Almost 3000 years after the Babylonians began recording motions in the heavens and 1300 years after Ptolemy's geometrical explanation of those motions, Nicolaus Copernicus revived Aristarchus's heliocentric concept launching astronomy, and consequently science, on a new road to understanding motion. On Copernicus's heels followed Tycho Brahe with new and better observations of the motions of planets and Johannes Kepler with his ingenious laws describing planetary motion. Briefly stated, Kepler's first two laws are that the orbits of the planets are ellipses, not circles as Plato had wanted, and their variable motion is due to their changing distances from the Sun. His third law establishes the relationship between the planets' orbital periods and their mean distances from the Sun. Kepler's laws are universal in that they apply to any two bodies gravitationally bound to each other, whether in the Solar System or elsewhere in the Universe.
Close behind Copernicus and Kepler came a fascinating personality and the architect of a new perception of motion in the person of Galileo Galilei. The 200-year period spanned by Copernicus, Kepler, and Galileo can be thought of as a dividing line between an old science, which had seen only minor changes since its invention by the Greeks, and a new science in which the methods of observation and experimentation would play an increasingly more profound role. There is, however, in this transition period no abandonment of the basic elements of scientific thinking as first laid down by the Greeks. Therefore let us begin this chapter by considering scientific concepts, elements on which physical laws and theories are based.
2.1. Nature and Role of Physical Concepts
In Supplement 1, several concepts, such as the "geocentric" concept or the concept of "a spherical Earth", were mentioned. Later we shall encounter additional concepts such as the concept of "the electron," the concept of "the photon," the concept of "a force," or the concept of "energy." Let us see if we can determine what is implied by such phrases. In doing so, our discussion of concepts, models, laws, theories-elements in the processes of science-in this and following chapters has been heavily shaped by Gerald Holton's and Stephan Brush's book (Introduction to Concepts and Theories in Physical Science, 2nd ed. Addison-Wesley, 1973). Their book, which we encourage you to consult for a more detail discussion, contains many references for additional reading on the conceptual foundations of physical science.
It is clear historically that scientists limit themselves to certain types of observations and thought processes. Their use of concepts as a reference frame for understanding, their rules for fashioning conceptual schemes, and their type of argument for seeking agreement are unlike those of nonscience. Science has accumulated since the time of the ancient Greeks a set of internationally accepted and reasonably enduring concepts, which are its medium of exchange. To a large degree the secret of science's successful harmony and continuity lies in the nature of these concepts, their definitions, and their ready acceptance as the "coin of the realm."
To make concepts more definite, let us adopt for our discussion the following definition of a physical concept taken in part from Holton:
Physical Concept: A physical concept is a general idea, or notion, or understanding as derived from specific instances or occurrences in the natural world. In general, physical concepts can be defined operationally and most possess a quantitative nature.
The characteristics of concepts-possessing operational definitions and quantitative natures-actually do more to define the notion than does the first sentence. Each of these characteristics aid in the unambiguous communication of problems and results, make possible unambiguous agreement on facts and their interpretations, and knit together the efforts of many independent scientists, even those who are widely separated in subject, time, and place. The renown physicist Albert Einstein (1879-1955) noted that, "The only justification for concepts is that they serve to represent the complex of our experiences." Let us take each characteristic in turn and amplify its meaning.
Some physical concepts are neither intuitive nor unambiguously understandable. In such cases, scientists believe that they can still work with such concepts by being able to operationally define them. The first characteristic mentioned in the definition is then the operational meaning of a concept. By an operational definition we signify that:
Operational Definition: One can prescribe an operational procedure, that is, develop a prescribed series of actions, whereby the concept or physical quantity can be measured.
Ideally, each concept used in physical science can be clarified by some such operational definition, and that perhaps is the most important mechanism by which mutual understanding among scientists is possible. For clearly it is more difficult to misinterpret rigorously performed actions than it is words.
Several points need to be considered regarding operational definitions. The first is that to most people everyday notions seem clear and scientific terms mysterious. Such a belief is more a product of our greater familiarity with everyday notions than with scientific terms. A little thought, however, should show that the opposite is actually true. The words of daily life are usually so flexible in definition, so open to emotional coloring, that they lend themselves to a variety of interpretations and contextual meanings. For science, this will not do. Nonscientists generally find it difficult to get used to a highly specific vocabulary and to the apparently picayune insistence by scientists on its rigorous use. However with out a rigorously defined set of terms, meaningful communication in science is impossible.
