5
5-1
The progress of science is generally regarded as a kind of clean,
rational advance along a straight line; in fact, it has followed a
zig-zag course, at times almost more bewildering than the evolution
of political thought. The history of cosmic theories, in
particular, may without exaggeration be called a history of
collective obsessions and controlled schizophrenias; and the manner
in which some of the most important individual discoveries were
arrived at reminds one more of a sleepwalker’s performance than an
electronic brain. Arthur Koestler, The Sleepwalkers, 1959
The Historical Quest to Model the
Solar System
Part 1 • Discovering the Science of Astronomy5-2
In this chapter we follow the way our knowledge of the layout of
the Solar System developed from the time of the Greek
Earth-centered view to the Sun-center view during the renaissance
in Europe. We will be interested in how the ideas that developed
in
one era were a natural outgrowth of the state of astronomical
observations and degree of sophistication of the times. In previous
chapters we briefly discussed some historical aspects of astronomy;
for example, we considered the idea of a spherical Earth, and how
this concept slowly came to be accepted by 100 b.c.e. by Greek
thinkers through the gradual accumulation of evidence from
observations of both the Earth and the heavens. Yet this concept
was established relatively rapidly when compared to the slow
acceptance of the idea of a Sun-centered Solar System.
5.1 Greek Astronomy Although we saw in the last chapter that a
variety of cultures were intimately involved with phenomena of the
sky, it was the Greek culture that made progress toward the models
we have today.
What the Greeks Inherited The Babylonians and Egyptians left the
Greeks an extensive body of astronomical knowledge. However, the
astronomy of both ancient Egypt and Babylonia was the province of a
priestly aristocracy; as a consequence, practical and political
considerations often took precedence over theoretical inquiries.
Egyptian astronomers, for example, had discovered that when the
bright star Sirius could just be seen rising in the east before the
Sun, the flooding of the Nile was imminent. Such knowledge gave
tremendous power to the priesthood and inevitably involved its
members closely with the state. Similarly, in Babylonia,
astronomer-priests had acquired a considerable amount of
information concerning the motions of the Moon, Sun, and planets
and had found certain regular cycles that enabled them to predict
some eclipses—a power that was frequently used for political
purposes.
The ancient Egyptians and Babylonians knew the length of the year
and the different types of calendars, both solar and lunar. The
Egyptians had learned the rudiments of simple mathematics, algebra,
and geometry. Sundials had been invented, and systems of
timekeeping were in existence. The Babylonians had made systematic
observations of the positions of heavenly objects and had practical
methods for predicting the positions of the Moon, Sun, and planets.
Figure 5-1, for example, shows a table of data for Jupiter. The
bottom part of the figure describes the method of calculation. The
Babylonian value for the length of the synodic
period of the Moon was not surpassed in accuracy until the end of
the nineteenth century. Both cultures had attempted to construct a
cosmology that placed Earth and humanity in their proper position
in relation to the universe and the gods, but they never attempted
to construct a truly consistent theoretical framework for their
cosmology in the way the Greeks did.
Aristotle The Greeks enjoyed philosophy, by which they meant a
broad attempt to understand all things. Aristotle (384–322 B.C.E.)
viewed the universe in the abstract and argued that simply thinking
logically about the universe was sufficient to provide an
understanding of it.
At the time of Aristotle, the thinking was that there were four
elements making up the universe — earth, air, fire and water. The
tendency of objects to fall to the ground was explained as follows:
objects made of “earthy material” fell to the ground because they
moved toward similar
Chapter opening photo: Galileo showing the heavens to the public in
Venice with one of his telescopes.
FIGURE 5-1. A clay tablet from ancient Mesopotamia containing
astronomical observations of Jupiter in the top part, and a
description of the method of calculation in the bottom part.
5-3Chapter 5 • The Historical Quest to Model the Solar System
these two time intervals were in fact nearly equal, implying that
rays of light from the Sun are arriving in nearly parallel lines
for all parts of the Moon’s orbit. From this one concludes that the
Sun must be many times farther away than the Moon.
Inquiry 5-1 What did Aristarchus assume about the Moon’s
orbital shape and motion?
Aristarchus was also able to estimate the relative sizes of the
Earth and Moon by timing the duration of lunar eclipses. Figure 5-3
shows the principle of this determination. The length of time it
takes the entire Moon to enter the Earth’s shadow depends directly
on (that is, is proportional to) the diameter of the Moon. The
length of time it takes for one part of the Moon — say, its leading
edge — to completely cross the Earth’s shadow is proportional to
the diameter of the Earth. By comparing these two times, we can
estimate the ratio between the diameters of the Earth and Moon, as
Aristarchus did.1
Inquiry 5-2 Suppose it takes an hour for the entire Moon to enter
the Earth’s shadow and four hours for the edge of the Moon to cross
the shadow. How many times larger than the Moon’s diameter is the
Earth’s diameter?
1 Actually, the diameter of the Earth’s shadow at the distance of
the Moon is slightly less than
the diameter of the Earth; however, this leads to only a small
error.
materials. Thus, an object’s “natural place” was close to the
Earth, to which it moved. Similarly, fire rises towards its
“natural place,” which is away from the “earthy materials,” which
is up.
In addition, he taught that continuous movement required continuous
contact between the mover and the moving object to perpetuate the
motion. And, as discussed later in this chapter, he argued that
planets would move in circular orbits. Such ideas came from
philosophy and did not need to be tested because logic said it was
right. These ideas were the mainstay of scientific knowledge until
the seventeenth century.
Aristarchus of Samos When Greek philosophical thinking was coupled
with their highly developed mathematical skills, the Greeks
advanced so rapidly in astronomy from 600 B.C.E. to 100 B.C.E. that
their era was without question one of the shining examples of
scientific discovery prior to the year 1500 C.E. Progress in
understanding the structure of our solar system came especially
quickly with the emergence of the Alexandrian school of Greek
astronomers, particularly the work of Aristarchus of Samos (c. 300
B.C.E.), who combined careful observations and sharp reasoning to
draw inferences about the relative sizes and distances of the
Earth, Moon, and Sun.
Aristarchus demonstrated that the Sun was many times farther from
the Earth than the Moon by using a reasoning process illustrated in
Figure 5-2. You can see that if the Sun were quite close to the
Earth–Moon system, then the time interval from first quarter Moon
to third quarter Moon would be longer than the time interval from
third quarter to first quarter. Aristarchus observed that
Sun
M1
Moon at first quarter
Moon at third quarter
FIGURE 5-2. Aristarchus’s method to show that the Sun is farther
from the Earth than the Moon. If the Sun were very near the
Earth-Moon system, the times between quarter phases would be a lot
different. However, if the Sun were much more distant than the
Moon, the lines S-M1 and S-M3 would be nearly parallel, and the
time interval between first and third quarters would be nearly
equal to the time interval between third and first quarters.
Earth's shadow
ABC
FIGURE 5-3. The principle of Aristarchus’s measurement of the
Moon’s diameter compared to that of Earth. The time it takes the
Moon to enter the Earth’s shadow (between points A and B) is
proportional to the Moon’s diameter. The time the Moon takes to
cross the Earth’s shadow, between A and C, is proportional to the
Earth’s shadow diameter, which is nearly that of the Earth.
Part 1 • Discovering the Science of Astronomy5-4
Finally, Aristarchus knew that during a total eclipse of the Sun,
the Moon was just able to cover the Sun, and therefore, the angular
sizes of the Moon and Sun in the sky were about the same. He could
then reason that the actual sizes of the Moon and Sun were
proportional to their distances, and in this way he could estimate
the diameter of the Sun (Figure 5-4). Unfortunately, although his
estimate of the distance to the Moon was quite good, his estimate
of the distance to the Sun was about 10 times too small. For this
reason, his estimate of the Sun’s diameter was also 10 times too
small. Nevertheless, he was able to show that the distance to the
Sun was considerably greater than that to the Moon, and that the
Sun’s diameter was much greater than the Earth’s.
Inquiry 5-3 Aristarchus was the first to propose that the Earth
goes around the Sun, rather than vice versa. Suggest one factor
that may have led him to this conclusion.
Eratosthenes Another classic experiment of antiquity was the
determination of the Earth’s circumference by Eratosthenes (c. 200
B.C.E.). The conclusions of Aristarchus concerning the relative
sizes and distances of the Earth, Moon, and Sun were all in terms
of the then-unknown size of the Earth. Their actual sizes could not
be known until the Earth’s size was known.
Eratosthenes had heard that at Syene, near the modern Aswan in
Egypt, there was a deep well, and that on a certain day of the year
the Sun stood directly overhead so that its reflection could be
seen in the water in the bottom of the well. Eratosthenes was also
able to observe that on that same day of the year in Alexandria,
where he lived, the Sun was not directly overhead but was 7° south
of the zenith. He determined this angle with
= 360
angle at C .
Inquiry 5-4 Assuming that the angle at C is 7° and that the
distance of Alexandria from Syene is 5000 stadia (the units used by
the ancient Greeks; stadia is the plural of stadium), what is the
diameter of the Earth in stadia? Although the exact length of a
stadium is unknown, it is thought to be about 0.16 of a kilometer
(one-tenth of a mile). Using this value, compute the radius of the
Earth.
