This chapter was copied with permission from Nick Strobel’s Astronomy Notes. Go to his site at www.astronomynotes.com for the updated and corrected version. This chapter has been edited for content. Text that has been altered from the original is denoted in green font.

History and Philosophy of Western Astronomy

I focus on the rise of modern science in Europe, from the ancient Greeks to Kepler.

4.1 Introduction

Now that you have some feeling for the scales of time and space that astronomy This chapter covers the development of western astronomy and modern science. I focus on the rise of modern science in Europe, from the ancient Greeks to Isaac Newton. Other cultures were also quite interested and skilled in astronomy (the Mayans; link will display in another window), Egyptians, peoples of India and China come immediately to mind), but the Greeks were the first ones to try to explain how the universe worked in a logical, systematic manner using models and observations. Modern astronomy (and all of science) has its roots in the Greek tradition. If you would like a more thorough discussion of the history of astronomy than what I will present here, please take a look at Science and the Human Prospect by Ronald Pine (links will display in another window). I will give dates of when certain persons lived and worked to give you some reference points in the long history of astronomy. Don’t worry about memorizing the dates. What is more important is to see the development of ideas and methods of modern science.

I include images of world atlases from different time periods in this chapter and the next as another way to illustrate the advances in our understanding of our world and the universe. Links to the sites from which the photographs came are embedded in the images. Select the picture to go to the site. The vocabulary terms are in boldface.

4.2 Philosophical Backdrop

Ancient Greece’s view of their world. Select the image to go to Jim Siebold’s ancient maps database from which this picture came (will display in another window). By the 7th Century B.C.E. a common viewpoint had arisen in Greece that the Universe is a rational place following universal, natural laws and we are able to figure out those laws. Open inquiry and critical evaluation was highly valued. The emphasis was on the process of learning about the universe rather than attaining the goal. But people eventually got tired of learning and wanted absolute answers. Science is not able to give absolute, certain answers. There was disagreement among the experts and there came to be a crisis in confidence that led to the rise of the Sophists.

The Sophists taught that an absolute truth and morality are myths and are relative to the individual. Since truth and morality were just cultural inventions for the Sophists, they said a person should conform to the prevailing views, rather than resolutely holding to some belief as an absolute one. Socrates (lived 470–399 B.C.E.) disagreed with the Sophists, teaching that we can attain real truth through collaboration with others. By exploring together and being skeptical about “common sense” notions about the way things are, we can get a correct understanding of how our world and society operate. This idea of being skeptical so that a truer understanding of nature can be found is still very much a part of modern science.

Socrates’ student, Plato (lived 427–347 B.C.E.), developed Socrates’ ideas further. Plato taught that there are absolute truths — mathematics is the key. While statements about the physical world will be relative to the individual and culture, mathematics is independent of those influences: 2 + 2 = 4 always, here on Earth or on the far side of the galaxy. Plato had Four Basic Points:

1. There is certainty.
2. Mathematics gives us the power of perception.
3. Though the physical applications of mathematics may change, the thoughts themselves are eternal and are in another realm of existence.
4. Mathematics is thought and, therefore, it is eternal and can be known by anyone. [Today we view mathematical ideas as free creations of the human mind. They are the tools we use to map the world. Experience is the key. Although absolute certainty is not possible, we can still attain accurate knowledge and reasonable beliefs about the world.]

Out of Plato’s teachings grew the belief that when one studies mathematics, one studies the mind of God. Mathematical symmetries are the language of universal design and harmony. Their faith in order caused the Greeks to try to find explanation for the seemingly unordered planets (particularly retrograde motion). Their faith in an ordered universe compelled them to make precise observations and they were sustained by their belief in the power of reason. In one form or another, modern scientists have this faith in an ordered universe and the power of human reason.