Second, operational definitions must seem to be just human convention and they do not necessarily tell us what concepts like "length," or "force," really are. Perfectly true, but therein lies their great utility. For we may make laboratory measurements suffice for discussion purposes until such time that questions concerning the deeper significance or reality of a concept can profitably be addressed.
Third, errors associated with measurements in an operational definition do not have the connotation to scientists of wrong, mistaken, or sinful acts, which exist in everyday usage of the word. Errors of measurement are merely departures from exactness. But in time new technology may reduce these departures, so that scientists can at least strife for greater, if not achieve, exactness in their measurements.
Finally, all physical concepts are not of equal importance. Galileo Galilei (1564-1642) pointed out that some concepts are more directly observable, and therefore can play a primary role in science. Other concepts which do not readily lend themselves to being defined operationally must play secondary roles.
The second characteristic of a meaningful concept is that most are quantitative in nature. The demand for quantitative concepts rests on an article of faith that is as ancient as it is revered. This article of faith is the belief that nature works according to mathematical laws and that observations are explained when we find the mathematical law relating observable quantities. Quoting Galileo, "Philosophy [i.e., science] is written in that great book which ever lies before our eyes-I mean the Universe-but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures [i.e., mathematical schemes] without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth."
To Galileo and his successors, mathematics provides the technique par excellence for comprehending and seeing the unity that exists in nature first perceived by the Greeks through geometry. On a practical level, understanding and manipulating concepts is made possible by the fact that mathematically formulated ideas can be expressed symbolically in equations. Thus mathematics is both a tool for science and a universal language through which scientists communicate.
Because of the quantitative nature of concepts, the important thing for scientists about various experimental apparatus, such as weights, springs, or meter sticks, is no longer their compositions, their colors, or their histories outside of the laboratory. Since all that really matters to scientists are those mathematical relationships involving experimental objects, then the objects cease to be individual entities and they become in the minds of the investigator mathematically-ideal bodies. Their real weight is totally forgotten, and in its place we envision "a point mass." In this sense, science does not deal directly with "real bodies" but with abstractions that exist and move in a hypothetical space of precise mathematical properties. In this mathematical world of the mind, scientists can manipulate the experiment at will, eliminating in their minds all air resistance, regarding the inclined plane as perfectly smooth, or changing one aspect and leaving all other factors untouched. As seen from the outside, this idealized world, rendered in the esoteric language of mathematics and filled with simplifications and exaggerations, is quite analogous to that of the modern painter, poet, or composer. To further our understanding, let us define those concepts which are mathematical ones as:
Mathematical Concepts: Mathematical concepts are those concepts taken from a mathematicised world of ideal and precise objects, and it is a realm in which actions exist as calculations. Such concepts are often justifiable by empirical evidence, but some can not be and they find justification as coherent parts of a fruitful theory. Mathematical concepts are taken seriously only insofar as they yield new understanding of the world of our experiences.
To many scholars mathematics has moved beyond being a tool and a language in science. Mathematical equations have become a way of knowing the world that is unavailable to us in any other way. Equations possess a reality that defies language and visual images. And, they can be just as real, if not more so, in the mind of the scientist as sensual experience.
Especially useful concepts recur in a great many descriptions and laws, often in areas far removed from the context in which they were initially formulated. As you progress through the book, keep in mind our comments on the characteristics of physical concepts, for they will help you to appreciate the outlook astronomers bring to their effort in trying to understand the cosmos.
2.2. Renaissance Revolution in Science
From the third century B.C. until the latter part of the seventeenth century, cosmological thought was pretty much that of the Greeks, with some mathematic refinements but no conceptual innovations. Halfway through the thirteenth century, knowledge of astronomy had spread throughout Europe as Greek manuscripts, having come by way of an earlier Arabic science, were translated into Latin in the newly founded European universities. The Renaissance blossomed in the next two centuries, ending the dominance of ecclesiastical concerns in medieval thought and beginning the development of a broader range of intellectual considerations. Renaissance scientists initiated a new era in picturing the physical world. In doing so they paved the way for important changes in scientific thought and outlook. For astronomy the most important advances were Kepler's, Galileo's, Descartes's, and Newton's concepts of motion, gravity, space, and time. After Copernicus, Kepler, and Galileo, Newton's conceptual framework introduced a a whole new era in cosmological thought. Let us examine these events leading up to Newton closer.
