion by a God who never 
afterwards shows His presence. Reflecting this attitude, the 
historian and philosopher of science, E. A. Burtt, has written 
scathingly that: 



This is, of course, a distortion of 
what scientists themselves believe 
— one of the wrong images of 
science discussed in "The Seven 
Images of Science" in Reader 3. 



. . . the great Newton's authority was squarely behind that 
view of the cosmos which saw in man a puny, irrelevant 
spectator (so far as being wholly imprisoned in a dark 
room can be called such) of the vast mathematical 
system whose regular motions according to mechanical 
principles constituted the world of nature. The gloriously 
romantic universe of Dante and Milton, that set no 
bounds to the imagination of man as it played over 
space and time, had now been swept away. Space was 
identified with the realm of geometry, time with the 
continuity of number. The world that people had thought 



Epilogue 119 

themselves living in -a world rich with color and sound, 
redolent with fragrance, filled with gladness, love and 
beauty, speaking everywhere of purposive harmony and 
creative ideals — was crowded now into minute comers in 
the brains of scattered organic beings. The really 
important world outside was a world hard, cold, 
colorless, silent, and dead; a world of quantity, a world of 
mathematically computable motions in mechanical 
regularity. The world of qualities as immediately 
perceived by man became just a curious and quite minor 
effect of that infinite machine beyond. 

Because in their view, the whole (whether it be a single human 
being or the entire universe) is pervaded by a spirit that cannot be 
rationally explained but can only be intuitively felt, the Romantics 
insisted that phenomena cannot meaningfully be analyzed and 
reduced to their separate parts by mechanistic explanations. 

Continental leaders of the Romantic movement, such as the 
German philosopher Friedrich Schelling (1775-1854) proposed a 
new way of doing scientific research, a new type of science called 
"Nature Philosophy." (This term is not to be confused with the 
older "natural philosophy," meaning mainly physics.) The Nature 
Philosopher does not analyze phenomena such as a beam of white 
light into separate parts or factors which he can measure 
quantitatively in his laboratory — or at least that is not his primary 
purpose. Instead, he tries to understand the phenomenon as a 
whole, and looks for underlying basic principles that govern all 
phenomena. The Romantic philosophers in Germany regarded 
Goethe as their greatest scientist as well as their greatest poet, and 
they pointed in particular to his theory of color, which flatly 
contradicted Newton's theory of light. Goethe held that white light 
does not consist of a mixture of colors but rather that the colors are 
produced by the prism acting on and changing the light which was 
itself pure. 

In the judgment of all modem physicists, Newton was right and 
Goethe wrong. Yet, in retrospect. Nature Philosophy was not simply 
an aberration. The general tendency of Nature Philosophy did 
encourage speculation about ideas which could never be tested by 
experiment; hence, Nature Philosophy was condemned by most 
scientists. But it is now generally agreed by historians of science 
that Nature Philosophy played an important role in the historical 
origins of some scientific discoveries. Among these was the general 
principle of conservation of energy, which is described in Chapter 
10. The recognition of the principle of conservation of energy came 
in part out of the viewpoint of Nature Philosophy, for it asserted that 
all the "forces of nature" -the phenomena of heat, gravity, 
electricity, magnetism, and so forth — are manifestations of one 
underlying "force" (which we now call energy). 

Much of the dislike which Romantics (like some modern artists 
and intellectuals) expressed for science was based on the mistaken 
notion that scientists claimed to be able to find a mechanistic 



120 Motion in the Heavens 

explanation for everything, including the human mind. If everything 
is explained by Newtonian science, then everything would also be 
determined in the way the motions of different parts of a machine 
are determined by its construction. Most modern scientists no 
longer believe this, but some scientists in the past have made 
statements of this kind. For example, the French mathematical 
physicist Laplace (1749-1827) said: 

We ought then to regard the present state of the universe 
as the effect of its previous state and as the cause of the 
one which is to follow. Given for one instant a mind 
which could comprehend all the forces by which nature 
is animated and the respective situation of the beings 
who compose it — a mind sufficiently vast to submit these 
data to analysis -it would embrace in the same formula 
the movements of the greatest bodies of the universe and 
those of the lightest atom; for it, nothing would be 
uncertain and the future, as the past, would be present 
to its eyes. 

Even the ancient Roman philosopher Lucretius (100-55 B.C.), 
who supported the atomic theory in his poem On the Nature of 
Things, did not go as far as this. In order to preserve some vestige 
of "free will" in the universe, Lucretius suggested that the atoms 
might swerve randomly in their paths. This was still unsatisfactory 
to the Romantics and also to some scientists such as Erasmus 
Darwin (grandfather of evolutionist Charles Darwin), who asked: 

Dull atheist, could a giddy dance 

Of atoms lawless hurl'd 
Construct so wonderful, so wise. 

So harmonised a world? 

The Nature Philosophers thought they could discredit the 
Newtonian scientists by forcing them to answer this question; to 
say "yes," they argued, would be absurd, and to say "no" would be 
disloyal to their own supposed beliefs. We shall see how successful 
the Newtonians were in explaining the physical world without 
committing themselves to any definite answer to Erasmus Darwin's 
question. Instead, they were led to the discovery of immensely 
SG 8.23, 8.24 powerful and fruitful laws of nature, discussed in the next units. 



8.1 The Project Physics learning materials 
particularly appropriate for Chapter 8 include: 

Experiment 

Stepwise Approximation to an Orbit 
Activities 

Model of the orbit of Halley's comet 

Other comet orbits 

Forces on a Pendulum 

Haiku 

Trial of Copernicus 

Discovery of Neptune and Pluto 
Reader Articles 

Newton and the Principia 

The Laws of Motion and Proposition I 

Universal Gravitation 

An Appreciation of the Earth 

The Great Comet of 1965 

Gravity Experiments 

Space the Unconquerable 

The Life Story of a Galaxy 

Expansion of the Universe 

Negative Mass 

The Dyson Sphere 
Flim Loops 

Jupiter Satellite Orbit 

Program Orbit I 

Program Orbit II 

Central forces -iterated blows 

Kepler's Laws 

Unusual Orbits 
Transparency 

Motion under central force 
8.2 In the table below are the periods and 
distances from Jupiter of the four large satellites, 
as measured by telescopic observations. Does 
Kepler's law of periods apply to the Jupiter system? 

SATELLITE PERIOD DISTANCE FROM JUPITER'S 

CENTER 
(in terms of Jupiter's radius, r) 



1 


1.77 days 


6.04 r 


II 


3.55 


9.62 


III 


7.15 


15.3 


IV 


16.7 


27.0 



8.3 Give some reasons why Descartes" theory of 
planetary motion might have been "a useful idea." 

8.4 On p. 100 it was claimed that the depen- 
dence of the gravitational force on the masses of 
both interacting bodies could be expressed as 

(a) Using a diagram similar to that for Q 13 on 
p. 101, show that this is correct. 

(b) To test alternatives to using the product, 
consider the possibilities that the force 
could depend upon the masses in either of 
two ways: 

(1) total force depends on (msun + "^pianet). or 

(2) total force depends on (msun/^pianet)- 
What would these relationships imply would 
happen to the force if either mass were reduced 
to zero? Would there still be a force even though 
there were onlv one mass left? Could you speak of 
a gravitational force when there was no body to 
be accelerated? 



8.5 Use the values for the mass and size of the 
moon (See table on page 123 ) to show that the 
"surface gravity " (acceleration due to gravity 
near the moon's surface) is only about j of what 
it is near the earth's. 

8.6 Complicated mathematics is necessary to 
find the exact force exerted by a spherical body, 
but it is not difficult to prove that the direction of 
the net force is toward the center. Newton's 
argument involved symmetry and considering 
tiny pieces of the whole body. Develop such an 
argument. 

8.7 Use the equation for centripetal force and 
the equation for gravitational force to derive an 
expression for the period of a Satallite orbiting 
around a planet in terms of the radius of the 
orbit and mass of the planet. 

8.8 The sun's mass is about 27,000.000 times 
greater than the moon's mass; the sun is about 
400 times further from the earth than the moon 
is. How does the gravitational force exerted on 
the earth by the sun compare with that exerted by 
the moon? 

8.9 By Newton's time, telescopic observations of 
Jupiter led to values for the orbital periods and 
radii of Jupiter's four large satellites. For 
example, the one named Callisto was found to 
have a period of 16.7 days and the radius of its 
orbit was calculated as 1 80 AU. 

(a) From these data calculate the value of 
^Jupiter- (First convert days to years.) 

(b) Show that Jupiter's mass is about 1/TOOO 
the mass of the sun. 

(c) How was it possible to have a value for the 
orbital radius of a satellite of Jupiter? 

8.10 What orbit radius must an earth satellite be 
given to keep it always above the same place on 
the earth — that is, in order to have a 24- hour 
period? (Hint: See SG 8.7) 

8.11 Calculate the mass of the earth from the 
fact that a 1kg object at the earths surface is 
attracted to the earth with a force of 9.8 newtons. 
The distance from the earths center to its 
surface is 6.4 x 10" meters. How many times 
greater is this than the greatest masses which 
you have had some experience in accelerating 
(for example, cars)? 

8.12 The mass of the earth can be calculated 
also from the distance and period of the moon. 
Show that the value obtained in this way agrees 
with the value calculated from measurements at 
the earth's surface. (See table on page 192.) 

8.13 Cavendish's value for G made it possible to 
calculate the mass of the earth, and therefore its 
average density. The "density" of water is 1000 
kg per cubic meter. That is, for any sample of 
water, dividing the mass of the sample by its 
volume gives 100 kg/m'. 

(a) What is the earth's average density? 

(b) The densest kind of rock known has a 
density of about 5000 kg m\ Most rock we 
find has a density of about 3000 kg m'. 

121 



STUDY GUIDE 



What do you conclude from this about the 
structure of the earth? 

8.14 The manned Apollo 8 capsule (1968) was 
put into a nearly circular orbit 1 12 km above the 
moon's surface. The period of the orbit was 120.5 
minutes. From these data calculate the mass 

of the moon. (The radius of the moon is 1740 km. 
Don't forget to use a consistent set of units.) 

8.15 Why do you suppose there is no reliable 
value for the mass of Pluto? 

8.16 Mars has two satellites. Phobos and Deimos 
— Fear and Panic. A science-fiction story was 
once written in which the natives of Mars showed 
great respect for a groove in the ground. The 
groove turned out to be the path of Mars' closest 
moon, "Bottomos." 

(a) If such an orbit were possible, what would 
the period be? 

(b) What speed would it need to have in order 
to go into such an orbit? 

(c) What difficulties do you see for such an 
orbit? 

8.17 Using the values given in the table on p. 
105 make a table of relative masses compared to 
the mass of the earth. 

8.18 The period of Halley's comet is about 75 
years. What is its average distance from the sun? 
The eccentricity of its orbit is 0.967. How far 
from the sun does it go? How close? 

8.19 Accepting the validity of F„av — GmtmJR', 
and recognizing that G is a universal constant, 
we are able to derive, and therefore to understand 
better, many particulars that previously seemed 
separate. For example, we can conclude: 

(a) That a^ for a body of any mass m„ should be 
constant at a particular place on earth. 

(b) That a^ might be different at places on 
earth at different distances from the earth's 
center. 

(c) That at the earth's surface the weight of a 
body be related to its mass. 

(d) That the ratio R^IT^ is a constant for all the 
satellites of a body. 

(e) That high tides occur about six hours apart. 
Describe briefly how each of these conclusions 
can be derived from the equation. 

8.20 The making of theories to account for 
observations is a major purpose of scientific study. 
Therefore some reflection upon the theories 
encountered thus far in this course will be useful. 
Comment in a paragraph or more, with examples 
from Units 1 and 2, on some of the statements 
below. Look at all the statements and select at 
least six, in any order you wish. 

(1) A good theory should summarize and not 
conflict with a body of tested observations. 



(For example, Kepler's unwillingness to 
explain away the difference of eight 
minutes of arc between his predictions and 
Tycho's observations.) 

(2) There is nothing more practical than a 
good theory. 

(3) A good theory should permit predictions of 
new observations which sooner or later 
can be made. 

(4) A good new theoiy should give almost the 
same predictions as older theories for the 
range of phenomena where they worked 
well. 

(5) Every theory involves assumptions. Some 
involve also esthetic preferences of the 
scientist. 

(6) A new theory relates some previously 
unrelated observations. 

(7) Theories often involve abstract concepts 
derived from observation. 

(8) Empirical laws or "rules" organize many 
observations and reveal how changes in 
one quantity vary with changes in another 
but such laws provide no explanation of 
the causes or mechanisms. 

(9) A theory never fits all data exactly. 

(10) Predictions from theories may lead to the 
observation of new effects. 

(11) Theories that later had to be discarded 
may have been useful because they 
encouraged new observations. 

(12) Theories that permit quantitative 
predictions are preferred to qualitative 
theories. 

(13) An "unwritten text" lies behind the 
statement of every law of nature. 

(14) Communication between scientists is an 
essential part of the way science grows. 

(15) Some theories seem initially so strange that 
they are rejected completely or accepted 
only very slowly. 

(16) Models are often used in the making of a 
theory or in describing a theory to 
people. 

(17) The power of theories comes from their 
generality. 

8.21 What happened to Plato's problem? Was it 
solved? 

8.22 Why do we believe today in a heliocentric 
system? Is it the same as either Copernicus' or 
Kepler's? What is the experimental evidence? Is 
the geocentric system disproved? 

8.23 Is Newton's work only of historical interest, 
or is it useful today? Explain. 

8.24 What were some of the major consequences 
of Newton's work on scientists' view of the 
world? 



122 



SATELLITES OF THE PLANETS 









AVERAGE RADIUS 


PERIOD 


OF 








DISCOVERY 


OF ORBIT 


REVOLUTION 


DIAMETER 


EARTH: 


Moon 




238,857 miles 


27d 


7h 


43m 


2160 miles 


MARS: 


Phobos 


1877, Hall 


5,800 





7 


39 


10? 




Delmos 


1877, Hall 


14,600 


1 


6 


18 


5? 


JUPITER: 


V 


1892, Barnard 


113,000 





11 


53 


150? 




1(10) 


1610, Galileo 


262,000 


1 


18 


28 


2000 




II (Europa) 


1610, Galileo 


417,000 


3 


13 


14 


1800 


III (Ganymede) 


1610, Galileo 


666,000 


7 


3 


43 


3100 


IV (Callisto) 


1610, Galileo 


1,170,000 


16 


16 


32 


2800 




VI 


1904, Perrine 


7,120,000 


250 


14 




100? 




VII 


1905, Perrine 


7,290,000 


259 


14 




35? 




X 


1938, Nicholson 


7,300,000 


260 


12 




15? 




XII 


1951, Nicholson 


13,000,000 


625 






14? 




XI 


1938, Nicholson 


14,000,000 


700 






19? 




VIII 


1908, Melotte 


14,600,000 


739 






35? 




IX 


1914, Nicholson 


14,700,000 


758 






17? 


SATURN: 


Mimas 


1789, Herschel 


115,000 





22 


37 


300? 




Enceladus 


1789, Herschel 


148,000 


1 


8 


53 


350 




Tethys 


1684, Cassini 


183,000 


1 


21 


18 


500 




Dione 


1684, Cassini 


234,000 


2 


17 


41 


500 




Rhea 


1672, Cassini 


327,000 


4 


12 


25 


1000 




Titan 


1655, Huygens 


759,000 


15 


22 


41 


2850 




Hyperion 


1848, Bond 


920,000 


21 


6 


38 


300? 




Phoebe 


1898, Pickering 


8,034,000 


550 






200? 




lapetus 


1671, Cassini 


2,210,000 


79 


7 


56 


800 


URANUS: 


Miranda 


1948, Kuiper 


81,000 


1 


9 


56 







Ariel 


1851,Lassell 


119,000 


2 


12 


29 


600? 




Umbriel 


1851,Lassell 


166,000 


4 


3 


28 


400? 




Titania 


1787, Herschel 


272,000 


8 


16 


56 


1000? 




Oberon 


1787, Herschel 


364,000 


13 


11 


7 


900? 


NEPTUNE: Triton 


1846, Lassell 


220,000 


5 


21 


3 


2350 




Nereid 


1949, Kuiper 


3,440,000 


359 


10 




200? 



THE SOLAR SYSTEM 









AVERAGE RADIUS 


PERIOD OF 




RADIUS 


MASS 


OF ORBIT 


REVOLUTION 


Sun 


6.95 X 10® meters 


1.98 X 10^° kilograms 


— 


— 


Moon 


1.74 X 10' 


7.34 X 10" 


3.8 X 10® meters 


2.36 X 10* seconds 


Mercury 


2.57 X 10* 


3.28 X 10'^ 


5.79 X 10'° 


7.60 X 10' 


Venus 


6.31 X 10' 


4.83 X 10^" 


1.08 X 10" 


1.94 X 10' 


Earth 


6.38 X 10' 


5.98 X 10'* 


1.49 X 10" 


3.16 X 10' 


Mars 


3.43 X 10' 


6.37 X 10" 


2.28 X10" 


5.94 X 10' 


Jupiter 


7.18 X 10^ 


1.90 X 10" 


7.78 X10" 


3.74 X 10® 


Saturn 


6.03 X 10^ 


5.67 X 10" 


1.43 X 10" 


9.30 X 10® 


Uranus 


2.67 X 10' 


8.80 X 10" 


2.87 X 10'' 


2.66 X 10' 


Neptune 


2.48 X 10' 


1.03 X 10" 


4.50 X 10" 


5.20 X 10' 


Pluto 


? 


? 


5.9 X 10" 


7.28.x 10' 



staff and Consultants 



Robert Gardner, Harvard University 
Fred Geis, Jr., Harvard University 
Nicholas J. Georgis, Staples High School, 

Westport, Conn. 
H. Richard Gerfin, Somers Middle School, 

Somers, N.Y. 
Owen Gingerich, Smithsonian Astrophysical 

Observatory, Cambridge, Mass. 

Stanley Goldberg, Antioch College, Yellow Springs, 

Ohio 
Leon Goutevenier, Paul D. Schreiber High School, 

Port Washington, N.Y. 
Albert Gregory, Harvard University 
Julie A. Goetze, Weeks Jr. High School, Newton, 

Mass. 
Robert D. Haas, Clairemont High School, San 

Diego, Calif. 
Walter G. Hagenbuch, Plymouth-Whitemarsh 

Senior High School, Plymouth Meeting, Pa. 
John Harris, National Physical Laboratory of 

Israel, Jerusalem 
Jay Hauben, Harvard University 
Peter Heller, Brandeis University, Waltham, Mass. 
Robert K. Henrich, Kennewick High School, 

Washington 
Ervin H. Hoffart, Raytheon Education Co., Boston 
Banesh Hoffmann, Queens College, Flushing, N.Y. 
Elisha R. Huggins, Dartmouth College, Hanover, 

N.H. 
Lloyd Ingraham, Grant High School, Portland, 

Ore. 
John Jared, John Rennie High School, Pointe 

Claire, Quebec 
Harald Jensen, Lake Forest College, 111. 
John C. Johnson, Worcester Polytechnic Institute, 

Mass. 
Kenneth J. Jones, Harvard University 
LeRoy Kallemeyn, Benson High School, Omaha, 

Neb. 
Irving Kaplan, Massachusetts Institute of 

Technology, Cambridge 
Benjamin Karp, South Philadelphia High School, 

Pa. 
Robert Katz, Kansas State University, Manhattan, 

Kans. 
Harry H. Kemp, Logan High School, Utah 
Ashok Khosla, Harvard University 
John Kemeny, National Film Board of Canada, 

Montreal 
Merritt E. Kimball, Capuchino High School, San 

Bruno, Calif. 
Walter D. Knight, University of California, 

Berkeley 
Donald Kreuter, Brooklyn Technical High School, 

N.Y. 
Karol A. Kunysz, Laguna Beach High School, 

Calif. 
Douglas M. Lapp, Harvard University 
Leo Lavatelli, University of Illinois, Urbana 



Joan Laws, American Academy of Arts and 

Sciences, Boston 
Alfred Leitner, Michigan State University, East 

Lansing 
Robert B. Lillich, Solon High School, Ohio 
James Lindblad, Lowell High School, Whittier, 

Calif. 
Noel C. Little, Bowdoin College, Brunswick, Me. 
Arthur L. Loeb, Ledgemont Laboratory, Lexington, 

Mass. 
Richard T. Mara, Gettysburg College, Pa. 
Robert H. Maybury, UNESCO, Paris 
John McClain, University of Beirut, Lebanon 
E. Wesley McNair, W. Charlotte High School, 

Charlotte, N.C. 
William K. Mehlbach, Wheat Ridge High School, 

Colo. 
Priya N. Mehta, Harvard University 
Glen Mervyn, West Vancouver Secondary School, 

B.C., Canada 
Franklin Miller, Jr., Kenyon College, Gambier, 

Ohio 
Jack C. Miller, Pomona College, Claremont, Calif. 
Kent D. Miller, Claremont High School, Cahf. 
James A. Minstrell, Mercer Island High School, 

Washington 
James F. Moore, Canton High School, Mass. 
Robert H. Mosteller, Princeton High School, 

Cincinnati, Ohio 
William Naison, Jamaica High School, N.Y. 
Henry Nelson, Berkeley High School, Calif. 
Joseph D. Novak, Purdue University, Lafayette, 

Ind. 
Thorir Olafsson, Menntaskolinn Ad, Laugarvatni, 

Iceland 
Jay Orear, Cornell University, Ithaca, N.Y. 
Paul O'Toole, Dorchester High School, Mass. 
Costas Papaliolios, Harvard University 
Jacques Parent, National Film Board of Canada, 

Montreal 
Father Thomas Pisors, C.S.U., Griffin High 

School, Springfield, 111. 
Eugene A. Platten, San Diego High School. Calif. 
L. Eugene Poorman, University High School, 

Bloomington, Ind. 
Gloria Poulos, Harvard University 
Herbert Priestley. Knox College, Galesburg, 111. 
Edward M. Purcell, Harvard University 
Gerald M. Rees, Ann Arbor High School, Mich. 
James M. Reid, J. W. Sexton High School. 

Lansing, Mich. 
Robert Resnick, Rensselaer Polytechnic Institute, 

Troy, N.Y. 
Paul I. Richards, Technical Operations, Inc., 

Burlington, Mass. 
John Rigden, Eastern Nazarene College, Quincy, 

Mass. 
Thomas J. Ritzinger, Rice Lake High School, Wise. 
Nickerson Rogers. The Loomis School, Windsor, Conn. 



124 



(continued on p. 198) 




The Project Physics Course 



Handbook 



2 



Motion in the Heavens 





Acknowledgments, Handbook Section 

P. 131 The table is reprinted jfrom Solar and 
Planetary Longitudes for Years -2500 to +2500 
prepared by William D. Stalman and Owen 
Gingerich (University of Wisconsin Press, 1963) 

Picture Credits, Handbook Section 

Cover: (lower left) Cartoon by Charles Gary 
Solin and reproduced by his permission only. 

Pp. 129, 153, 155, 160, 169, 170, 175, 179, 
191 Mount Wilson and Palomar Observatories 
photographs. 

Pp. 130, 132, 134, 139, 142, 148, 154, 164 
(cartoons). By permission of Johnny Hart and 
Field Enterprises, Inc. 

Pp. 132, 137, 161 National Aeronautics and 
Space Administration photograph. 

P. 135 Lick Observatory photograph. 

P. 140 Fig. 5-13. Photograph courtesy of 
Damon Corporation, Educational Division. 

P. 145 Map courtesy of Rand McNally and Co. 

P. 148 Sun film strip photographs courtesy of 
the U.S. Naval Observatory. 

P. 153 Venus, Yerkes Observatory photograph. 

P. 155 Sun spot drawings courtesy of Robert 
A. Pinkham. 

P. 156 Nubbin Cartoon, King Features 
Syndicate. 

P. 162 Lowell Observatory photograph. 

P. 183 By permission of United Features 
Syndicate. 

All photographs used with film loops courtesy 
of National Film Board of Canada. 

Photographs of laboratory equipment and of 
students using laboratory equipment were 
supplied with the cooperation of the Project 
Physics staff and Damon Corporation. 