The Greeks were guided by a paradigm that was first articulated by Pythagoras (lived ca. 569–475 B.C.E., picture above) before Socrates’ time. A paradigm is a general consensus of belief of how the world works. It is a mental framework we use to interpret what happens around us. It is what could be called “common sense.” The Pythagorean Paradigm had three key points about the movements of celestial objects:

1. The planets, Sun, Moon and stars move in perfectly circular orbits;
2. The speed of the planets, Sun, Moon and stars in their circular orbits is perfectly uniform;
3. The Earth is at the exact center of the motion of the celestial bodies.

4.3 Plato’s Homework Problem

Plato gave his students a major problem to work on. Their task was to find a geometric explanation for the apparent motion of the planets, especially the strange retrograde motion. One key observation: as a planet undergoes retrograde motion (drifts westward with respect to the stars), it becomes brighter. Plato and his students were, of course, also guided by the Pythagorean Paradigm. This meant that regardless of the scheme they came up with, the Earth should be at the unmoving center of the planet motions. One student named Aristarchus violated that rule and developed a model with the Sun at the center. His model was not accepted because of the obvious observations against a moving Earth.

Some of the observations that convinced the Greeks that the Earth was not moving are

1. The Earth is not part of the heavens. Today the Earth is known to be just one planet of nine that orbit an average star in the outskirts of a large galaxy, but this idea gained acceptance only recently when telescopes extended our vision.
2. The celestial objects are bright points of light while the Earth is an immense, nonluminous sphere of mud and rock. Modern astronomers now know that the stars are objects like our Sun but very far away and the planets are just reflecting sunlight.
3. The Greeks saw little change in the heavens — the stars are the same night after night. In contrast to this, they saw the Earth as the home of birth, change, and destruction. They believed that the celestial bodies have an immutable regularity that is never achieved on the corruptible Earth. Today astronomers know that stars are born and eventually die (some quite spectacularly!)—the length of their lifetimes are much more than a human lifetime so they appear unchanging. Also, modern astronomers know that the stars do change positions with respect to each other over, but without a telescope, it takes hundreds of years to notice the slow changes.
4. Finally, our senses show that the Earth appears to be stationary! Air, clouds, birds, and other things unattached to the ground are not left behind as they would be if the Earth was moving. There should be a strong wind if the Earth were spinning as suggested by some radicals. There is no strong wind. If the Earth were moving, then anyone jumping from a high point would hit the Earth far behind from the point where the leap began. Furthermore, they knew that things can be flung off an object that is spinning rapidly. The observation that rocks, trees, and people are not hurled off the Earth proved to them that the Earth was not moving. Today we have the understanding of inertia and forces that explains why this does not happen even though the Earth is spinning and orbiting the Sun. That understanding, though, developed about 2000 years after Plato.

Plato taught that since an infinite number of theories can be constructed to account for the observations, we can never empirically answer what the universe is really like. He said that we should adopt an instrumentalist view: scientific theories are just tools or calculation devices and are not to be interpreted as real. Any generalizations we make may be shown to be false in the future and, also, some of our false generalizations can actually work—an incorrect theory can explain the observations (see the scientific method page for background material on this).

Aristotle (lived 384–322 B.C.E.) was a student of Plato and had probably the most significant influence on many fields of studies (science, theology, philosophy, etc.) of any single person in history. He thought that Plato had gone too far with his instrumentalist view of theories. Aristotle taught a realist view: scientific, mathematical tools are not merely tools—they characterize the way the universe actually is. At most one model is correct. The model he chose was one developed by another follower of Plato, Eudoxus. The planets and stars were on concentric crystalline spheres centered on the Earth. Each planet, the Sun, and the Moon were on their own sphere. The stars were placed on the largest sphere surrounding all of the rest.

Aristotle chose this model because most popular and observational evidence supported it and his physics and theory of motion necessitated a geocentric (Earth-centered) universe. In his theory of motion, things naturally move to the center of the Earth and the only way to deviate from that is to have a force applied to the object. So a ball thrown parallel to the ground must have a force continually pushing it along. This idea was unchallenged for almost two thousand years until Galileo showed experimentally that things will not move or change their motion unless a force is applied. Also, the crystalline spheres model agreed with the Pythagorean paradigm of uniform, circular motion (see the previous section).