Scientific ideas, including astronomy, proliferated after the 1450s, when the printing press was invented. Although the Ptolemaic system had been immensely successful in describing general aspects of planetary motion for over 13 centuries, the discrepancies between the observed and predicted positions of some planets had become easily recognizable by the fifteenth century. Such discrepancies prompted some thinkers to reconsider the details of Ptolemy's geocentric system. However, about the time the New World was being discovered, Nicolaus Copernicus, a Polish canon of ecclesiastical law and astronomer, questioned whether some other configuration for the planetary system might not be simpler, more reasonable, and more aesthetically pleasing than Ptolemy's geocentric one.
Around 1514 Copernicus resurrected Aristarchus's heliocentric concept and devised a new cosmology based on it. After nearly three decades of study, Copernicus's monumental book On the Revolutions of the Heavenly Orbs was published in the year of his death, 1543. Dedicating the work to Pope Paul III, Copernicus died without seeing his theory accepted. In the universities, Copernicus's book gradually became the focus of thought-provoking study, but the public reception to what was much later a revolution in the concept of the Universe was generally indifference.
Because Copernicus still believed in the Greek idea that heavenly bodies must move in perfect circles, he accounted for deviations from uniform motion by postulating a number of epicycles and other mathematical structures. His system, therefore, was not much more accurate or simpler than Ptolemy's, but the Copernican system ultimately proved to be a tremendous step in cosmological thought. The heliocentric model was as good at explaining retrograde motion and all the observed motions as was the geocentric model. As a result, in the next century this change led to acceptance of the concept that celestial physics was not a separate consideration from, but rather an extension of terrestrial physics; Isaac Newton was later to make this clear.
Appearing at an opportune time, the right man for the next advance in astronomy was Danish nobleman-astronomer Tycho Brahe. With financial help from King Frederick II, Brahe constructed in 1582 an observatory on the island of Hveen, about 32 km northeast of Copenhagen. There, with the most accurate pretelescopic observing instruments ever designed, Brahe determined positions with a precision of 1 minute of arc, far surpassing any previous measurements. He observed the Sun, Moon, planets, and stars with regularity instead of haphazardly, as others had in the past. An uninterrupted record of their movements over many years was thus available for future study.
Brahe had reservations about adopting Copernicus's heliocentric theory. He accepted the idea that the five planets revolved around the Sun, but not the idea that the heavy and sluggish Earth moved. Earth's motion would be felt, he argued-and besides, a moving Earth was contrary to scriptural belief. Moreover, he was unable to detect the Earth's orbital motion by parallactic shifts in the positions of the brighter stars. Consequently, Brahe's cosmological system was a compromise: The planets orbited the Sun; the Sun and Moon, in turn, orbited a fixed Earth. There was relatively little interest in Brahe's cosmology, and it never really won a place in cosmological thought.
In the years just prior to 1600, the Renaissance and the Reformation were coming to an end. Copernicus's work was read by a few astronomers who recognized the computational advantages of the Copernican system but were not willing to take seriously its philosophical and physical implications. However, Johannes Kepler (Figure 2.1), the German assistant and successor to Tycho Brahe, was a devoted Copernican from his twenties on, and was destined to bring about acceptance of the heliocentric concept.
The life-long question that concerned Kepler was the nature of the clockwork that governed the celestial machinery, for he was firmly convinced that mathematical relations existed that could make sense of the planetary system. He saw the planetary system operating like a mechanical model according to its own set of mathematical laws. After years of labor, during which he rejected many ideas because they did not fit Brahe's observations, Kepler published his first two laws of planetary motion in 1609 in a book entitled New Astronomy; his third law was published in The Harmonies of the World in 1619. As Thales can be thought of as an initiator of the early period in science, Kepler can be seen in many respects to mark the beginnings of what we call modern science. Kepler developed his empirical laws from Brahe's data on Mars: "By the study of the orbit of Mars," he said, "we must either arrive at the secrets of astronomy or forever remain in ignorance of them." However, in what proved to be a revolutionary step, Kepler then generalized saying that his laws applied to all the planets, including the Earth, without ever actually verifying that this was indeed true. The expectation that the mathematical laws of science are universal in character is so readily accepted today that it is difficult to imagine just how important to science Kepler's actions were.