Contents 



HANDBOOK SECTION 



Chapter 5 Where is the Earth?— 
The Greeks' Answer 

Experiments 

14. Naked-eye Astronomy 128 

(A) One Day of Sun Observations 128 

(B) A Year of Sun Observations 128 

(C) Moon Observations 129 

(D) Locating the Planets 130 

(E) Graphing the Positions of the Planets 130 

15. Size of the Earth 132 

16. The Height of Piton, a Mountain on the Moon 135 
Activities 

Making Angular Measurements 138 
Epicycles and Retrograde Motion 139 
Celestial Sphere Model 141 
How Long is a Sidereal Day? 143 
Scale Model of the Solar System 143 
Build a Sundial 144 
Plot an Analemma 144 
Stonehenge 144 
Moon Crater Names 144 
Literature 144 

The Size of the Earth — SimpUfied Version 145 
Film Strip: Retrograde Motion of Mars 146 
Film Loops 
Film Loop lOA: Retrograde Motion of Mars 

and Mercury 147 
Film Loop 10: Retrograde Motion — 

Geocentric Model 147 

Chapter 6 Does the Earth Move?— The Work of 
Copernicus and Tycho 

Experiments 

17. The Shape of the Earth's Orbit 148 

18. Using Lenses to Make a Telescope 151 
Activities 

Two Activities on Frames of Reference 156 
Film Loop 
Film Loop 1 1 : Retrograde Motion- 
Heliocentric Model 157 



Chapter 7 A New Universe Appears— the Work of 
Kepler and Galiieo 

Experiments 

19. The Orbit of Mars 158 

20. The Orbit of Mercury 162 
Activities 

Three-Dimensional Model of Two Orbits 165 

IncUnation of Mars' Orbit 165 

Demonstrating Satelhte Orbits 167 

Gahleo 168 

Conic-sections Models 168 

Challenging Problem: Finding Earth-Sun Distance 

from Venus Photos 168 
Measuring Irregular Areas 168 



Chapter 8 Unity of Earth and Sky — 
the Work of Newton 

Experiment 

21. Stepwise Approximation to an Orbit 170 
Activities 

Model of the Orbit of Halley's Comet 176 

Other Comet Orbits 179 

Forces on a Pendulum 180 

Haiku 181 

Trial of Copernicus 181 

Discovery of Neptune and Pluto 181 

How to Find the Mass of a Double Star 181 
Film Loops 

Film Loop 12: Jupiter Satelhte Orbit 184 
Interesting Features of the Film 185 
Measurements 185 

Film Loop 13^ Program Orbit I 186 

Film Loopj 13:i'rogram Orbit II 187 
Orbit Prdffam 187 

Film Loop 15: Central Forces — 

Iterated Blows 187 

Film Loop 16: Kepler's Laws 189 

Film Loop 17: Unusual Orbits 190 



127 



Chapter w Where is the Earth? — The Greek's Answers 



EXPERIMENT 14 NAKED-EYE ASTRONOMY 
(Continued from Unit 1, Experiment 1) 

Weather permitting, you have been watch- 
ing events in the day and night sky since this 
course started. Perhaps you have followed the 
sun's path, or viewed the moon, planets, or 
stars. 

From observations much like your own, 
scientists in the past developed a remarkable 
sequence of theories. The more aware you are 
of the motions in the sky and the more you 
interpret them yourself, the more easily you 
can follow the development of these theories. 
If you do not have your own data, you can use 
the results provided in the following sections. 

A. One Day of Sun Observations 

One student made the following observa- 
tions of the sun's position on September 23. 



Eastern Daylight 
Time (EDT) 



Sun's 
Altitude 



Sun's 
Azimuth 



7:00 A.M. 

8:00 

9:00 
10:00 
11:00 
12:00 

1:00 P.M. 

2:00 

3:00 

4:00 

5:00 

6:00 

7:00 



08° 

19 

29 

38 

45 

49 

48 

42 

35 

25 

14 

03 



097° 

107 

119 

133 

150 

172 

197 

217 

232 

246 

257 

267 



If you plot altitude (vertically) against 
azimuth (horizontally) on a graph and mark 
the hours for each point, it will help you to 
answer these questions. 

1. What was the sun's greatest altitude during 
the day? 

2. What was the latitude of the observer? 

3. At what time (EDT) was the sun highest? 

4. When during the day was the sun's direction 
(azimuth) changing fastest? 




iS^^M^^f 



5. When during the day was the sun's altitude 
changing fastest? 

6. At what time of day did the sun reach its 
greatest altitude? How do you explain the fact 
that it is not exactly at 12:00? (Remember 
that daylight time is an hour ahead.) 

B. A Year of Sun Observations 

One student made the following monthly 
observations of the sun through a full year. (He 
had remarkably clear weather!) 



Dates 



Sun's 

Noon 

Altitude 



Sunset 
Azimuth 



Time Between 

Noon and 

Sunset 



Jan 1 
Feb 1 
Mar 1 
Apr 1 
May 1 
Jun 1 
Jul 1 
Aug 1 
Sep 1 
Oct 1 
Nov 1 
Dec 1 

*h = hours, m 



20° 
26 
35 
47 
58 
65 
66 
61 
52 
40 
31 
21 
= minutes. 



238° 

245 

259 

276 

291 

300 

303 

295 

282 

267 

250 

239 



4h25, 

4 50 

5 27 

6 15 

6 55 

7 30 
7 40 
7 13 
6 35 
5 50 
5 00 
4 30 



Use these data to make three plots (differ- 
ent colors or marks on the same sheet of graph 



Experiment 14 129 



40° - 






-t^^ — \ — 


— — ' r^n — ' 1 1 ! 1 — n ' — ' ' ' 




























































































^0^ 


















l" ' 










































































i i 




1 


-i- " 4I xi 




± _ ' _n:: 




X ' 


: ±___ ± ±___± 



90° 



110° 



150° 



150° 



70° 

Fig. 5-1 

paper) of the sun's noon altitude, direction at 
sunset, and time of sunset after noon. Place 
these data on the vertical axis and the dates on 
the horizontal axis. 

1. What was the sun's noon altitude at the 
equinoxes (March 21 and September 23)? 

2. What was the observer's latitude? 

3. If the observer's longitude was 71°W, what 
city was he near? 

4. Through what range (in degrees) did his 
sunset point change during the year? 

5. By how much did the observer's time of 
sunset change during the year? 

6. If the time from sunrise to noon was always 
the same as the time between noon and sunset, 
how long was the sun above the horizon on the 
shortest day? on the longest day? 

C. Moon Observations 

During October 1966 a student in Las 
Vegas, Nevada made the following observa- 



170° 
Azimuth 



190° 



210° 



230° 



250° 



270° 



tions of the moon at sunset when the sun had 
an azimuth of about 255°. 





Angle from 


Moon 


Moon 


Date 


Sun to Moon 


Altitude 


Azimuth 


Oct. 








16 


032° 


17° 


230° 


18 


057 


25 


205 


20 


081 


28 


180 


22 


104 


30 


157 


24 


126 


25 


130 


26 


147 


16 


106 


28 


169 


05 


083 



1. Plot these positions of the moon on a chart 
such as in Fig. 5-1. 

2. From the data and your plot, estimate the 
dates of new moon, first quarter moon, and full 
moon. 

3. For each of the points you plotted, sketch 
the shape of the lighted area of the moon. 



Phases of tfie moon: (1) 23 days, (2) 26 days, (3) 17 days, (4) 5 days, (5) 3 days after new moon. 




130 Experiment 14 



B.C. 



By John Hart 




D. Locating the Planets 

Table 1, Planetary Longitudes lists the 
position of each major planet along the ecliptic. 
The positions are given, accurate to the nearest 
degree, for every ten-day interval. By interpo- 
lation you can find a planet's position on any 
given day. 

The column headed "J.D." shows the cor- 
responding Julian Day calendar date for each 
entry. This calendar is simply a consecutive 
numbering of days that have passed since an 
arbitrary "Juhan Day 1" in 4713 B.C.: Sep- 
tember 30, 1970, for example, is the same as 
J.D. 2,440,860. 

Julian dates are used by astronomers for 
convenience. For example, the number of days 
between March 8 and September 26 of this 
year is troublesome to figure out, but it is easy 
to find by simple subtraction if the Julian dates 
are used instead. 

Look up the sun's present longitude in the 
table. Locate the sun on your SC-1 Constella- 
tion Chart: The sun's path, the ecliptic, is the 
curved line marked off in 360 degrees of 
longitude. 

A planet that is just to the west of the sun's 
position (to the right on the chart) is "ahead of 



the sun," that is, it rises and sets just before 
the sun does. One that is 180° from the sun 
rises near sundown and is in the sky all night. 
When you have decided which planets may 
be visible, locate them along the ecliptic shown 
on your sky map SC-1. Unhke the sun, they are 
not exactly on the ecliptic, but they are never 
more than 8° from it. Once located on the Con- 
stellation Chart you know where to look for a 
planet among the fixed stars. 

E. Graphing the Position of the Planets 

Here is a useful way to display the infor- 
mation in Table I, Planetary Longitudes. On 
ordinary graph paper, plot the sun's longitude 
versus time. Use Julian Day numbers along 
the horizontal axis, beginning close to the 
present date. The plotted points should fall 
on a nearly straight line, sloping up toward the 
right until they reach 360° and then starting 
again at zero. 



3()0f 



CD 

o 



tt'me 

How long will it be before the sun again 
has the same longitude it has today? Would the 
answer to that question be the same if it were 
asked three months from now? What is the 
sun's average angular speed (in degrees per 
day) over a whole year? When is its angular 
speed greatest? 

Plot Mercury's longitudes on the same 
graph (use a different color or shape for the 
points). According to your plot, how far (in 
longitude) does Mercury get from the sun? 
(This is Mercury's "maximum elongation.") 
At what time interval does Mercury pass the 
sun? 



Experiment 14 131 



Table 1 Planet Longitudes at 10-Day Intervals 



Yr. 


Date 


J.D. 


Sun 


Mer 


Ven 


Mors 


Jup 


Sot 


Yr. 


Dote 


J D 


Sun 


Mer 


Ven 


Mors 


Jup 


Sol 


Yr. 


Date 


J.D. 


Sun 


Mer 


Ven 


Mors 


Jup 


Sot 


1969 


Nov 24 


0550 


242 


246 


227 


314 


206 


32 


1972 


Jon 23 


1340 


303 


286 


338 


18 


267 


58 


1974 


Mar 23 


2130 


2 


335 


316 


73 


333 


88 


1969 


Dec 4 


0560 


252 


262 


240 


322 


207 


31 


1972 


Feb 2 


1350 


313 


302 


350 


24 


269 


58 


1974 


Apr 2 


2140 


12 


346 


325 


79 


335 


89 


1969 


Dec 14 


0570 


262 


278 


252 


329 


209 


31 


1972 


Feb 12 


1360 


323 


319 


2 


31 


271 


59 


1974 


Apr 12 


2150 


22 


1 


335 


85 


337 


89 


1969 


Dec 24 


0580 


272 


292 


265 


337 


211 


31 


1972 


Feb 22 


1370 


333 


337 


14 


38 


273 


59 


1974 


Apr 22 


2160 


32 


18 


346 


91 


340 


90 


1970 


Jon 3 


0590 


283 


300 


277 


344 


212 


31 


1972 


Mor 3 


1380 


343 


356 


26 


44 


274 


60 


1974 


Moy 2 


2170 


42 


38 


357 


97 


342 


91 


1970 


Jon 13 


0600 


293 


293 


290 


352 


214 


32 


1972 


Mor 13 


1390 


353 


12 


37 


51 


276 


61 


1974 


May 12 


2180 


51 


61 


8 


103 


343 


93 


1970 


Jon 23 


0610 


303 


284 


303 


359 


215 


32 


1972 


Mor 23 


1400 


3 


16 


48 


57 


277 


61 


1974 


May 22 


2190 


61 


80 


19 


109 


345 


94 


1970 


Feb 2 


0620 


313 


288 


315 


6 


215 


32 


1972 


Apr 2 


1410 


13 


10 


59 


64 


278 


62 


1974 


Jun 1 


2200 


70 


94 


31 


115 


346 


95 


1970 


Feb 12 


0630 


323 


298 


327 


13 


216 


33 


1972 


Apr 12 


1420 


23 


4 


69 


70 


278 


64 


1974 


Jun 11 


2210 


80 


102 


42 


121 


347 


96 


1970 


Feb 22 


0640 


333 


312 


340 


21 


216 


34 


1972 


Apr 22 


1430 


32 


6 


78 


77 


279 


65 


1974 


Jun 21 


2220 


89 


103 


54 


128 


347 


97 


1970 


Mor 4 


0650 


343 


327 


353 


28 


216 


35 


1972 


May 2 


1440 


42 


15 


86 


84 


279 


66 


1974 


Jul 1 


2230 


99 


97 


66 


134 


348 


99 


1970 


Mar 14 


0660 


353 


345 


5 


35 


215 


36 


1972 


May 12 


1450 


52 


29 


91 


90 


278 


67 


1974 


Jul 11 


2240 


109 


94 


77 


140 


348 


100 


1970 


Mor 24 


0670 


3 


4 


17 


42 


214 


37 


1972 


May 22 


1460 


61 


46 


95 


97 


278 


69 


1974 


Jul 21 


2250 


118 


97 


89 


146 


347 


102 


1970 


Apr 3 


0680 


13 


25 


30 


49 


213 


38 


1972 


Jun 1 


1470 


71 


66 


94 


103 


277 


70 


1974 


Jul 31 


2260 


128 


110 


101 


152 


348 


103 


1970 


Apr 13 


0690 


23 


42 


42 


56 


212 


39 


1972 


Jun 11 


1480 


80 


89 


89 


109 


276 


72 


1974 


Aug 10 


2270 


137 


129 


114 


159 


347 


104 


1970 


Apr 23 


0700 


33 


52 


55 


63 


211 


41 


1972 


Jun 21 


1490 


90 


108 


83 


115 


274 


73 


1974 


Aug 20 


2280 


147 


150 


126 


165 


345 


105 


1970 


May 3 


0710 


42 


52 


67 


70 


209 


42 


1972 


Jul 1 


1500 


99 


124 


77 


122 


273 


74 


1974 


Aug 30 


2290 


157 


169 


139 


171 


344 


106 


1970 


May 13 


0720 


52 


45 


80 


77 


208 


43 


1972 


Jul 11 


1510 


109 


136 


76 


128 


271 


75 


1974 


Sep 9 


2300 


166 


185 


151 


177 


342 


106 


1970 


May 23 


0730 


62 


42 


92 


83 


207 


45 


1972 


Jul 21 


1520 


119 


142 


80 


134 


270 


76 


1974 


Sep 19 


2310 


176 


200 


163 


184 


341 


107 


1970 


Jun 2 


0740 


71 


47 


104 


90 


207 


46 


1972 


Jul 31 


1530 


128 


140 


85 


141 


269 


77 


1974 


Sep 29 


2320 


186 


212 


176 


191 


339 


108 


1970 


Jun 12 


0750 


81 


58 


115 


97 


206 


47 


1972 


Aug 10 


1540 


138 


133 


93 


147 


269 


78 


1974 


Oct 9 


2330 


196 


220 


188 


197 


338 


108 


1970 


Jun 22 


0760 


90 


74 


127 


103 


206 


48 


1972 


Aug 20 


1550 


147 


130 


101 


153 


268 


79 


1974 


Oct 19 


2340 


206 


219 


201 


204 


338 


109 


1970 


Jul 2 


0770 


100 


94 


139 


109 


206 


49 


1972 


Aug 30 


1560 


157 


139 


111 


160 


268 


79 


1974 


Oct 29 


2350 


216 


207 


213 


211 


337 


109 


1970 


Jul 12 


0780 


109 


116 


150 


116 


206 


50 


1972 


Sep 9 


1570 


167 


157 


121 


166 


269 


80 


1974 


Nov 8 


2360 


226 


207 


226 


218 


338 


109 


1970 


Jul 22 


0790 


119 


135 


162 


122 


207 


50 


1972 


Sep 19 


1580 


176 


176 


132 


173 


269 


80 


1974 


Nov 18 


2370 


236 


218 


238 


224 


338 


109 


1970 


Aug 1 


0800 


128 


152 


173 


129 


208 


51 


1972 


Sep 29 


1590 


186 


194 


143 


179 


270 


80 


1974 


Nov 28 


2380 


246 


234 


251 


231 


339 


109 


1970 


Aug 11 


0810 


138 


165 


184 


136 


209 


52 


1972 


Oct 9 


1600 


196 


210 


154 


186 


271 


80 


1974 


Dec 8 


2390 


256 


250 


264 


238 


340 


108 


1970 


Aug 21 


0820 


148 


175 


194 


142 


211 


53 


1972 


Oct 19 


1610 


206 


225 


166 


192 


273 


80 


1974 


Dec 18 


2400. 


266 


266 


276 


245 


341 


108 


1970 


Aug 31 


0830 


157 


178 


204 


149 


212 


53 


1972 


Oct 29 


1620 


216 


239 


178 


198 


274 


81 


1974 


Dec 28 


2410 


276 


281 


289 


253 


342 


107 


1970 


Sep 10 


0840 


167 


172 


213 


155 


214 


53 


1972 


Nov 8 


1630 


226 


249 


190 


205 


276 


80 


1975 


Jon 7 


2420 


287 


298 


301 


260 


344 


105 


1970 


Sep 20 


0850 


177 


164 


222 


161 


216 


53 


1972 


Nov 18 


1640 


236 


253 


202 


212 


278 


79 


1975 


Jon 17 


2430 


297 


314 


314 


267 


346 


104 


1970 


Sep 30 


0860 


187 


168 


229 


167 


217 


52 


1972 


Nov 28 


1650 


246 


241 


214 


218 


280 


78 


1975 


Jon 27 


2440 


307 


325 


326 


274 


348 


103 


1970 


Oct 10 


0870 


197 


184 


234 


174 


219 


52 


1972 


Dec 8 


1660 


257 


237 


227 


225 


282 


77 


1975 


Feb 6 


2450 


317 


322 


339 


281 


350 


103 


1970 


Oct 20 


0880 


206 


201 


236 


180 


221 


51 


1972 


Dec 18 


1670 


267 


246 


239 


232 


284 


76 


1975 


Feb 16 


2460 


327 


312 


351 


289 


353 


102 


1970 


Oct 30 


0890 


216 


218 


234 


186 


224 


50 


1972 


Dec 28 


1680 


277 


259 


252 


238 


287 


75 


1975 


Feb 26 


2470 


337 


312 


4 


296 


355 


102 


1970 


Nov 9 


0900 


227 


234 


230 


192 


226 


49 


1973 


Jon 7 


1690 


287 


274 


264 


245 


289 


74 


1975 


Mor 8 


2480 


347 


320 


16 


304 


357 


102 


1970 


Nov 19 


0910 


237 


250 


224 


199 


228 


48 


1973 


Jan 17 


1700 


297 


290 


277 


252 


291 


73 


1975 


Mor 18 


2490 


357 


332 


28 


312 





102 


1970 


Nov 29 


0920 


247 


265 


220 


205 


231 


47 


1973 


Jon 27 


1710 


307 


306 


289 


259 


294 


73 


1975 


Mar 28 


2500 


7 


347 


41 


319 


2 


102 


1970 


Dec 9 


0930 


257 


278 


222 


212 


233 


46 


1973 


Feb 6 


1720 


317 


324 


302 


266 


296 


73 


1975 


Apr 7 


2510 


17 


5 


53 


327 


5 


103 


1970 


Dec 19 


0940 


267 


285 


226 


218 


235 


45 


1973 


Feb 16 


1730 


328 


342 


315 


273 


298 


73 


1975 


Apr 17 


2520 


27 


25 


65 


334 


7 


103 


1970 


Dec 29 


0950 


277 


276 


233 


224 


237 


44 


1973 


Feb 26 


1740 


338 


356 


327 


280 


300 


73 


1975 


Apr 27 


2530 


36 


46 


76 


342 


9 


104 


1971 


Jon 8 


0960 


288 


268 


242 


231 


238 


44 


1973 


Mor 8 


1750 


348 


358 


340 


287 


303 


74 


1975 


May 7 


2540 


46 


65 


88 


349 


12 


105 


1971 


Jon 18 


0970 


298 


273 


251 


237 


240 


44 


1973 


Mor 18 


1760 


358 


349 


352 


294 


305 


74 


1975 


May 17 


2550 


56 


78 


99 


357 


14 


106 


1971 


Jon 28 


0980 


308 


285 


261 


243 


242 


44 


1973 


Mor 28 


1770 


8 


345 


4 


301 


306 


75 


1975 


May 27 


2560 


65 


83 


110 


4 


17 


107 


1971 


Feb 7 


0990 


318 


299 


272 


249 


243 


45 


1973 


Apr 7 


1780 


17 


350 


17 


308 


308 


76 


1975 


Jun 6 


2570 


75 


81 


120 


12 


18 


108 


1971 


Feb 17 


1000 


328 


315 


283 


256 


244 


46 


1973 


Apr 17 


1790 


27 





29 


315 


309 


77 


1975 


Jun 16 


2580 


84 


75 


131 


19 


20 


no 


1971 


Feb 27 


1010 


338 


332 


294 


262 


245 


46 


1973 


Apr 27 


1800 


37 


14 


41 


322 


310 


78 


1975 


Jun 26 


2590 


94 


74 


140 


27 


21 


111 


1971 


Mar 9 


1020 


348 


351 


306 


268 


246 


48 


1973 


May 7 


1810 


47 


32 


54 


329 


311 


79 


1975 


Jul 6 


2600 


103 


82 


148 


33 


22 


112 


1971 


Mor 19 


1030 


358 


10 


318 


274 


246 


49 


1973 


May 17 


1820 


56 


52 


66 


336 


312 


81 


1975 


Jul 16 


2610 


113 


96 


155 


40 


24 


114 


1971 


Mar 29 


1040 


8 


27 


330 


280 


247 


50 


1973 


Moy 27 


1830 


66 


75 


79 


343 


312 


82 


1975 


Jul 26 


2620 


123 


115 


160 


47 


24 


115 


1971 


Apr 8 


1050 


18 


34 


341 


286 


246 


51 


1973 


Jun 6 


1840 


75 


94 


91 


350 


312 


83 


1975 


Aug 5 


2630 


132 


137 


162 


54 


24 


116 


1971 


Apr 18 


1060 


28 


31 


353 


292 


246 


52 


1973 


Jun 16 


1850 


85 


109 


103 


357 


312 


85 


1975 


Aug 15 


2640 


142 


156 


160 


60 


25 


117 


1971 


Apr 28 


1070 


37 


24 


5 


297 


245 


53 


1973 


Jun 26 


1860 


94 


119 


116 


4 


312 


86 


1975 


Aug 25 


2650 


151 


172 


154 


66 


25 


118 


1971 


May 8 


1080 


47 


23 


17 


303 


244 


55 


1973 


Jul 6 


1870 


104 


123 


128 


10 


311 


87 


1975 


Sep 4 


2660 


161 


187 


149 


72 


24 


119 


1971 


May 18 


1090 


57 


31 


29 


308 


242 


56 


1973 


Jul 16 


1880 


114 


120 


140 


16 


310 


89 


1975 


Sep 14 


2670 


171 


198 


145 


77 


23 


120 


1971 


Moy 28 


1100 


66 


43 


41 


314 


241 


58 


1973 


Jul 26 


1890 


123 


113 


152 


22 


308 


89 


1975 


Sep 24 


2680 


181 


205 


146 


82 


22 


121 


1971 


Jun 7 


1110 


76 


60 


53 


318 


240 


59 


1973 


Aug 5 


1900 


133 


114 


164 


28 


306 


90 


1975 


Oct 4 


2690 


190 


202 


150 


86 


21 


122 


1971 


Jun 17 


1120 


85 


80 


66 


320 


239 


60 


1973 


Aug 15 


1910 


142 


125 


176 


33 


305 


91 


1975 


Oct 14 


2700 


200 


190 


156 


90 


19 


122 


1971 


Jun 27 


1130 


95 


103 


78 


322 


238 


61 


1973 


Aug 25 


1920 


152 


143 


188 


37 


304 


92 


1975 


Oct 24 


2710 


210 


192 


164 


92 


18 


123 


1971 


Jul 7 


1140 


104 


122 


90 


323 


237 


62 


1973 


Sep 4 


1930 


162 


163 


200 


39 


303 


93 


1975 


Nov 3 


2720 


220 


205 


174 


93 


16 


123 


1971 


Jul 17 


1150 


114 


138 


103 


324 


237 


63 


1973 


Sep 14 


1940 


171 


181 


211 


40 


302 


94 


1975 


Nov 13 


2730 


230 


221 


184 


92 


15 


123 


1971 


Jul 27 


1160 


123 


151 


115 


323 


236 


64 


1973 


Sep 24 


1950 


181 


198 


223 


40 


302 


94 


1975 


Nov 23 


2740 


240 


237 


194 


89 


14 


123 


1971 


Aug 6 


1170 


133 


159 


127 


320 


237 


65 


1973 


Oct 4 


1960 


191 


213 


235 


88 


302 


94 




















1971 


Aug 16 


1180 


143 


160 


140 


315 


237 


65 


1973 


Oct 14 


1970 


201 


226 


246 


36 


302 


95 




















1971 


Aug 26 


1190 


152 


153 


152 


313 


238 


66 


1973 


Oct 24 


1980 


211 


235 


257 


32 


303 


94 




















1971 


Sep 5 


1200 


162 


147 


164 


312 


239 


66 


1973 


Nov 3 


1990 


221 


236 


268 


27 


304 


94 




















1971 


Sep 15 


1210 


172 


154 


177 


312 


240 


66 


1973 


Nov 13 


2000 


231 


224 


278 


25 


305 


94 


Courtesy of Will 


om D, 


Stohlmc 


n and 


Owen 


Gingeri 


:h. Solo 


r and 


1971 


Sep 25 


1220 


182 


170 


189 


314 


242 


67 


1973 


Nov 23 


2010 


241 


222 


288 


24 


306 


93 


Planetary Longiti 


des for 


Yeorj - 


-2500 to +2500 by lOdoy Intervals, 


1971 


Oct 5 


1230 


191 


189 


202 


315 


243 


67 


1973 


Dec 3 


2020 


251 


232 


297 


25 


308 


93 


the U 


diversity of Wiicons 


in Pres 


, Modi 


son. 