A slight digression: Another conclusion drawn from Aristotle’s teachings was that the Earth was unique with its own set of physical laws that were different from how things worked in the heavens. The Earth was a world and filled with change and decay while the planets, Moon, and Sun were perfect, unchanging and essentially ornaments on the sky, not worlds that could be explored. (Imagine the transformation of our viewpoint when we discovered using telescopes that those wandering points of light are worlds like the Earth and then later discovered other planets orbiting other stars!) Now, to return to the motions of the planets…

Astronomers continued working on models of how the planets moved. In order to explain the retrograde motion some models used epicycles—small circles attached to larger circles centered on the Earth. The planet was on the epicycle so it executed a smaller circular motion as it moved around the Earth. This meant that the planet’s distance from us changed and if the epicyclic motion was in the same direction (e.g., counter-clockwise) as the overall motion around the Earth, the planet would be closer to the Earth as the epicycle carried the planet backward with respect to the usual eastward motion. This explained why planets are brighter as they retrogress.

Ptolemy’s view of the world. Select the image to go to Jim Siebold’s ancient maps database from which this picture came (will display in another window).

4.3.1 Ptolemy’s geocentric universe

Ptolemy (lived 85–165 C.E.) set out to finally solve the problem of the planets motion. He combined the best features of the geocentric models that used epicycles with the most accurate observations of the planet positions to create a model that would last for nearly 1500 years. He added some refinements to explain the details of the observations: an “eccentric” for each planet that was the true center of its motion (not the Earth!) and an “equant” for each planet moved uniformly in relation to (not the Earth!). See the figure below for a diagram of this setup.

Select image to show animation of retrograde motion. These refinements were incompatible with Aristotle’s model and the Pythagorean paradigm — a planet on an epicycle would crash into its crystalline sphere and the motion is not truly centered on the Earth. So Ptolemy adopted an instrumentalist view — this strange model is only an accurate calculator to predict the planet motions but the reality is Aristotle’s model. This apparent contradiction between reality and a calculation device was perfectly fine in his time. Our modern belief that models must characterize the way the universe actually is is a tribute to the even longer-lasting influence of Aristotle’s realism. Ptolemy was successful in having people adopt his model because he gathered the best model pieces together, used the most accurate observations and he published his work in a large 13-volume series called the Almagest, ensuring that his ideas would last long after he died.

4.4 Renaissance

Our view of the history of astronomy will now skip almost 1500 years to the next major advances in astronomy. Europe was beginning to emerge from a long period of instability in the Middle Ages. During the Middle Ages the Islamic civilization had flourished in the Arabic countries. They had preserved and translated the Greek writings and adopted the Greek ideals of logic and rational inquiry. Islamic astronomers were careful observers of the sky and created accurate star catalogs and tables of planet motions. Many of the names of the bright stars in our sky have Arabic names (e.g., Deneb, Alberio, Aldebaran, Rigel to name a few). For more about this, see the database of Islam and astronomy (will display in another window). However, advances in the explanations of the motions of the stars and planets were made by astronomers in Europe starting in the 16th century.

Martin Waldseemüller’s world map of 1507. Select the image to go to Jim Siebold’s Renaissance maps database from which this picture came (will display in another window). This is the first map that identified the land across the Atlantic Ocean as “America”. Exploration of that land was still quite new.

By the 16th century the following paradigm had developed: Man is God’s special creation of the physical universe; the Earth is the center of a mathematically-planned universe and we are given the gift of reading this harmony. The Greek ideal of finding logical, systematic explanations to physical events was rediscovered and celebrated to trading with islamic nations. Along with this came an unbounded faith in the power of reason to solve physical problems. This time in history is called the Renaissance (french for “rebirth”).