Kepler's work put to rest any notion that planets move in perfectly circular orbits because nature has decreed that the heavenly bodies must show perfection in their movements. Although Kepler never knew why planets move by these empirical relationships, he diligently sought a cause of which his three laws were the effect. As he stated, "I am much occupied with the investigation of physical causes. My aim in this is to show that the celestial machine is to be likened not to a divine organism, but rather a clockwork..." Kepler vaguely sensed that bodies have a natural "magnetic" affinity for each other and guessed that the Sun has an attractive force. However, it remained for Newton, half a century later, to formulate a unified theory of motion that invoked gravity as the cause of planetary motion.
Galileo Galilei (1564-1642), shown in Figure 2.2, was a contemporary of Johannes Kepler. Yet for some reason, he seems not to have been significantly influenced in his work by Kepler or Kepler's three laws of planetary motion. Galileo's approach to understanding motion in the cosmic world was through the study of terrestrial motion, especially falling bodies.
The dominant concepts of motion during the Renaissance were still those of Aristotle, who had defined motion as either natural or forced. A rock falling toward the ground was an instance of "natural motion," or the tendency of earthly materials to return to their natural place-the Earth. No cause was needed to assist such motion; it was a natural tendency. However, a thrown rock required a cause both to set it in motion and to continue it in motion. This was an unnatural tendency-referred to by Aristotle as "forced motion."
Galileo did not formulate the principle of gravity that we recognize today; Newton did that later. However, Galileo did conceptualize a force as something that brings about a change in the motion of bodies, and he saw the Earth as exerting an attractive force (that is, gravity) that influences falling bodies. He also recognized the tendency of bodies either at rest or in motion to resist a change in their motion. Thus to Galileo and his scientific contemporary in France, Rene Descartes (1596-1650), rest and uniform motion were a natural state of affairs. To change such a state, it was necessary to have a force act on the body regardless of whether it was falling straight down or moving across the surface of the Earth. A departure from uniformity in motion is now referred to as accelerated motion.
Although mechanics was possibly his most significant accomplishment, Galileo also revolutionized astronomy in 1609 by designing a telescope. As the first telescopic explorer of the heavens, though not the builder of the first telescope, he established his place in history through such discoveries as Jupiter's four large satellites, craters and mountains on the Moon, the phases of Venus, and individual stars in the Milky Way. (The frontispiece to this chapter reproduces Galileo's drawings of the Moon.) Kepler had utilized Copernicus's heliocentric system as the basis for a dramatic new understanding of planetary motion, and Galileo gave the Copernican theory observational support. For example, he observed that on a smaller scale Jupiter's satellites moving around the planet were analogous to the planets orbiting the Sun. Here obviously were heavenly bodies not in orbit about the Earth, and here also was evidence disputing Aristotle's contention that a moving Earth would leave the Moon behind. Since Jupiter retains its satellites, then logically the Earth can move around the Sun without losing its satellite.
Theological hostility loomed over Galileo for supporting Copernican cosmology. Pope Paul V instructed his emissary, Cardinal Bellarmine, to warn Galileo against teaching or upholding Copernican doctrine, and from the Holy Office in February 1616 came the following internal memo made public later at Galileo's trial in 1633:
The following propositions are to be censured: (1) that the Sun is at the center of the world and the Universe....Unanimously, this proposition has been declared stupid and absurd as a philosophy, and formally heretical because it contradicts in express manner sentences in the Holy Scripture....(2) that the Earth is not the center of the world and motionless, but changes its place entirely according to its diurnal movement. Unanimously, this proposition is declared false as a philosophy....
When the more liberal Pope Urban VIII took office, however, Galileo obtained permission to discuss both the Ptolemaic and Copernican systems; he was, though, to present the latter as an unproved alternative. Encouraged by this opportunity, Galileo began work on a masterly astronomical commentary, that passed censorship and was published in 1632 as The Dialogues of Galileo Galilei on the Two Principal Systems of the World: The Ptolemaic and Copernican. Powerful enemies soon convinced the Pope that Galileo had cast the Ptolemaic system in an unfavorable light. As a result, the book was officially banned, and in the year 1633 the great scientist was publicly humiliated before a papal tribunal in which he recanted his Copernican views.