1971 


Oct 15 


1240 


201 


206 


214 


318 


245 


66 


1973 


Dec 13 


2030 


261 


246 


304 


26 


310 


92 




















1971 


Oct 25 


1250 


211 


222 


227 


322 


247 


66 


1973 


Dec 23 


2040 


272 


262 


309 


28 


312 


91 




















1971 


Nov 4 


1260 


221 


238 


239 


328 


249 


65 


1974 


Jon 2 


2050 


282 


278 


313 


32 


314 


90 




















1971 


Nov 14 


1270 


231 


252 


252 


333 


251 


64 


1974 


Jan 12 


2060 


292 


294 


311 


37 


317 


89 




















1971 


Nov 24 


1280 


241 


264 


264 


340 


253 


63 


1974 


Jon 22 


2070 


302 


311 


306 


41 


319 


88 




















1971 


Dec 4 


1290 


252 


269 


277 


346 


256 


61 


1974 


Feb 1 


2080 


312 


328 


300 


46 


321 


87 




















1971 


Dec 14 


1300 


262 


259 


289 


352 


258 


60 


1974 


Feb 11 


2090 


322 


341 


296 


51 


324 


87 




















1971 


Dec 24 


1310 


272 


252 


302 


358 


261 


59 


1974 


Feb 21 


2100 


332 


340 


297 


57 


326 


87 




















1972 


Jon 3 


1320 


282 


259 


314 


5 


263 


59 


1974 


Mar 3 


2110 


342 


330 


301 


63 


329 


87 




















1972 


Jon 13 


1330 


292 


272 


326 


12 


265 


59 


1974 


Mar 13 


2120 


352 


327 


308 


68 


331 


88 





















132 Experiment 14 



B.C. 



By John Hart 



OF CELesnAUTHRJUS. 




^^<r 



By permission of John Hart and Field Enterprises, Inc. 

Plot the positions of the other planets using 
a different color for each one. The data on the 
resulting chart is much like the data that puz- 
zled the ancients. In fact, the table of longi- 
tudes is just an updated version of the tables 
that Ptolemy, Copernicus, and Tycho had 
made. 



The graph contains a good deal of useful 
information. For example, when will Mercury 
and Venus next be close enough to each other 
so that you can use bright Venus to help you 
find Mercury? Where are the planets, relative 
to the sun, when they go through their retro- 
grade motions? 



A "full earth" photo- 
graph from 22,300 
miles In space. 




Experiment 15 133 



EXPERIMENT 15 SIZE OF THE EARTH 

People have been telling you for many 
years that the earth has a diameter of about 
8000 miles and a circumference of about 
25,000 miles. You've believed what they told 
you. But suppose someone challenged you to 
prove it? How would you go about it? 

The first recorded calculation of the size of 
the earth was made a long time ago — in the 
third century B.C., by Eratosthenes. He com- 
pared the lengths of shadows cast by the sun 
at two different points in Egypt. The points 
were rather far apart, but nearly on a north- 
south line on the earth's surface. The experi- 
ment you do here uses a similar method. 
Instead of measuring the length of a shadow, 
you will measure the angle between the verti- 
cal and the sight line to a star or to the sun. 

You will need a colleague at least 200 
miles away, due north or south of your position, 
to take simultaneous measurements. The two 
of you will need to agree in advance on the 
star, the date, and the time for your observa- 
tions. See how close you can come to calculat- 
ing the actual size of the earth. 

Assumptions and Theory of the Experiment 

The experiment is based on the assump- 
tions that 

1. the earth is a perfect sphere, 

2. a plumb line points towards the center of 
the earth, and 



3. the distance from the earth to the stars and 
sun is very great compared with the earth's 
diameter. 

The two observers must be located at 
points nearly north and south of each other. 
Suppose they are at points A and B, separated 
by a distance s, as shown in Fig. 5-2. The ob- 
server at A and the observer at B both sight on 
the same star at the prearranged time, when 
the star is on or near their meridian, and mea- 
sure the angle between the vertical of the 
plumb line and the sight line to the star. 

Light rays from the star reaching locations 
A and B are parallel (this is implied by assump- 
tion 3). 

You can therefore relate the angle ^a at A 
to the angle ^b at B, and to the angle <A between 
the two radii, as shown in Fig. 5-3. 

In the triangle ABO 



</> = (^A - ^b) 



(1) 



If C is the circumference of the earth, and 
s is an arc of the meridian, then you can make 
the proportion 



(2) 



C 360° 
Combining equations (1) and (2), you have 
C= 360° , 

^A ~ ^B 

where ^a and ^b are measured in degrees. 




Fig. 5-2 



Fig. 5-3 



134 Experiment 15 



Doing the Experiment 

For best results, the two locations A and B 
should be directly north and south of each 
other, and the observations should be made 
when the star is near its highest point in the 
sky. 

You will need some kind of instrument to 
measure the angle 0. Such an instrument is 
called an astrolabe. If your teacher does not 
have an astrolabe, you can make one fairly 
easily from a protractor, a small sighting tube, 
and a weighted string assembled according to 
the design in Fig. 5-4. 

Align your astrolabe along the north-south 
line and measure the angle from the vertical 
to the star as it crosses the north-south line. 

If the astrolabe is not aligned along the 
north-south line or meridian, the star will be 
observed before or after it is highest in the sky. 
An error of a few minutes from the time of 
crossing the meridian will make little differ- 
ence in the angle measured. 

An alternative method would be to mea- 
sure the angle to the sun at local noon. This 
means the time when the sun is highest in the 
sky, and not necessarily 12 o'clock by standard 
time. (Remember that the sun, seen from the 
earth, is itself ^° wide.) 



Uock 




Fig. 5-4 






DO NOT TRY SIGHTING DIRECTLY AT THE 
SUN. You may damage your eyes. Instead, have 
the sighting tube of your astrolabe pierce the 
center of a sheet of cardboard so that sunlight 
going through the sighting tube makes a bright 
spot on a shaded card that you hold up behind 
the tube. 



An estimate of the uncertainty in your 
measurement of 6 is important. Take several 
measurements on the same star (at the same 
time) and take the average value of 6. Use the 
spread in values of 6 to estimate the uncer- 
tainty of your observations and of your result. 

Your value for the earth's circumference 
depends also in knowing the over-the-earth 
distance between the two points of observation. 
You should get this distance from a map, using 
its scale of miles. For a description of what 
earth measurements over the years have dis- 
closed about the earth's shape, see: "The Shape 
of the Earth," Scientific American, October, 
1967, page 67. 

Ql How does the uncertainty of the over-the- 
earth distance compare with the uncertainty 
in your value for 6? 

Q2 What is your calculated value for the cir- 
cumference of the earth and what is the uncer- 
tainty of your value? 

Q3 Astronomers have found that the average 
circumference of the earth is about 24,900 
miles (40,000 km). What is the percentage 
error of your result? 

Q4 Is this acceptable, in terms of the uncer- 
tainty of your measurement? 



B.C. 



By John Hart 




LIKE, WE'RE RI&HTHERE! 

y 



now; -WESAIUALLTHE 
WAV AROUND TO HBRE - 



By permission o£ John Hart and Field Enterprl 




Experiment 16 135 



EXPERIMENT 16 THE HEIGHT OF 
PITON, A MOUNTAIN ON THE MOON 

Closeup photographs of the moon's surface 
have been radioed back to earth from Lunar 
Orbiter spacecraft (Fig. 5-10) and from Sur- 
veyor vehicles that have made "soft landings" 
on the moon (Fig. 5-11, p. 137), and carried back 
by the Apollo astronauts. Scientists are discov- 
ering a great deal about the moon from such 
photographs, as well as from the landings 
made by astronauts in Apollo spacecraft. 

But long before the space age, indeed since 
Galileo's time, astronomers have been learning 
about the moon's surface without even leaving 
the earth. In this experiment, you will use a 
photograph (Fig. 5-5) taken with a 36-inch 
telescope in California to estimate the height 
of a mountain on the moon. You will use a 
method similar in principle to one used by 
Galileo, although you should be able to get a 
more accurate value than he could working 
with his small telescope (and without photo- 
graphs!). 




The photograph of the moon in Fig. 5-5 
was taken at the Lick Observatory very near 
the time of the third quarter. The photograph 
does not show the moon as you see it in the sky 
at third quarter because an astronomical tele- 
scope gives an inverted image — reversing top- 
and-bottom and left-and-right. (Thus north is 
at the bottom.) Fig. 5-6 is a lOX enlargement of 
the area within the white rectangle in Fig. 5-5. 

Why Choose PIton? 

Piton, a mountain in the moon's Northern 
Hemisphere, is fairly easy to measure because 
it is a slab-like pinnacle in an otherwise fairly 
fiat area. When the photograph was made, 
with the moon near third quarter phase, Piton 
was quite close to the line separating the 
lighted portion from the darkened portion of 
the moon. (This line is called the terminator.) 

You will find Piton to the south and west of 
the large, dark-floored crater, Plato (numbered 
230 on your moon map) which is located at a 
longitude of —10° and a latitude of +50°. 




Fig. 5-5 



Fig. 5-6 



136 Experiment 16 



Assumptions and Relations 

Fig. 5-7 represents the third-quarter moon 
of radius r, with Piton P, its shadow of length 
I, at a distance d from the terminator. 

The rays of light from the sun can be con- 
sidered to be parallel because the moon is a 
great distance from the sun. Therefore, the 

Soltth foi-£ 




Fig. 5-7 

angle at which the sun's rays strike Piton will 
not change if, in imagination, we rotate the 
moon on an axis that points toward the sun. In 
Fig. 5-8, the moon has been rotated just enough 
to put Piton on the lower edge. In this position 
it is easier to work out the geometry of the 
shadow. 




Fig. 5-9 shows how the height of Piton can 
be found from similar triangles, h represents 
the height of the mountain, I is the apparent 
length of its shadow, d is the distance of the 
mountain from the terminator; r is a radius of 
the moon (drawn from Piton at P to the center 
of the moon's outline at O.) 

It can be proven geometrically (and you 
can see from the drawing) that the small tri- 
angle BPA is similar to the large triangle PCO. 
The corresponding sides of similar triangles 
are proportional, so we can write 

h^d 

I r 



and h 



ixd 



All of the quantities on the right can be mea- 
sured from the photograph. 




Fig. 5-8 



Fig. 5-9 

The curvature of the moon's surface intro- 
duces some error into the calculations, but as 
long as the height and shadow are very small 



Experiment 16 137 



compared to the size of the moon, the error is 
not too great. 

Measurements and Calculations 

Unless specifically instructed by your 
teacher, you should work on a tracing of the 
moon picture rather than in the book itself. 
Trace the outline of the moon and the location 
of Piton. If the photograph was made when the 
moon was exactly at third quarter phase, then 
the moon was divided exactly in half by the 
terminator. The terminator appears ragged 
because highlands cast shadows across the 
lighted side and peaks stick up out of the 
shadow side. Estimate the best overall straight 
line for the terminator and add it to your trac- 
ing. Use a cm ruler to measure the length of 
Piton's shadow and the distance from the 
terminator to Piton's peak. 




It probably will be easiest for you to do all 
the calculations in the scale of the photograph, 
find the height of Piton in cm, and then finally 
change to the real scale of the moon. 
Ql How high is Piton in cm on the photograph 
scale? 

Q2 The diameter of the moon is 3,476 km. 
What is the scale of the photograph? 
Q3 What value do you get for the actual height 
of Piton? 

Q4 Which of your measurements is the least 
certain? What is your estimate of the uncer- 
tainty of your height for Piton? 
Q5 Astronomers, using more complicated 
methods than you used, find Piton to be about 
2.3 km high (and about 22 km across at its 
base). Does your value differ from the accepted 
value by more than your experimental uncer- 
tainty? If so, can you suggest why? 




Fig. 5-10 A fifty square mile area of the moon's surface 
near the large crater, Goclenius. An unusual feature of 
this crater is the prominent rille that crosses the crater 
rim. 



Fig. 5-11 A four-inch rock photographed on the lunar 
surface by Surveyor VII in 1968. 



138 



ACTIVITIES 



MAKING ANGULAR MEASUREMENTS 

For astronomical observations, and often 
for other purposes, you need to estimate the 
angle between two objects. You have several 
instant measuring devices handy once you 
calibrate them. Held out at arm's length in 
front of you, they include: 

1. Your thumb, 

2. Your fist not including thumb knuck- 
les, 

3. Two of your knuckles, and 

4. The span of your hand from thumb-tip 
to tip of little finger when your hand is 
opened wide. 

For a first approximation, your fist is about 
8° and thumb-tip to little finger is between 
15° and 20°. 

However, since the length of people's arms 
and the size of their hands vary, you can cali- 
brate yours using the following method. 

To find the angular size of your thumb, fist, 
and hand span at your arm's length, you make 
use of one simple relationship. An object 
viewed from a distance that is 57.4 times its 
diameter covers an angle of 1°. For example, 
a one-inch coin viewed from 57.4 inches away 
has an angular size of 1°. 

Set a 1" ruler on the blackboard chalk tray. 
Stand with your eye at a distance of 572^" from 
the scale. From there observe how many 
inches of the scale are covered by your thumb, 
etc. Make sure that you have your arm straight 
out in front of your nose. Each inch covered 
corresponds to 1°. Find some convenient mea- 
suring dimensions on your hand. 



A J inch diameter object placed at a distance 
of 27.7 inches (call it 28 inches) from your eye, 
would cut off an angle of 1°. At this same dis- 
tance, 28 inches, a 1 inch diameter object 
would cut off 2° and a 27 inch object 5°. 

Now you can make a simple device that 
you can use to estimate angles of a few degrees. 




Fig. 5-12 



Cut a series of step-wise slots as shown in 
Fig. 5-12, in the file card. Make the slots ^inch 
for 1°, one inch for 2°, and 22" inches for 5°. 
Mount the card vertically at the 28 inch mark 
on a yard stick. (If you use a meter stick, put 
the card at 57 cm and make the slots 1 cm wide 
for 1°, 2 cm for 2°, etc.) Cut flaps in the bot- 
tom of the card, fold them to fit along the stick 
and tape the card to the stick. Hold the zero 
end of the stick against your upper lip — and 
observe. (Keep a stiff upper lip!) 



A Mechanical Aid 

You can use a 3" x 5" file card and a meter 
stick or yard stick to make a simple instrument 
for measuring angles. Remember that when 
an object with a given diameter is placed at a 
distance from your eye equal to 57.4 times its 
diameter, it cuts off an angle of 1°. This means 
that a one inch object placed at a distance of 
57.4 inches from your eye would cut off an 
angle of 1°. An instrument 57.4 inches long 
would be a bit cumbersome, but we can scale 
down the measurements to a convenient size. 




Things to Observe 

1. What is the visual angle between the 
pointers of the Big Dipper? 

2. What is the angular length of Orion's Belt? 



Activities 



139 



B.C. 



By John Hart 




By permission of John Hart and Field Enterprises, Inc. 



3. How many degrees away from the sun is 
the moon? Observe on several nights at sunset. 

4. What is the angular diameter of the moon? 
Does it change between the time the moon 
rises and the time when it is highest in the sky 
on a given day? To most people, the moon 
seems larger when near the horizon. Is it? 
See: "The Moon Illusion," Scientific Ameri- 
can, July 1962, p. 120. 



-T^. 



x 
^ 



and 1 to 2 (one loop per two revolutions). To 
change the ratio, simply slip the drive band to 
another set of pulleys. The belt should be 
twisted in a figure 8 so the deferent arm (the 
long arm) and the epicycle arm (the short arm) 
rotate in the same direction. 



AX/S 

I 

1 



jno 



i--^.. 






EPICYCLES AND RETROGRADE MOTION 

The hand-operated epicycle machine al- 
lows you to explore the motion produced by two 
circular motions. You can vary both the ratio 
of the turning rates and the ratio of the radii 
to find the forms of the different curves that 
may be traced out. 

The epicycle machine has three possible 
gear ratios: 2 to 1 (producing two loops per 
revolution). 1 to 1 (one loop per revolution) 



Tape a light source (pen-light cell, holder 
and bulb) securely to one end of the short, 
epicycle arm and counter-weight the other end 
of the arm with, say, another (unlit) light 
source. If you use a fairly high rate of rotation 
in a darkened room, you and other observers 
should be able to see the light source move in 
an epicycle. 

The form of the curve traced out depends 
not only on the gear ratio but also on the rela- 
tive lengths of the arms. As the light is moved 
closer to the center of the epicycle arm, the 
epicycle loop decreases in size until it becomes 



140 



Activities 




Fig. 5-13 

a cusp. When the hght is very close to the 
center of the epicycle arm, as it would be for 
the motion of the moon around the earth, the 
curve will be a slightly distorted circle. (Fig. 
5-14). 

To relate this machine to the Ptolemaic 
model, in which planets move in epicycles 
around the earth as a center, you should really 
stand at the center of the deferent arm (earth) 
and view the lamp against a distant fixed back- 
ground. The size of the machine, however, 
does not allow you to do this, so you must view 
the motion from the side. (Or, you can glue a 
spherical glass Christmas-tree ornament at 
the center of the machine; the reflection you 
see in the bulb is just what you would see if 
you were at the center.) The lamp then goes 
into retrograde motion each time an observer 




in front of the machine sees a loop. The retro- 
grade motion is most pronounced with the 
light source far from the center of the epicycle 
axis. 



Photographing Epicycles 

The motion of the light source can be 
photographed by mounting the epicycle ma- 
chine on a turntable and holding the center 
pulley stationary with a clamp (Fig. 5-15). 
Alternatively, the machine can be held in 
a burette clamp on a ringstand and turned 
by hand. 

Running the turntable for more than one 
revolution may show that the traces do not 
exactly overlap (Fig. 5-13). (This probably 
occurs because the drive band is not of uni- 
form thickness, particularly at its joint, or 
because the pulley diameters are not in exact 
proportion.) As the joining seam in the band 
runs over either pulley, the ratio of speeds 
changes momentarily and a slight displace- 
ment of the axes takes place. By letting the 
turntable rotate for some time, the pattern will 
eventually begin to overlap. 

A time photograph of this motion can re- 
veal very interesting geometric patterns. You 
might enjoy taking such pictures as an after- 
class activity. Figures 5-16, a through d, show 
four examples of the many diff'erent patterns 
that can be produced. 




Fig. 5-14 



Fig. 5-15 

turntable. 



An epicycle dennonstrator connected to a 



Activities 141 





Fig. 5-1 6a 



Fig. 5-16b 




Fig. 5-1 6c 

CELESTIAL SPHERE MODEL* 

You can make a model of the celestial 
sphere with a round-bottom flask. With it, you 
can see how the appearance of the sky changes 
as you go northward or southward and how the 
stars appear to rise and set. 

To make this model, you will need, in ad- 
dition to the round-bottom flask, a one-hole 
rubber stopper to fit its neck, a piece of glass 
tubing, paint, a fine brush (or grease pencil), 
a star map or a table of star positions, and 
considerable patience. 

On the bottom of the flask, locate the point 
opposite the center of the neck. Mark this point 
and label it "N" for north celestial pole. With 
a string or tape, determine the circumference 
of the flask — the greatest distance around it. 
This will be 360° in your model. Then, starting 
at the north celestial pole, mark points that 



*Adapted from You and Science, by Paul F. Brandwein, et 
al., copyright 1960 by Harcourt, Brace and World, Inc. 




Fig. 5-1 6d 

are j of the circumference, or 90°, from the 
North Pole point. These points lie around the 
flask on a line that is the celestial equator. You 
can mark the equator with a grease pencil 
(china marking pencil), or with paint. 



eouafor 




Fig. 5-17 



By John Hart 
Tl^e COtJSTEU-ATiOWS. 




By permission of John Hart and Field Enterprises, Inc. 



To locate the stars accurately on your 
"globe of the sky," you will need a coordinate 
system. If you do not wish to have the coordi- 
nate system marked permanently on your 
model, put on the lines with a grease pencil. 

Mark a point 23-|-° from the North Pole 
(about i of 90°). This will be the pole of the 
ecliptic marked E.P. in Fig. 5-17. The ecliptic 
(path of the sun) will be a great circle 90° from 
the ecliptic pole. The point where the ecliptic 
crosses the equator from south to north is 
called the vernal equinox, the position of the 
sun on March 21. All positions in the sky, are 
located eastward from this point, and north or 
south from the equator. 

To set up the north-south scale, measure 
off eight circles, about 10° apart, that run east 
and west in the northern hemisphere parallel 
to the equator. These lines are like altitude on 
the earth but are called declination in the sky. 
Repeat the construction of these lines of de- 
clination for the southern hemisphere. 

A star's position, called its right ascension, 
is recorded in hours eastward from the vernal 
equinox. To set up the east-west scale, mark 
intervals of l/24th of the total circumference 
starting at the vernal equinox. These marks 
are 15° apart (rather than 10°) — the sky turns 
through 15° each hour. 

From a table of star positions or a star 
map, you can locate a star's coordinates, then 
mark the star on your globe. All east-west 
positions are expressed eastward, or to the 
right of the vernal equinox as you face your 
globe. 



To finish the model, put the glass tube into 
the stopper so that it almost reaches across the 
flask and points to your North Pole point. Then 
put enough inky water in the flask so that, 
when you hold the neck straight down, the 
water just comes up to the line of the equator. 
For safety, wrap wire around the neck of the 
flask and over the stopper so it will not fall 
out (Fig. 5-18). 

Now, as you tip the flask you have a model 
of the sky as you would see it from different 
latitudes in the Northern Hemisphere. If you 
were at the earth's North Pole, the north celes- 
tial pole would be directly overhead and you 
would see only the stars in the northern half of 
the sky. If you were at latitude 45°N, the north 




€^ 



u ft tor 



Fig. 5-18 



celestial pole would be halfway between the 
horizon and the point directly overhead. You 
can simulate the appearance of the sky at 45°N 
by tipping the axis of your globe to 45° and 
rotating it. If you hold your globe with the axis 
horizontal, you would be able to see how the 
sky would appear if you were at the equator. 



HOW LONG IS A SIDEREAL DAY? 

A sidereal day is the time interval it takes a 
star to travel completely around the sky. To 
measure a sidereal day you need an electric 
clock and a screw eye. 

Choose a neighboring roof, fence, etc., to- 
wards the west. Then fix a screw-eye as an 
eye-piece in some rigid support such as a post 
or a tree so that a bright star, when viewed 
through the screw-eye will be a little above the 
roof (Fig. 5-19). 

Record the time when the star viewed 
through the screw-eye just disappears behind 
the roof, and again on the next night. How long 
did it take to go around? What is the uncer- 
tainty in your measurement? If you doubt your 
result, you can record times for several nights 
in a row and average the time intervals; this 
should give you a very accurate measure of a 
sidereal day. (If your result is not exactly 24 
hours, calculate how many days would be 
needed for the error to add up to 24 hours.) 



A scale model of the solar system 













Sample 


Object 


Solar 


Distance 


Diameter 


Object 




AU 


Model 


km 


Model 








(ft) 


(approx.) ( 


inches) 




Sun 






1 ,400,000 


3 


tennis ball 


Mercury 


0.39 




4,600 






Venus 


0.72 




12,000 






Earth 


1.00 


27 


13,000 




pinhead 


Mars 


1.52 




6.600 






Jupiter 


5.20 




140,000 






Saturn 


9.45 




120,000 






Uranus 


19.2 




48,000 






Neptune 


30.0 




45,000 






Pluto 


39.5 




6,000 






Nearest 












star 


2.7 X 


10^ 










SCALE MODEL OF THE SOLAR SYSTEM 

Most drawings of the solar system are 
badly out of scale, because it is impossible to 
show both the sizes of the sun and planets and 
their relative distances on an ordinary-sized 
piece of paper. Constructing a simple scale 
model will help you develop a better picture 
of the real dimensions of the solar system. 

Let a three-inch tennis ball represent the 
sun. The distance of the earth from the sun is 
107 times the sun's diameter, or for this model, 
about 27 feet. (You can confirm this easily. In 
the sky the sun has a diameter of half a degree 
— about half the width of your thumb when 
held upright at arm's length in front of your 
nose. Check this, if you wish, by comparing 
your thumb to the angular diameter of the 
moon, which is nearly equal to that of the sun; 
both have diameters of j°. Now hold your 
thumb in the same upright position and walk 
away from the tennis ball until its diameter is 
about half the width of your thumb. You will 
be between 25 and 30 feet from the ball!) 
Since the diameter of the sun is about 1,400,000 
kilometers (870,000 mUes), in the model one 
inch represents about 464,000 kilometers. 
From this scale, the proper scaled distances 
and sizes of all the other planets can be derived. 

The moon has an average distance of 
384,000 kilometers from the earth and has a 
diameter of 3,476 kilometers. Where is it on 
the scale model? How large is it? Completion 
of the column for the scale-model distances in 
the table to the left will yield some surprising 
results. 



144 Activities 



The average distance between the earth 
and sun is called the "astronomical unit" (AU). 
This unit is used for describing distance within 
the solar system. 

BUILD A SUNDIAL 

If you are interested in building a sundial, 
there are numerous articles in the Amateur 
Scientist section of Scientific American that 
you will find helpful. See particularly the 
article in the issue of August 1959. Also see the 
issues of September 1953, October 1954, 
October 1959, or March 1964. The book Sun- 
dials by Mayall and Mayall (Charles T. Bran- 
ford, Co., publishers, Boston) gives theory and 
building instructions for a wide variety of 
sundials. Encyclopedias also have helpful 
articles. 

PLOT AN ANALEMMA 

Have you seen an analemma? Examine a 
globe of the earth, and you will usually find a 
graduated scale in the shape of a figure 8, 
with dates on it. This figure is called an ana- 
lemma. It is used to summarize the changing 
positions of the sun during the year. 

You can plot your own analemma. Place a 
small square mirror on a horizontal surface so 
that the reflection of the sun at noon falls on a 
south-facing wall. Make observations each day 
at exactly the same time, such as noon, and 
mark the position of the reflection on a sheet 
of paper fastened to the wall. If you remove 
the paper each day, you must be sure to replace 
it in exactly the same position. Record the date 
beside the point. The north-south motion is 
most evident during September-October and 
March- April. You can find more about the 
east-west migration of the marks in astronomy 
texts and encyclopedias under the subject 
Equation of Time. 