Inquiry 5-5 What does Eratosthenes’ experiment assume about the
shape of the Earth?
Inquiry 5-6 If the Earth were flat, what would be the value of the
angle at C?
Small, nearby object Large, distant object
FIGURE 5-4. Two objects with the same angular sizes have diameters
that are proportional to their distances.
FIGURE 5-5. Eratosthenes’s measurement of the diameter of the
Earth. The difference in altitude of the Sun is proportional to the
distance between the two points, which is known. This allows the
length of one degree on the Earth’s surface to be determined.
7
To zen
5-5Chapter 5 • The Historical Quest to Model the Solar System
Hipparchus Perhaps the greatest of all ancient astronomers was
Hipparchus (c. 150 B.C.E.). Many of the conclusions he drew were so
sophisticated that it takes some knowledge of astronomy to
appreciate how great his contributions were. He built an
observatory, constructed the best astronomical instruments up to
that time, and established a program of careful and systematic
observations that resulted in the compilation of a great star
catalog with 850 entries. His catalog used a celestial coordinate
system similar to our modern one for cataloging the sky. It was
Hipparchus who originated a system, which is still in use today in
modified form, for estimating the brightness of stars. In addition,
he used the older Babylonian observations and deduced Earth’s
precession (Section 4.6), which is so slow that it takes almost
26,000 years for it to complete one cycle. Finally, he greatly
developed trigonometry, which was, and still is, a useful tool for
astronomy.
Other Deductions of the Greek Astronomers: The Distances of the
Planets The Greeks estimated the relative distances of the planets
from Earth using principles still in use today for determining
distances to astronomical objects. They reasoned that the more
distant a planet was, the more slowly it would appear to move
across the sky. The effect is similar to what happens when we
compare the apparent motion of a high-flying airplane with that of
one that is flying low. The distant airplane appears to move slowly
across the sky, whereas the low-flying one is seen for only a short
time and then is gone. In the same way, the Greeks could put most
of the naked-eye planets in order of their distance from Earth by
assuming that increasing distances corresponded to slower apparent
motions. The argument fails with Mercury and Venus, however,
because it places Mercury closer to Earth.
Inquiry 5-7 What assumption results in Mercury being placed closer
to Earth than Venus?
We have another, independent determination of relative planetary
distances from their brightnesses.
We use an analogy: when you are driving at night on a two-lane road
and wish to pass the car in front of you, you pass only if the
headlights of an oncoming car are faint. When you do this, you are
making an implicit assumption: all car headlights have about the
same intrinsic brightness, with their apparent brightness depending
on the distance. Similarly, if we assume that all planets have the
same intrinsic brightness, then their apparent brightness as seen
from Earth would depend on their distances from us. But, all the
planets do not have the same intrinsic brightness, because their
differences in size, distance from the Sun, surfaces, and
atmospheric properties affect the amount of light they reflect in
our direction. However, even allowing for these uncertainties, it
is still possible to use this principle to rank the planets
approximately in order of distance from the Earth.
The Apparent Motions of the Planets in Relation to the Stars Three
additional observations of planetary motion were important in
determining the details of the models the Greeks developed for the
Solar System. We now summarize these observations, which were
discussed in detail in the previous chapter:
Because the planets are considerably closer to us than the fixed
stars, they appear to move against the starry background.
Observations of Mars, Jupiter, and Saturn showed them to move
generally eastward on the celestial sphere. In other words, stars
move with the celestial sphere whereas the Sun, Moon, and planets
move on the celestial sphere.
Occasionally, however, as discussed in Chapter 4, a planet’s motion
becomes retrograde in which it changes from eastward to westward
for up to several months before it slows down and again reverses
its direction, resuming its normal easterly motion (see Figure
4-28).
Venus and Mercury are never more than 48° and 28°, respectively,
from the Sun.
The Geocentric Model of the Solar System Where should the center of
the system be? There were really only two obvious candidates — the
Earth and the Sun. This question was considered carefully by Greek
philosophers, and the fact that ultimately they reached an
incorrect conclusion provides an interesting example of why the
scientific method is not the simple turn-the-crank-and-the-answers-
fall-out process that some sources describe it to be.
Part 1 • Discovering the Science of Astronomy5-6
If the Sun is at the center of the Solar System, then Earth moves
around it in space. Such a hypothesis provides a testable
prediction. As shown in Figure 5-6, some of the stars ought to
shift their apparent positions in the sky as the Earth moves from
one side of its orbit to the other. Such parallax effects, as they
are called, were looked for by many Greek observers, including
Hipparchus, but were never found. The Greeks therefore concluded
that the Earth was stationary in space.
Aristarchus, however, apparently supported the hypothesis that the
Earth orbits the Sun, if surviving works of Archimedes and Plutarch
are correct. Unfortunately, the work in which he put forth his
hypothesis is lost, and apparently no other Greek astronomers held
to this opinion. The model of a geocentric (Earth-centered) system
easily won out over the heliocentric (Sun-centered) one.
Inquiry 5-8 What hypothesis might explain why stellar parallaxes
were not observed, even though the Earth does in fact orbit the
Sun?
The Hypothesis of Circular Motion A second feature of the Greek
models of the Solar System provides another excellent example of
the way in which scientific models can go astray — inflexible
assumptions. Ever since the time of Pythagoras (c. 570–500 b.c.e.)
and Aristotle, the
circle and the sphere had been considered to be the most perfect
geometrical figures. Such perfection was expected in the natural
world. (Even today, symmetry and simplicity in scientific thought
are important concepts.) Thus, the great astronomer Hipparchus
assumed the widely accepted idea that celestial objects, being
perfect, could move only in circular orbits. In addition, this
circular motion had to be uniform: an object moved the same angular
distance in its orbit each day. However, the observations of
retrograde motion made it impossible to model the Solar System with
a simple geocentric system. Hipparchus cleverly got around this
problem by extending the idea of uniform motion to include motion
that was a combination of circular motions. In this way, as we will
now see, it was eventually possible to model the retrograde motion
of the planets using uniform circular motion.
The idea of combining circles geometrically is due to Apollonius of
Perga (c. 265–190 b.c.e.), but Hipparchus was the first to apply
the idea to actual celestial objects when he proposed a hypothesis
to explain the motion of the Sun and Moon. However, he did not have
enough data to apply it to the planets; this final step was carried
out by the astronomer Ptolemy (whom we discuss shortly).
The explanation of retrograde motion by means of combinations of
circular motion is illustrated in Figure 5-7a. The planet moves
around a small circle called an epicycle, and the center of the
epicycle moves around on a larger circle called the deferent.
Because one can adjust the relative sizes of the epicycle and
deferent, and the speeds with which motion takes place on each, it
is not difficult for the model to produce complicated apparent
motions. In particular, if the planet moves around the epicycle
faster than the epicycle moves around the deferent, retrograde
motion will be observed at certain points along the orbit, as shown
in Figure 5-7b.
An explanation for the observation that Mercury and Venus were
never far from the Sun (28° for Mercury and 48° for Venus) became
possible by adding one more feature to the epicycle-deferent model.
If the center of the epicycle of Venus, for example, is attached
firmly to the line joining the Earth and Sun, then Venus will
always remain near the Sun, as shown in Figure 5-8. In a similar
fashion, Mercury’s epicycle is attached to the Earth- Sun line.
Now, as the Sun moves along its deferent at a rate of about 1° per
day, it carries Mercury and
Orbit o
s
FIGURE 5-6. The prediction of stellar parallax in a
heliocentric universe. The apparent position of a star with respect
to the background stars would appear to change if the Earth went
around the Sun. (Not to scale.)
Venus along with it and the model explains the available
observations.
Inquiry 5-9 What phases of Venus are possible for the model shown
in Figure 5-8? (Hint: Make an enlarged drawing and shade the parts
of Venus that are not illuminated by the Sun.)
Ptolemy The last of the great ancient astronomers was Claudius
Ptolemy (c. 150 C.E.). In fact, the geocentric system that was
previously discussed was passed down bearing his name — the
Ptolemaic system. Ptolemy completed the explanation of planetary
motion in terms of combinations of circles and added a number of
complex refinements to Hipparchus’ system to improve the model’s
agreement with observations.
Ptolemy’s aim was to produce a model that correctly predicted
observations, which remains a key goal of science today. In that
sense, he succeeded admirably. The Greeks believed that a true
knowledge of reality was confined to the gods alone, and the best
that humans could do was to produce descriptions of the observed
world that would correctly predict the results of observation and
experiment. Leaving aside the gods, these ideas are close to the
spirit of modern science, in which even everyday concepts such as
force and mass, which modern physicists think they understand quite
well, have meaning only in reference to models and the measuring
processes that are appropriate to them.
Models: Revisited Models are powerful tools not only to interpret
observations, but to suggest possible new observations. In fact, an
incorrect model can often lead one to ask the wrong questions, and
thus can delay understanding. For example, Kapteyn’s model for the
universe (described in Section 3.5) was wrong and delayed our
understanding of Earth’s position in the Galaxy. In astronomy,
models necessarily become more and more uncertain as the objects
they represent become more distant, because we have less and less
data on which to base the model. Despite this, as we will see
throughout this book, models play a central role in our quest to
understand the universe.