Galileo spent the last 9 years before his death in his villa in Arcetri, some distance from Florence, under strict house arrest. He was forbidden to publish or discuss the forbidden philosophy, although he did finish Two New Sciences and have it published in Leiden in the Netherlands in 1638. By then he was 74 and totally blind-about which he writes, "...this Universe which by my remarkable observations and clear demonstrations I have enlarged a hundred, nay a thousand fold beyond the limits universally accepted by the learned men of all previous ages, are now shrivelled up for me into such a narrow compass as is filled by my bodily sensations." The silencing of Galileo acted to silence Catholic scientists in the south of Europe, and from there, consequently, the scientific revolution moved to northern Europe. Galileo died in the same year, 1642, that Isaac Newton was born in England.
If, as Galileo believed, nature works according to mathematical laws and observations of nature are explained when we find the mathematical law relating observable quantities, we must be more specific in what we mean by a scientific law and to distinguish it from a physical concept? Let us depart once again from our historical narrative to consider the laws of science.
In 1687, Isaac Newton (1642-1727) published a treatise entitled The Mathematical Principles of Natural Philosophy, commonly known as the Principia. This monument in intellectual thought contains a remarkable passage on the rules of reasoning. There are four rules, which collectively reflect his profound faith in the unity of nature, and they were intended by Newton to guide scientists in the scientific process.
The first rule is called the principle of parsimony, and it says that scientists should make no more assumptions or assume no more causes than are absolutely necessary to explain their observations. The principle of parsimony is also known as Occum's razor, after William of Occum (1285?-?1349), who stated his principle of economy of thought in the phrase, "a plurality must not be asserted without necessity." The second rule is the principle of cause and effect, or the belief that what occurs in nature is the result of cause-and-effect relationships, and where similar effects are seen then the same cause must be operating. The third rule is the principle of universal qualities or the belief that those qualities, such as mass or length, that describe bodies exposed to our immediate experience also describe bodies removed from our immediate experience, such as stars and galaxies. The final rule is the principle of induction. Induction is the process of deriving conclusions about a class of objects by examining a few of them-reasoning from the particular to the more general (deduction is the process of reasoning from the general to the more specific). The rule states that concepts, hypotheses, laws, and theories arrived at by induction should be assumed as universal both in time and place until new evidence proves the contrary to be true, as Kepler did in developing his laws of planetary motion.
These rules for reasoning are fundamental to the process of discovery of natural or scientific laws. To be more concrete in what we mean by a scientific law, let us adopt the following definition:
Scientific Laws: As formulate by human beings, natural or scientific laws are rules, preferably mathematical rules, by which we believe nature operates, and such laws can be classified as being either empirical, definitional, or derived laws.
In their observations and experiments, scientists often synthesize their observations of phenomena by developing empirical laws, a general statement which identifies a regularity in many observations without offering a theoretical explanation for it. A good example of empirical laws are Kepler's laws.
Definitional laws are a second level of physical law, so named because these laws usually involve the definition of fundamentally important concepts. Examples of such laws are Newton's second law and the law of conservation of energy to be discussed in Supplement 3.
Finally, there are the derived laws which are derived from some underlying theory, such as Newton's law of universal gravitation, which is derived from Kepler's laws, Newton's three laws of motion, and the concept of "action-at-a-distance."
The scientific laws of nature are usually thought of as inexorable and inescapable, in part because the word "law" suggests an erroneous analogy with divine law. Scientific laws, being built on concepts, hypotheses, and experiments, are only as trustworthy as those concepts are complete and as those experiments are accurate. Since humans formulate scientific laws, they are neither eternally true nor unchangeable. In fact with the advance of knowledge and experience, many laws of science prove, sooner or later, to be too limited or too inaccurate. An example is the law of conservation of mass, which today we recognize as having only limited applicability.
Since its origin in Greek thought, the larger goal of science has been to explain the intricacies of nature as rationally and coherently as possible. Such an explanation does not necessarily mean attributing a motivating agent, such as God, to events, but it does mean discovering, if possible, mathematical laws between observable quantities. But how do scientists find explanation by discovering mathematical laws? Such laws may aid utilization, control, and direction, but how is anything explained thereby?
For human beings the only tools for understanding physical phenomena are the pictures, allusions, and analogies involving the primitive mechanical events of everyday life that dwell in our imaginations. Over the course of history as physical science has moved toward problems more removed from the realm of common experience, it has been necessary to supplement those mental tools with which we grasp and comprehend phenomena with mathematical concepts and laws.