STONEHENGE 

Stonehenge (pages 1 and 2 of your Unit 2 
Text) has been a mystery for centuries. Some 
scientists have thought that it was a pagan 
temple, others that it was a monument to 
slaughtered chieftains. Legends invoked the 



power of Merlin to explain how the stones were 
brought to their present location. Recent 
studies indicate that Stonehenge may have 
been an astronomical observatory and eclipse 
computer. 

Read "Stonehenge Physics," in the April, 
1966 issue of Physics Today: Stonehenge 
Decoded, by G. S. Hawkins and J. B. White; or 
see Scientific American, June, 1953. Make a 
report and/or a model of Stonehenge for your 
class. 



MOON CRATER NAMES 

Prepare a report about how some of the 
moon craters were named. See Isaac Asimov's 
Biographical Encyclopedia of Science and 
Technology for material about some of the 
scientists whose names were used for craters. 



LITERATURE 

The astronomical models that you read 
about in Chapters 5 and 6, Unit 2, of the Text 
strongly influenced the Elizabethan view of 
the world and the universe. In spite of the ideas 
of Galileo and Copernicus, writers, philoso- 
phers, and theologians continued to use Aris- 
totelian and Ptolemaic ideas in their works. In 
fact, there are many references to the crystal- 
sphere model of the universe in the writings 
of Shakespeare, Donne, and Milton. The refer- 
ences often are subtle because the ideas were 
commonly accepted by the people for whom 
the works were written. 

For a quick overview of this idea, with 
reference to many authors of the period, read 
the paperbacks The Elizabethan World Pic- 
ture, by E. M. W. Tillyard, Vintage Press, or 
Basil Willey, Seventeenth Century Back- 
ground, Doubleday. See also the articles by 
H. Butterfield and B. Willey in Project Physics 
Reader 1. 

An interesting specific example of the 
prevailing view, as expressed in literature, is 
found in Christopher Marlowe's Doctor Faus- 
tus, when Faustus sells his soul in return for 
the secrets of the universe. Speaking to the 
devil, Faustus says: 



Activities 145 



". . . Come, Mephistophilis, let us 

dispute again 
And argue of divine astrology. 
Tell me, are there many heavens 

above the moon? 
Are all celestial bodies but one 

globe. 
As is the substance of this centric 

earth? . . . 

THE SIZE OF THE EARTH— SIMPLIFIED 
VERSION 

Perhaps, for lack of a distant colleague, 
you were unable to determine the size of the 
earth as described in Experiment 13. You may 
still do so if you measure the maximum alti- 
tude of one of the objects on the following list 
and then use the attached data as described 
below. 

In Santiago, Chile, Miss Maritza Campu- 
sano Reyes made the following observations 
pf the maximum altitude of stars and of the 



sun: (all were observed north of her zenith) 

Antares (Alpha Scorpio) 83.0° 

Vega (Alpha Lyra) 17.5 

Deneb (Alpha Cygnus) 11.5 

Altair (Alpha Aquila) 47.5 

Fomalhaut (Alpha Pisces Austr.) 86.5 

Sun: October 1 59.4° 
15 64.8° 
November 1 70.7° 
15 74.8° 

Since Miss Reyes made her observations when 
the objects were highest in the sky, the values 
depend only upon her latitude and not upon 
her longitude or the time at which the observa- 
tions were made. 

From the map below, find how far north 
you are from Santiago. Next, measure the 
maximum altitude of one or more of these 
objects at your location. Then using the reason- 
ing in Experiment 13, calculate a value for the 
circumference of the earth. 




."W*-^ 



11 O* N * ' ' = ' ^ ■' " i*"«»^' /'<2*'«*cM«? »'»«" ' 






R T W"- 



::;;--;-A-.W,;h 



JAPAN 






TmfM •/ Cmmt#t 



P 'Vftj" 1. ►•• 












«/•«•:. 



•i<l«!06 






' Gun 









'*ijLJ»» • 



H 

K A 2 I L 









llitii^ 



■M^ 



"f«t 



'*•!*• 












J^LI A 



"S'#r 






Trtflt Iff t rn f lmn 



.<.*oo.rAME ft I c A-'"v ^ 






^'.jpTKM Aires 



S <> u ( h 






•nv?' 






■f« .XJil-i 



^. 



146 Activities 



FILM STRIP 

RETROGRADE MOTION OF MARS 

Photographs of the positions of Mars, from 
the files of the Harvard College Observatory, 
are shown for three oppositions of Mars, in 
1941, 1943, and 1946. The first series of twelve 
frames shows the positions of Mars before and 
after the opposition of October 10, 1941. The 
series begins with a photograph on August 3, 
1941 and ends with one on December 6, 1941. 
The second series shows positions of Mars 
before and after the opposition of December 5, 
1943. This second series of seven photographs 
begins on October 28, 1943 and ends on Febru- 
ary 19, 1944. 

The third set of eleven pictures, which 
shows Mars during 1945-46, around the opposi- 
tion of January 14, 1946, begins with October 
16, 1945 and ends with February 23, 1946. 

The film strip is used in the following way: 

1. The star fields for each series of frames 
have been carefully positioned so that the star 
positions are nearly identical. If the frames of 
each series can be shown in rapid succession, 
the stars will be seen as stationary on the 
screen, while the motion of Mars among the 
stars is quite apparent. This would be like 
viewing a flip-book. 

2. The frames can be projected on a paper 
screen where the positions of various stars and 
of Mars can be marked. If the star pattern for 
each frame is adjusted to match that plotted 
from the first frame of that series, the positions 
of Mars can be marked accurately for the 
various dates. A continuous line through these 
points will be a track for Mars. The dates of 
the turning points (when Mars begins and ends 
its retrograde motion) can be estimated. From 
these dates, the duration of the retrograde 



motion can be found. By use of the scale (10°) 
shown on one frame, the angular size of the 
retrograde loop can also be derived. 

During 1943-44 and again in 1945-46, 
Mars and Jupiter came to opposition at ap- 
proximately the same time. As a result, Jupiter 
appears in the frames and also shows its retro- 
grade motion. Jupiter's oppositions were: 
January 11, 1943; February 11, 1944; March 
13, 1945; and April 13, 1946. Jupiter's position 
can also be tracked, and the duration and size 
of its retrograde loop derived. The durations 
and angular displacements can be compared 
to the average values listed in Table 5.1 of the 
Unit 2 Text. This is the type of observational 
information which Ptolemy, Copernicus and 
Kepler attempted to explain by their theories. 

The photographs were taken by the routine 
Harvard Sky Patrol with a camera of 6-inch 
focal length and a field of 55°. During each 
exposure, the camera was driven by a clock- 
work to follow the daily western motion of the 
stars and hold their images fixed on the photo- 
graphic plate. Mars was never in the center of 
the field and was sometimes almost at the edge 
because the photographs were not made es- 
pecially to show Mars. The planet just hap- 
pened to be in the star fields being photo- 
graphed. 

The images of the stars and planets are 
not of equal brightness on all pictures because 
the sky was less clear on some nights and the 
exposures varied somewhat in duration. Also, 
the star images show distortions from limita- 
tions of the camera's lens. Despite these 
limitations, however, the pictures are adequate 
for the uses described above. 

From a purely artistic point of view, some 
of the frames show beautiful pictures of the 
Milky Way in Taurus (1943) and Gemini (1945). 



147 



FILM LOOPS 



FILM LOOP 10A RETROGRADE 
MOTION OF MARS AND MERCURY 

To illustrate the retrograde motions of all 
the planets, the retrograde motions of Mercury 
and Mars are shown. The changing positions 
of each planet against the background of stars 
are shown during several months by animated 
drawings. Stars are represented by small disks 
whose sizes are proportional to the brightness 
of the stars. 

Mercury first moves eastward, stops, and 
moves westward in retrograde motion. During 
the retrograde motion, Mercury passes be- 
tween the earth and the sun, which has been 
moving steadily eastward. Mercury stops its 
westward motion and resumes its eastward 
motion following the sun. Time flashes appear 
for each five days. The star field includes por- 
tions of the constellations of Aries and Taurus; 
the familiar cluster of stars known as the 
Pleiades is in the upper left part of the field. 

Mars similarly moves eastward across the 
star field, stops, moves westward in retrograde 
motion, stops, and resumes its eastward mo- 
tion. Time flashes appear for each ten days. 
The star field included parts of the constella- 
tions of Leo and Cancer. The open star cluster 
at the upper right is Praesepe (the Beehive) 
which is faintly visible on a moonless night 
(and beautiful in a small telescope). 

An angular scale (10°) allows the magni- 
tude of the retrograde motions to be measured, 
while the time flashes permit a determination 
of the duration of those motions. The disks 
representing the planets change in brightness 
in the same manner as observed for the planets 
in the sky. 

FILM LOOP 10 RETROGRADE MOTION 
-GEOCENTRIC MODEL 

The film illustrates the motion of a planet 



such as Mars, as seen from the earth. It was 
made using a large "epicycle machine," as a 
model of the Ptolemaic system. 

First, from above, you see the character- 
istic retrograde motion during the "loop" when 
the planet is closest to the earth. Then the 
studio lights go up and you see that the motion 
is due to the combination of two circular mo- 
tions. One arm of the model rotates at the end 
of the other. 

The earth, at the center of the model, is 
then replaced by a camera that points in a fixed 
direction in space. The camera views the mo- 
tion of the planet relative to the fixed stars (so 
the rotation of the earth on its axis is being 
ignored). This is the same as if you were look- 
ing at the stars and planets from the earth 
toward one constellation of the zodiac, such as 
Sagittarius. 

The planet, represented by a white globe, 
is seen along the plane of motion. The direct 
motion of the planet, relative to the fixed stars, 
is eastward, toward the left (as it would be if 
you were facing south). A planet's retrograde 
motion does not always occur at the same 
place in the sky, so some retrograde motions 
are not visible in the chosen direction of obser- 
vation. To simulate observations of planets 
better, an additional three retrograde loops 
were photographed using smaller bulbs and 
slower speeds. 

Note the changes in apparent brightness 
and angular size of the globe as it sweeps close 
to the camera. Actual planets appear only as 
points of light to the eye, but a marked change 
in brightness can be observed. This was not 
considered in the Ptolemaic system, which fo- 
cused only on positions in the sky. 

Another film loop, described on page 33, 
Chapter 6 of this Handbook, shows a similar 
model based on a heliocentric theory. 



Chapter 



6 



Does the Earth Move? — The Work of Copernicus and Tycho 



EXPERIMENT 17 

THE SHAPE OF THE EARTH'S ORBIT 

Ptolemy and most of the Greeks thought 
that the sun revolved around the earth. But 
after the time of Copernicus the idea gradually 
became accepted that the earth and other 
planets revolve around the sun. Although you 
probably believe the Copemican model, the 
evidence of your senses gives you no reason to 
prefer one model over the other. 

With your unaided eyes you see the sun 
going around the sky each day in what ap- 
pears to be a circle. This apparent motion of 
the sun is easily accounted for by imagining 
that it is the earth wiiich rotates once a day. 
But the sun also has a yearly motion with 
respect to the stars. Even if we argue that the 
daily motion of objects in the sky is due to the 
turning of the earth, it is still possible to think 
of the earth as being at the center of the uni- 
verse, and to imagine the sun moving in a 
year-long orbit around the earth. Simple mea- 
surements show that the sun's angular size 
increases and decreases slightly during the 
year as if it were alternately changing its 
distance from the earth. An interpretation that 
fits these observations is that the sun travels 
around the earth in a slightly off-center circle. 

The purpose of this laboratory exercise is 
to plot the sun's apparent orbit with as much 
accuracy as possible. 



Plotting the Orbit 

You know the sun's direction on each date 
that the sun is observed. From its observed 
diameter on that date you can find its relative 
distance from the earth. So, date by date, you 
can plot the sun's direction and relative dis- 
tance. When you connect your plotted points by 
a smooth curve, you will have drawn the sun's 
apparent orbit. 




Fig. 6-1 Frame 4 of the Sun Filmstrip. 

For observations you will use a series of 
sun photographs taken by the U.S. Naval Ob- 
servatory at approximately one-month inter- 
vals and printed on a film strip. Frame 4, in 
which the images of the sun in January and in 
July are placed adjacent to each other, has 
been reproduced in Fig. 6-1 so you can see how 



B.C. 



By John Hart 




By permission of John Hart and Field Enterprises, Inc. 



Experiment 17 149 



much the apparent size of the sun changes 
during the year. Then note in Fig. 6-2 how the 
apparent size of an object is related to its dis- 
tance from you. 




Fig. 6-2 When an object is closer to your eye, it looks 
bigger; it fills a larger angle as seen by your eye. In fact, 
the angles 0.^ and ^b are inversely proportional to the 
distances EA and EB: 

0B^EA 
dj, EB 

In this drawing EB = f EA, so angle ^b = I angle 0^- 



Procedure 

On a large sheet of graph paper (16" x 20", 
or four 82"" X 11 "pieces taped together) make a 
dot at the center to represent the earth. It is 
particularly important that the graph paper be 
this large if you are going on to plot the orbit of 
Mars (Experiments 17 and 19) which uses the 
results of the present experiment. 

Take the 0° direction (toward a reference 



point among the stars) to be along the graph- 
paper lines toward the right. This will be the 
direction of the sun as seen from the earth on 
March 21. (Fig. 6-3) The dates of all the photo- 
graphs and the directions to the sun, measured 
counterclockwise from this zero direction, are 
given in the table below. Use a protractor in 
order to draw accurately a fan of lines radiat- 
ing from the earth in these different directions. 




March 21 
April 6 
May 6 
June 5 
Julys 
Aug. 5 
Sept. 4 



I 



^mrci^ d ^ 






Direction 

from earth 

to sun 



Date 



Direction 

from earth 

to sun 



000° 

015 

045 

074 

102 

132 

162 



Oct. 4 
Nov. 3 
Dec. 4 
Jan. 4 
Feb. 4 
March 7 



191° 

220 

250 

283 

315 

346 



I 




150 Experiment 17 



Measure carefully the diameter of the pro- 
jected image on each of the frames of the 
film strip. The apparent diameter of the sun 
depends inversely on how far away it is. You 
can get a set of relative distances to the sun 
by choosing some constant and then dividing it 
by the apparent diameters. An orbit with a 
radius of about 10 cm will be a particularly 
convenient size for later use. If you measure 
the sun's diameter to be about 50 cm, a con- 
venient constant to choose would be 500, since 
-^-= 10. A larger image 51.0 cm in diameter 
leads to a smaller earth- sun distance: 



500 
51.0 



= 9.8 cm. 



Make a table of the relative distances for 
each of the thirteen dates. 

Along each of the direction lines you have 
drawn, measure off the relative distance to 
the sun for that date. Through the points lo- 
cated in this way draw a smooth curve. This 
is the apparent orbit of the sun relative to the 
earth. (Since the distances are only relative, 
you cannot find the actual distance in miles 
from the earth to the sun from this plot.) 
Ql Is the orbit a circle? If so, where is the 
center of the circle? If the orbit is not a circle, 
what shape is it? 

Q2 Locate the major axis of the orbit through 
the points where the sun passes closest to and 
farthest from the earth. What are the approxi- 
mate dates of closest approach and greatest 
distance? What is the ratio of the largest dis- 
tance to the smallest distance? 

A Heliocentric System 

Copernicus and his followers adopted the 
sun-centered model because they believed that 
the solar system could be described more 
simply that way. They had no new data that 
could not be accounted for by the old model. 

Therefore, you should be able to use the 
same data to turn things around and plot the 
earth's orbit around the sun. Clearly there's 
going to be some similarity between the two 
plots. 



You already have a table of the relative 
distances between the sun and the earth. The 
dates of largest and smallest distances from 
the earth won't change, and your table of 
relative distances is still valid because it 
wasn't based on which body was moving, only 
on the distance between them. Only the direc- 
tions used in your plotting will change. 

To figure out how the angles will change, 
remember that when the earth was at the 
center of the plot the sun was in the direction 
0° (to the right) on March 21. 
Q3 This being so, what is the direction of the 
earth as seen from the sun on that date? See if 
you can't answer this question for yourself 
before studying Fig. 6-4. Be sure you under- 
stand it before going on. 




direct) an 
of vet^nal 



March 



Fig. 6-4 



At this stage the end is in sight. Perhaps 
you can see it already without doing any more 
plotting. But if not, here is what you can do. 

If the sun is in the 0° direction from the 
earth, then from the sun the earth will appear 
to be in just the opposite direction, 180° away 
from 0°. You could make a new table of data 
giving the earth's apparent direction from the 
sun on the 12 dates, just by changing all the 
directions 180° and then making a new sun- 
centered plot. An easier way is to rotate your 
plot until top and bottom are reversed; this will 
change all of the directions by 180°. Relabel 
the 0° direction; since it is toward a reference 
point among the distant stars, it will still be 
toward the right. You can now label the center 
as the sun, and the orbit as the earth's. 



Experiment 18 151 



EXPERIMENT 18 

USING LENSES TO MAKE A TELESCOPE 

In this experiment, you will first examine 
some of the properties of single lenses. Then, 
you will combine these lenses to form a tele- 
scope, which you can use to observe the moon, 
the planets, and other heavenly (as well as 
earth-bound) objects. 

The Simple Magnifier 

You certainly know something about 
lenses already — for instance, that the best way 
to use a magnifier is to hold it immediately in 
front of the eye and then move the object you 
want to examine untU its image appears in 
sharp focus. 

Examine some objects through several 
different lenses. Try lenses of various shapes 
and sizes. Separate the lenses that magnify 
from those that don't. Describe the difference 
between lenses that magnify and those that 
do not. 

Ql Arrange the lenses in order of their magni- 
fying powers. Which lens has the highest 
magnifying power? 

Q2 What physical feature of a lens seems to 
determine its power or ability to magnify — is it 
diameter, thickness, shape, the curvature of 
its surface? To vary the diameter, simply put 
pieces of paper or cardboard with various sizes 
of holes in them over the lens. 

Sketch side views of a high-power lens, of 
a low-power lens, and of the highest-power and 
lowest-power lenses you can imagine. 

Real Images 

With one of the lenses you have used, 
project an image of a ceiling light or an out- 



door scene on a sheet of paper. Describe all the 
properties of the image that you can observe. 
An image that can be projected is called a real 
image. 

Q3 Do all your lenses form real images? 
Q4 How does the size of the image depend on 
the lens? 

Q5 If you want to look at a real image without 
using the paper, where do you have to put your 
eye? 

Q6 The image (or an interesting part of it) may 
be quite small. How can you use a second lens 
to inspect it more closely? Try it. 
Q7 Try using other combinations of lenses. 
Which combination gives the greatest mag- 
nification? 

Making a Telescope 

With two lenses properly arranged, you can 
magnify distant objects. Figure 6-5 shows a 
simple assembly of two lenses to form a tele- 
scope. It consists of a large lens (called the 
objective) through which light enters and 
either of two interchangeable lenses for eye- 
pieces. 

The following notes will help you assemble 
your telescope. 

1. If you lay the objective down on a fiat clean 
surface, you will see that one surface is more 
curved than the other. The more curved sur- 
face should face the front of the telescope. 

2. Clean dust, etc., off the lenses (using lens 
tissue or clean handkerchief) before assembl- 
ing and try to keep fingerprints off it during 
assembly. 

3. Wrap rubber bands around the slotted end 
of the main tube to give a convenient amount 



OgJBCT 







K&Al^ 



O' R/fV6r 







LOUJ-P00^6.^ 

^yc piece 



Fig. 6-5 



152 Experiment 18 



of friction with the draw-tube — tight enough 
so as not to move once adjusted, but loose 
enough to adjust without sticking. Focus by 
shding the draw tube with a rotating motion, 
not by moving the eyepiece in the tube. 

4. To use high power satisfactorily, a steady 
support (a tripod) is essential. 

5. Be sure that the lens lies flat in the high- 
power eyepiece. 

Use your telescope to observe objects in- 
side and outside the lab. Low power gives 
about 12X magnification. High power gives 
about 30X magnification. 

Mounting the Telescope 

If no tripod mount is available, the tele- 
scope can be held in your hands for low-power 
observations. Grasp the telescope as far for- 
ward and as far back as possible (Fig. 6-6) and 
brace both arms firmly against a car roof, 
telephone pole, or other rigid support. 

With the higher power you must use a 
mounting. If a swivel-head camera tripod is 
available, the telescope can be held in a 
wooden saddle by rubber bands, and the saddle 
attached to the tripod head by the head's stan- 
dard mounting screw. Because camera tripods 
are usually too short for comfortable viewing 
from a standing position, it is strongly recom- 
mended that you be seated in a reasonably 
comfortable chair. 




Fig. 6-6 

Aiming and Focusing 

You may have trouble finding objects, 
especially with the high-power eyepiece. One 
technique is to sight over the tube, aiming 
slightly below the object, and then to tilt the 



tube up slowly while looking through it and 
sweeping left and right. To do this well, you 
will need some practice. 

Focusing by pulling or pushing the sliding 
tube tends to move the whole telescope. To 
avoid this, rotate the sliding tube while moving 
it as if it were a screw. 

Eyeglasses will keep your eye farther from 
the eyepiece than the best distance. Far- 
sighted or near-sighted observers are gen- 
erally able to view more satisfactorily by 
removing their glasses and refocusing. Ob- 
servers with astigmatism have to decide 
whether or not the distorted image (without 
glasses) is more annoying than the reduced 
field of view (with glasses). 

Many observers find that they can keep 
their eye in line with the telescope while aim- 
ing and focusing if the brow and cheek rest 
lightly against the forefinger and thumb. (Fig. 
6-6) When using a tripod mounting, remove 
your hands from the telescope while actually 
viewing to minimize shaking the instrument. 

Limitations of Your Telescope 

You can get some idea of how much fine 
detail to expect when observing the planets by 
comparing the angular sizes of the planets 
with the resolving power of the telescope. For 
a telescope with a 1 inch diameter object lens, 
to distinguish between two details, they must 
be at least 0.001° apart as seen from the loca- 
tion of the telescope. The low-power Project 
Physics eyepiece may not quite show this 
much detail, but the high power will be more 
than sufficient. 

The angular sizes of the planets as viewed 
from the earth are: 



Venus: 


0,003° 


(minimum) 




0.016 


(maximum) 


Mars: 


0.002 


(minimum) 




0.005 


(maximum) 


Jupiter: 


0.012 


(average) 


Saturn: 


0.005 


(average) 


Uranus: 


0.001 


(average) 



Galileo's first telescope gave 3x magnifi- 
cation, and his "best" gave about 30x magni- 
fication. (But, he used a different kind of 



Experiment 18 153 



eyepiece that gave a much smaller field of 
view.) You should find it challenging to see 
whether you can observe all the phenomena 
he saw which are mentioned in Sec. 7.7 of 
the Text. 

Observations You Can Make 

The following group of suggested objects 
have been chosen because they are (1) fairly 
easy to find, (2) representative of what is to 
be seen in the sky, and (3) very interesting. You 
should observe all objects with the low power 
first and then the high power. For additional 
information on current objects to observe, see 
the paperback New Handbook of the Heavens, 
or the last few pages of each monthly issue of 
the magazines Sky and Telescope, Natural 
History, or Science News. 
Venus: No features will be visible on this 
planet, but you can observe its phases, as 
shown in the photographs below (enlarged to 
equal sizes) and on page 72 of the Unit 2 Text. 
When Venus is very bright you may need to 
reduce the amount of light coming through the 
telescope in order to tell the true shape of the 
image. A paper lens cap with a round hole in 
the center will reduce the amount of light (and 
the resolution of detail!) You might also try 
using sunglasses as a filter. 




Venus, photographed at Yerkes Observatory with the 
82-inch reflector telescope. 

Saturn: The planet is so large that you can 
resolve the projection of the rings beyond the 
disk, but you probably can't see the gap be- 
tween the rings and the disk with your 30x 




Saturn photographed with the 100-inch telescope at 
Mount Wilson. 

telescope. Compare your observations to the 
sketches on page 73 of the Text. 
Jupiter: Observe the four satellites that Galileo 
discovered. Observe them several times, a few 
hours or a day apart, to see changes in their 
positions. By keeping detailed data over 
several months time, you can determine the 
period for each of the moons, the radii of their 
orbits, and then the mass of Jupiter. (See the 
notes for the Film Loop, "Jupiter Satellite 
Orbit," in Chapter 8 of this Handbook for direc- 
tions on how to analyze your data.) 

Jupiter is so large that some of the detail 
on its disk — like a broad, dark, equatorial cloud 
belt — can be detected (especially if you know 
it should be there!) 




Jupiter photographed with the 200-inch telescope at 
Mount Palomar. 



Moon: Moon features stand out mostly because 
of shadows. Best observations are made about 
the time of half-moon, that is, around the first 
and last quarter. Make sketches of your ob- 
servations, and compare them to Galileo's 
sketch on page 66 of your Text. Look carefully 
for walls, mountains in the centers of craters, 
bright peaks on the dark side beyond the 
terminator, and craters in other craters. 



154 Experiment 18 
B.C. 



By John Hart 




By permission of John Hart and Field Enterprises, I 



The Pleiades: A beautiful little star cluster, 
this is located on the right shoulder of the 
bull in the constellation Taurus. These stars 
are almost directly overhead in the evening 
sky in December. The Pleiades were among the 
objects Galileo studied with his first telescope. 
He counted 36 stars, which the poet Tennyson 
described as "a swarm of fireflies tangled in a 
sUver braid." 

The Hyades: This group of stars is also in Tau- 
rus, near the star Aldebaran, which forms the 
bull's eye. Mainly, the Hyades look like a "v." 
The high-power may show that several stars 
are double. 