Inquiry 5-10 What are some models, in the sense just described,
that are used in everyday life? (You may want to think about such
fields as politics, economics, education, philosophy, psychology,
etc.)
5.2 Astronomy during the Middle Ages No significant
astronomical advances were made for the next 13 centuries. Greek
civilization declined and the Greeks became subservient to the
Romans. (This had already taken place by the time of Ptolemy.) If
the Greeks were scientists, the
FIGURE 5-7. A geocentric explanation of retrograde motion. (a) The
planet moves on an epicycle, which moves about the Earth on a
deferent. (b) The motion resulting from a planet moving about an
epicycle as it moves around the deferent.
FIGURE 5-8. An explanation of the motion of Venus, using epicycles.
According to this model, Venus is always closer to Earth than the
Sun is, and the angular separation of Venus and the Sun is never
large.
Deferent Epicycle
Earth Planet
Deferent Epicycle
Part 1 • Discovering the Science of Astronomy5-8
Romans were engineers, doing little basic science. Progress in
science came to a virtual standstill in the West.
A different story emerged in the Middle East. The astronomical
knowledge that originated in Greece reached the Arab world, and
scientists and scholars there continued the pursuit of knowledge.
There was a flowering of literature, art, and science throughout
the Middle East. One of the most important events was the
preservation of much of the scientific knowledge of the ancient
world, at least in translation. Ptolemy’s great work, for example,
survived as a book called the Almagest, Arabic for “the greatest.”
In this form the Arabs eventually transmitted Ptolemy’s writings
back to the West.
A considerable amount of original work was done in the Islamic
world. Much of it was directed toward practical matters, such as
navigation. Beautiful and intricate instruments such as the
astrolabe (Figure 5-9) were perfected. Much theoretical work was
accomplished, particularly in mathematics, aided by the Arabic
system of numbers (that had come from India), as well as by
perfection of the methods of algebra (the word algebra is Arabic).
Imagine doing your income taxes using Roman numerals!
At the same time in China, an indigenous tradition of astronomy
made important progress during long periods of political stability.
Astronomy
was always closely connected with the state, and changes in dynasty
were frequently accompanied by calendar reforms, providing secure
employment for court astronomers. Early on, the Chinese developed
the view that space was infinite in extent and that the stars
floated independently in it. They were tireless and systematic
observers of heavenly events; often our only records of novae and
comets are Chinese. They invented advanced instrumentation for
observing the sky and measuring time, including elaborate
water-driven clocks. In many ways their science was more advanced
than what the Greeks had accomplished. The influence of Chinese
science on the West has been shown to have been much greater than
has been generally recognized.
In Europe, by contrast, intellectual decadence was widespread. The
principles of Greek science, received from the Arabs, were adopted
as dogma by the increasingly powerful Roman Catholic Church. It was
argued that there was only a finite amount of knowledge, and that
the Greeks had discovered it all! Discovery was no longer
necessary; humans had only to memorize what the Greeks had found to
be true. The Ptolemaic picture became frozen into a rigid image of
the universe with no room for independent thought and inquiry that
would challenge this view of the universe.
5.3 The Heliocentric Hypothesis Once the renaissance in Europe
began in the 14th century, changes began to occur in most fields of
human endeavor: architecture, art, literature, philosophy,
religion, and science. We will now look at the contribution of
three scientists whose work challenged the status quo and helped
move Western thought to a Sun-centered view.
Nicolaus Copernicus During the Middle Ages, tinkering with the
Ptolemaic model had continued, but in an uncreative way — for
example, by adding more epicycles. In fact, this model had become
so complicated that King Alfonso of Castile is said to have
remarked, when having the contemporary version of the Ptolemaic
hypothesis explained to him, that if he had been around when the
world was created he could have taught the Creator a thing or
two.
FIGURE 5-9. A Persian astrolabe. The small leaves mark the
positions of bright stars. By hanging the astrolabe by the ring and
sighting along the movable sight, we can measure the altitude of a
star.
5-9Chapter 5 • The Historical Quest to Model the Solar System
The human mind cannot be contained forever. By the early sixteenth
century the Renaissance was shaking the antique castles of Western
intellectualism. Nicolaus Copernicus (1473– 1543), a Polish prelate
with a strong mathematical bent as well as a new vision of the
heavens (Figure 5-10), found himself in this changing atmosphere.
As a student in Italy, Copernicus read of Aristarchus’s
heliocentric hypothesis. Wherever he got the idea of placing the
Sun at the center of the universe (Figure 5-11), there is no doubt
that he felt this was the proper place, for as he himself says,
“For who would place this lamp of a very beautiful temple in
another or better place than this wherefrom it can illuminate
everything at the same time?” An important benefit of the
heliocentric model is that it allowed the sidereal orbital periods
and relative distances of the planets to be determined from the
observed synodic periods of the planets (as discussed at the end of
Section 4.8).
Moreover, in a heliocentric model, the apparent retrograde motion
of a planet like Mars (or any planet more distant from the Sun than
the Earth) is a natural consequence of the relative motions of the
Earth and Mars. Figure 5-12 shows how this comes about. At some
time Mars will appear in direction 1. As each planet orbits the
Sun, Mars’s line-of-sight position in relation to the
background
stars will move toward the east, as indicated by lines 1 through 4.
However, between times 4 and 6 Mars will appear to move backward,
in a retrograde direction, toward the west. At point 6 Mars will
have resumed its normal easterly track. As you can see, Mars shows
retrograde motion whenever the Earth laps it in its orbit. The
greater orbital speed of the Earth results in the apparent backward
motion of Mars. One sees a similar effect when a fast automobile
passes a slower one on the highway. Viewed from the faster car, the
slower one appears to move backward when seen against the distant
landscape.
FIGURE 5-10. Nicolaus Copernicus (1473–1543) proposed the daring
idea that the Sun, not the Earth, was at the center of the
universe.
FIGURE 5-11. The Copernican model of the Solar System.
FIGURE 5-12. Retrograde motion is a natural consequence of the
heliocentric hypothesis and occurs as Earth approaches and then
passes the planet. The loop appears because the orbital planes of
the Earth and the planet are not identical.
Earth
Mars
Jupiter
Saturn
Mercury
Venus
Sun
Part 1 • Discovering the Science of Astronomy5-10
Inquiry 5-11 What would be the relative positions of the Earth,
Sun, and Mars for the retrograde motion to occur?
Despite its natural and simple explanation of retrograde motion and
elegant positioning of the Sun in the center of the Solar System,
many objections were raised against the Copernican hypothesis. For
example, some people contended that if the Earth moved it would
leave the Moon behind, an argument that was hard to answer because
the concept of gravity had not yet been developed. More seriously,
Copernicus’s hypothesis in this simple form could not explain the
observations as accurately as Ptolemy’s hypothesis. For this reason
it was necessary to introduce a large number of small epicycles
into Copernicus’s heliocentric model to explain numerous small
deviations of the observations from the idealized circular motion.
However, Copernicus was opposed to these complicated refinements,
feeling that they detracted from the perfection of uniform circular
motion and were unworthy of celestial objects. The end result was
that he actually needed more epicycles than Ptolemy to get
equivalent accuracy, and this marred the aesthetic simplicity of
his basic heliocentric hypothesis, which used circular
orbits.
Inquiry 5-12 Employing the principle of Occam’s razor (the simplest
model is preferred), from the point of view of astronomers of the
time, which model of the Solar System would have been preferable,
Ptolemy’s or Copernicus’s?
Unquestionably, Copernicus was aware of the difficulties with his
model. For example, the model predicted that parallax should be
observed for nearby stars, which it was not. However, he realized
that the lack of parallax meant the stars were far away and the
parallax was too small to be observed with the instruments then
available. In addition, he was a rather timid man, and it is
possibly for these reasons that publication of his hypotheses was
delayed until he was literally on his deathbed. His work appeared
in the year 1543 titled De Revolutionibus Orbium Cœlestium, which
means On the Revolutions of the Heavenly Spheres. The book
contained an unauthorized preface by the
Lutheran cleric Andreas Osiander stating that the book expounded a
mathematical model and was not to be construed as a representation
of reality. It is doubtful that Copernicus would have sanctioned
its inclusion had he known of it.
Copernicus hypothesized a heliocentric universe; he did not prove
it. Validation of his ideas was to require input from additional
scientists over the following decades and centuries.
Tycho Brahe In 1546, a Danish nobleman was born who was to become
the first great observational astronomer of the modern era — Tycho
Brahe (pronounced Tee-ko Bra-he) (Figure 5-13). Tycho became
interested in astronomy as a teenager when he observed a predicted
eclipse of the Sun. Later, in 1572, he observed a supernova, an
exploding star, that was so bright it could be observed in daytime.
He favored the heliocentric hypothesis, but the lack of supporting
observations caused him to rethink his position. Through careful
observations, Tycho showed that the daily rotation of the Earth
caused no parallactic shifts in the supernova’s position — in fact,
the star did not move at all during the many months it was visible.