Holton suggests that, "`to explain' means to reduce to the familiar, to establish a relationship between what is to be explained and the (correctly or incorrectly) unquestioned preconceptions." Modern scientists, like their ancient and medieval counterparts, do bring preconceptions to the scientific process; for example, just what we have been discussing, that nature works according to simple models or mathematical schemes. To modern science, scientific laws are an explanation of nature in that they allow scientists to incorporate mathematically the unfamiliar into the body of familiar experience. Experience shows that it requires training and repeated personal success in solving physical problems to be satisfied with and to believe that a mathematical answer explains nature.
2.4. Kepler's Laws of Planetary Motion
2.4.1. Kepler's First Law, Elliptical Orbits
Kepler's first law of planetary motion can be stated as follows:
Kepler's First Law (Law of Elliptic Orbits): Each planet moves in an elliptic orbit around the Sun, with the Sun occupying one of the two foci of the ellipse.
The ellipse, part of a "family" of mathematical curves known since the second century B.C., is important in understanding the orbit of one body about another, not just planets. Roughly speaking, an ellipse is a circle with the opposite ends of a diameter pulled outward, which distorts the circle into an oval-shaped figure. The long axis of the ellipse is known as the major axis, with half the major axis being called the semimajor axis, and perpendicular to it through the center of the figure is the minor axis. The size of an elliptic orbit is set by the length of the semimajor axis. There are two points on the major axis, called the foci (singular form is focus), about which the figure is roughly symmetrical. In a planet's orbit, the Sun occupies one focus; the other one is empty. Since the sum of the distances from each of the foci to every point on an ellipse is a constant, this suggests a means of drawing an ellipse: Loop a piece of string around two tacks (the foci), and wield a pencil as shown in Figure 2.3.
The farther the foci are from each other, the more elongated the
ellipse; the closer together they are, the more nearly circular
the ellipse. Thus the ratio of the distance of a focus from the
center of the ellipse to the length of the semimajor axis, known
as the eccentricity, determines the shape of an elliptical
orbit. When the ratio is zero, the foci and center coincide,
and the ellipse degenerates into a circle. The more elongated
an elliptical orbit is, the nearer the eccentricity is to 1.
The eccentricities of the planets' orbits are given in Table 3.2.
They vary from near zero for Venus to around 0.2 for Mercury
and Pluto. Thus planetary orbits are not very elongated but very
nearly circular, which is why it was not obvious to Kepler or
his predecessors that the orbits of the planets are not Plato's
perfect circles.
2.4.2. Kepler's Second Law, Orbital Speed of the Planets
If the planets were orbiting the Earth in circular orbits, they would traverse equal angles on the celestial sphere in equal intervals of time anywhere along their orbits. In fact as pointed out earlier, they do not actually do this, but rather traverse variable angles in equal intervals of time. In the elliptical orbits determined by Kepler, the planets' distances from the Sun, which occupies one focus, vary with their position in the orbit. Therefore, the speeds in the orbits vary from one position to another such that planets move fastest when closest to the Sun and slowest when farthest away. This variation in speed is a consequence of Kepler's second law of planetary motion, which may be stated as follows:
Kepler's Second Law (Law of Areas): The imaginary line connecting any planet to the Sun sweeps over equal areas of the ellipse in equal intervals of time.
The mean distance of a planet from the Sun is the average of the
distance between the point of closest approach (called perihelion,
which is located at one end of the major axis; point 1 in Figure 2.3) and the most distant point of the orbit (called aphelion,
which is located at the other end of the major axis; point 7).
The average is one-half the length of the major axis, or the
semimajor axis, as shown in the figure. The alternate gray and
colored sectors in the figure are of equal area. Therefore, according
to Kepler`s second law, a planet passes through the numbered positions
in equal intervals of time.
2.4.3. Kepler's Third Law, a Yardstick for the Solar System
Kepler's third law is important in that it provided a means of determining the relative size of the Solar System in units of the mean Earth-Sun distance, the astronomical unit (AU). Second, it gave the mathematical relationship between orbit size and sidereal period, which was important in suggesting to Newton that gravity varied as the inverse-square of distance, as he would later demonstrate. The law may be stated as:
Kepler's Third Law (Harmonic Law): The square of any planet's orbital period (its sidereal period) is proportional to the cube of its mean distance (the length of the semimajor axis) from the Sun.