The Great Nebula in Orion: Look about halfway 
down the row of stars that form the sword of 
Orion. It is in the southeastern sky during 
December and January. Use low power. 

Algol: This famous variable star is in the 
constellation Perseus, south of Cassiopeia. 



Algol is high in the eastern sky in December, 
and nearly overhead during January. Generally 
it is a second-magnitude star, like the Pole 
Star. After remaining bright for almost 2j days, 
Algol fades for 5 hours and becomes a fourth- 
magnitude star, like the faint stars of the Little 
Dipper. Then, the variable star brightens dur- 
ing 5 hours to its normal brightness. From one 
minimum to the next, the period is 2 days, 20 
hours, 49 minutes. 

Great Nebula in Andromeda: Look high in the 
western sky in the early evening in December 
for this nebula, for by January it is low on the 
horizon. It will appear like a fuzzy patch of 
light, and is best viewed with low power. The 
light you see from this galaxy has been on its 
way for two million years. 

The Milky Way: This is particularly rich in 
Cassiopeia and Cygnus (if air pollution in 
your area allows it to be seen at all). 



B.C. 



I THINK TWe MOOH 
AND TUB «JnJ are 
GOlN6> TO OCASH . 



yepI ..•THey are. 



By John Hart 




By permission of John Hart and Field Enterprises, Inc. 



Experiment 18 155 




Fig. 6-7 Observing sunspots with a telescope. 

Observing sunspots: DO NOT LOOK AT THE 
SUN THROUGH THE TELESCOPE. THE 
SUNLIGHT WILL INJURE YOUR EYES. 

Figure 6-7 shows an arrangement of a tripod, 
the low-power telescope, and a sheet of paper 
for projecting sunspots. Cut a hole in a piece 
of cardboard so it fits snugly over the object 
end of the telescope. This acts as a shield so 
there is a shadow area where you can view the 
sunspots. First focus the telescope, using the 
high-power eyepiece, on some distant object. 
Then, project the image of the sun on a piece 
of white paper a couple of feet behind the eye- 
piece. Focus the image by moving the draw- 
tube slightly further out. When the image is 
in focus, you may see some small dark spots 
on the paper. To tell marks on the paper from 
sunspots, jiggle the paper back and forth. How 




The sunspots of April 7, 1947. 

can you tell that the spots aren't on the lenses? 
By focusing the image farther from the tele- 
scope, you can make the image larger and not 
so bright. It may be easier to get the best focus 
by moving the paper rather than the eyepiece 
tube. 





Drawings of the projected image of the sun on Aug. 26 and Aug. 27, 1966, drawn by 
an amateur astronomer in Walpole, Mass. 



156 



ACTIVITIES 



TWO ACTIVITIES ON FRAMES OF 
REFERENCE 

1. You and a classmate take hold of opposite 
ends of a meter stick or a piece of string a 
meter or two long. If you rotate about on one 
fixed spot so that you are always facing him 
while he walks around you in a circle, you will 
see him moving around you against a back- 
ground of walls and furniture. But, how do you 
appear to him? Ask him to describe what he 
sees when he looks at you against the back- 
ground of walls and furniture. How do your 
reports compare? In what direction did you see 
him move — toward your left or your right? In 
which direction did he see you move — toward 
his left or his right? 

2. The second demonstration involves a 
camera, tripod, blinky, and turntable. Mount 
the camera on the tripod (using motor-strobe 
bracket if camera has no tripod connection) 
and put the blinky on a turntable. Aim the 
camera straight down. 

Take a time exposure with the camera at 
rest and the blinky moving one revolution in a 
circle. If you do not use the turntable, move 
the blinky by hand around a circle drawn 
faintly on the background. Then take a second 
print, with the blinky at rest and the camera 
on time exposure moved steadily by hand 
about the axis of the tripod. Try to move the 
camera at the same rotational speed as the 
blinky moved in the first photo. 

Can you tell, just by looking at the photos 
whether the camera or the blinky was moving? 

Nubbin 












HOW 

HUMIUAtlNS/ 
jVg ^^^N MAKING 
A COMf^BTg FOOL 

OP MVggUF/ 








157 



FILM LOOP 




FILM LOOP 11 RETROGRADE MOTION 
—HELIOCENTRIC MODEL 

This film is based on a large heliocentric 
mechanical model. Globes represent the earth 
and a planet moving in concentric circles 
around the sun (represented by a yellow globe). 
The earth (represented by a light blue globe) 
passes inside a slower moving outer planet 
such as Mars (represented by an orange globe). 

Then the earth is replaced by a camera 
having a 25° field of view. The camera points 
in a fixed direction in space, indicated by an 
arrow, thus ignoring the daily rotation of the 
earth and concentrating on the motion of the 
earth relative to the sun. 

The view from the moving earth is shown 
for more than 1 year. First the sun is seen in 
direct motion, then Mars comes to opposition 
and undergoes a retrograde motion loop, and 
finally you see the sun again in direct motion. 

Scenes are viewed from above and along 
the plane of motion. Retrograde motion occurs 
whenever Mars is in opposition, that is, when- 
ever Mars is opposite the sun as viewed from 
the earth. But not all these oppositions take 
place when Mars is in the sector the camera 
sees. The time between oppositions averages 
about 2.1 years. The film shows that the earth 
moves about 2.1 times around its orbit between 
oppositions. 

You can calculate this value. The earth 
makes one cycle around the sun per year and 
Mars makes one cycle around the sun every 



1.88 years, 
motion are: 



So the frequencies of orbital 



fearth = 1 cyc/yr and fn,ars = 1 cyc/1.88 yr 

= 0.532 cyc/yr 

The frequency of the earth relative to Mars is 
f — f 

■■•earth •■^mars' 

fearth " fmars "" 1-00 cyc/yr - 0.532 cyc/yr 
= 0.468 cyc/yr 

That is, the earth catches up with and passes 
Mars nnrp pvprv 



Mars once every 



0.468 



= 2.14 years. 



Note the increase in apparent size and 
brightness of the globe representing Mars 
when it is nearest the earth. Viewed with the 
naked eye. Mars shows a large variation in 
brightness (ratio of about 50:1) but always 
appears to be only a point of light. With the 
telescope we can see that the angular size also 
varies as predicted by the model. 

The heliocentric model is in some ways 
simpler than the geocentric model of Ptolemy, 
and gives the general features observed for the 
planets: angular position, retrograde motion, 
and variation in brightness. However, de- 
tailed numerical agreement between theory 
and observation cannot be obtained using 
circular orbits. 

A film of a similar model for the geocentric 
theory of Ptolemy is described on page 147, 
Chapter 5 of this Handbook. 



Chapter 



A New Universe Appears — the Work of Kepler and Galileo 



EXPERIMENT 19 THE ORBIT OF MARS 

In this laboratory activity you will derive 
an orbit for Mars around the sun by the same 
method that Kepler used in discovering that 
planetary orbits are elliptical. Since the obser- 
vations are made from the earth, you will need 
the orbit of the earth that you developed in 
Experiment 17, "The Shape of the Earth's 
Orbit." Make sure that the plot you use for this 
experiment represents the orbit of the earth 
around the sun, not the sun around the earth. 

If you did not do the earth-orbit experi- 
ment, you may use, for an approximate orbit, 
a circle of 10 cm radius drawn in the center of 
a large sheet of graph paper (16" x 20" or four 
82" X 11" joined). Because the eccentricity of 
the earth's orbit is very small (0.017) you can 
place the sun at the center of the orbit without 
introducing a significant error in this experi- 
ment. 

From the sun (at the center), draw a line to 
the right, parallel to the grid of the graph 
paper (Fig. 7-1). Label the line 0°. This line is 
directed toward a point on the celestial sphere 
called the vernal equinox and is the reference 
direction from which angles in the plane of the 
earth's orbit (the ecliptic plane) are measured. 
The earth crosses this line on September 23. 
When the earth is on the other side of its orbit 
on March 21, the sun is between the earth and 
the vernal equinox. 



Marcb 




of I'eyntl 



Fig. 7-1 

Photographic Observations of Mars 

You will use a booklet containing sixteen 
enlarged sections of photographs of the sky 
showing Mars among the stars at various dates 
between 1931 and 1950. All were made with 



the same small camera used for the Harvard 
Observatory Sky Patrol. In some of the photo- 
graphs Mars was near the center of the field. 
In many other photographs Mars was near the 
edge of the field where the star images are 
distorted by the camera lens. Despite these 
distortions the photographs can be used to 
provide positions of Mars that are satisfactory 
for this study. Photograph P is a double ex- 
posure, but it is still quite satisfactory. 

Changes in the positions of the stars 
relative to each other are extremely slow. Only 
a few stars near the sun have motions large 
enough to be detected after many years obser- 
vations with the largest telescopes. Thus you 
can consider the pattern of stars as fixed. 



Finding Mars' Location 

Mars is continually moving among the 
stars but is always near the ecliptic. From 
several hundred thousand photographs at the 
Harvard Observatory sixteen were selected, 
with the aid of a computer, to provide pairs of 
photographs separated by 687 days - the period 
of Mars around the sun as determined by 
Copernicus. Thus, each pair of photographs 
shows Mars at one place in its orbit. 

During these 687 days, the earth makes 
nearly two full cycles of its orbit, but the in- 
terval is short of two full years by 43 days. 
Therefore, the position of the earth, from 
which we can observe Mars, will not be the 
same for the two observations of each pair. If 
you can determine the direction from the earth 
towards Mars for each of the pairs of observa- 
tions, the two sight lines must cross at a point 
on the orbit of Mars. (See Fig. 7-2.) 

Coordinate System Used 

When you look into the sky you see no 
coordinate system. Coordinate systems are 
created for various purposes. The one used 
here centers on the ecliptic. Remember that 
the ecliptic is the imaginary line on the celes- 
tial sphere along which the sun appears to 
move. 



Experiment 19 159 




r> 






Fig. 7-2 Point 2 is the position of the earth 687 days 
after leaving point 1. In 687 days. Mars has made ex- 
actly one revolution and so has returned to the same 
point on the orbit. The intersection of the sight lines 
from the earth determines that point on Mars' orbit. 

Along the ecliptic, longitudes are always 
measured eastward from the 0° point (the 
vernal equinox). This is toward the left on star 
maps. Latitudes are measured perpendicular 
to the ecliptic north or south to 90°. (The small 
movement of Mars above and below the 
ecliptic is considered in the Activity, "The 
Inclination of Mars' Orbit.") 

To find the coordinates of a star or of Mars 
you must project the coordinate system upon 
the sky. To do this you are provided with trans- 
parent overlays that show the coordinate sys- 
tem of the ecliptic for each frame, A to P. The 
positions of various stars are circled. Adjust 
the overlay until it fits the star positions. Then 
you can read off the longitude and latitude of 
the position of Mars. Figure 7-3 shows how you 
can interpolate between marked coordinate 
lines. Because you are interested in only a 
small section of the sky on each photograph, 
you can draw each small section of the ecliptic 
as a straight line. For plotting, an accuracy of 
2"° is satisfactory. 

In a chart like the one shown in Figure 7-4, 
record the longitude and latitude of Mars for 
each photograph. For a simple plot of Mars' 
orbit around the sun you will use only the first 
column -the longitude of Mars. You will use 
the columns for latitude. Mars' distance from 
the sun, and the sun-centered coordinates if 



A^ 



• T 



e^ 



Fig. 7-3 Interpolation between cooromate lines. In 
the sketch, Mars (M), is at a distance y° from the 170° 
line. Take a piece of paper or card at least 10 cm long. 
Make a scale divided into 10 equal parts and label 
alternate marks ), 1,2, 3, 4, 5. This gives a scale in j° 
steps. Notice that the numbering goes from right to 
left on this scale. Place the scale so that the edge passes 
through the position of Mars. Now tilt the scale so that 
the and 5 marks each fall on a grid line. Read off the 
value of y from the scale. In the sketch, y = lf°, so that 
the longitude of M is ^7^J°. 

you do the Activity on the inclination, or tilt, of 
Mars' orbit on page 165. 

Finding Mars' Orbit 

When your chart is completed for all eight 
pairs of observations, you are ready to locate 
points on the orbit of Mars. 
1. On the plot of the earth's orbit, locate the 
position of the earth for each date given in the 



Frame 


Fig. 7-4 ( 

Date 

Mar. 21, 1931 
Feb 5, 1933 


Dbse 

Geoce 
Long 


rved 

ntric 
Lat. 


Position 

Mars 
to 
Earth 
Distance 


s of Mar 

Mars 

to 

Sun 

Distance 


S 

Heiioce 
Long 


sntric 
Lat. 


A 






B 

















Apr. 20, 1933 
Mar. 8, 1935 














D 














E 


May 26. 1935 
Apr. 12, 1937 














F 














G 


Sept.16, 1939 
Aug. 4. 1941 














H 














1 


Nov. 22, 1941 
Oct. 11, 1943 














J 














K 


Jan. 21, 1944 
Dec 9, 1945 














L 














M 


Mar. 19, 1946 
Feb. 3, 1948 














N 

















Apr. 4, 1948 
Feb. 21, 1950 














P 















160 Experiment 19 




Fig. 7-5 



16 photographs. You may do this by interpo- 
lating between the dates given for the earth's 
orbit experiment. Since the earth moves 
through 360° in about 365 days, you may use 
±1° for each day ahead or behind the date 
given in the previous experiment. For example, 
frame A is dated March 21. The earth was at 
166° on March 7: fourteen days later on March 
21, the earth will have moved 14° from 166° to 
180°. Always work from the earth-position date 
nearest the date of the Mars photograph. 
2. Through each earth-position point draw a 
"0° line" parallel to the line you drew from the 
sun toward the vernal equinox (the grid on the 



graph paper is helpful). Use a protractor and a 
sharp pencil to mark the angle between the 
0° line and the direction to Mars on that date 
as seen from the earth (longitude of Mars). The 
two lines drawn from the earth's positions for 
each pair of dates will intersect at a point. This 
is a point on Mars' orbit. Figure 7-5 shows one 
point on Mars' orbit obtained from the data of 
the first pair of photographs. By drawing the 
intersecting lines from the eight pairs of posi- 
tions, you establish eight points on Mars' orbit. 
3. You will notice that there are no points in 
one section of the orbit. You can fill in the 
missing part because the orbit is symmetrical 
about its major axis. Use a compass and, by 
trial and error, find a circle that best fits the 
plotted points. Perhaps you can borrow a 
French curve or long spline from the mechani- 
cal drawing or mathematics department. 

Now that you have plotted the orbit, you 
have achieved what you set out to do: you have 
used Kepler's method to determine the path of 
Mars around the sun. 

If you have time to go on, it is worthwhile 
to see how well your plot agrees with Kepler's 
generalization about planetary orbits. 

Kepler's Laws from Your Plot 

Ql Does your plot agree with Kepler's con- 
clusion that the orbit is an ellipse? 



Photographs of Mars made with a 60 inch reflecting telescope (Mount Wilson and Palomar Observatories) during clos- 
est approach to the earth in 1956. Left: August 10; right: Sept. 11. Note the shrinking of the polar cao 





Experiment 19 161 



Q2 What is the average sun-to-Mars distance 
in AU? 

Q3 As seen from the sun, what is the direction 
(longitude) of Mars' nearest and farthest 
positions? 

Q4 During what month is the earth closest to 
the orbit of Mars? What would be the mini- 
mum separation between the earth and Mars? 
Q5 What is the eccentricity of the orbit of 
Mars? 

Q6 Does your plot of Mars' orbit agree with 
Kepler's law of areas, which states that a line 
drawn from the sun to the planet sweeps out 
areas proportional to the time intervals? From 
your orbit, you see that Mars was at point B' 
on February 5, 1933, and at point C on April 
20, 1933, as shown in Fig. 7-6. There are eight 
such pairs of dates in your data. The time 
intervals are different for each pair. 

Connect these pairs of positions with a 



line to the sun (Fig. 7-6). Find the areas of 
squares on the graph paper (count a square 
when more than half of it lies within the area). 
Divide the area (in squares) by the number of 
days in the interval to find an "area per day" 
value. Are these values nearly the same? 

Q7 How much (by what percentage) do they 
vary? 

Q8 What is the uncertainty in your area mea- 
surements? 

Q9 Is the uncertainty the same for large areas 
as for small? 

QIO Do your results bear out Kepler's law of 
areas? 

This is by no means all that you can do 
with the photographs you used to make the 
plot of Mars' orbit. If you want to do more, look 
at the Activity, "The Inclination of Mars' 
Orbit." 



Fig. 7-6 In this example, the time interval is 74 days. 



^/KRTi^'^ o/?g/: 








APRIL ao. 1^33 



Television picture of a 40 x 50 mile area just below Mars' 
equator, radioed from the Mariner 6 Mars probe during 
its 1969 fly-by. 




162 Experiment 20 




Fig. 7-7 Mercury, first quarter phase, taken June 7, 1 934 at the Lowell Observatory, Flagstaff, Ariz. 



EXPERIMENT 20 

THE ORBIT OF MERCURY 

Mercury, the innermost planet, is never 
very far from the sun in the sky. It can be seen 
only close to the horizon, just before sunrise 
or just after sunset, and viewing is made dif- 
ficult by the glare of the sun. (Fig. 7-7.) 

Except for Pluto, which differs in several 
respects from the other planets. Mercury has 
the most eccentric planetary orbit in our solar 
system (e = 0.206). The large eccentricity of 
Mercury's orbit has been of particular impor- 
tance, since it has led to one of the tests for 
Einstein's General Theory of Relativity. For a 
planet with an orbit inside the earth's, there 
is a simpler way to plot the orbit than by the 
paired observations you used for Mars. In this 
experiment you will use this simpler method to 
get the approximate shape of Mercury's orbit. 

Mercury's Elongations 

Let us assume a heliocentric model for the 
solar system. Mercury's orbit can be found 
from Mercury's maximum angles of elonga- 
tion east and west from the sun as seen from 
the earth on various known dates. 



^|\R.lH'^.pR8,j. 




BARJH 
Fig. 7-8 The greatest western elongation of Mercury. 
May 25, 1964. The elongation had a value of 25' West. 

The angle (Fig. 7-8), between the sun and 
Mercury as seen from the earth, is called the 
"elongation." Note that when the elongation 
reaches its maximum value, the sight lines 



Experiment 20 163 



from the earth are tangent to Mercury's orbit. 
Since the orbits of Mercury and the earth 
are both elhptical, the greatest value of the 
elongation varies from revolution to revolution. 
The 28° elongation given for Mercury on page 
14 of the Text refers to the maximum value. 
Table 1 gives the angles of a number of these 
greatest elongations. 

TABLE 1 SOME DATES AND ANGLES OF GREATEST 
ELONGATION FOR MERCURY (from the 
American Ephemeris and Nautical Almanac) 



Date 


963 


Elongation 


Jan. 4, 1 


19° E 


Feb. 14 




26 W 


Apr. 26 




20 E 


June 13 




23 W 


Aug. 24 




27 E 


Oct. 6 




18 W 


Dec. 18 




20 E 


Jan. 27, 


1964 


25 W 


Apr. 8 




19 E 


May 25 




25 W 



Plotting the Orbit 

You can work from the plot of the earth's 
orbit that you established in Experiment 17. 
Make sure that the plot you use for this experi- 
ment represents the orbit of the earth around 
the sun, not of the sun around the earth. 

If you did not do the earth's orbit experi- 
ment, you may use, for an approximate earth 
orbit, a circle of 10 cm radius drawn in the 
center of a sheet of graph paper. Because the 
eccentricity of the earth's orbit is very small 
(0.017) you can place the sun at the center of 
the orbit without introducing a significant 
error in the experiment. 

Draw a reference line horizontally from 
the center of the circle to the right. Label the 
line 0°. This line points toward the vernal 
equinox and is the reference from which the 
earth's position in its orbit on different dates 
can be established. The point where 0° line 
from the sun crosses the earth's orbit is the 
earth's position in its orbit on September 23. 

The earth takes about 365 days to move 
once around its orbit (360°). Use the rate of 



approximately 1° per day, or 30° per month, to 
establish the position of the earth on each of 
the dates given in Table 1. Remember that the 
earth moves around this orbit in a counter- 
clockwise direction, as viewed from the north 
celestial pole. Draw radial lines from the sun 
to each of the earth positions you have located. 

Now draw sight lines from the earth's 
orbit for the elongation angles. Be sure to note, 
from Fig. 7-8, that for an eastern elongation, 
Mercury is to the left of the sun as seen from 
the earth. For a western elongation. Mercury 
is to the right of the sun. 

You know that on a date of greatest elonga- 
tion Mercury is somewhere along the sight 
line, but you don't know exactly where on the 
line to place the planet. You also know that the 
sight line is tangent to the orbit. A reasonable 
assumption is to put Mercury at the point along 
the sight line closest to the sun. 

You can now find the orbit of Mercury by 
drawing a smooth curve through, or close to, 
these points. Remember that the orbit must 
touch each sight line without crossing any of 
them. 



Finding R^,. 

The average distance of a planet in an el- 
liptical orbit is equal to one half the long 
diameter of the ellipse, the "semi-major axis." 
To find the size of the semi-major axis a of 
Mercury's orbit, relative to the earth's semi- 
major axis, you must first find the aphelion and 
perihelion points of the orbit. You can use a 
drawing compass to find these points on the 
orbit farthest from and closest to the sun. 



X 




y 



164 Experiment 20 

Measure the greatest diameter of the orbit 
along the Hne perihehon-sun-aphelion. Since 
10.0 cm corresponds to one AU (the semi- 
major axis of the earth's orbit) you can now 
obtain the semi-major axis of Mercury's orbit 
in AU's. 

Calculating Orbital Eccentricity 

Eccentricity is defined as e = da (Fig. 7-9). 
Since c, the distance from the center of Mer- 
cury's ellipse to the sun, is small on your plot, 
you lose accuracy if you try to measure it 
directly. 




From Fig. 7-9, you can see that c is the 
difference between Mercury's perihelion dis- 
tance Rp and the semi-major axis a. That is: 



a-Rr 



So 



c 

e = — 

a 



a 
a 



You can measure Rp and a with reason- 
able accuracy from your plotted orbit. Compute 
e, and compare your value with the accepted 
value, e == 0.206. 

Kepler's Second Law 

You can test Kepler's equal-area law on 
your Mercury orbit in the same way as that 
described in Experiment 19, The Orbit of Mars. 
By counting squares you can find the area 
swept out by the radial line from the sun to 
Mercury between successive dates of observa- 
tion, such as January 4 to February 14, and 
June 13 to August 24. Divide the area by the 
number of days in the interval to get the "area 
per day." This should be constant, if Kepler's 
law holds for your plot. Is it constant? 



Fig. 7-9 



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By permission of John Hart and Field Enterprises, Inc. 



ACTIVITIES 



165 



THREE-DIMENSIONAL MODEL OF TWO 
ORBITS 

You can make a three-dimensional model 
of two orbits quickly with two small pieces of 
cardboard (or 3" x 5" cards). On each card draw 
a circle or ellipse, but have one larger than the 
other. Mark clearly the position of the focus 
(sun) on each card. Make a straight cut to the 
sun, on one card from the left, on the other 
from the right. Slip the cards together until 
the sun-points coincide. (Fig. 7-10) Tilt the two 
cards (orbit planes) at various angles. 



^CU9T\c9Uiv\\^ \ 




Fig. 7-10 

INCLINATION OF MARS' ORBIT 

When you plotted the orbit of Mars in Ex- 
periment 17, you ignored the slight movement 
of the planet above and below the ecliptic. 
This movement of Mars north and south of the 
ecliptic shows that the plane of its orbit is 
slightly inclined to the plane of the earth's 
orbit. In this activity, you may use the table of 
values for Mars latitude (which you made in 
Experiment 17) to determine the inclination 
of Mar's orbit. 

Do the activity, "Three-dimensional model 
of two orbits," just before this activity, to see 
exactly what is meant by the inclination of 
orbits. 

Theory 

From each of the photographs in the set 
of 16 that you used in Experiment 17, you can 
find the observed latitude (angle from the 
ecliptic) of Mars at a particular point in its 
orbital plane. Each of these angles is mea- 
sured on a photograph taken from the earth. 
As you can see from Fig. 7-11, however, it is 
the sun, not the earth, which is at the center 







."^JAU 



ill AU 



d = /., 



Fig. 7-11 



of the orbit. The inclination of Mars' orbit 
must, therefore, be an angle measured at the 
sun. It is this angle (the heliocentric latitude) 
that you wish to find. 

Figure 7-11 shows that Mars can be repre- 
sented by the head of a pin whose point is 
stuck into the ecliptic plane. We see Mars from 
the earth to be north or south of the ecliptic, 
but we want the N-S angle of Mars as seen 
from the sun. The following example shows 
how you can derive the angles as if you were 
seeing them from the sun. 

In Plate A (March 21, 1933), Mars was 
about 3.2° north of the ecliptic as seen from 
the earth. But the earth was considerably 
closer to Mars on this date than the sun was. 
Can you see how the angular elevation of Mars 
above the ecliptic plane as seen from the sun 
will therefore be considerably less than 3.2°? 

For very small angles, the apparent angu- 
lar sizes are inversely proportional to the 
distances. For example, if the sun were twice 
as far from Mars as the earth was, the angle 
at the sun would be 7 the angle at the earth. 

Measurement on the plot of Mars' orbit 
(Experiment 17) gives the earth-Mars distance 
as 9.7 cm (0.97 AU) and the distance sun-Mars 
as 17.1 cm (1.71 AU) on the date of the photo- 



166 Activities 



graph. The heUocentric latitude of Mars is 
therefore 



9.7 
17.1 



X 3.2°N = 1.8°N 



You can check this value by finding the helio- 
centric latitude of this same point in Mars' 
orbit on photograph B (February 5, 1933). The 
earth was in a different place on this date so 
the geocentric latitude and the earth-Mars 
distance will both be different, but the helio- 
centric latitude should be the same to within 
your experimental uncertainty. 