He therefore concluded that it must be well beyond the Moon, and
probably as distant as the stars. This caused instant problems for
the older understanding of the universe, which had assumed that the
heavens were unchangeable. Five years later, Tycho proved that the
comet of 1577 was also more distant than the Moon. Because comets
had been believed to be “exhalations of the Earth” this posed
additional problems for the older ideas.
The fortunate combination of Tycho’s noble birth and his
astronomical talents soon earned him the finest observatory in the
Western world up to that time. It was financed from the coffers of
King Frederick II of Denmark and was located on an island off the
Danish coast. Since optical instruments had not yet been invented,
the instruments were sighting devices similar to the quadrant,
protractor and cross-staff, but larger in size and constructed with
the greatest possible precision. Tycho’s principal instrument was
the great mural quadrant (Figure 5-14). By sighting along the
movable pointers in a manner not unlike sighting a gun, he could
aim the pointers at two objects and read the angle between them
with great accuracy for the time (about 0.5 minute of arc).
5-11Chapter 5 • The Historical Quest to Model the Solar
System
Tycho distinguished himself by the great care he brought to his
observations. This included making multiple observations and
averaging them together to minimize random uncertainties. So
painstaking was his work that his observations had unprecedented
accuracy and precision, not to be improved upon until the invention
of the telescope. Another valuable aspect of his observations was
that he observed continuously and systematically over many years,
as Hipparchus had done. As a result, he amassed a large body of
data of consistently high accuracy that was suitable for detailed
analysis. Why an aristocrat such as Tycho could become as obsessed
with astronomy as he was is something of a mystery, and it is
miraculous that his obsession led him to produce such excellent
astronomical data. Another fortunate historical accident is that
this body of work eventually fell into the hands of Johannes Kepler
(1571–1630), who, as it turned out, was uniquely equipped to put it
to good use.
Tycho developed an alternative Solar System model with the Earth at
the center, the Sun orbiting the Earth, and the other planets
orbiting the Sun (Figure 5-15). This Tychonic system had the
advantages of reproducing the observed motions of the planets while
predicting no stellar parallax. It
became the preferred model in the late 16th and 17th centuries and
was, in fact, accepted by the Roman Catholic Church until some
critical observations, discussed later, were made.
Johannes Kepler Although Tycho was a great observer, he realized
that he was not a strong mathematician, and he longed for a
collaborator who could properly analyze and interpret his high
quality observations. His accumulated observations on the positions
of Mars posed the greatest difficulties for theoretical
interpretation because Mars deviated from its predicted position
more than any other planet. Having moved to Prague toward the end
of his life, Tycho acquired a young assistant who, by his
brilliance in mathematics, transcended his lower- class origins.
Johannes Kepler (Figure 5-16) had sufficient genius to overthrow
the sterile hypotheses of the geocentric universe and perfect
circular motion and to substitute a new and truer description of
the Solar System.
In his book The Sleepwalkers, Arthur Koestler describes the
contrast between Tycho and Kepler. Tycho was an aristocrat by birth
and arrogant by
FIGURE 5-13. Tycho Brahe (1546–1601) was the greatest naked- eye
observer in the history of astronomy.
FIGURE 5-14. Tycho’s great mural quadrant, his principal instrument
for measuring star positions.
Part 1 • Discovering the Science of Astronomy5-12
nature, accustomed to power and privilege. He ran his observatory
like a court, arriving for the night in full formal dress and
arrogantly ordering his assistants about. Always sure of himself,
in his college days he had lost his nose in a duel over a point of
mathematics, and ever afterward he wore a false nose made of silver
or bronze. He had a dwarf servant named Jepp, who followed his
master about like a pet and received scraps of food at the supper
table. Kepler, by contrast, had no advantages of birth and reached
a position of eminence by virtue of brilliance and sheer tenacity.
He had little in common with Tycho, other than irascibility and an
interest in astronomy. During much of his life, he had to earn his
living by casting horoscopes, and in fact he cast his own horoscope
every day. He was a compulsive individual who kept meticulous notes
on everything he thought and did. A hypochondriac as well, Kepler
recorded an hour-by-hour chronicle of his physical maladies. But
because his diary also included his scientific efforts, we have
been left a detailed account of the paths through which he wandered
in making his momentous discoveries.
Kepler took Tycho’s observations of Mars (literally took them,
because Tycho’s heirs had other plans for the data!) and set out to
find a geometric curve that would represent its motion accurately.
(Recent evidence indicates that Tycho may have been poisoned. Was
this done by Kepler to obtain the data?) The calculations were
extremely tedious, and he made many mistakes. Without calculators,
every calculation had to be done by long multiplication or
division. Yet, something in Kepler’s character kept him working
persistently at the problem. Time after time he rejected solutions
that had taken him months, even years, to work out, because they
failed to agree with the observations as accurately as he knew they
should. At last, after a total of eight years, he boldly rejected
the hallowed ancient idea that planetary motion must take
place
on circular paths, thus ending two millennia of tradition. He
describes in his diaries the fear and trembling he suffered in his
mind when he took this step, which you will learn about next.
5.4 Kepler’s Laws of Planetary Motion Kepler’s laws of
planetary motion are not laws in the sense that we use the word
today, because there was at that time no conception of the physical
forces that caused these motions. His laws would be better
described as empirical descriptions of planetary motions, meaning
they were derived solely on the basis of observation with no
theoretical underpinning. However, for the first time they gave a
description that was as accurate as the best available data
allowed. Although the word “law” is a misnomer, it has been in use
for many years and we are stuck with it.
Kepler found that Mars moved around the Sun on a mathematical curve
called an ellipse. Figure 5-17 shows a simple way to draw this
curve. The two points located by pins are called the foci of the
ellipse (foci is the plural of focus). Ellipses
FIGURE 5-16. Johannes Kepler (1571–1630) refined Copernicus’s
heliocentric hypothesis with the laws of elliptical motion.
Figure 5-15. The Tychonic model of the solar system, in which the
Earth is at the center, the Sun orbits it, and the other planets
orbit the Sun.
5-13Chapter 5 • The Historical Quest to Model the Solar
System
are characterized by the length of their longest dimension (called
the major axis — B in the figure) and their degree of
non-circularity. If the two foci coincide, we have a circle. The
further apart the foci, the greater the flattening, or
eccentricity, of the ellipse.
Kepler’s first law can be stated in this way: The orbits of the
planets are ellipses, with the Sun at one focus.
Figure 5-18 illustrates Kepler’s first law. The flattening of the
ellipse is much exaggerated — no planet has an orbit this
elliptical. In fact, if one were to observe these orbits from
outside the Solar System, it would be difficult using the naked eye
to distinguish most of them from circles. This is the principal
reason why it took so long to discover the true shapes of the
planetary orbits. As shown, the second focus is empty. Eccentric
ellipses are also found; comets are examples of objects that move
in highly eccentric ellipses.
The distance of a planet from the Sun varies as the planet moves.
For example, Earth is almost 5 million kilometers closer to the Sun
in January than in July. Kepler found that as Mars’s distance from
the Sun varied, its orbital speed also varied, being greatest when
the planet was closest to the Sun and least when it was farthest
away. He found even more: there is a definite relationship between
a planet’s distance from the Sun and its speed in its orbit.
Figure 5-19 indicates four positions on an elliptical orbit. During
the interval of time between points A and B, a line from the planet
to the Sun sweeps out the long, skinny area shaded on the left.
Between the equal time interval from C to D, the line from the
planet to the Sun sweeps out the fat area on the right. Kepler
found that this area is equal in size to the area swept out between
times A and B. In other words, because the time intervals from A to
B and C to D are equal, and because the distance from C to D is
greater than that from A to B, the planet must be moving faster
from C to D than from A to B. For example, the motion of the Earth
is more rapid in January than in July.
Kepler’s second law of planetary motion can be formulated as
follows:
The line joining a planet and the Sun sweeps out equal areas in
equal amounts of time.
Kepler’s third law was discovered much later than the first two. It
appears, almost as an afterthought, in a rambling, mystical work of
Kepler’s titled Mysterium Cosmographicum, or Cosmic Mystery. The
emphasis of this work lay in
FIGURE 5-17. Drawing an ellipse with two pins and a loop of string.
The pins mark the position of the two foci of the ellipse. The
eccentricity of an ellipse is defined to be e = F/B, where F is the
distance between the two foci and B is the longest axis (the major
axis) of the ellipse. It characterizes how far the ellipse deviates
from being a perfect circle.
FIGURE 5-19. Kepler’s second law: If the planet takes the same
amount of time to go from A to B as it does to go from C to D, then
the two shaded areas are equal.
FIGURE 5-18. Kepler’s first law: Planets move in ellipses with the
Sun at one focus.
Drawn ellipse
Part 1 • Discovering the Science of Astronomy5-14
mystical speculation on the cosmos rather than on the mathematical
relationship expressed in the third law.
Kepler’s third law is a relationship between the average distance
of a planet from the Sun (which equals half the length of the major
axis, or semi- major axis) and the planet’s sidereal period, the
length of time it takes for the planet to orbit once around the
Sun. The third law is most easily expressed mathematically. It
states that if P is the orbital period of a planet measured in
years and A its average distance from the Sun measured in
astronomical units, then
The period squared equals the average distance cubed.