The orbital period is the planet's sidereal period; that is, the time to move through 360o revolution about the Sun. The sidereal period cannot be measured directly, but as pointed out in Supplement 1, it could be calculated once the synodic period was measured. Kepler believed that the constant of proportionality necessary to turn his law into an equality was indeed constant and did not depend on the planet in any way. If this were so, when the sidereal period is expressed in units of years (the Earth's sidereal period), Kepler's third law would allow the computation of the mean planet-Sun distance (semimajor axis) in astronomical units. The tabular part of Figure 2.3 illustrates the results of such a computation.
Newton later showed that Kepler's assumption about the constant was incorrect (see Section 3.3.4).
Early in this chapter, we discussed the quantitative nature of
concepts in which Galileo's work played a significant role. Galileo
did much to clarify quantitative concepts of motion, but most
importantly he united physics and mathematics in the pursuit of
an understanding of nature. It was in Galileo's consideration
of freely falling bodies and accelerated motion that he revealed
to physical science a new attitude toward experimentation. It
was the mental ability to sweep aside those impediments in motion,
such as air resistance, revealing an idealized world of motion
that underlies quantitative experiences in the laboratory or the
everyday world. Before we can discuss Newton's unification of
the concepts of motion through his laws, we should consider these
concepts individually.
2.5.1. Force, The Cause of Changes in Motion
Force is a push or a pull that causes a body to change its state of motion (a state of motion is the concept of a quantitative description of motion). We should note that rest (or no apparent motion) is only one possibility among many for a state of motion and should not be thought of as special. Most familiar to us are mechanical forces; these are exerted by one body in contact with another, such as a bat striking a baseball. Fields of force make up the other class of forces, and they are our response to a belief in the concept of action-at-a-distance. That is, we envision that bodies exert forces on one another without any physical contact between the bodies. Such a concept was difficult for renaissance scientists to accept, and it continues to strike us as not intuitively obvious. Action-at-a-distance is an example where defining a concept operationally proves the value of operational definitions, as we shall show in Newton's second law in Supplement 3.
Science recognizes four types of force fields: strong and
weak nuclear forces, which act on the subatomic scale and
are responsible for structure on the nuclear scale; electromagnetic
forces, which are either attractive or repulsive and give
form to the world of our immediate existence; and finally gravitational
forces, which are attractive, but never repulsive and are
responsible for structure on the astronomical scale. Empty space
between matter is no barrier to fields of force, as experience
shows in the case of gravitational forces.
2.5.2. Mass, A Measure of Inertia
Every material body possesses a property called inertia, the resistance it offers to a change in its state of motion. The more matter a body has, the greater its inertia. Mass measures the amount of matter a body contains; therefore, mass is a measure of inertia. Massive bodies are more resistant to a change in their state of motion than less massive ones, as we know from common experience. Newton's attempts in the Principia to define mass were less than precise. However as we noted earlier, because of the operational nature of concepts, those mathematical concepts in a new theory that are not easily defined in terms of more fundamental concepts are still quite usable.
A material body has the same mass regardless of where in the Universe it is located, but its weight depends on its position relative to various attracting masses. Therefore weight is a measure of the gravitational force that an attracting object exerts on a body. For example, a person weighing 90 kilograms (kg) on the Earth's surface would weigh 15 kg on the Moon's surface because the Moon's gravitational pull is one-sixth that of Earth. However, the individual's mass is the same on both the Moon and Earth (Figure 2.4).
An important concept related to mass is that of compactness or
density. The mass density, or just density, of
matter is defined as its mass per unit volume. For example, water
has a density of 1 gram per cubic centimeter (g/cm3), while that
of lead is 11.3 g/cm3. Bodies may have the same mass but quite
different densities, such as those of a feather pillow and a book.
If matter is distributed unevenly throughout a body-as it is
in the Earth-then the mass divided by the volume yields a mean
density.
2.5.3. Describing Motion, Velocity and Acceleration
Distance is familiar in everyday experience as a measure of the space that separates material objects. However, to develop a useful description of motion, we must specify the origin from which and direction in which distance is being measured. The place where we observe and measure motion is called a frame of reference; it possesses an origin from which distances can be measured relative to some reference direction. As an example, let's consider the distances to the planets: To measure these, we must first select a reference frame, such as that with the Sun as the origin (Figure 2.5), and as our reference direction, we might choose the direction toward a given star lying along the ecliptic. This is only one of several possible reference frames that could be used to measure the distances to the planets.