Making the Measurements 

Turn to the table you made that is like 
Fig. 7-4 in Experiment 17, on which you re- 
corded the geocentric latitudes \g of Mars. On 
your Mars' orbit plot from Experiment 17, mea- 
sure the corresponding earth-Mars and sun- 
Mars distances and note them in the same table. 

From these two sets of values, calculate 
the heliocentric latitudes as explained above. 
The values of heliocentric latitude calculated 
from the two plates in each pair (A and B, C 
and D, etc.) should agree within the limits of 
your experimental procedure. 

On the plot of Mars' orbit, measure the 
heliocentric longitude K for each of the eight 
Mars positions. Heliocentric longitude is mea- 
sured from the sun, counterclockwise from the 
0° direction (direction toward vernal equinox), 
as shown in Fig. 7-12. 

Complete the table given in Fig. 7-4, 
Experiment 17, by entering the earth-to-Mars 




Fig. 7-12 On February 5, the heliocentric longitude 
(\h) of Point B on Mars' orbit is 150°; the geocentric 
longitude (\,) measured from the earth's position 
is 169°. 



and sun-to-Mars distances, the geocentric and 
heliocentric latitudes, and the geocentric and 
heliocentric longitudes for all sixteen plates. 
Make a graph, like Fig. 7-13, that shows 
how the heliocentric latitude of Mars changes 
with its heliocentric longitude. 




310' 



LOVtflTUOe 



5 

Fig. 7-13 Change of Mars' heliocentric latitude with 
heliocentric longitude. Label the ecliptic, latitude, 
ascending node, descending node and inclination of 
the orbit in this drawing. 



From this graph, you can find two of the 
elements that describe the orbit of Mars with 
respect to the ecliptic. The point at which Mars 
crosses the ecliptic from south to north is 
called the ascending node. (The descending 
node, on the other side of the orbit, is the point 
at which Mars crosses the ecliptic from north 
to south.) 

The angle between the plane of the earth's 
orbit and the plane of Mars' orbit is the inclina- 
tion of Mars' orbit, i. When Mars reaches its 
maximum latitude above the ecliptic, which 
occurs at 90° beyond the ascending node, the 
planet's maximum latitude equals the inclina- 
tion of the orbit, i. 

Elements of an Orbit 

Two angles, the longitude of the ascending 
node, n, and the inclination, i, locate the plane 
of Mars' orbit with respect to the plane of the 
ecliptic. One more angle is needed to orient the 
orbit of Mars in its orbital plane. This is the 
"argument of perihelion" a>. shown in Fig. 7-14 
which is the angle in the orbit plane between 
the ascending node and perihelion point. On 
your plot of Mars' orbit measure the angle from 
the ascending node H to the direction of peri- 



Experiment 21 171 



J 



Fig. 8-2a 



Fig. 8-2b 



J 







Fig. 8-2c 



You can now proceed to plot an approxi- 
mate comet orbit if you will make these addi- 
tional assumptions: 

1. The force on the comet is an attraction 
toward the sun. 

2. The force of the blow varies inversely with 
the square of the comet's distance from the 
sun. 

3. The blows occur regularly at equal time 
intervals, in this case, 60 days. The magnitude 
of each brief blow is assumed to equal the 
total effect of the continuous attraction of the 
sun throughout a 60-day interval. 

Effect of the Central Force 

From Newton's second law you know that 
the gravitational force will cause the comet 
to accelerate toward the sun. If a force F acts 
for a time interval At on a body of mass m, you 
know that 

F =ma = TM-r— and therefore 
At 

At;=— At 
m 

This equation relates the change in the 
body's velocity to its mass, the force, and the 
time for which it acts. The mass m is constant. 
So is At (assumption 3 above). The change in 
velocity is therefore proportional to the force, 
Ai^ SE f But remember that the force is not 
constant in magnitude; it varies inversely with 
the square of the distance from comet to sun. 
Q4 Is the force of a blow given to the comet 
when it is near the sun greater or smaller than 
one given when the comet is far from the sun? 
Q5 Which blow causes the biggest velocity 
change? 



In Fig. 8-2a the vector v^q represents the 
comet's velocity at the point A. During the first 
60 days, the comet moves from A to B (Fig. 
8-2b). At B a blow causes a velocity change 
AiJ!, (Fig. 8-2c). The new velocity after the blow 
is t7, = i/q + Ax/j, and is found by completing the 
vector triangle (Fig. 8-2d). 

The comet therefore leaves point B with 
velocity z^, and continues to move with this 
velocity for another 60-day interval. Because 
the time intervals between blows are always 
the same (60 days), the displacement along the 
path is proportional to the velocity, "v. You 
therefore use a length proportional to the 
comet's velocity to represent its displacement 
during each time interval. (Fig. 8-2e.) 

Each new velocity is found, as above, by 
adding to the previous velocity the Az^given by 
the blow. In this way, step by step, the comet's 
orbit is built up. 




172 Experiment 21 



Scale of the Plot 

The shape of the orbit depends on the ini- 
tial position and velocity, and on the force 
acting. Assume that the comet is first spotted 
at a distance of 4 AU from the sun. Also as- 
sume that the comet's velocity at this point is 
v^2 AU per year (about 20,000 miles per hour) 
at right angles to the sun-comet distance R. 

The following scale factors will reduce the 
orbit to a scale that fits conveniently on a 
16" X 20" piece of graph paper. (Make this up 
from four S"!"" x 11" pieces if necessary.) 

1. Let 1 AU be scaled to 2.5 inches (or 6.5 cm) 
so that 4 AU becomes 10 inches (or about 
25 cm). 

2. Since the comet is hit every 60 days, it is 
convenient to express the velocity in AU per 
60 days. Suppose you adopt a scale factor in 
which a velocity vector of 1 AU/60 days is 
represented by an arrow 2.5 inches (or 6.5 cm) 
long. 

The comet's initial velocity of 2 AU per 
year can be given as 2/365 AU per day, or 2/365 
X 60 = 0.33 AU per 60 days. This scales to an 
arrow 0.83 inches (or 2.1 1 cm) long. This is the 
displacement of the comet in the first 60 days. 

Computing Ai^ 

On the scale and with the 60-day iteration 
interval that has been chosen, the force field 
of the sun is such that the Av given by a blow 
when the comet is 1 AU from the sun is 1 
AU/60 days. 



To avoid computing At; for each value of i?, 
you can plot Az; against i? on a graph. Then for 
any value of R you can immediately find the 
value of At;. 

Table 1 gives values of R in AU and in 
inches and in centimeters to fit the scale of 
your orbit plot. The table also gives for each 
value of R the corresponding value of Ai» in 
AU/60 days and in inches and in centimeters 
to fit the scale of your orbit plot. 





Table 1 Scales for R and Av 




Distance from 


sun, R 


Change in 


speed, 


^v 


AU 


inches 


cm 


AU/60 days 


inches 


cm 


0.75 


1.87 


4.75 


1.76 


4.44 


11.3 


0.8 


2.00 


5.08 


1.57 


3.92 


9.97 


0.9 


2.25 


5.72 


1.23 


3.07 


7.80 


1.0 


2.50 


6.35 


1.00 


2.50 


6.35 


1.2 


3.0 


7.62 


0.69 


1.74 


4.42 


1.5 


3.75 


9.52 


0.44 


1.11 


2.82 


2.0 


5.0 


12.7 


0.25 


0.62 


1.57 


2.5 


6.25 


15.9 


0.16 


0.40 


1.02 


3.0 


7.50 


19.1 


0.11 


0.28 


0.71 


3.5 


8.75 


22.2 


0.08 


0.20 


0.51 


4.0 


10.00 


25.4 


0.06 


0.16 


0.41 



Graph these values on a separate sheet of 
paper at least 10 inches long, as illustrated in 
Fig. 8-3, and carefully connect the points with 
a smooth curve. 






vi 2 

> 



U 



4-- 



Fig. 8-3 




Experiment 21 173 



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::■; :r.z 




^\\\\^^X\\\\^\ 






^4=^'A ce/sn^ OP FoRc e i l-i-i ! II , ! 
Fig. 8-4 



You can use this curve as a simple graphi- 
cal computer. Cut off the bottom margin of the 
graph paper, or fold it under along the R axis. 
Lay this edge on the orbit plot and measure 
the distance from the sun to a blow point (such 
as B in Fig. 8-4). With dividers or a drawing 
compass pick off the value of Av corresponding 
to this R and lay off this distance along the 
radius line toward the sun (see Fig. 8-4). 

Making the Plot 

1. Mark the position of the sun S halfway up 
the large graph paper (held horizontally) and 
12 inches (or 30 cm) from the right edge. 

2. Locate a point 10 inches (or 25 cm), 4 AU, 
that is, to the right from the sun S. This is point 
A where you first find the comet. 



Vn 



3. To represent the comet's initial velocity 
draw vector AB perpendicular to SA. B is the 
comet's position at the end of the first 60-day 
interval. At B a blow is struck which causes a 
change in velocity Av^. 

4. Use your Ai; graph to measure the distance 
of B from the sun at S, and to find Ax;, for this 
distance (Fig. 8-4). 

5. The force, and therefore the change in ve- 
locity, is always directed toward the sun. From 
B lay off Av'i toward S. Call the end of this 
short line M. » 




6. Draw the line BC, which is a continuation 
of AB and has the same length as AB. That is 



174 Experiment 21 



where the comet would have gone in the next 
60 days if there had been no blow at B. 
7. The new velocity after the blow is the vector 
sum of the old velocity (represented by BC) 
and At! (represented by BM). To find the new 
velocity v'l draw the line C'C parallel to BM 




and of equal length. The line BC represents 
the new velocity vector t?i, the velocity with 
which the comet leaves point B. 




i A 



8. Again the comet moves with uniform 
velocity for 60 days, arriving at point C. Its 
displacement in that time is Ad, =Ty, x 60 days, 
and because of the scale factor chosen, the 
displacement is represented by the line BC. 

9. Repeat steps 1 through 8 to establish point 
D and so forth, for at least 14 or 15 steps (25 
steps gives the complete orbit). 

10. Connect points A, B, C . . . with a smooth 
curve. Your plot is finished. 

Prepare for Discussion 

Since you derived the orbit of this comet, you 
may name the comet. 



Q6 From your plot, find the perihelion distance. 

Q7 Find the center of the orbit and calculate 

the eccentricity of the orbit. 

Q8 What is the period of revolution of your 

comet? (Refer to Text, Sec. 7.3.) 

Q9 How does the comet's speed change with 

its distance from the sun? 

If you have worked this far, you have 
learned a great deal about the motion of this 
comet. It is interesting to go on to see how well 
the orbit obtained by iteration obeys Kepler's 
laws. 

QIO Is Kepler's law of ellipses confirmed? 
(Can you think of a way to test your curve to 
see how nearly it is an ellipse?) 
Qll Is Kepler's law of equal areas confirmed? 

To answer this remember that the time 
interval between blows is 60 days, so the comet 
is at positions B, C, D . . . , etc., after equal time 
intervals. Draw a line from the sun to each of 
these points (include A), and you have a set of 
triangles. 

Find the area of each triangle. The area A 
of a triangle is given by A = jab where a and b 
are altitude and base, respectively. Or you can 
count squares to find the areas. 



More Things to Do 

1. The graphical technique you have prac- 
ticed can be used for many problems. You can 
use it to find out what happens if different ini- 
tial speeds and/or directions are used. You may 
wish to use the 1/R- graph, or you may con- 
struct a new graph. To do this, use a different 
law (for example, force proportional to 1/R^, or 
to 1/R or to R) to produce different paths; ac- 
tual gravitational forces are not represented 
by such force laws. 

2. If you use the same force graph but reverse 
the direction of the force to make it a repulsion, 
you can examine how bodies move under such 
a force. Do you know of the existence of any 
such repulsive force? 



Experiment 21 175 




Spiral nebula in the constellation Leo. photographed by the 200-inch telescope at 
Mount Palomar. 



176 



ACTIVITIES 

MODEL OF THE ORBIT OF HALLEY'S 
COMET 

Halley's comet is referred to several times 
in your Text. You will find that its orbit has a 
number of interesting features if you construct 
a model of it. 

Since the orbit of the earth around the sun 
lies in one plane and the orbit of Halley's comet 
lies in another plane intersecting it, you will 
need two large pieces of stiff cardboard for 
planes, on which to plot these orbits. 



The Earth's Orbit 

Make the earth's orbit first. In the center 
of one piece of cardboard, draw a circle with a 
radius of 5 cm (1 AU) for the orbit of the 
earth. On the same piece of cardboard, also 
draw approximate (circular) orbits for Mercury 
(radius 0.4 AU) and Venus (radius 0.7 AU). For 
this plot, you can consider that all of these 
planets lie roughly in the one plane. Draw a 
line from the sun at the center and mark this 
line as 0° longitude. 

The table on page 149 of this Handbook 
lists the apparent position of the sun in the sky 
on thirteen dates. By adding 180° to each of the 
tabled values, you can get the position of the 
earth in its orbit on those dates. Mark .these 
positions on your drawing of the earth's orbit. 
(If you wish to mark more than those thirteen 
positions, you can do so by using the technique 
described on page 160.) 

The Comet's Orbit 

Figure 8-9 shows the positions of Halley's 
comet near the sun in its orbit, which is very 
nearly a parabola. You will construct your own 
orbit of Halley's comet by tracing Fig. 8-9 and 
mounting the tracing on stiff cardboard. 

Combining the Two Orbits 

Now you have the two orbits, the comet's 
and the earth's in their planes, each of which 
contains the sun. You need only to fit the two 
together in accordance with the elements of 
orbits shown in Fig. 7-1 that you may have 
used in the activity on the "Inclination of Mars 
Orbit" in Chapter 7. 



The line along which the comet's orbital 
plane cuts the ecliptic plane is called the "line 
of nodes." Since you have the major axis 
drawn, you can locate the ascending node, in 
the orbital plane, by measuring w, the angle 
from perihelion in a direction opposite to the 
comet's motion (see Fig. 8-9). 

To fit the two orbits together, cut a narrow 
slit in the ecliptic plane (earth's orbit) along 
the line of the ascending node in as far as the 
sun. The longitude of the comet's ascending 
node n was at 57° as shown in Fig. 8-5. Then 
slit the comet's orbital plane on the side of the 
descending node in as far as the sun (see Fig. 
8-6). Slip one plane into the other along the 
cuts until the sun-points on the two planes 
come together. 




:r€fTjt3 






Si- 




^ O' 



Fig. 8-6 



hJOLJ& 



Activities 177 



To establish the model in three dimen- 
sions you must now fit the two planes together 
at the correct angle. Remember that the in- 
clination i, 162°, is measured upward (north- 
ward) from the ecliptic in the direction of 
n + 90° (see Fig. 8-7). When you fit the two 
planes together you will find that the comet's 
orbit is on the underside of the cardboard. The 
simplest way to transfer the orbit to the top of 
the cardboard is to prick through with a pin at 
enough points so that you can draw a smooth 
curve through them. Also, you can construct 
a small tab to support the orbital plane in the 
correct position. 

Halley's comet moves in the opposite sense 
to the earth and other planets. Whereas the 
earth and planets move counterclockwise 
when viewed from above (north of) the ecliptic, 
Halley's comet moves clockwise. 

If you have persevered this far, and your 
model is a fairly accurate one, it should be easy 
to explain the comet's motion through the sky 
shown in Fig. 8-8. The dotted line in the figure 
is the ecliptic. 

With your model of the comet orbit you can 
now answer some very puzzling questions 
about the behavior of Halley's comet in 1910. 

1. Why did the comet appear to move west- 
ward for many months? 

2. How could the comet hold nearly a station- 
ary place in the sky during the month of April 
1910? 

Fig. 8-8 Motion of 

NORTH 




Fig. 8-7 

3. After remaining nearly stationary for a 
month, how did the comet move nearly half- 
way across the sky during the month of May 
1910? 

4. What was the position of the comet in space 
relative to the earth on May 19th? 

5. If the comet's tail was many millions of 
miles long on May 19th, is it likely that the 
earth passed through part of the tail? 

6. Were people worried about the effect a 
comet's tail might have on life on the earth? 
(See newspapers and magazines of 1910!) 

7. Did anything unusual happen? How dense 
is the material in a comet's tail? Would you 
expect anything to have happened? 

Halley's Comet in 1909-10. 




llh lOh 



2^h 23h 



178 Activities 




The elements of Halley's comet are, approximately: 

a (semi-major axis) 17.9 AU 

e (eccentricity) 0.967 

i (inclination Forbit plane) 162° 

il (longitude of ascending node) 057° 

&> (angle to perihelion) 112° 

T (perihelion date) April 20, 1910 

From these data we can calculate that the period is 76 years, and is 0.59 AU 
the perihelion distance. 



Activities 179 



OTHER COMET ORBITS 

If you enjoyed making a model of the orbit 
of Halley's comet, you may want to make 
models of some other comet orbits. Data are 
given below for several others of interest. 

Encke's comet is interesting because it 
has the shortest period known for a comet, 
only 3.3 years. In many ways it is representa- 
tive of all short-period comet orbits. All have 
orbits of low inclination and pass near the 
orbit of Jupiter, where they are often strongly 
deviated. The full ellipse can be drawn at the 
scale of 10 cm for 1 AU. The orbital elements 
for Encke's comet are: 
a = 2.22 AU 
e = 0.85 
i=15° 



a = 335° 

aj= 185° 

From these data we can calculate that the 

perihelion distance R^ is 0.33 AU and the 

aphelion distance Ra is 4.11 AU. 

The comet of 1680 is discussed extensively 
in Newton's Principia, where approximate 
orbital elements are given. The best parabolic 
orbital elements known are: 

T = Dec. 18, 1680 

w = 350.7° 

a = 272.2° 
i = 60.16° 
Ro = 0.00626 AU 




April ^b April 27 April 30 May 2- 



May 3 



May 4 



May b 




..May 15 



May 25 



May ze> June 5 June fo Jonc '^ June 11 



180 Activities 




M. Babinet prevenu par sa portiere de la visite de la 
comete. A lithograph by the French artist Honore 
Daumier (1808-1879) Museum of Fine Arts, Boston. 



Note that this comet passed very close to the 
sun. At periheHon it must have been exposed 
to intense destructive forces Hke the comet 
of 1965. 

Comet Candy (1960N) had the following 
parabolic orbital elements: 

T = Feb. 8, 1961 

0)= 136.3° 

n= 176.6 
i= 150.9 
Rp= 1.06 AU 

FORCES ON A PENDULUM 

If a pendulum is drawn aside and released 
with a small sideways push, it will move in an 
almost elliptical path. This looks vaguely like 



Fig. 8-10 




the motion of a planet about the sun. but there 
are some differences. 

To investigate the shape of the pendulum 
orbit and see whether the motion follows the 
law of areas, you can make a strobe photo with 
the setup shown in Fig. 8-10. Use either an 
electronic strobe flashing from the side, or use 
a small light and AA battery cell on the pen- 
dulum and a motor strobe disk in front of the 
lens. If you put the tape over one slot of a 12- 
slot disk to make it half as wide as the rest, it 
will make every 12th dot fainter giving a 
handy time marker, as shown in Fig. 8-11. 
You can also set the camera on its back on the 
floor with the motor strobe above it, and sus- 
pend the pendulum overhead. 

Are the motions and the forces similar for 
the pendulum and the planets? The center of 
force for planets is located at one focus of the 
ellipse. Where is the center of force for the 



Fig. 8-11 




Activities 183 



(mi + 7n2)t 



n \-/R 



R. 



The arithmetic is greatly simplified if we 
take the periods in years and the distances in 
astronomical units (A.U.) which are both units 
for the earth. The period of Kruger 60 is about 
45 years. The mean distance of the com- 
ponents can be found in seconds of arc from 
the diagram above. The mean separation is 

max + min 3.4 seconds +1.4 seconds 



4.8 seconds 



= 2.4 seconds. 



Earlier we found that the distance from the 
sun to the pair is nearly 8.7 x 10^ A.U. Then the 
mean angular separation of 2.4 seconds equals 



2.4 X 8.7 X 10^ A.U. 
2.1 X 10^ 



10 A.U. 



or the stars are separated from each other by 
about the same distance as Saturn is from the 
sun. 

Now, upon substituting the numbers into 
the equation we have 



(ttIi + m^X 



''pair 

m,„n 45 



1 ^ 10^ 
1 



1000 
2025 



= 0.50, 



or, the two stars together have about half the 
mass of the sun. 

We can even separate this mass into the 
two components. In the diagram of motions 
relative to the center of mass we see that one 
star has a smaller motion, and we conclude 
that it must be more massive. For the positions 
of 1970 (or those observed a cycle earlier in 

Peanuts 




E h «° „ • 



SECONDS OF ARC 



Kruger 60's components trace elliptical orbits, indicated 
by dots, around their center of mass, marked by a 
double circle. For the years 1932 to 1975, each dot is 
plotted on September 1. The outer circle is calibrated in 
degrees, so the position angle of the companion may be 
read directly, through the next decade. (Positions after 
1965 by extrapolation from data for 1932 to 1965). 

1925) the less massive star is 1.7 times farther 
than the other from the center of mass. So the 
masses of the two stars are in the ratio 1.7:1. 
Of the total mass of the pair, the less massive 
star has 



1 + 1.7 



X 0.5 = 0.18 



the mass of the sun. while the other star has 
0.32 the mass of the sun. The more massive 
star is more than four times brighter than the 
smaller star. Both stars are red dwarfs, less 
massive and considerably cooler than the sun. 

By Charles M. Schuiz 




© 1961 United Features Syndicote, Inc. 



184 



FILM LOOPS 



FILM LOOP 12 

JUPITER SATELLITE ORBIT 

This time-lapse study of the orbit of Jupi- 
ter's satellite, lo, was filmed at the Lowell 
Observatory in Flagstaff, Arizona, using a 
24-inch refractor telescope. 

Exposures were made at 1-minute in- 
tervals during seven nights in 1967. An almost 
complete orbit of lo is reconstructed using all 
these exposures. 

The film first shows a segment of the orbit 
as photographed at the telescope; a clock 
shows the passage of time. Due to small er- 
rors in guiding the telescope, and atmospheric 
turbulence, the highly magnified images of 
Jupiter and its satellites dance about. To 
remove this unsteadiness, each image — over 
2100 of them! -was optically centered in the 
frame. The stabilized images were joined to 
give a continuous record of the motion of lo. 
Some variation in brightness was caused by 
haze or cloudiness. 

The four Galilean satellites are listed in 
Table 1. On Feb. 3, 1967, they had the config- 
uration shown in Fig. 8-12. The satellites move 
nearly in a plane which we view almost edge- 
on; thus they seem to move back and forth 
along a line. The field of view is large enough 
to include the entire orbits of I and II, but III 
and IV are outside the camera field when they 
are farthest from Jupiter. 

The position of lo in the last frame of the 
Jan. 29 segment matches the position in the 




Business end of the 24-inch refractor at Lowell Ob- 
servatory. 

first frame of the Feb. 7 segment. However, 
since these were photographed 9 days apart, 
the other three satellites had moved varying 
distances, so you see them pop in and out while 
the image of lo is continuous. Lines identify 
lo in each section. Fix your attention on the 
steady motion of lo and ignore the comings and 
goings of the other satellites. 







TABLE 1 










SATELLITES OF JUPITER 










RADIUS 


ECCEN- 








OF 


TRICITY 








ORBIT 


OF 


DIAMETER 




NAME 


PERIOD (miles) 


ORBIT 


(miles) 


1 


lo 


1d igh 28"^ 262.000 


0.0000 


2,000 


II 


Europa 


3" ^3^ 14"^ 417,000 


0.0003 


1,800 


III 


Ganynnede 


jd 31, 43m 666,000 


0.0015 


3.100 


IV 


Callisto 


16" 16" 32"^ 1,171,000 


0.0075 


2,800 



Film Loops 185 







Fig. 8-12 



Interesting Features of the Film 

1. At the start lo appears almost stationary at 
the right, at its greatest elongation; another 
satellite is moving toward the left and over- 
takes it. 

2. As lo moves toward the left (Fig. 8-13), it 
passes in front of Jupiter, a transit. Another 
satellite, Ganymede, has a transit at about the 
saroe^time. Another satellite moves toward the 

^,«ght-apd disappears behind Jupiter, an occu- 
^ lation: It is a very active scene! If you look 
Nclo^eiy during the transit, you may see the 




r 



\ 




Fig. 8-13 Still photograph from Film Loop 12 showing 
the positions of three satellites of Jupiter at the start of 
the transit and occultation sequence. Satellite IV is out 
of the picture, far to the right of Jupiter. 



Fig. 8-14 



shadow of Ganymede and perhaps that of lo, 
on the left part of Jupiter's surface. 

3. Near the end of the film, lo (moving toward 
the right) disappears; an occulation begins. 
Look for Jo's reappearance — it emerges from 
an eclipse and appears to the right of Jupiter. 
Note that lo is out of sight part of the time be- 
cause it is behind Jupiter as viewed from the 
earth and part of the time because it is in Jupi- 
ter's shadow. It cannot be seen as it moves 
from O to E in Fig. 8-14. 

4. Jupiter is seen as a flattened circle because 
its rapid rotation period (9 h 55 m) has caused 
it to flatten at the poles and bulge at the 
equator. The effect is quite noticeable: the 
equatorial diameter 89,200 miles and the polar 
diameter is 83,400 miles. 