Scientists use a guiding principle called Occam’s Razor to choose between two or more models that accurately explain the observations. This principle, named after the English philosopher, William of Occam, who stated this principle in the mid-1300’s, says: the best model is the simplest one—the one requiring the fewest assumptions and modifications in order to fit the observations. Guided by Occam’s Razor some scientists began to have serious doubts about Ptolemy’s geocentric model in the early days of the Renaissance.

Copernicus’ heliocentric universe

One such astronomer, Nicolaus Copernicus (lived 1473–1543 C.E.), found many deficiencies in the Ptolemaic model. He felt that any model of the planet motions must account for the observations and have circular, uniform motion. The Ptolemaic model did not do that. Also, the Ptolemaic model was not elegant and, therefore, “un-Godlike”. During the years between Ptolemy and Copernicus, many small epicycles had been added to the main epicycles to make Ptolemy’s model agree with the observations. By Copernicus’ time, the numerous sub-epicycles and offsets had made the Ptolemaic model very complicated. Surely, God would have made a cleaner more elegant universe!

Copernicus was strongly influenced by neoplatonism (beliefs that combined elements of Christianity and Platonism) in developing a model to replace Ptolemy’s. This led him to believe that the Sun is a material copy of God—God is the creative force sustaining life and the Sun gives us warmth and light. He adopted Aristarchus’ heliocentric (Sun-centered) model because he felt that God should be at the center of the universe. Copernicus’ model had the same accuracy as the revised Ptolemaic one but was more elegant.

Copernicus retained the Aristotelian notion that planets fulfill the goal of perfect (circular) motion. His model still used small epicycles to get the details of the retrograde loops correct, though they were only a minor feature. He used trigonometry to describe the distances of the planets from the Sun relative to the astronomical unit (average Earth-Sun distance), but he did not know the numerical value of the astronomical unit. He found that the planets farther from the Sun move slower. The different speeds of the planets around the Sun provided a very simple explanation for the observed retrograde motion.

Retrograde motion is the projected position of a planet on the background stars as the Earth overtakes it (or is passed by, in the case of the inner planets). The figure below illustrates this. Retrograde motion is just an optical illusion! You see the same sort of effect when you pass a slower-moving truck on the highway. As you pass the truck, it appears to move backward with respect to the background trees and mountains. If you continue observing the truck, you will eventually see that the truck is moving forward with respect to the background scenery. The relative geometry of you and the other object determines what you see projected against some background.

Select image to show animation of retrograde motion.Copernicus thought his model was reality but other people used his model as a more convenient calculation device only. If the Earth were moving around the Sun, then the stars should appear to shift due to our looking at them from different vantage points in our orbit (a “parallactic shift”). The parallax effect can be illustrated when you look at your thumb at arm’s length with one eye and then the other—your thumb appears to shift position!

 

Now imagine that the Earth at opposite points in its orbit is your left and right eye and the nearby star is your thumb and you have the situation illustrated below.

 

 

However, no parallactic shift was observed in the stars. If there was actually a very small parallactic shift, then the stars would have to be very far away. Copernicus’ contemporaries felt that God would not waste that much space! They argued that, therefore, there must be no parallactic shift at all—the Earth is not in motion. Astronomers now know that the stars are indeed very far away and telescopes must be used to detect the small parallactic shifts.

Try out the Solar System Models module of the University of Nebraska-Lincoln’s Astronomy Education program for checking your understanding of the geocentric vs. heliocentric model explanation of the planet motions (link will appear in a new window).

Tycho Brahe’s excellent observations

Tycho Brahe (lived 1546–1601 C.E.) revived Heroclides’ model that had the all of the planets, except the stationary Earth, revolving around the Sun. Because Brahe was not a neoplatonist, he believed that the Sun, Moon, and stars revolved around the Earth. Tycho’s model was mathematically equivalent to Copernicus’ model but did not violate Scripture and common sense.

Tycho calculated that if the Earth moved, then the stars are at least 700 times farther away from Saturn than Saturn is from Sun. Since Tycho felt that God would not waste that much space in a harmonious, elegant universe, he believed that the Earth was at the center of the universe. Astronomers now know that the nearest star is over 28,500 times farther away than Saturn is from the Sun!