Time is equally familiar from our everyday experience. Our intuitive concept of time is based on changing patterns and events in our lives-which seems different from our intuitive concept of distance, the separation between tables and chairs, for example. Time, however, like distance, can be measured from an arbitrarily chosen origin, such as a historical event. Although we can move forward and backward over a distance in space, in human experience we can only move forward in time, never backward.
For a moving body, the distance traversed divided by elapsed time is the speed of the body. If we take account of the direction of motion as well as the speed, we define the velocity of the body. A change in the velocity is known as acceleration. The change can be either in the speed or the direction of motion or both. Thus acceleration is measured as the rate of increase or decrease of a body's speed or as the rate at which its direction of motion changes. Since distance depends on a frame of reference, so also will velocity and acceleration. Figure 2.6 is a space-time diagram in which we couple spatial and temporal measurements to show the graphical meaning of velocity.
An airplane's velocity, for example, is usually measured relative
to the surface of the Earth, say, 600 miles per hour. We could
measure it relative to the Sun, in which case the Earth's rotational
and orbital velocities would have to be added to that of the airplane.
Any fixed point on the surface of the Earth is continuously changing
velocity-that is, accelerating-because of its rotation and revolution
with the Earth. What we really want to specify in the orbital
motion of planets is the velocity at one moment of time, since
an instant later the velocity would be different owing to acceleration.
This concept of velocity is called instantaneous velocity,
and when we use the word velocity in this book, we will mean instantaneous
velocity.
2.5.4. Momentum, a Measure of the Quantity of Motion
Simple observation shows us that there is more to defining the motion of a body than finding its velocity. For example, in the collision of two billiard balls moving with different velocities, they may simply exchange velocities so that the total quantity of motion is conserved and just redistributed between them. However, in the case of two balls of different masses moving with the same speed, the more massive one is able to transfer a greater quantity of motion in a collision than the ball with a small mass. Thus the concept of quantity of motion, or momentum, depends on both the velocity and the mass of the body, as illustrated in Figure 2.7. To find momentum, we multiply a body's mass by its velocity. Because momentum depends explicitly on the concepts of velocity and mass, and implicitly on the reference frame, it represents the precise mathematical definition of motion that we desire in the concept of a state of motion. Therefore, one can use these terms interchangeably.
It is not difficult to visualize philosophically the possibility that the total quantity of motion, or momentum, in the Universe is a constant. This can be stated in the form of a natural law:
Law of Conservation of Momentum (Quantity of Motion): The total momentum of the Universe is conserved, i.e., momentum can neither be created nor destroyed, even though the interaction of bodies with each other continuously redistributes the total momentum among the individual bodies of the Universe.
This concept of conservation of momentum was first suggested by
the French philosopher-physicist Rene' Descartes in his book Principles
of Philosophy published in 1644. Remembering our earlier
comments about the importance of mathematical laws, it becomes
clear why laws of constancy, such as the conservation of momentum,
are so highly prized in science. Such laws combine the most successful
features of science with its most persistent preoccupation-the
mathematical formulation of concepts that aid in the discovery
of unchanging patterns in the chaos of experience.
2.5.5. Rotation and Angular Momentum
In addition to a quantity of motion involved in straight line motion (translational motion), our experiences tell us that there is obviously a quantity of motion involved in rotation also. This observation gives rise to the concept of angular momentum, which is calculated as the product of the translational momentum and the distance from the axis of rotation. Reflection on your part should suggest that just as translational momentum can be conserved in the interaction of moving bodies, such as in the collision of billiard balls, so can rotational momentum be conserved. A simple consequence of the conservation of angular momentum is the observation that the spinning ice skater spins faster or slower when she draws her arms in or extends them out, respectively. The quantity of all motion in the Universe in the form of rotational motion is comparable to that in translational motion. For example, the planets rotate, stars rotate, galaxies rotate, etc. In fact, the orbital motion of the planets can be thought of as rotation about an external axis, so that orbiting planets, orbiting stars, and orbiting galaxies contribute to the rotational momentum of the Universe.