Measurements 

1. Period of orbit. Time the motion between 
transit and occulation (from B to D in Fig. 
8-14), half a revolution, to find the period. The 
film is projected at about 18 frames/sec, so that 
the speed-up factor is 18 x 60, or 1080. How 
can you calibrate your projector more accur- 
ately? (There are 3969 frames in the loop.) 
How does your result for the period compare 
with the value given in the table? 

2. Radius of orbit. Project on paper and mark 
the two extreme positions of the satellite, 



186 Film Loops 



farthest to the right (at A) and farthest to the 
left (at C). To find the radius in miles, use 
Jupiter's equatorial diameter for a scale. 
3) Mass of Jupiter. You can use your values 
for the orbit radius and period to calculate the 
mass of Jupiter relative to that of the sun (a 
similar calculation based on the satellite 
Callisto is given in SG 8.9 of the Text). How 
does your experimental result compare with 
the accepted value, which is mj/ms = 1/1048? 

FILM LOOP 13 PROGRAM ORBIT I 

A student (right, Fig. 8-15) is plotting the 
orbit of a planet, using a stepwise approxima- 
tion. His teacher (left) is preparing the com- 
puter program for the same problem. The 
computer and the student follow a similar 
procedure. 




Fig. 8-15 



The computer "language" used was 
FORTRAN. The FORTRAN program (on a 
stack of punched cards) consists of the "rules 
of the game": the laws of motion and of gravi- 
tation. These describe precisely how the cal- 
culation is to be done. The program is trans- 
lated and stored in the computer's memory 
before it is executed. 

The calculation begins with the choice of 
initial position and velocity of the planet. The 
initial position values of X and Y are selected 
and also the initial components of velocity 
XVEL and YVEL. (XVEL is the name of a 
single variable, not a product of four variables 
X, V, E, and L.) 

Then the program instructs the computer 



to calculate the force on the planet from the 
sun from the inverse-square law of gravitation. 
Newton's laws of motion are used to calculate 
how far and in what direction the planet moves 
after each blow. 

The computer's calculations can be dis- 
played in several ways. A table of X and Y 
values can be typed or printed. An X-Y plotter 
can draw a graph from the values, similar to 
the hand-constructed graph made by the 
student. The computer results can also be 
shown on a cathode ray tube (CRT), similar to 
that in a television set, in the form of a visual 
trace. In this film, the X-Y plotter was the mode 
of display used. 

The dialogue between the computer and 
the operator for trial 1 is as follows. The nu- 
merical values are entered at the computer 
typewriter by the operator after the computer 
types the messages requesting them. 
Computer: GIVE ME INITIAL POSITION IN 

AU . . . 
Operator: X = 4 

Y = 
Computer: GIVE ME INITIAL VELOCITY IN 

AU/YR . . . 
Operator: XVEL - 

YVEL = 2 
Computer: GIVE ME CALCULATION STEP 

IN DAYS . . . 
Operator: 60. 
Computer: GIVE ME NUMBER OF STEPS 

FOR EACH POINT PLOTTED . . . 
Operator: 1. 

Computer: GIVE ME DISPLAY MODE . . . 
Operator: X-Y PLOTTER. 

You can see that the orbit displayed on the 
X-Y plotter, like the student's graph, does not 
close. This is surprising, as you know that the 
orbits of planets are closed. Both orbits fail to 
close exactly. Perhaps too much error is intro- 
duced by using such large steps in the step-by- 
step approximation. The blows may be too 
infrequent near perihelion, where the force is 
largest, to be a good approximation to a con- 
tinuously acting force. In the Film Loop, 
"Program Orbit II," the calculations are based 
upon smaller steps, and you can see if this ex- 
planation is reasonable. 



Film Loops 187 



FILM LOOP 14 PROGRAM ORBIT II 

In this continuation of the film "Program 
Orbit I," a computer is again used to plot a 
planetary orbit with a force inversely propor- 
tional to the square of the distance. The com- 
puter program adopts Newton's laws of 
motion. At equal intervals, blows act on the 
body. We guessed that the orbit calculated in 
the previous film failed to close because the 
blows were spaced too far apart. You could 
calculate the orbit using many more blows, but 
to do this by hand would require much more 
time and effort. In the computer calculation 
we need only specify a smaller time interval 
between the calculated points. The laws of 
motion are the same as before, so the same 
program is used. 

A portion of the "dialogue" between the 
computer and the operator for trial 2 is as 
follows : 
Computer: GIVE ME CALCULATION STEP 

IN DAYS . . . 
Operator: 3. 

Computer: GIVE ME NUMBER OF STEPS 
FOR EACH POINT PLOTTED . . . 
Operator: 7. 

Computer: GIVE ME DISPLAY MODE . . . 
Operator: X-Y PLOTTER. 
Points are now calculated every 3 days (20 
times as many calculations as for trial 1 on the 
"Program Orbit I" film), but, to avoid a graph 
with too many points, only 1 out of 7 of the 
calculated points is plotted. 

The computer output in this film can also 
be displayed on the face of a cathode ray tube 
(CRT). The CRT display has the advantage of 
speed and flexibility and we will use it in the 
other loops in this series, Film loops 15, 16 and 
17. On the other hand, the permanent record 
produced by the X-Y plotter is sometimes very 
convenient. 

Orbit Program 

The computer program for orbits is written 
in FORTRAN II and includes "ACCEPT" 
(data) statements used on an IBM 1620 input 
typewriter. (Example at the right.) 

With slight modification it worked on a 
CDC 3100 and CDC 3200, as shown in the film 



PROORAM oRan 

HAHVARD PROJECT PHYSICS ORBIT PROGRAM, 
tMPIRlCAL VERIFICATION OF HEPLtRS LAWS 
FROM NtWTONS LAW OF UNIVERSAL ORAVITATION. 

G='>0. 
« CALL HAHKF(0.«O.I 

6 PRINT 7 

7 FORMAT (9HG1VE ME Y ) 
X=0. 

ACCEPT 5,Y 
PRINT B 

8 FORMAT (12HGIVE ME XVELl 
5 FORMAT(F 10.6) 

ACCEPT 5.XVEL 
YVEL=0. 
PRINT 9 

9 FORMAT CgHGIVE ME DELTA IN OAYSt AND NUMBER BETWEEN PRINTS) 
ACCEPT 5. DELTA 

DELTA=DELTA/365.25 
ACCEPT 5. PRINT 
IPRINT = PRINT 
INDEX = 
NFALLS = 
13 CALL MARKF (X.Y) 
PRINT 10, X.Y 

15 IFlSENSb SWITCH 3) 20,16 

20 PRINT 21 

10 FORMAT(2F7.3) 

NFALLS = NFaLLS ♦ IPRINT 

21 FORMAT (23HTURN OFF SFnSE SWITCH 3 ) 

22 CONTINUE 

IF(SLNSE SWITCH 3) 22,'. 

16 RADIUS = S0RTF(X»X ♦ Y«Y) 
ACCEL = -6/(kA0IUS»RADIUS) 
XACCtL = (X/RADIUS)«ACCEL 
YACCEL = (Y/KADIUS)»ACCEL 

FIRST TIME THROUGH WE WANT TO GO ONLY 1/2 DELTA 
IF(lMbEX) 17,17,18 

17 XVEL = XVEL ♦ 0.5 « XACCEL • JELTA 
YVLL = YVLL ♦ 0.5 » YACC> L » DELTA 
GO TO 19 

DELTA V = ACCELLHATION TIMES DELTA T 

18 XVEL = XVEL ♦ XACCEL » DELTA 
YVEL = YVEL ♦ YACCEL « DELTA 

DELTA X = XVELOCITY TIMtb DELTA T 

19 X = X ♦ XVEL • DELTA 
Y = Y ♦ YVtL » DELTA 
INDEX = INDEX ♦ 1 
IFdNDEX - NFhLLS) 15,15.13 
cND 



loops 13 and 14, "Program Orbit I" and "Pro- 
gram Orbit II." With additional slight modifi- 
cations (in statement 16 and the three suc- 
ceeding statements) it can be used for other 
force laws. The method of computation is the 
scheme used in Project Physics Reader 1 
"Newton's Laws of Dynamics." A similar pro- 
gram is presented and explained in FORTRAN 
for Physics (Alfred M. Bork, Addison-Wesley, 
1967). 

Note that it is necessary to have a sub- 
routine MARK. In our case we used it to plot 
the points on an X-Y plotter, but MARK could 
be replaced by a PRINT statement to print 
the X and Y coordinates. 

FILM LOOP 15 CENTRAL FORCES- 
ITERATED BLOWS 

In Chapter 8 and in Experiment 19 and 
Film Loop 13 on the stepwise approximation 
or orbits we find that Kepler's law of areas 
applied to objects acted on by a central force. 
The force in each case was attractive and was 
either constant or varied smoothly according 



188 Film Loops 



to some pattern. But suppose the central force 
is repulsive; that is, directed away from the 
center? or sometimes attractive and some- 
times repulsive? And what if the amount of 
force applied each time varies unsystemati- 
cally? Under these circumstances would the 
law of areas still hold? You can use this film 
to find out. 

The film was made by photographing the 
face of a cathode ray tube (CRT) which dis- 
played the output of a computer. It is important 
to realize the role of the computer program in 
this film: it controlled the change in direction 
and change in speed of the "object" as a re- 
sult of a "blow." This is how the computer 
program uses Newton's laws of motion to pre- 
dict the result of applying a brief impulsive 
force, or blow. The program remained the 
same for all parts of the loop, just as Newton's 
laws remain the same during all experiments 
in a laboratory. However, at one place in the 
program, the operator had to specify how he 
wanted the force to vary. 




Fig. 8-16 

Random Blows 

The photograph (Fig. 8-16) shows part of 
the motion of the body as blows are repeatedly 



applied at equal time intervals. No one decided 
in advance how great each blow was to be. 
The computer was programmed to select a 
number at random to represent the magni- 
tude of the blow. The directions toward or 
away from the center were also selected at 
random, although a slight preference for 
attractive blows was built in so the pattern 
would be likely to stay on the face of the CRT. 
The dots appear at equal time intervals. The 
intensity and direction of each blow is repre- 
sented by the length of line at the point of the 
blow. 

Study the photograph. How many blows 
were attractive? How many were repulsive? 
Were any blows so small as to be negligible? 

You can see if the law of areas applies to 
this random motion. Project the film on a 
piece of paper, mark the center and mark the 
points where the blows were applied. Now 
measure the areas of the triangles. Does the 
moving body sweep over equal areas in equal 
time intervals? 

Force Proportional to Distance 

If a weight on a string is pulled back and 
released with a sideways shove, it moves in an 
elliptical orbit with the force center (lowest 
point) at the center of the ellipse. A similar 
path is traced on the CRT in this segment of 
the film. Notice how the force varies at dif- 
ferent distances from the center. A smooth 
orbit is approximated by the computer by 
having the blows come at shorter time in- 
tervals. In 2(a), 4 blows are used for a full or- 
bit; in 2(b) there are 9 blows, and in 2(c), 20 
blows which give a good approximation to 
the ellipse that is observed with this force. 
Geometrically, how does this orbit differ from 
planetary orbits? How is it different physically? 

Inverse-square Force 

A similar program is used with two planets 
simultaneously, but with a force on each 
varying inversely as the square of the distance 
from a force center. Unlike the real situation, 
the program assumes that the planets do not 
exert forces on one another. For the resulting 
ellipses, the force center is at one focus (Kep- 



Film Loops 189 



ler's first law), not at the center of the elHpse 
as in the previous case. 

In this film, the computer has done thou- 
sands of times faster what you could do if you 
had enormous patience and time. With the 
computer you can change conditions easily, 
and thus investigate many different cases and 
display the results. And, once told what to do, 
the computer makes fewer calculation errors 
than a person! 

FILM LOOP 16 KEPLER'S LAWS 

A computer program similar to that used in 
the film "Central forces — iterated blows" 
causes the computer to display the motion of 
two planets. Blows directed toward a center 
(the sun), act on each planet in equal time in- 
tervals. The force exerted by the planets on one 
another is ignored in the program; each is at- 
tracted only by the sun, by a force which varies 
inversely as the square of the distance from 
the sun. 

Initial positions and initial velocities for 
the planets were selected. The positions of the 
planets are shown as dots on the face of the 
cathode ray tube at regular intervals. (Many 
more points were calculated between those 
displayed.) 

You can check Kepler's three laws by 
projecting on paper and marking successive 
positions of the planets. The law of areas can 
be verified by drawing triangles and measur- 
ing areas. Find the areas swept out in at least 
three places: near perihelion, near aphelion, 
and at a point approximately midway between 
perihelion and aphelion. 

Kepler's third law holds that in any given 
planetary system the squares of the periods 
of the planets are proportional to the cubes of 
their average distances from the object around 
which they are orbiting. In symbols, 

T2 oc R 3 

where T is the period and i?av is the average 
distance. Thus in any one system, the value of 
T^/Rav^ ought to be the same for all planets. 
We can use this film to check Kepler's law 
of periods by measuring T and for each of the 
two orbits shown, and then computing T^/Rgv' 




P 



Fig. 8-17 The mean distance Ra,. of a planet P orbiting 
about the sun is (Rp + F\J/2. 



for each. To measure the periods of revolution, 
use a clock or watch with a sweep second 
hand. Another way is to count the number of 
plotted points in each orbit. To find Rav for each 
orbit, measure the perihelion and aphelion 
distances (JRp and R^) and take their average 
(Fig. 8-17). 

How close is the agreement between your 
two values of T-jR^^r'? Which is the greater 
source of error, the measurement of T or of Rav? 

To check Kepler's first law, see if the orbit 
is an ellipse with the sun at a focus. You can 
use string and thumbtacks to draw an ellipse. 
Locate the empty focus, symmetrical with 
respect to the sun's position. Place tacks in a 




Fig. 8-18 



190 Film Loops 



board at these two points. Make a loop of string 
as shown in Fig. 8-18. 

Put your pencil in the string loop and draw 
the ellipse, keeping the string taut. Does the 
ellipse match the observed orbit of the planet? 
What other methods can be used to find if a 
curve is a good approximation to an ellipse? 

You might ask whether checking Kepler's 
laws for these orbits is just busy-work, since 
the computer already "knew" Kepler's laws 
and used them in calculating the orbits. But 
the computer was not given instructions for 
Kepler's laws. What you are checking is 
whether Newton's laws lead to motions that 
fit Kepler's descriptive laws. The computer 
"knew" (through the program we gave it) 
only Newton's laws of motion and the inverse- 
square law of gravitation. This computation 
is exactly what Newton did, but without the 
aid of a computer to do the routine work. 

FILM LOOP 17 UNUSUAL ORBITS 

In this film a modification of the computer 
program described in "Central forces — iterated 
blows" is used. There are two sequences: the 
first shows the effect of a disturbing force on 
an orbit produced by a central inverse-square 
force; the second shows an orbit produced by 
an inverse-cube force. 



The word "perturbation" refers to a small 
variation in the motion of a celestial body 
caused by the gravitational attraction of an- 
other body. For example, the planet Neptune 
was discovered because of the perturbation it 
caused in the orbit of Uranus. The main force 
on Uranus is the gravitational pull of the sun, 
and the force exerted on it by Neptune causes 
a perturbation which changes the orbit of 
Uranus very slightly. By working backward, 
astronomers were able to predict the position 
and mass of the unknown planet from its small 
effect on the orbit of Uranus. This spectacular 
"astronomy of the invisible" was rightly re- 
garded as a triumph for the Newtonian law of 
universal gravitation. 

Typically a planet's entire orbit rotates 
slowly, because of the small pulls of other 
planets and the retarding force of friction due 
to dust in space. This effect is called "advance 
of perihelion." (Fig. 8-19.) Mercury's perihelion 
advances about 500 seconds of arc, (j°) per 
century. Most of this was explained by per- 
turbations due to the other planets. However, 
about 43 seconds per century remained unex- 
plained. When Einstein reexamined the nature 
of space and time in developing the theory of 
relativity, he developed a new gravitational 
theory that modified Newton's theory in cru- 





Flg. 8-19 



Film Loops 191 



cial ways. Relativity theory is important for 
bodies moving at high speeds or near massive 
bodies. Mercury's orbit is closest to the sun 
and therefore most affected by Einstein's 
extension of the law of gravitation. Relativity 
was successful in explaining the extra 43 
seconds per century of advance of Mercury's 
perehelion. But recently this "success" has 
again been questioned, with the suggestion 
that the extra 43 seconds may be explained 
instead by a slight bulge of the sun at its 
equator. 

The first sequence shows the advance of 
perihelion due to a small force proportional to 
the distance R, added to the usual inverse- 
square force. The "dialogue" between operator 
and computer starts as follows: 
PRECESSION PROGRAM WILL USE 

ACCEL = GKR*R) + P*R 
GIVE ME PERTURBATION P. 



P= 0.66666. 
GIVE ME INITIAL POSITION IN AU 

X = 2. 

Y = 0. 
GIVE ME INITIAL VELOCITY IN AU/YR 

XVEL = 0. 

YVEL = 3. 
The symbol * means multiplication in the 
Fortran language used in the program. Thus 
GKR*R) is the inverse-square force, and P*R 
is the perturbing force, proportional to R. 

In the second part of the film, the force is 
an inverse-cube force. The orbit resulting from 
the inverse-cube attractive force, as from 
most force laws, is not closed. The planet 
spirals into the sun in a "catastrophic" orbit. 
As the planet approaches the sun, it speeds up, 
so points are separated by a large fraction of a 
revolution. Different initial positions and 
velocities would lead to quite different orbits. 




Man in observation chamber of the 200-inch reflecting telescope on Mt. Palomar. 



SATELLITES OF THE PLANETS 







AVERAGE RADIUS 


PERIOD 


OF 






DISCOVERY 


OF ORBIT 


REVOLUTION 


DIAMETER 


EARTH: Moon 




238,857 miles 


27d 


7h 


43 m 


2160 miles 


MARS: Phobos 


1877, Hall 


5,800 





7 


39 


10? 


Deimos 


1877, Hall 


14,600 


1 


6 


18 


5? 


JUPITER: V 


1892, Barnard 


113,000 





11 


53 


150? 


1(10) 


1610, Galileo 


262,000 


1 


18 


28 


2000 


II (Europa) 


1610, Galileo 


417,000 


3 


13 


14 


1800 


III (Ganymede) 


1610, Galileo 


666.000 


7 


3 


43 


3100 


IV (Callisto) 


1610, Galileo 


1,170,000 


16 


16 


32 


2800 


VI 


1904, Perrine 


7,120,000 


250 


14 




100? 


VII 


1905, Perrine 


7,290,000 


259 


14 




35? 


X 


1938, Nicholson 


7,300,000 


260 


12 




15? 


XII 


1951, Nicholson 


13,000,000 


625 






14? 


XI 


1938, Nicholson 


14,000,000 


700 






19? 


VIII 


1908, Melotte 


14,600,000 


739 






35? 


IX 


1914, Nicholson 


14,700,000 


758 






17? 


SATURN: Mimas 


1789, Herschel 


115,000 





22 


37 


300? 


Enceiadus 


1789, Herschel 


148,000 


1 


8 


53 


350 


Tethys 


1684, Cassini 


183,000 


1 


21 


18 


500 


Dione 


1684, Cassini 


234,000 


2 


17 


41 


500 


Rhea 


1672, Cassini 


327,000 


4 


12 


25 


1000 


Titan 


1655, Huygens 


759,000 


15 


22 


41 


2850 


Hyperion 


1848, Bond 


920,000 


21 


6 


38 


300? 


Phoebe 


1898, Pickering 


8,034,000 


550 






200? 


lapetus 


1671, Cassini 


2,210,000 


79 


7 


56 


800 


URANUS: Miranda 


1948, Kuiper 


81,000 


1 


9 


56 




Ariel 


1851,Lassell 


119,000 


2 


12 


29 


600? 


Umbriel 


1851, Lassell 


166,000 


4 


3 


28 


400? 


Titania 


1787, Herschel 


272,000 


8 


16 


56 


1000? 


Oberon 


1787, Herschel 


364,000 


13 


11 


7 


900? 


NEPTUNE: Triton 


1846, Lassell 


220,000 


5 


21 


3 


2350 


Nereid 


1949, Kuiper 


3,440,000 


359 


10 




200? 



THE SOLAR SYSTEM 









AVERAGE RADIUS 


PERIOD OF 




RADIUS 


MASS 


OF ORBIT 


REVOLUTION 


Sun 


6.95 X 10* meters 


1.98 X 10^° kilograms 


— 


— 


Moon 


1.74 X 10* 


7.34 X 10" 


3.8 X 10« meters 


2.36 X 10* seconds 


Mercury 


2.57 X 10* 


3.28 X 10'^ 


5.79 X 10'° 


7.60 X 10* 


Venus 


6.31 X 10* 


4.83 X 10'^ 


1.08 X 10" 


1.94 X 10' 


Earth 


6.38 X 10* 


5.98 X 10'" 


1.49 X 10" 


3.16 X 10' 


Mars 


3.43 X 10* 


6.37 X 10" 


2.28 X 10" 


5.94 X 10' 


Jupiter 


7.18 X 10' 


1.90 X 10^' 


7.78 X 10" 


3.74 X 10" 


Saturn 


6.03 X 10' 


5.67 X 10" 


1.43 X 10'^ 


9.30 X 10* 


Uranus 


2.67 X 10' 


8.80 X 10" 


2.87 X 10'' 


2.66 X 10' 


Neptune 


2.48 X 10' 


1.03 X 10" 


4.50 X 10" 


5.20 X 10' 


Pluto 


? 


? 


5.9 X 10" 


7.28 X 10' 



192 



INDEX/TEXT 



Acceleration, 90, 94, 99, 106-107 

Adams, John, 44 

Age of Enlightenment, 77 

Age of Reason, 4 

Alexander the Great, 3 

Almagest (Ptolemy), 21, 31, 45 

Aphelion, 69 

Aquinas, Thomas, 3, 17, 26 

Archimedes, 19 

Aristarchus, 19, 20, 21, 31, 32, 38 

Aristotle, 3, 17, 18, 19 

Astronomia Nova (Kepler), 56, 57, 

66 
Astronomical Unit (AU), 35 
Astronomy, 1-4 

Babylonians, 3, 7 
Bacon, Francis, 39 
Bayeux tapestry, 108 
Bible, 41, 76 
Black plague, 86 
Brahe, Tycho, see Tycho 
Bruno, Giordano, 76 
Brutt, E. A., 118 

Caesar, Julius, 8 
Calendar 
Egyptian, 8 
Gregorian, 8 
JuUan, 8 
Cardinal Barberini, 75 
Cardinal Baronius, 76 
Cavendish, Henry 

his apparatus for determining 
value of G, 104 
Celestial equator, 10 
Central force, 65, 91-94 
Centripetal acceleration, 90 
"Clockwork," the world seen as, 68 
Comets, 45-46, 107 
Halley's, 107-108 
orbit of, 107 
Conic sections, 63 
Constant of Universal Gravitation 

(G), 104, 105 
Constellations, 8-9, 10 
Copernicus, Nicholas, 3, 29, 44, 
55, 67, 95 
arguments against his system, 

39-41, 43 
arguments for his system, 32, 

35-36, 38 
assumptions, 29, 31-32 
calculation of periods of the 
planets around the sun, 
33, 34 
finds distance between planets 
and sun, 33 



On the Revolutions of the 

Heavenly Spheres (De Rev- 
olutionibus), 28, 29, 31, 34, 
41,43, 45 

rejection of equants, 31-32 

sketch of his system, 32 

time line, 30 
Cosmographia (Apianus), 18 

Darwin, Charles, 5, 120 
Deferent circle, 23, 32, 33, 36, 37 
De Revolutionibus (Copernicus), 
28, 29, 31, 34. 41. 43, 45 
condemned by Jewish commu- 
nity, 41 
condemned by Luther, 41 
condemned by Papacy, 41 
Descartes, Rene, 91, 96 
Dialogue Concerning the Two 
Chief World Systems (Gali- 
leo), 73 
Discourses Concerning Two New 

Sciences, (Galileo), 73 
Divine Comedy (Dante), 18 

Earth ( see geocentric view of uni- 
verse) 
Eclipse 

of moon, 12 

of sun, 12 
Ecliptic (path of sun), 10, 12, 13 
Eccentric, 23, 31 
Eccentricity, 61, 66 

defined, 60 
Egyptians, 7, 8 
Einstein, Albert, 66, 76 
Elements 

four basic, 16, 18 
Ellipse, 60,61,63, 91 

and conic sections, 63 

eccentricity, 56, 60, 61, 66 

focus, 60 
Epicycle, 23, 24, 31, 33, 36, 37, 41, 

56 
Epicyclic motion, 23 
Equant, 23, 24, 25 

rejection by Copernicus, 31-32 
Equator 

celestial, 10 
Ethers, 16 
Eudoxus 

system of spheres, 18, 26 

Focus, of eclipse, 60, 61 

Fontenelle, 115 

Frederick II of Denmark, 45 

G, see Constant of Universal Grav- 
itation, 104-105 



Galileo, 69, 77 

and the Scholastics, 74 
builds telescopes, 69, 70-71 
Dialogue Concerning the Two 
Chief World Systems, 73 
Discourse Concerning Two New 

Sciences, 73 
discovery of sunspots, 72 
Inquisition, 75-76 
observation of the moon and 

fixed stars, 71 
relationship with Kepler, 69 
satellites of Jupiter, 71, 72 
Sidereal Messenger (Sidereus 