Though Tycho’s beliefs of the universe did not have that much of an effect on those who followed him, his exquisite observations came to play a key role in determining the true motion of the planets by Johannes Kepler. Tycho was one of the best observational astronomers who ever lived. Without using a telescope, Tycho was able to measure the positions of the planets to within a few arc minutes—a level of precision and accuracy that was at least ten times better than anyone had obtained before!

Sections Review

Vocabulary

• astronomical unit
• epicycle
• geocentric
• heliocentric
• instrumentalism
• Occam’s Razor
• paradigm
• Pythagorean paradigm
• realism
• retrograde motion

Review Questions 1

1. What two basic kinds of models have been proposed to explain the motions of the planets?
2. What is the Ptolemaic model? What new things did Ptolemy add to his model?
3. Why are epicycles needed in Ptolemy’s model?
4. Why was the Ptolemaic model accepted for more than 1000 years?
5. In what ways was the Ptolemaic model a good scientific model and in what ways was it not?
6. What is the Copernican model and how did it explain retrograde motion?
7. Why did Copernicus believe in his model?
8. Why did Copernicus not know the absolute distance between various planets and the Sun in his model? Explain what he would have needed to know to get the absolute distances.
9. What important contributions did Tycho Brahe make to astronomy?

4.5 Battle with the Church

In the 16th century the hierarchical structure of the Church’s authority was inextricably bound with the geocentric cosmology. “Up” meant ascension to greater perfection and greater control. God and heaven existed outside the celestial sphere. There was a gradation of existence and control from perfect existence to the central imperfect Earth. God delegated power to angels to control the planet movements and to guide the various earthly events. Plants and animals existed to serve humans and humans were to serve God through the ecclesiastical hierarchy of the Church.

Giordano Bruno (lived 1548–1600 C.E.) revived Democritus’ (a contemporary of Socrates) view that the Sun was one of an infinite number of stars. This infinite sphere was consistent with the greatness of God. Bruno believed in a heliocentric universe. He believed that God gave each of us an inner source of power equal to all others, so there was no justification for domination and servitude. His model had definite political ramifications that threatened the Church’s political authority.

4.5.1 Galileo

Select the image to go to the Galileo Project’s homepage at Rice University (will display in another window).

One of Galileo’s telescopes. Select the image to go to Joseph Dauben’s homepage for The Art of Renaissance Science.

Galileo Galilei (1564–1642 C.E.) was the first person we know of that used the telescope for astronomical observations (starting in 1609). The telescope was originally used as a naval tool to assess the strength of the opponent’s fleet from a great distance. He found many new things when he looked through his telescope:

Galileo’s drawing of the Moon as seen through one of his telescopes.

1. The superior light-gathering power of his telescope over the naked eye enabled him to see many, many new fainter stars that were never seen before. This made Bruno’s argument more plausible.
2. The superior resolution and magnification over the naked eye enabled him to see pits and craters on the Moon and spots on the Sun. This meant that the Earth is not only place of change and decay!
   Galileo’s drawing of the Moon as seen through one of his telescopes.
3. With his superior “eye” he discovered four moons orbiting Jupiter. These four moons (Io, Europa, Ganymede, and Callisto) are called the Galilean satellites in his honor. In this system Galileo saw a mini-model of the heliocentric system. The moons are not moving around the Earth but are centered on Jupiter. Perhaps other objects, including the planets, do not move around the Earth.
4. He also made the important discovery that Venus goes through a complete set of phases. The gibbous and full phases of Venus are impossible in the Ptolemaic model but possible in Copernican model (and Tychonic model too!). In the Ptolemaic model Venus was always approximately between us and the Sun and was never found further away from the Earth than the Sun. Because of this geometry, Venus should always be in a crescent, new, or quarter phase. The only way to arrange Venus to make a gibbous or full phase is to have it orbiting the Sun so that, with respect to our viewpoint, Venus could get on the other side of, or behind, the Sun further away from us than the Sun. This was possible only if Venus orbited the Sun (see the figure at the end of the planetary motions section of the previous chapter).