Expressed symbolically:
P 2 = A 3.
This equation clearly works for planet Earth. Earth’s orbital
period is 1 year and its average distance from the Sun is 1 AU, so
P = 1 and A = 1, and 12 = 13.
A = P 23 .
P = 9 9 = 9 × 3= 27 years.
A = P 23 = 823 = 643 = 4 AU
because 4 × 4 × 4 = 64. Problems using Kepler’s third law are more
easily solved with an electronic calculator.
Inquiry 5-13 The average distance of Mars from the Sun is 1.52 AU.
To the nearest tenth of a year, what is its orbital period?
Inquiry 5-14 Uranus’s orbital period is 84 years. To the nearest
tenth of an AU, what is its average distance from the Sun? (Hint:
20 × 20 × 20 = 8000.)
The sidereal period is not a directly observable quantity because
we are on a moving Earth. What is observable, however, is the
planet’s synodic period, which is the time interval between
successive occurrences of a given planetary configuration
(described in Section 4.8). For example, the synodic period is the
time interval between, say, two successive oppositions or
conjunctions.
To clarify the difference between synodic and sidereal periods, we
draw an analogy using two runners on a track. The sidereal period
is the time from the starting line back to the starting line. The
synodic period is the time for one runner to lap the other one —
that is, to move ahead of and then catch-up to the other runner
again. Because both runners are moving, one runner cannot directly
measure the sidereal period of the other. However, each runner
could easily determine the time it takes to lap the other runner —
the synodic period. If you then know your own sidereal period, you
can compute the other runner’s sidereal period. (The mathematics
and some examples are given in Appendix A6.)
In summary, we observe an object’s synodic period, compute the
sidereal period, plug it into Kepler’s third law, and determine the
planet’s semi- major axis. Figure 5-20 illustrates the procedure
and the logic involved.
Kepler’s three laws, as described here, are exact only in the
idealized case of two isolated objects. In the real Solar System,
the large number of objects that interact with one another causes
deviations from the ideal. Nevertheless, the laws are accurate
enough to describe the motions of the planets and agree with the
best naked-eye observations, which happened to be Tycho’s. Had
Tycho’s observations been somewhat less accurate, or had Kepler
been less exacting in his criteria for agreement between hypothesis
and observation, Kepler’s three laws might not have been discovered
until much later in history.
Kepler’s three laws overthrew the earlier Greek ideas; no longer
did we believe that the planets moved with uniform speeds in
circular orbits. These laws are general in that they apply to any
two objects circling each other under the influence of
5-15Chapter 5 • The Historical Quest to Model the Solar
System
gravity. They may be applied to natural or artificial satellites
orbiting planets, stars orbiting stars, stars orbiting galaxies, or
even galaxies orbiting galaxies. Thus, we will return to Kepler’s
laws at many places throughout our study of astronomy.
5.5 The Search for Underlying Laws Kepler’s works are purely
empirical, meaning they were derived solely on the basis of
observation with no theoretical underpinning. Nevertheless,
Galileo’s experimental work on the motion of objects and Newton’s
beautiful and general theoretical development of his laws of motion
show the validity of Kepler’s work.
Galileo Galilei Galileo Galilei (1564–1642), shown in the chapter
opening illustration, was a contemporary of Kepler’s, was an early
convert to the heliocentric hypothesis, and argued vigorously for
it. In his later years this brought him into conflict with the
Inquisition, and he was forced to recant his beliefs and live under
a form of house arrest — a pitiful, partially blind old man,
fearful of the instruments of torture that apparently had been
shown to him.
Kepler’s laws, successful as they were in predicting the positions
of the planets in the sky,
were no proof that the Earth moves. Skeptics could reasonably claim
that the three laws were merely mathematical tricks that happened
to give accurate predictions and not in any way the truth about the
physical nature of the universe.
Galileo clung to the hypothesis of circular motion for heavenly
objects while conducting experiments in dynamics that eventually
overthrew it. Remember, the ideas of Aristotle had been the
accepted ones for centuries. And, they were what the Catholic
Church, as the arbiter of authority, said was truth.
Galileo, in a revolutionary move, decided to let observation, not
human reason, be the ultimate arbiter of physical reality. He
performed experiments to decide whether a particular hypothesis was
true. Galileo’s insistence on making and using observations to find
scientific “truth” led the way to the modern approach to science
with the use of inductive logic. For example, he investigated the
motions of objects under various conditions by rolling balls down
inclined planes, thus showing that a state of motion was as natural
as a state of rest, and that to change either state required an
outside influence. In another famous case, it had been claimed by
believers in Aristotelian physics that an object that was twice as
heavy as a second one would fall twice as fast. Perhaps by actual
demonstration, or maybe only with thought experiments, Galileo
showed that such objects accelerated at nearly the same rate and
hypothesized that only the resistance of the air kept the rates
from being identical. The truth of Galileo’s position was
demonstrated dramatically when the Apollo 15 astronauts dropped a
feather and a hammer together in the vacuum of the Moon. The two
objects reached the lunar surface simultaneously. You, too, can
perform Galileo’s experiment by simultaneously dropping, say, a
paper clip and a heavy shoe.
In performing his experiments, Galileo formulated the concept of
inertia, which is an object’s tendency to resist a change in its
motion. The amount of inertia is measured by what we call mass,
which is, loosely, a measure of the amount of matter in an object.
The greater the mass, the more inertia, and the harder it is to set
an object into motion. Furthermore, the greater the mass, the more
inertia, and the harder it is to change an object’s motion.
Galileo’s greatest fame comes from his discoveries in astronomy,
even though his
Observations
Which, along with
Results in the
Figure 5-20. The procedure for finding a planet’s semi-major axis
given an observation of its synodic period.
Part 1 • Discovering the Science of Astronomy5-16
contributions to physics were probably equally important. Upon
learning of the invention of the telescope, he constructed one of
his own and turned it to the heavens. His most important
astronomical discoveries can be divided into three groups: those
that argued most strongly for the heliocentric model (numbers 1 and
2 below), those that showed the universe is not perfect and
unchangeable (numbers 3 and 4), and other observations (numbers 5
and 6).
1. He found that the planets show phases, as the Moon does. This
was particularly important in the case of Venus, which exhibits all
phases. In the Ptolemaic model, Venus could only show new and
crescent phases, because it would always be closer to Earth than is
the Sun and show only a small fraction of the illuminated side of
the planet (Figure 5-21a). Only if Venus were sometimes farther
from the Earth than the Sun could a larger fraction of the
illuminated side be seen from Earth and thus exhibit a gibbous
phase (Figure 5-21b). This discovery showed that the Ptolemaic
picture of the universe must be incorrect. An observation such as
this is an example of a critical observation, which is one that, by
itself, is sufficient to favor one model over another. In this
case, however, preconceived ideas won out over observation, and the
true meaning of this observation was discounted.
2. He discovered the four brightest moons of Jupiter by following
the variations in their positions from night to night and showing
that they always remained close to the moving planet.
3. He examined the Moon in detail, and discussed and named many of
its surface features — its mountains, the dark areas called maria
(“seas”), craters, and the like.
4. He studied sunspots by observing the Sun at sunrise or sunset
when the Sun is not too bright and not so harmful to the eyes. By
following sunspots across the visible disk of the Sun, he showed
that the Sun spins. Together with his observations of the Moon,
this discovery struck a blow to the older assumptions of astronomy
that had maintained that the heavenly objects were perfect, without
blemishes. The existence of sunspots, mountains, and craters showed
that these bodies were neither perfect nor without blemishes.
5. He found that the diffuse band of light across the sky, known as
the Via Lactea or the Milky Way, actually consists of myriad stars
too faint to be seen by the naked eye. This discovery was the first
step toward a modern view of the nature of our galaxy.
6. While observing with a telescope, he discovered that the planets
had a visible disk, whereas the stars remained infinitesimal points
even at the highest magnification.
Inquiry 5-15 What might be the reason for Galileo’s sixth
observation?
Inquiry 5-16 One of the arguments against Copernicus’s hypothesis
was that the Earth could not be moving because it would leave the
Moon behind. How does Galileo’s second observation help disprove
that contention?
Isaac Newton As important as the contributions of Galileo and
Kepler were in altering long-held ideas about astronomy and
physics, they did not create a single comprehensive system that
unified the physics of the heavens with the physics of the Earth.
Such
(b)
(a)
FIGURE 5-21. (a) Venus’s phases under the Ptolemaic model, as
viewed from Earth. The path shown is Venus’s epicycle; see Figure
5-8 for another view. Note that only crescent phases are seen (b)
Venus’s phases in a heliocentric model, as viewed from Earth.
Because Venus can go behind the Sun, a full set of phases including
gibbous phases is possible.