Nuncius), 70, 72 
study of Saturn, 72 
study of Venus, 72 
Geocentric view of universe, 17, 
18, 36, 38, 39 
and Ptolemy, 21-25 
Gilbert, Wilham, 68 
Goethe, Johann Wolfgang von, 118 
Gravitational constant (G), 102- 

103, 104-105 
Gravitational force, 68, 90, 94-95, 

99 
Gravitational interaction, 96 
Greeks, 15, 16, 26 

their explanation of motion in 

the sky, 3, 7 
their idea of explanation, 16-17 

Halley, Edmund 
his comet, 107-108 
and Newton, 87 
Harmonices Mundi (Kepler), 54, 

66 
Heliocentric view of the universe, 

19-20, 26, 28, 36, 38, 43, 44, 

55 
Heracleides, 19 
Herschel, WUliam, 111 

Index Expurgatorius (Index of 

Forbidden Books), 41, 75 
Inquisition, 75-76 
Inverse-square law, 90-91 

Johnson, Samuel, 118 
Jupiter, 12, 14, 99 

Galileo's observation of, 71-72 

satellites of, 71 

Kepler, Johannes, 4, 77 

derives the orbit of the earth, 58 
Dioptrice, 67 
five perfect solids, 54 
Harmony of the World (Har- 
monices Mundi), 54, 66 



193 



heliocentric theory, 55 
Law of Areas, 57-59, 90, 93 
Law of EUiptical Orbits, 60-61, 

63 
Law of Periods (Harmonic 

Law), 66-67, 90, 91 
Laws of Planetary Motion, 100 
model explaining planetary 

orbits, 56 
New Astronomy (Astronomia 

Nova), 56, 57, 66 
new concept of physical law, 68 
notion of world as "clockwork," 

68 
relationship with Galileo, 69 
Rudolphine Tables, 67 
study of the orbit of Mars, 56 
use of logarithms, 67 

Laplace, Marquis de, 119 
Law of Areas, 57-59, 90, 93 
Law of Elliptical Orbits, 60-61, 63 
Law of Periods, 66-67, 90, 91 
Law of Universal Gravity, 89, 91, 

94, 100-101 
Leap years, 8 
Le Clerc, Sebastian, 116 
Locke, John, 115 
Logarithms, 67 
Lucretius, 120 
Luther, Martin, 41 

Mars, 12 

orbit derived, 58, 59 

retrograde motion of, 12-13, 14 
Mass-points, 99 
Mercury, 12, 14 

orbit of, 61 

retrograde motion of, 12, 13 
Milton, John 

Paradise Lost, 51, 69 
Moon 

and month, 1 

motions of, 11-12 

path in sky, 12 

phases of, 1 1 

and tides, 106 
Motion 

epicylic, 23 

natural, 89-90 

under a central force, 93 
Muslims, 3, 15 

Neptune, 12, 110 
Newton, Sir Isaac, 4, 18, 52, 1 14, 
115, 117 
and the apple, 94 
concept of gravitational force, 
68 



Constant of Universal Gravita- 
tion (G), 102-104, 108-109 
and eclipses, 91 
and gravitational force, 86, 94, 

95 
and hypotheses, 96-97 
Law of Universal Gravitation, 

89,91,94, 100-101 
magnitude of planetary force, 

98-99 
mechanics, 112, 113 
and natural motion, 89-90 
and planetary motion, 102-104 
Principia, 77, 83, 87, 88, 89, 97, 

106, 111, 112, 117 
reflecting telescope, 85 
relationship with Halley, 87 
"Rules of Reasoning in Philos- 
ophy," 88-89 
study of optics, 87 
synthesis of earthly and heav- 
enly motions, 95, 114—120 
Theory of Light and Colors, 87 
and the tides, 106-107 
North Celestial Pole, 9 
North star (see Polaris) 

On the Revolutions of the Heav- 
enly Spheres (Copernicus), 29 

Opposition, 12 

Orbit eccentricity periods, 61 

Orbit shape, 61 

Orbital speed, a general explana- 
tion of, 64-65 

Origins of Modern Science (But- 
terfield),41 

Origin of Species (Darwin), 5 

Pantheism, 76 

Parallax, 20 

Pepys, Samuel, 81 

Perihelion, 69 

Phenomena, observation of, 17 

Planets 

angular distance from sun, 14 

brightness of, 14 

eccentricities of orbits, 61 

masses of, 104, 105-106 

opposition to sun, 12 

periods around the sun, 33, 34 

retrograde motion of, 12-14 

size and shape of their orbits, 35 
Planetary force, 98-99 
Planetary motion, 102-104 
Plato, 18, 19, 23, 29, 76 

explanation of the moon's 
phases, 12 

four elements, 16 

problem, 15-16 



Pluto, 12, 14, 110 

Polaris, 9 

Pope, Alexander, 89, 118 

Pope Gregory, 8 

Pope Urban VIII, 75 

Principia (Newton), 77, 83, 87, 

88, 89, 97, 106, 111, 112, 117 
Principle of Parsimony, 88 
Ptolemy, Claudius, 23, 24, 40, 95 

Almagest, 21, 31, 45 

assumptions, 21 

differences between his system 
and Plato's, 21-22 

disagrees vidth Aristarchus, 21- 
22 

geocentric system, 21-25 

limitations of his model, 26 

successes of his model, 25-26 

system sketched, 24 

Pythagoreans, 17 

Quadrant, 48 

Quantum mechanics, 113 

Quintessence, 16 

Refraction of light, 49 
Relativity theory, 113 
Renaissance movement, 3, 5 
Retrograde motion, 12, 14, 20, 32, 
37 
created by epicycle machine, 23 
drawings of, 13 
and epicycle, 24 
and heliocentric view of uni- 
verse, 19 
Rudolph II, 67 

Rules of Reasoning in Philosophy 
(Newtons), 88-89 

Sabatelh, Luigi, 74 
Saturn, 12, 14 

retrograde motion of, 12-13 
Schelling, Friedrich, 119 
Scholastics, 74 
Scientific Revolution, 4 
Sidereus Niincius (Galileo), 70 
Sizzi, Francesco, 74 
Solar Ephemeris, 41-42 
Starry Messenger (see Sidereus 

Nuncius) 
Stars 

fixed, 9, 10 

movement of, 8-10 
Stroboscopic photographs, 23 
Stonehenge, 2 

Sun, see also heliocentric view of 
universe 



194 



its motion, 7, 10-11 Trinity College, Cambridge Uni- Universal gravitation, 89, 91, 94, 

path (ecliptic), 10, 13 versity, 85 100-101 

solar year, defined, 8 Tycho Brahe, 36, 44, 45, 46, 47, Uraniborg, 41 

Sunspots, 72 48, 57, 61, 67 instruments there, 46 

calibration of his instruments, Uranus, 12, 110 

Telescope, 20, 45, 47, 50 47-48,49 

Vpniit; 19 14 

adapted by Galileo, 69, 70-71 compromise system, 49, 51 veiiu:,, i^, i^ 

reflecting, 85 discovers a new star, 45 v i •' in 

Tides, and universal gravitation, observatory, 46 vernal equmox, 10 

106-107 observes comet, 45, 47 Wilson, James, 117 

Triangle, area of, 64 quadrant, 48 William the Conqueror, 108 



195 



INDEX/HANDBOOK 



Activities 

build a sundial, 144 

celestial sphere model, 141 

conic-sections models, 168 

demonstrating satellite orbits, 167 

discovery of Neptune and Pluto, 181 

epicycles and retrograde motion, 139 

finding earth-sun distance from Venus photos, 
168 

forces on a pendulum, 179 

Galileo, 168 

haiku, 181 

how long is a sidereal day?, 143 

how to find the mass of a double star, 181-83 

inclination of Mars' orbit, 165 

literature, 144 

making angular measurements, 138 

measuring irregular areas, 168 

model of the orbit of Halley's comet, 176 

moon crater names, 144 

other comet orbits, 179 

plot an analemma, 144 

scale model of the solar system, 143 

size of the earth — simplified version, 145 

Stonehenge, 144 

three-dimensional model of two orbits, 165 

trial of Copernicus, 181 

two activities on frames of reference, 156 
Algol, 154 

Analemma, plotting of, 144 
Andromeda, Great Nebula, 154 
Angular measurements, 138-39 
Astrolabe, 133-34 
Astronomical unit, 143 

Calendar, JuUan day, 130 
Celestial sphere model, 141-43 
Comets, orbit of, 170-74, 176-79 
Constellations, 146 
Computer program for orbits, 186-91 
Conic-section models, 168 
Coordinate system, 158-59 
Copemican model, 148 
Copernicus, mock trial of, 181 
Crystal-sphere model of universe, 144 

Declination, 142 

Double star, mass of, 181-83 

Earth, circumference of, 132-34 

orbit, 148-50, 176 

shape of, 134 

size of, 145 
Earth-sun distance, 168 
Ecliptic, plane, 158 

pole of, 141-42 
Ellipse, pendulum motion in shape of orbit, 180 
Epicycle machine, 139-40 



Epicycles, photographing of, 140-41 

Eratosthenes, 132 

Experiments 

height of Piton, a mountain on the moon, 135 

naked-eye astronomy, 129 

orbit of Mars, 158 

orbit of Mercury, 162 

shape of the earth's orbit, 148 

size of the earth, 132 

stepwise approximation to an orbit, 170 

using lenses to make a telescope, 151 

Film Strip : Retrograde Motion of Mars, 146 
Film Loops 

Central Forces — Iterated Blows, 187 

Jupiter Satelhte Orbit, 184 

Kepler's Laws, 189 

Program Orbit I, 186 

Program Orbit II, 187 

Retrograde Motion — Geocentric Model, 147 

Retrograde Motion — Heliocentric Model, 157 

Retrograde Motion of Mars and Mercury, 147 

Unusual Orbits, 190 
Force 

central, iterated blows, 187 

inverse-square, 188 
Frames of reference, 156 

Galileo 

moon observations, 135 

Brecht's play, 167 
Geocentric latitude and longitude, 166 
Geocentric theory of Ptolemy, 157 

Haiku, 181 

Halley's comet, orbit, 176-78 

Heliocentric latitude and longitude, 165-66 

Heliocentric mechanical model, 157 

Heliocentric system, 150 

Irregular areas, measurement of, 168 
Iteration of orbits, 170 

Julian day calendar, 130 
Jupiter, 147, 153 

satelhte orbit, 184-86 

Kepler's Laws 

computer program for, 189-190 
from orbital plot, 160-61 
of areas, 161 
second law, 164 

Latitudes, measurement of , 159 
Law of areas, Kepler's, 161 
Laws of motion, Newton's, 170 
Lenses, 151 
Longitudes, measurement of, 159 



196 



Mars, oppositions of, 147 

orbit, 158-61 

photographic observations of, 158 

retrograde motion of, 146-47, 157 
Mercury, elongations, 162-63 

longitudes, 130 

orbit, 162-63 

retrograde motion of, 146-47 
MilkyWay, 147, 154 
Moon, 153 

crater names, 144 

height of a mountain on, 135 

observations of, 129 

phases of, 129 

photographs of, 135, 137 

surface of, 135-37 

Naked-eye astronomy, 128-32 
Neptune, predicting existence of, 181 
Newton, laws of motion, 170 
second law, 171 

Objective lens, telescope, 151 
Observations, moon, 129 

sun's position, 128-29 
Occulta tion, 185 
Oppositions of Mars, 147 
Orbit(s), comets, 170-80 

computer program for, 186-91 

earth, 148-50, 176 

elements of, 166-67 

Halley's comet, 176-79 

Jupiter satellite, 184-86 

Mars, 158-61 

Mercury, 162-63 

pendulum, 180-81 

planetary, 158 

satellite, 167-68, 170-76 

sun, 148-50 

three-dimensional model of , 165 



Orbital eccentricity, calculation of, 164 

Pendulum, orbit, 180-81 

Piton, height of, 135-38 

Planetary Longitudes Table, 130-31 

Planet(s), location and graphing of, 130-32 

orbits, 158 
Pleiades, 146, 153 
Pluto, predicting existence of, 181 
Proper motion, 182 
Ptolemaic model, 140 
Ptolemy, geocentric theory of, 157 

R^,., for an orbit, 163 
Real image, 151 
Retrograde motion, 139-40, 157 
of Mars and Mercury, 146-47 
Right ascension, 142 

Satellite orbits, 167, 170-76 

Saturn, 153 

Semi-major axis, 163 

Sidereal day, 143 

Solar system, scale model of, 143 

Stonehenge, 144 

Sun, measurement of angle, 133-34 

observations of position, 128-29 

plotting orbit of, 148-50 
Sundial, building of, 144 
Sunspots, observation of, 154-55 

Telescope, making and use of, 151-55 

observations, 152-55 
Terminator, 135 
Transit, 185 

Venus, 153, 168 
Vernal equinox, 142 



197 



staff and Consultants 



Sidney Rosen, University of Illinois, Urbana 
John J. Rosenbaum, Livermore High School, 

Calif. 
William Rosenfeld, Smith College, Northampton, 

Mass. 
Arthur Rothman, State University of New York, 

Buffalo 
Daniel Rufolo, Clairemont High School, San 

Diego, Calif. 
Bemhard A. Sachs, Brooklyn Technical High 

School, N.Y. 
Morton L. Schagrin, Denison University, Granville, 

Ohio 
Rudolph Schiller, Valley High School, Las Vegas, 

Nev. 
Myron O. Schneiderwent, Interlochen Arts 

Academy, Mich. 
Guenter Schwarz, Florida State University, 

Tallahassee 
Sherman D. Sheppard, Oak Ridge High School, 

Tenn. 
William E. Shortall, Lansdowne High School, 

Baltimore, Md. 
Devon Showley, Cypress Junior College, Calif. 
William Shurcliff, Cambridge Electron 

Accelerator, Mass. 
Katherine J. Sopka, Harvard University 
George I. Squibb, Harvard University 
Sister M. Suzanne Kelley, O.S.B., Monte Casino 

High School, Tulsa, Okla. 
Sister Mary Christine Martens, Convent of the 

Visitation, St. Paul, Minn. 
Sister M. Helen St. Paul, O.S.F., The Catholic 

High School of Baltimore, Md. 



M. Daniel Smith, Earlham CoUege, Richmond, 

Ind. 
Sam Standring, Santa Fe High School, Santa Fe 

Springs, Calif. 
Albert B. Stewart, Antioch CoUege, YeUow 

Springs, Ohio 
Robert T. SuUivan, Burnt Hills-Ballston Lake 

Central School, N.Y. 
Loyd S. Swenson, University of Houston, Texas 
Thomas E. Thorpe, West High School, Phoenix, 

Ariz. 
June Goodfield Toulmin, Nuffield Foundation, 

London, England 
Stephen E. Toulmin, Nuffield Foundation, London, 

England 
Emily H. Van Zee, Harvard University 
Ann Venable, Arthur D. Little, Inc., Cambridge, 

Mass. 
W. O. Viens, Nova High School, Fort Lauderdale, 

Fla. 
Herbert J. Walberg, Harvard University 
Eleanor Webster, WeUesley College, Mass. 
Wayne W. Welch, University of Wisconsin, 

Madison 
Richard Weller, Harvard University 
Arthur Western, Melbourne High School, Fla. 
Haven Whiteside, University of Maryland, CoUege 

Park 
R. Brady Williamson, Massachusetts Institute of 

Technology, Cambridge 
Stephen S. Winter, State University of New York, 

Buffalo 



198 



Answers to End-of-Section Questions 



Chapter 5 

Q1 The sun would set 4 minutes later each day. 

Q2 Calendars were needed to schedule agricultural 

activities and religious rites. 

Q3 The sun has a westward motion each day, an 

eastward motion with respect to the fixed stars and 

a north-south variation. 

Q4 



fsxil 



|s+.^a'^r"N 









than the planet. So the planet would appear to us to 
be moving westward. 

Q18 The direction to the stars should show an 
annual shift — the annual parallax. (This involves a 
very small angle and so could not be observed with 
instruments available to the Greeks. It was first 
observed in 1836 A. D.) 

Q19 Aristarchus was considered to be impious 
because he suggested that the earth, the abode of 
human life, might not be at the center of the universe. 
His system was neglected for a number of reasons: 
^1) "Religious" — it displaced man from the center 
of the universe. 

(2) Scientific — stellar parallax was not observed. 

(3) Practical — it predicted celestial events no better 
than other, less offensive, theories. 



3*^ onoritr 

Q5 Eclipses do not occur each month, because 

the moon and the earth do not have the same planes 

of orbit. 

Q6 Mercury and Venus are always found near the 

sun, either a little ahead of it or a little behind it. 

Q7 When in opposition, a planet is opposite the 

sun; therefore the planet would rise at sunset and be 

on the north-south line at midnight. 

Q8 After they have been farthest east of the sun 

and are visible in the evening sky. 

Q9 When they are near opposition. 

Q10 No, they are always close to the ecliptic. 

Q11 How may the irregular motions of the planets 

be accounted for by combinations of constant speeds 

along circles? 

Q12 Many of their written records have been 

destroyed by fire, weathering and decay. 

Q13 Only perfect circles and uniform speeds were 

suitable for the perfect and changeless heavenly 

bodies. 

Q14 A geocentric system is an earth-centered 

system. The yearly motion of the sun is 

accounted for by assuming that it is attached to a 

separate sphere which moves contrary to the motion 

of the stars. 

Q15 The first solution, as proposed by Eudoxus, 

consisted of a system of transparent crystalline 

spheres which turned at various rates around various 

axes. 

Q16 Aristarchus assumed that the earth rotated 

daily — which accounted for all the daily motions 

observed in the sky. He also assumed that the earth 

revolved around the sun — which accounted for the 

many annual changes observed in the sky. 

Q17 When the earth moved between one of these 

planets and the sun (with the planet being observed 

in opposition), the earth would be moving faster 



Chapter 6 

Q1 The lack of uniform velocity associated with 
equants was (1) not sufficiently absolute, (2) not 
sufficiently pleasing to the mind. 
Q2 (a) P, C 

(b) P, C 

(c)P 

(d)C 

(e) P. C 

(f) C 

Q3 The relative size of the planetary orbits as 

compared with the distance between the earth and 

the sun. These were related to the calculated periods 

of revolution about the sun. 

Q4 (b) and (d) 

Q5 2° in both cases 

Q6 No; precise computations required more small 

motions than in the system of Ptolemy. 

Q7 Both systems were about equally successful in 

explaining observed phenomena. 

Q8 The position of man and his abode, the earth, 

were important in interpreting the divine plan of the 

universe. 

Q9 They are equally valid; for practical purposes 

we prefer the Copernican for its simplicity. 

Q10 He challenged the earth-centered world 

outlook of his time and opened the way for later 

modifications and improvements by Kepler, Galileo, 

and Newton. 

Q11 The appearance in 1572 of a "new star" of 

varying brightness. 

Q12 It included expensive equipment and facilities 

and involved the coordinated work of a staff of 

people. 

Q13 They showed that comets were distant 

astronomical objects, not local phenomena as had 

been believed. 

Q14 He made them larger and sturdier and devised 

scales with which angle measurements could be 

read more precisely. 



199 



Q15 He analyzed the probable errors inherent in 
each piece of his equipment; also he made 
corrections for the effects of atmospheric refraction. 
Q16 He kept the earth fixed as did Ptolemy and he 
had the planets going around the sun as did 
Copernicus. 

Chapter 7 

Q1 Finding out the correct motion of Mars through 
the heavens. 

Q2 By means of circular motion, Kepler could not 
make the position of Mars agree with Tycho Brahe's 
observations. (There was a discrepancy of 8 minutes 
of arc in latitude.) 

Q3 By means of triangulation, based on observa- 
tions of the directions of Mars and the sun 687 days 
apart, he was able to plot the orbit of the earth. 
Q4 A line drawn from the sun to a planet sweeps 
out equal areas during equal time intervals. 
Q5 Where it is closest to the sun. 
Q6 Mars has the largest eccentricity of the planets 
Kepler could study. 
Q7 (a) E 

(b)A 

(c) A + E (+ date of passage of perihelion, for 

example) 
Q8 The square of the period of any planet is 
proportional to the cube of its average distance to 
the sun. 

Q9 Kepler based his laws upon observations, and 
expressed them in a mathematical form. 
Q10 Popular language, concise mathematical 
expression. 

Q11 Both the heliocentric and Tychonic theories. 
Q12 The sunspots and the mountains on the moon 
refuted the Ptolemaic assertion that all heavenly 
bodies were perfect spheres. 
Q13 Galileo's observations of the satellites of 
Jupiter showed that there could be motions around 
centers other than the earth. This contradicted basic 
assumptions in the physics of Aristotle and the 
astronomy of Ptolemy. Galileo was encouraged to 
continue and sharpen his attacks on those earlier 
theories. 

Q14 No, they only supported a belief which he 
already held. 

Q15 Some believed that distortions in the telescope 
(which were plentiful) could have caused the 
peculiar observations. Others believed that 
established physics, religion, and philosophy far 
outweighed a few odd observations. 
Q16 b, c (d is not an unreasonable answer since it 
was by writing in Italian that he stirred up many 
people.) 



Chapter 8 

Q1 The forces exerted on the planets are always 

directed toward the single point where the sun is 

located. 

Q2 The formula for centripetal acceleration 

Q3 That the orbit was circular 

Q4 No, he included the more general case of all 

conic sections (ellipses, parabolas and hyperbolas 

as well as circles). 

Q5 That one law would be sufficient to account for 

both. 

Q6 He thought it was magnetic and acted 

tangentially. 

Q7 The physics of motion on the earth and in the 

heavens under one universal law of gravitation. 

Q8 No, he thought it was sufficient to simply 

describe and apply it. 

Q9 An all pervasive ether transmitted the force 

through larger distances. 

Q10 He did not wish to use an hypothesis which 

could not be tested. 

Q11 Phenomenological and thematic 

Q12 (a) The forces are equal. 

(b) The accelerations are inversely propor- 
tional to the masses. 
Q13 (a) 2F 

(b)3F 

(c)6F 
Q14 (b)F,3 = 4F^^ 

Q15 The values of the constant in Kepler's third law 
T-/R^ = k as applies to satellites of each of the two 
planets to be compared. 
Q16 The numerical value of G 
Q17 Fp^3^., m^, m,, R 

Q18 The period of the moon and the distance 
between the centers of the earth and the moon or 
the ratio T^/R^. 

Q19 Similar information about Saturn and at least 
one of its satellites. 

Q20 1/1000; that is, inversely proportional to the 
masses. 

Q21 On the near side the water is pulled away 
from the solid earth; on the far side the solid 
earth is pulled away from the water. Since F « 1 /R- 
the larger R is, the smaller the corresponding F. 
Q22 All of them 

Q23 As the moon orbits its distance to the sun is 
continually changing, thus affecting the net force on 
the moon due to the sun and the earth. Also the 
earth is not a perfect sphere. 
Q24 (a), (b), and (c) 
Q25 Influenceof sun and shape of earth 
Q26 Comets travel on very elongated ellipses. 
Q27 No 



200 



Brief Answers to Study Guide Questions 



Chapter 5 

5.1 Information 

5.2 Discussion 

5.3 (a) 674 seconds 
(b) 0.0021 % 

5.4 Table 

5.5 Discussion 

5.6 Discussion 

5.7 Discussion 

5.8 102°, 78°, 78°, 102° starting with 
the upper right quadrant. 

5.9 (a) 15° 

(b) Geometric proof and calcu- 
lation; about 8000 miles. 

5.10 a, b, c, d,e, f 

5.11 Discussion 

5.12 Discussion 



Chapter 6 


6.1 


Information 


6.2 


Diagram construction 


6.3 


Discussion 


6.4 


11 times; derivation 


6.5 


Discussion 


6.6 


Discussion 


6.7 


2.8 X 10-^ AU 


6.8 


Discussion 


6.9 


Discussion 


6.10 


Discussion 


6.11 


Discussion 


6.12 


Discussion 



Chapter 7 

7.1 Information 

7.2 About 1/8 of a degree; about 
1/100thof an inch; roughly 1 /20th 
of a degree. 

7.3 Discussion 

7.4 4% 

7.5 Discussion 

7.6 a + c 

7.7 Discussion 

7.8 0.209 

7.9 .594/1 

7.10 Analysis 

7.11 (a)17.9AU 
(b) 35.3 AU 



(c)0.54AU 
(d) 66/1 

7.12 7 = 249 years 

7.13 K = 1.0 for all three planets 

7.14 Discussion 

7.15 (a) sketch 

(b) fta^ : 3.4 mm, 5.2 mm, 
8.2 mm, 4.6 mm 

T: 44", 84", 168", 384" 

(c) K: 485, 495, 501, 
470 Hr-/mm' 

7.16 Discussion 

7.17 Discussion 

7.18 Discussion 

Chapter 8 

8.1 Information 

8.2 Yes, to about 1 % agreement 

8.3 Discussion 

8.4 Discussion 

8.5 Derivation 

8.7 About 170 times as great 

8.8 Discussion 

8.9 (a) 1.05 X 103 daysVAU3 

(b) discussion 

(c) discussion 

8.10 26,500mi, or 42,600 km 

8.11 5.98X102* kg 

8.12 6.04 X 10-* kg 

8.13 (a)5.52 X 103 kg/m3 
(b) discussion 

8.14 7.30 X 1022 kg 

8.15 Pluto has no known satellite 

8.16 (a) 5.99 X 103 sec, or 1.66 hours 

(b) 3.55 km/sec 

(c) collisions 

8.17 Table 

8.18 1 7.7 AU, 0.60 AU, 34.8 AU 

8.19 Derivations 

8.20 Discussion 

8.21 Discussion. No. 

8.22 Discussion 

8.23 It is useful today. 

8.24 Discussion 



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