For Galileo the clear observations of a heliocentric universe was a powerful weapon against the hierarchical structure of the seventeenth century Church. Galileo argued that the heliocentric model is not a mere instrument but is reality. He stated that his observations showed that. From the discussion in science methods chapter, you can argue that Galileo went too far in what conclusions he could draw from his observations. A scientific model cannot be proven correct, only disproven. A model that survives repeated tests is one that is consistent with the available data. His observations were consistent with the heliocentric model, but could also be explained with a geocentric model like Tycho’s. But for Galileo, the observations were enough—he was convinced of the heliocentric system before he used his telescope and his observations confirmed his belief. More convincing evidence of the Earth’s motion around the Sun would have to wait until 1729 when James Bradley (lived 1693–1762 C.E.) discovered that a telescope has to be slightly tilted because of the Earth’s motion, just as you must tilt an umbrella in front of you when walking quickly in the rain to keep the rain from hitting your face. The direction the telescope must be tilted constantly changes as the Earth orbits the Sun. Over a century later, Friedrich W. Bessel (lived 1784–1846 C.E.) provided further evidence for the Earth’s motion by measuring the parallax of a nearby star in the late 1830s. The measurements of Bradley and Bessel required technology and precision beyond that of Galileo’s time. The telescope tilt angle is less than half of an arc minute and the parallax angle of even the nearest star is less than one arc second. Recall from the angles section in the previous chapter how tiny is an arc second, so you can understand why the people of Copernicus’ time or Galileo’s time could easily discount the heliocentric idea.

Galileo also made advances in understanding how ordinary objects move here on the Earth. He set up experiments to see how things move under different circumstances. He found that Aristotle’s view of how things move was wrong. Galileo’s observations contradicted the long-unchallenged physics of Aristotle, who taught that in order for something to keep moving at even a constant speed, a force must be continually applied. Aristotle also thought that something falling will fall at a constant speed and that heavier things will always fall more quickly than lighter things. Galileo discovered that an object’s motion is changed only by having a force act on it. He also discovered that objects falling to the ground will accelerate as they fall and that all objects, regardless of size, would fall with thesame acceleration in the absence of air drag.

Galileo’s studies of how forces operate also provide the foundation to prove that Earth spins on its axis. Although the stars and Sun appear to rise and set every night or day, they are actually stationary. Evidence of the Earth’s rotation (from west to east) is seen with the deflection of objects moving in north-south direction caused by the differences in the linear speed of the rotation at different latitudes. All parts of the Earth take 23 hours 56 minutes to turn once, but the higher latitudes are closer to the Earth’s rotation axis, so they do not need to rotate as fast as regions nearer the equator. A moving object’s west-east speed will stay at the original value it had at the start of its motion (unless some force changes it). If the object is also changing latitudes, then its west-east speed will be different than that for the part of the Earth it is over. Therefore, moving objects appear to be deflected to the right in the northern hemisphere and to the left in the southern hemisphere. This is called the coriolis effectafter Gustave-Gaspard Coriolis (lived 1792–1843 C.E.) who deduced the effect in 1835 to explain why cannonballs shot long distances kept missing their target if the cannon was aimed directly at its target. See the energy flow section of Chapter 9 for applications (and illustrations) of the Coriolis effect to planet atmospheres.

Jean-Bernard-Léon Foucault (lived 1819–1868) gave the first laboratory demonstration of the Earth’s spin in 1851. A large mass suspended from a long wire mounted so that its perpendicular plane of swing is not confined to a particular direction, will rotate in relation to the Earth’s surface. The only forces acting on the ball are gravity and the wire tension and they lie in the plane of oscillation.