5-17Chapter 5 • The Historical Quest to Model the Solar
System
unification awaited the genius of Isaac Newton (1642–1727), who was
born in the year of Galileo’s death (Figure 5-22). Newton provided
us with a system of physical laws — a theory of motion — that is so
fundamental that even today they are the basis for much scientific
activity. It would be difficult here to give an adequate idea of
the power of Newton’s intellect and the importance of his
accomplishments; we can only scratch the surface. As a young man,
Newton invented the tools of the branch of mathematics known as
calculus, did fundamental work on optics and color vision, and
developed the reflecting telescope, as well as proposing his famous
laws of motion and gravity. He made such an impression on his
contemporaries that the poet Alexander Pope was moved to write:
“Nature and Nature’s laws lay hid in night:/God said Let Newton be!
and all was light.”
To begin to understand Newton’s contributions to the study of
moving objects we first define some terms. Velocity describes the
change in position of an object divided by the time interval over
which the change occurs. Velocity describes not only the speed of
an object but its direction. For example, while a car traveling 100
km per hour north moves at the same speed as a car traveling 100 km
per hour east, their velocities are different because their
directions of motion are different.2 Motion is rarely constant in
either amount or direction. Any change in an object’s velocity, in
either its speed or direction, is called acceleration. Acceleration
is the term used even if you are slowing down. Even when traveling
at a constant speed on a Ferris wheel, you are accelerating because
the direction is continuously changing.
Inquiry 5-17 Use the definition of acceleration to list the
accelerators in a standard passenger car.
Newton is most renowned for his three laws of motion and his law of
gravity. His three laws of motion, first published in his
Philosophiae Naturalis Principia Mathematica (The Mathematical
Principles of Natural Philosophy), are generalized statements — a
theory — concerning the motions and interactions of objects. The
first law concerns inertia and is often referred to as the law of
inertia. 2 Velocity is an example of a vector, a quantity having
size and direction. speed, on the other hand, is called a scalar
because it has only a size associated with it.
In the absence of an outside influence, an object at rest will stay
at rest, while an object in motion will continue to move at a
constant speed in a straight line.
As an example consider a box that is not tied down in the back of a
pick-up truck. If the box is located toward the front of the truck
bed and the truck accelerates quickly forward, the box will move
toward the rear because it “wants” to stay at rest in relation to
the street as the truck moves forward. On the other hand, if the
box is toward the rear of the truck bed and the brakes are applied
rapidly, the box will move toward the front, since it continues
moving in the original direction. The more mass the box has, the
more inertia it has, and the more the box will resist motion and
tend to stay in its original location.
Newton’s first law, the law of inertia, was revolutionary. Since
the time of Aristotle, it had been assumed that an object required
some continual action on it to remain in motion, unless that motion
was a part of the natural motion of heavy or light substances. The
philosophers of the Middle Ages had refined Aristotle’s view by
asserting that an object was imparted a certain
FIGURE 5-22. Sir Isaac Newton (1642–1727) laid a firm theoretical
basis in mathematics and physics for planetary motions, which would
eventually reign unchallenged until Einstein’s theories in the
early 1900’s.
Part 1 • Discovering the Science of Astronomy5-18
quantity of “impetus” by the hand or bow string that launched it,
and that it would continue to move until it had used up its
impetus. Impetus, however, was an example of what Newton referred
to as an occult quality, for it could not be perceived in any
fashion. It was merely supposed to exist in order to preserve
Aristotle’s ideas. Thus, all motion seemed to require the
invocation of an unknown substance to explain it. Newton’s simple
statement swept away all that chaos and mental clutter, replacing
it with brilliant simplicity.
Inquiry 5-18 According to Aristotle, the planets continue to move
because that is their nature. What reason would Newton have
given?
What happens if there is an outside influence? In answering this
question, Newton called any outside influence acting on an object
and causing it to accelerate a force. Thus the second law of
motion:
The total force on an object is equal to the product of the
object’s mass times its acceleration.
This expresses the famous formula F = Ma, which is the basis for
studying the motions of particles of all sizes, from submicroscopic
atoms to stars in their orbits in the Galaxy. This concept turned
out to be enormously powerful in stating that forces produce
accelerations, and the size of the acceleration depends on the
mass. Furthermore, an object can accelerate only if there is a
force acting on it; remove the force and the object no longer
accelerates, but continues moving at the same speed and in the same
direction. Finally, because acceleration has size and direction, so
does force.
If an object accelerates, either its speed or its direction of
motion changes, or both change. If an object accelerates in the
direction of motion, speed increases. Acceleration in the direction
opposite to the direction of motion decreases the speed. A sideways
acceleration changes its direction of motion. Any acceleration
(forward, backward, or sideways) requires an external force that
points in the direction of the acceleration. Further, the greater
an object’s mass, the larger the force required to produce a given
acceleration.
Inquiry 5-19 As an example of Newton’s second law, consider a game
of kick-the-can played with two cans — one empty and one filled
with concrete. Which can has the greater mass? If someone came
along and kicked the two cans with exactly the same force, which
can would have the greater acceleration? Explain this in terms of
what you have learned.
In the third law, Newton shows that forces always come in
pairs.
For every force that an object exerts on a second object, there is
an equal and opposite force exerted by the second object on the
first.
As an example of pairs of forces, when someone touches you, you
also touch that person; one touch cannot occur without the other.
If you and a friend are in rowboats on a lake and one of you pushes
against the other with an oar, both boats will move, but in
opposite directions. The simple act of sitting in a chair
illustrates this law: you exert a downward force on the chair, and
the chair exerts an equal and opposite force on you. While a
falling brick exerts a force on the air molecules it encounters,
pushing them ahead of it, the air molecules exert an equal and
opposite force on the brick, thus slightly slowing its motion. A
final example is a rocket. If one force is the rocket engine
pushing gases backward, the opposite force is that of the gases
pushing on the rocket engine in the forward direction. That is why
rockets work in the vacuum of space; rockets do not operate because
of the exhaust gases pushing back on the Earth.
Inquiry 5-20 In the kick-the-can game in Inquiry 5-19, what does
this third law predict about the effect on the foot that kicks the
two cans?
Newton’s Law of Gravity Newton’s three laws, especially the second,
enable us to calculate the acceleration of an object and
5-19Chapter 5 • The Historical Quest to Model the Solar
System
hence its motion, but we must calculate the forces first. Heavenly
objects such as the Sun, Moon, and planets cannot exert forces on
each other directly, since they do not touch each other. Newton
proposed that they exert an attractive force on each other at a
distance, that is, across empty space. He called this force
gravitation. His universal law of gravitation expresses the
gravitational force of attraction between any two objects. What he
discovered is that gravity is a characteristic of any object that
has mass. This means that you and the person next to you have a
gravitational attraction between one another.
d2
where F is the force exerted by an object of mass M1 on an object
of mass M2, d is the distance between the centers of the two
objects, and G is a numerical constant called the universal
gravitational constant. The value of the universal constant depends
on the units used to express the other quantities in the equation.
From this formula, it is possible to calculate how the force
between the objects varies with the distance between them, and
therefore what the acceleration is at each moment.
Inquiry 5-21 If the distance between mass M
1 and mass M 2
is made four times greater, by what factor is the gravitational
force between them changed? Is it increased or decreased?
FIGURE 5-23. The motion of the Moon around the Earth according to
Newton. The Moon “falls” toward the Earth just enough to keep it on
a curved elliptical path around the planet.
Inquiry 5-22 Which force is greater: Earth’s gravitational pull on
the Moon or the Moon’s gravitational pull on the Earth?
Although the idea that objects of different masses fall at the same
rate was experimentally shown by Galileo to be true, it is not
intuitive. However, it is consistent with Newton’s universal law of
gravitation and his second law, as shown mathematically in Appendix
A7.
A measure of Newton’s genius is that he proposed that his laws of
motion and gravity applied not only to objects on Earth but also to
all objects in the universe. His insight occurred when he compared
the force of gravity exerted by the Earth on an object, allegedly a
falling apple, with the gravitational force required to keep the
Moon in its orbit around the Earth. As shown in Figure 5-23, if
there were no gravitational force exerted by the Earth on the Moon,
it would travel in a straight line past the Earth in accordance
with Newton’s first law. To make the Moon travel around the Earth
requires a force toward the Earth. The Earth’s gravitational force
on the Moon causes it to deviate from a straight line and follow a
curved path around the Earth. For each kilometer the Moon moves in
its orbit, the Moon must fall 0.0014 m (0.14 cm) toward the Earth
in order to stay in its elliptical orbit. In other words, the
Earth’s force of gravity causes the Moon to accelerate just enough
to maintain its distance from the Earth. For this reason, the
Moon’s orbital motion can be described as resulting from the Moon’s
falling toward the Earth’s center! From such considerations
Newton
Earth
Force of gravity
Part 1 • Discovering the Science of Astronomy20
found that the actual force required to hold the Moon in its orbit
around the Earth agreed well with the value he computed
theoretically. Thus, observations showed his description to be a
valid scientific theory. It is one that can be applied throughout
the universe anywhere and at any time.
Weight Weight is the force one object exerts on another object. In
particular, your weight is the gravitational force exerted by the
Earth’s mass on your body. Your weight depends on the mass of your
body, the mass of the Earth, and your distance from the Earth’s
center, which is the Earth’s radius. For this reason, your weight
would be different should you travel to a different planet having a
different mass and size than the Earth; your mass, however, would
be the same. For example, a person weighing 120 lbs on Earth would
weigh 20 lbs on the Moon but 305 lbs on Jupiter. Interested readers
can find weight expressed mathematically in Appendix A8.
To understand why an astronaut floats in the International Space
Station even
though there is still gravity in space, you should do Discovery 5-1
at this time.
Momentum You probably have some intuitive feel for the word
momentum. A train moving at 30 km per hour has more momentum than a
bicycle moving at the same speed; it would have more impact and
effect if it ran into something! For an object moving in a straight
line, linear momentum is defined as the object’s mass times its
velocity.
Newton’s first law can also be expressed as the principle of the
conservation of linear momentum. This says that for an isolated
system of objects without any outside force acting on them, the sum
of the linear momenta of all objects in a system is always the
same. For example, when a billiard ball slows down after hitting
another ball, the struck ball must speed up by the same amount
(since the masses are equal).
Most objects move in curved paths, and the concept of angular
momentum comes into play. The amount of angular momentum possessed
by a planet in a circular orbit around the Sun is given by:
Angular momentum = Mass × Speed × Distance from planet to
Sun.
Expressed symbolically, if M is the mass of an object and v its
speed when at a distance d, the angular momentum is given by
Angular momentum = Mvd.
Like linear momentum, angular momentum is conserved for an isolated
system. If the mass of an orbiting planet in an elliptical orbit
remains constant, the only way for angular momentum to remain
constant is for the speed of the planet to increase as the distance
decreases, and vice versa. This is exactly what Kepler’s second law
says. The concept of angular momentum conservation is important in
understanding not only the motions of planets but also such diverse
subjects as the formation of the Solar System, the motions of
stars, and the shapes of galaxies. As applied to the orbits of the
planets, Kepler’s second law states that the angular momentum of an
orbiting planet remains constant.
A spinning object also has angular momentum. In this case, d
represents the distance from the rotation axis to the surface — in
other words, the radius. As a spinning gas cloud collapses to form
a star, conservation of angular momentum requires that it increase
its rotation speed, just as an ice skater increases rotation speed
as the arms are brought in (Figure 5-24).
For simplicity we have defined angular momentum in terms of
circular orbits, but the
concept can be extended to elliptical orbits.
Inquiry 5-23 Suppose a planet’s farthest distance from the Sun is
twice as great as its nearest distance. How many times greater is
its orbital speed when the planet is nearest the Sun than when it
is farthest away?
Newton’s Generalization of Orbital Motion While Kepler showed that
planets moved in elliptical rather than circular orbits, Newton was
able to show that moving objects could be in other types of orbits,
too. The other possible orbital shapes
21Chapter 5 • The Historical Quest to Model the Solar System
are the parabola and the hyperbola. (Objects in such orbits are
said to be in parabolic and hyperbolic orbits.) The various orbital
shapes are best visualized geometrically and then understood by
looking at the energy of the orbiting object.
Circles, ellipses, parabolas, and hyperbolas are all shapes known
as conic sections because they are most readily visualized as
formed from a cone (Figure 5-25). Conic sections result where a
plane intersects the cone. For example, the curve formed by the
intersection between a plane perpendicular to the cone’s axis and
the cone will be a circle, as shown in purple. If the plane is not
perpendicular to the axis the intersection will be an ellipse
(red). If the plane is parallel to the cone’s axis, the
intersecting shape is a hyperbola (yellow). Finally, if the plane
is parallel to the cone’s sides, the shape will be a parabola
(orange).
Thus, objects moving in elliptical and circular orbits are in
closed, repeating orbits; planets in elliptical orbits around the
Sun are gravitationally bound to the Sun, which governs its motion.
Objects in parabolic or hyperbolic orbits, however, are not bound
but are in open orbits; the motions are not repeated and the
orbiting object will continue moving away from the Sun forever.
Examples of such objects are the Voyager spacecrafts that are
escaping from the Solar System and the non- periodic comets
(Chapter 7).
The type of orbit a body has is determined by the amount of energy
the orbiting body has. Orbital motion includes two types of
energy.
The first is potential energy. A rock sitting on a cliff has
gravitational potential energy because it has the potential for
motion if it were to fall off. Orbiting bodies have potential
energy because of the gravitational attraction of the attracting
body.
The second type of energy is kinetic energy, which is energy of
motion. All moving bodies have kinetic energy. The more massive the
body the greater the kinetic energy, and the faster it is moving
the greater the kinetic energy. Thus, even a small asteroid moving
at, say, 20 km/s contains significant kinetic energy such that if
it collided with Earth it would make a big dent in the
surface.
The total energy is the sum of the kinetic and potential energies.
The total energy is constant, which is the concept called the
conservation of energy. Thus, since the potential energy decreases
as an orbiting body approaches the Sun, its kinetic energy (that
is, its speed) must increase.
Objects in elliptical orbits have more potential energy than
kinetic energy and will therefore stay bound. If there is more
kinetic energy than
FIGURE 5-24. An ice skater’s spin rate increases due to
conservation of angular momentum when the arms are pulled in.
Figure 5-25. Defining the ellipse, parabola, and hyperbola from in
terms of sections of a cone
(a) (b)
Part 1 • Discovering the Science of Astronomy22
potential energy, the orbit is open and the orbiting body will
escape.
We can summarize this discussion by considering the motion of a
rocket. A rocket with the right amount of fuel would go into a
bound orbit around the Earth (a circular or elliptical orbit). With
just the right amount of additional energy, the rocket would travel
in an open, parabolic orbit with exactly the right speed to escape
from Earth. Additional energy beyond that amount would cause the
rocket to be in an open hyperbolic orbit and to escape Earth’s
gravity entirely.
The ideas presented here will be revisited in numerous contexts,
from the orbits of comets, to molecules in planetary atmospheres,
to the escape of stars, and to galaxies in collisions.
Newton’s Form of Kepler’s Third Law Newton was able to show, using
the calculus that he invented to solve the problem of orbital
motion, that Kepler’s empirical laws of planetary motion were
direct consequences of his own more fundamental theory of motion.
For example, he was able to show mathematically that a planet
governed by a gravitational force that varies inversely with the
square of the distance from the Sun would follow an elliptical
orbit. This was a major accomplishment! In addition, using his laws
he found Kepler’s third law was even more general than originally
proposed. In Newton’s formulation, it becomes
(M1 + M2)P 2 = A 3
where again P is measured in years and A in astronomical units, and
the additional factor (M1 + M2) is the sum of the masses of the two
objects in orbit about each other, expressed in terms of the mass
of the Sun.
When applied to the planets, (M1 + M2) is almost precisely equal to
1, because the mass of the Sun is so much greater than that of any
of the planets. (This is why Kepler did not discover the mass
factor.) The form in which Newton wrote Kepler’s third law turns
out to be much more useful than Kepler’s form. For example, if you
observe the orbital period and separation of a planet’s moon, you
can compute the mass of the planet; or, by observing the size of a
double star’s orbit and the
period of the orbiting stars, you can calculate the total mass of
the stars in the binary system. The revised ideas even apply to
galaxies in orbit around each other. In fact, much of the
observational data we have on the masses of objects in the universe
come from applying Newton’s form of Kepler’s third law.3
As an example of its use, if two stars identical to the Sun (so the
mass of each in solar units is 1) orbit about each other at a
distance of 0.1 AU, we can use Newton’s form of Kepler’s third law
as follows:
(1 + 1)P 2 = (0.1) 3 P 2 = 1/2 × 10 –3
P = 2.2 × 10 –2 years = 8.2 days. However, if the binary system had
a star identical
to the Sun and one six times as massive, the period would be 4.4
days.
In this example, the masses of the two objects are the same or not
much different. If one object is many times more massive than the
other (e.g., a star orbiting a galaxy), the sum of the two masses
is
basically the same as the more massive object.
Inquiry 5-24 Suppose a double star is observed to have a period of
10 years and an average distance between the two stars of 8 AU.
What is the sum of the masses of the two stars in the system?
Inquiry 5-25 Consider a 10 solar mass star orbiting a galaxy whose
mass is 10 11 solar masses. What is the value of the sum of the
masses? How long should it take such a star to orbit the galaxy if
it is located 10,000 parsecs from the galaxy’s center?
G A3
where G is the Universal gravitational constant from Newton’s
gravitation law and has a value dependent on the units chosen for
the masses, P and A. If the value in the back of the book is used,
then the masses are in kilograms, P is in seconds and A is in
meters. This form must be used if the units of the variables are
not in terms of the Sun’s mass, years, and AU.
23Chapter 5 • The Historical Quest to Model the Solar System
as universal as Newton had claimed. Newton himself had difficulty
explaining the motion of the Moon with accuracy comparable to
observations because the gravitational force of the Sun, as well as
that of the Earth, was important and altered the orbit away from
purely elliptical motion. Eventually, however, the mathematical
problems of computing the Moon’s motion were solved satisfactorily.
In a similar way, as telescopes and observational techniques
improved, the gravitational effects of one planet on another
explained observed deviations from purely elliptical motion.