There are no forces acting on the ball perpendicular to the oscillation plane, so the oscillation direction in space does not change. However, it does rotate relative to the Earth’s surface because the Earth is rotating under the swinging pendulum. The pendulum appears to rotate westward with a period that depends on the latitude: rotation period = (23^h 56^m)/sin(latitude), where “sin” is the trigonometric sine function. The coriolis effect and the Foucault pendulum are both based on Galileo’s discovery that an object’s motion (speed and/or direction) are changed only if there is a force acting on it.

Galileo is often considered the father of modern science because his ideas were not derived from thought and reason alone. He used the guidance of nature (experiments). This marked a revolutionary change in science—observational experience became the key method for discovering nature’s rules. His arguments for the heliocentric model and the critical role of objective observation of nature in science got him into trouble with the Church.

The struggle between Galileo and the Church was not a battle between science and religion but was part of a larger battle over different conceptions of the proper routes to knowledge, God, and world view.

Galileo’s intent was to improve the Church by giving a truer understanding of how God actually worked in the physical universe and by allowing greater access to God for more people. Galileo loved to debate and had the bad habit of ridiculing those he disagreed with. Some of those he ridiculed were powerful political figures in the Church. He wrote a book detailing the arguments for and against his model of the universe in a way that ridiculed the official view of the Church. It was written in Italian (the language of everyday discourse) rather than the scholarly Latin, so even non-scholars were exposed to his scathing arguments against the geocentric universe. He may have had more success in getting greater acceptance of his different views of God and research if his style was different but perhaps his ideas needed just such a champion at that time.

This section is titled “Battle with the Church” but that is perhaps a bit misleading when you take a careful look at what really happened between Galileo and church authorities. The disagreement between Galileo and some church figures is often cast as a battle between science (reason, the “good guy”) and religion (faith, the “bad guy”) in the popular media today but science historians know that the debate was not quite so clear cut—both sides were right on some points and wrong on others; reality is often messier than what is portrayed in the media. For example, Galileo’s argument that his observations showed that the heliocentric model is correct and that the Earth cannot be the center of the universe was only partly correct. While some reactionary church officials thought Galileo was a heretic because of their narrow more literal interpretation of scripture, other church officials knew that Galileo’s observation disproved Ptolemy’s geocentric model but not something like Tycho’s geocentric model. Those church officials knew the philosophical difference between disproving one model vs. proving another one correct. Other examples of the messy reality vs. the simplistic media view of the “Galileo affair” is found in Lawrence Principe’s “Science and Religion” lectures (see the fifth and especially the sixth lectures in the series).

Review Questions 2

1. What important contributions did Galileo make to modern science?
2. What were his astronomical discoveries and why was he able to make those discoveries?
3. Why did he get into political hot water? (Also reference the “Galileo and the Inquisition” activity to answer this question.)
4. What observation finally disproved the Ptolemaic model?
5. Why is Galileo sometimes referred to as the first “modern scientist”?
6. What evidence is there that the Earth is rotating and that it is revolving around the Sun?

4.6 Kepler’s Laws of Planetary Motion

Johaness Kepler (lived 1571–1630 C.E.) was hired by Tycho Brahe to work out the mathematical details of Tycho’s version of the geocentric universe. Kepler was a religious individualist. He did not go along with the Roman Catholic Church or the Lutherans. He had an ardent mystical neoplatonic faith. He wanted to work with the best observational data available because he felt that even the most elegant, mathematically-harmonious theories must match reality. Kepler was motivated by his faith in God to try to discover God’s plan in the universe — to “read the mind of God.” Kepler shared the Greek view that mathematics was the language of God. He knew that all previous models were inaccurate, so he believed that other scientists had not yet “read the mind of God.”