An early triumph of Newton’s theory of motion and gravity was the
discovery by Edmund Halley, a contemporary of Newton, that several
comets observed in the past were in fact one and the same object.
Halley predicted that it would reappear in 1758. Although Halley
died in 1742, its reappearance right on time electrified Europe and
led to a great flood of comet hunting. The comet, of course, was
named for Halley. It returned again, right on schedule during late
1985; its next appointment near the Sun will be in the year 2061.
Due to gravitational perturbations by the planets, the exact date
cannot yet be determined.
In 1781, another discovery was made that at first led to one of
Newtonian theory’s greatest challenges and later to one of its
greatest triumphs. Sir William Herschel, during a routine survey of
the sky, discovered a new object that, unlike the stars, showed a
clearly visible disk in his telescope. It was soon apparent that
the object was moving with respect to the stars, and when the
theory of gravity was used to establish its orbit, it was found to
be a new planet farther from the Sun than any previously known.
Eventually the new planet was named Uranus, after the Greek god of
the heavens.
The discovery of a new planet at first seemed to be a confirmation
of Newtonian physics. But after many years of observations, it
became evident that something was wrong. No matter what was done,
the orbit of Uranus could not be brought into agreement with the
predictions of the law of gravity. In late 1845 and early 1846, two
young mathematicians simultaneously proposed that there was
another, as yet undiscovered planet that was affecting Uranus’s
orbit and causing it to deviate from the predictions of Newton’s
laws. The English astronomer John Couch Adams sent
his calculations to the Astronomer Royal, who failed to appreciate
the importance of the young man’s work. The Frenchman U. J. J.
Leverrier had better luck. His predictions were sent to the Berlin
Observatory, where the new planet was found almost immediately,
close to the predicted position. Both Adams and Leverrier now share
the honor of Neptune’s discovery.4
In conclusion, the point is that Newton’s theory makes wide-ranging
testable predictions that observations over hundreds of years
verify.
5.6 Evidence in Favor of the Heliocentric Hypothesis You have
now learned about the history of the ideas leading up to the
development of the heliocentric model of our Solar System. In
addition, you have learned the concepts that govern all motions in
the universe. The question now is, what observational evidence
shows the heliocentric model is valid? All that is required to
validate it is clear evidence that the Earth moves. Astronomers now
sought such observational evidence.
It took so long to demonstrate the Earth’s motion because it is not
so easy to do. Although one might argue that the observation of
different constellations at different times of the year
unambiguously demonstrates the Earth’s motion, one could argue that
the presence of different constellations results from a motion of
the celestial sphere rather than the Earth. Or one might argue that
the presence of seasons proves the Earth’s motion. However,
hypothesizing a north-south motion of the Sun on the celestial
sphere could explain what produces the seasons. What we need are
authoritative arguments that cannot be explained away like these
two can.
It was not until the year 1729 that definitive evidence of Earth’s
motion was first observed. In that year James Bradley, the British
Astronomer Royal, was trying to observe stellar parallax (see
Figure 5-6). However, rather than observing parallax, he discovered
instead a phenomenon called the aberration of light. (The name does
not imply that there is anything wrong with light.) The aberration
of light is an apparent shift in a star’s position as a result of
the motion of the Earth and the finite velocity of light. It is
readily understood 4 Galileo recorded an object in his notes in the
year 1613 that turns out to have been Neptune, according to Charles
Kowal of Palomar Observatory. Of course, Galileo did not recognize
that he had seen a planet, so the honor of Neptune’s discovery
rightfully belongs to Adams and Leverrier.
Part 1 • Discovering the Science of Astronomy24
by means of an analogy. Imagine that you are in a rainstorm with
the raindrops coming straight down. To catch the rain in a tall
bucket, you hold it pointing upward, as shown in Figure 5-26a, and
the raindrops will fall straight down to the bottom of the bucket
without hitting the sides. On the other hand, if you run with the
bucket, you will have to tilt it in the direction of your motion to
keep the drops from hitting the bucket’s side (Figure 5-26b).
In the same way, a telescope on the moving Earth must be tilted
very slightly in the direction of its motion if light entering the
top of the telescope tube is to pass all the way through and arrive
at the eyepiece centered at the bottom. The effect is not large:
during the course of the year, a star’s observed position deviates
a maximum of 20 seconds of arc on either side of its true position.
However, eighteenth-century telescopes were sufficiently accurate
to enable this effect to be detected. Because the deviation depends
directly on the speed of the Earth in relation to the speed of
light, a measurement of the deviation gives a numerical value for
the speed of the Earth as it moves through space in its orbit
around the Sun. The value is about one ten-thousandth of the speed
of light, or 30 km/s (19 miles/s). Thus, the first actual
verification of the moving Earth came in 1729 with the discovery of
the aberration of starlight.
The discovery of parallax, predicted by the Greeks and searched for
in vain by Tycho Brahe, was delayed for yet another century. In the
year 1838 the great astronomer Friedrich Wilhelm Bessel finally
succeeded in measuring the parallax of the nearby faint star 61
Cygni. Why did parallax take so long to find? The answer is that
the stars are so far away that the amount of their parallactic
shift is extremely small. In the case of 61 Cygni, for example, the
total shift is only about 0.6 " (0.00017 degrees), less than a
thirtieth of the effect of aberration. This is approximately the
angular size of a U.S. penny viewed from six kilometers away — a
very small angle!
Figure 5-27a illustrates the effect of parallax for a star that is
located in the plane of the Earth’s orbit (the easiest case to
visualize). When the Earth is at point A, the nearby star will be
seen at position A ' with respect to the more distant background
stars. Six months later, when the Earth is at point B, the nearby
star appears to be located at point B '. The parallactic shift in
the position of the star is the angle A-star-B. Clearly, the more
distant the star, the smaller the parallactic shift will be.
If we were to take photographs of the star field when the Earth is
at points A and B in its orbit, they might look like Figure 5-27b.
The nearby star would appear to shift its position in relation to
the more distant stars, as shown. Assuming that the background
stars are much more distant than the nearby stars, the angle
A-star-B of Figure 5-27a will be practically equal to the angular
shift in position shown in Figure 5-27b.
Using parallax to find the distance to stars will be discussed in
Chapter 15.
Inquiry 5-26 If the parallax to the nearby star in Figure 5-27 was
observed from a planet farther from the Sun than the Earth, would
the parallactic shift be larger or smaller than when observed from
the Earth? If observed from Mars, which has a distance of 1.5 AU,
what would be the parallactic shift to the same nearby star?
(a)
(b)
FIGURE 5-26. (a) A stationary bucket in the rain. Raindrops fall
straight to the bottom. (b) Moving a bucket in the rain. It must be
tilted to keep the drops from hitting the side.
25Chapter 5 • The Historical Quest to Model the Solar System
5.7 Observational Evidence of the Earth’s Rotation Although
every schoolchild can tell you that the Earth spins, few college
graduates can explain how we know this simplest of all astronomical
facts. As evidence, most people would cite the observation of the
Sun and stars rising in the east and setting in the west. One could
argue, as the Greeks did, that what is observed is simply the
motion of a star-studded crystalline sphere around a stationary
Earth once every 24 hours, with the Sun moving on its own sphere at
its own slightly slower rate! Thus, just as in the case of the
Earth’s motion, unique and authoritative observations are
required.
Foucault Pendulum A famous demonstration of the rotation of the
Earth is the pendulum experiment first performed by the
nineteenth-century French physicist Jean- Bernard-Léon Foucault. In
this experiment, a massive pendulum is hung from a long wire and
set in motion. Pegs placed in a circle around the pendulum are
successively knocked down over time due to the apparently changing
plane in which the pendulum swings (Figure 5-28). To construct the
simplest possible explanation of what is observed, imagine that we
suspend a pendulum on a long cable from a support that has been
placed above the North Pole (Figure 5-29). We make the coupling
between the cable and the support as
friction-free as possible. If we set the pendulum to swinging back
and forth, it will swing along a line that we can mark by driving
pegs into the ice. Because of Newton’s first law of motion, the
plane defined by the swinging cable will be fixed in space in
relation to the distant stars. If the Earth were not rotating, the
line along which the pendulum would swing would not change.
However, because the Earth turns beneath the pendulum, the line of
the pendulum swing will appear to change (as viewed on the turning
Earth) in a direction that is opposite to the Earth’s rotation.
Thus, the observed change in the direction of the pendulum swing
over time is actually caused by Earth’s rotation.
At other latitudes, such as the latitude of Paris where the
experiment was originally performed, the rotation will be slower
but still evident. Only at the equator does the pendulum swing not
change.
Although the motion of Foucault’s pendulum is a demonstration of
Earth’s rotation on its axis, the concepts involved in the
explanation depend on theories and models from other areas, in
particular Newton’s first law. Without it, the Foucault pendulum
experiment would have made no sense.
Inquiry 5-27 Why doesn’t the plane of oscillation of the pendulum
change for an observer at the equator? (Hint: Consider what the
pendulum’s apparent direction of swing would be in the Southern
Hemisphere.)
FIGURE 5-27. (a) The effect of parallax on a nearby star (not to
scale). The apparent direction toward the star