Since an infinite number of models are possible (see Plato’s Instrumentalism above), he had to choose one as a starting point. Although he was hired by Tycho to work on Tycho’s geocentric model, Kepler did not believe in either Tycho’s model or Ptolemy’s model (he thought Ptolemy’s model was mathematically ugly). His neoplatonic faith led him to choose Copernicus’ heliocentric model over his employer’s model

Kepler tried to refine Copernicus’ model. After years of failure, he was finally convinced with great reluctance of an revolutionary idea: God uses a different mathematical shape than the circle. This idea went against the 2,000 year-old Pythagorean paradigm of the perfect shape being a circle! Kepler had a hard time convincing himself that planet orbits are not circles and his contemporaries, including the great scientist Galileo, disagreed with Kepler’s conclusion. He discovered that planetary orbits are ellipses with the Sun at one focus. This is now known as Kepler’s 1st law.

An ellipse is a squashed circle that can be drawn by punching two thumb tacks into some paper, looping a string around the tacks, stretching the string with a pencil, and moving the pencil around the tacks while keeping the string taut. The figure traced out is an ellipse and the thumb tacks are at the two foci of the ellipse. An oval shape (like an egg) is not an ellipse: an oval tapers at one end, but an ellipse is tapered at both ends (Kepler had tried oval shapes but he found they did not work).

There are some terms I will use frequently in the rest of this book that are used in discussing any sort of orbit. Here is a list of definitions:

1. Major axis — the length of the longest dimension of an ellipse.
2. Semi-major axis — one half of the major axis and equal to the distance from the center of the ellipse to one end of the ellipse. It is also the average distance of a planet from the Sun at one focus.
3. Minor axis — the length of the shortest dimension of an ellipse.
4. Perihelion — point on a planet’s orbit that is closest to the Sun. It is on the major axis.
5. Aphelion — point on a planet orbit that is farthest from the Sun. It is on the major axis directly opposite the perihelion point. The aphelion + perihelion = the major axis. The semi-major axis then, is the average of the aphelion and perihelion distances.
6. Focus— one of two special points along the major axis such that the distance between it and any point on the ellipse + the distance between the other focus and the same point on the ellipse is always the same value. The Sun is at one of the two foci (nothing is at the other one). The Sun is NOT at the center of the orbit!As the foci are moved farther apart from each other, the ellipse becomes more eccentric (skinnier). See the figure below. A circle is a special form of an ellipse that has the two foci at the same point (the center of the ellipse).
7. The eccentricity (e) of an ellipse is a number that quantifies how elongated the ellipse is. It equals 1 – (perihelion)/(semi-major axis). Circles have an eccentricity = 0; very long and skinny ellipses have an eccentricity close to 1 (a straight line has an eccentricity = 1). The skinniness an ellipse is specified by the semi-minor axis. It equals the semi-major axis × Sqrt[(1 – e²)].

Planet orbits have small eccentricities (nearly circular orbits) which is why astronomers before Kepler thought the orbits were exactly circular. This slight error in the orbit shape accumulated into a large error in planet positions after a few hundred years. Only very accurate and precise observations can show the elliptical character of the orbits. Tycho’s observations, therefore, played a key role in Kepler’s discovery and is an example of a fundamental breakthrough in our understanding of the universe being possible only from greatly improved observations of the universe.

Most comet orbits have large eccentricities (some are so eccentric that the aphelion is around 100,000 AU while the perihelion is less than 1 AU!). The figure above illustrates how the shape of an ellipse depends on the semi-major axis and the eccentricity. The eccentricity of the ellipses increases from top left to bottom left in a counter-clockwise direction in the figure but the semi-major axis remains the same. Notice where the Sun is for each of the orbits. As the eccentricity increases, the Sun’s position is closer to one side of the elliptical orbit, but the semi-major axis remains the same.

To account for the planets’ motion (particularly Mars’) among the stars, Kepler found that the planets must move around the Sun at a variable speed. When the planet is close to perihelion, it moves quickly; when it is close to aphelion, it moves slowly. This was another break with the Pythagorean paradigm of uniform motion! Kepler discovered another rule of planet orbits: a line between the planet and the Sun sweeps out equal areas in equal times. This is now known as Kepler’s 2nd law.

Later, scientists found that this is a consequence of the conservation of angular momentum. The angular momentum of a planet is a measure of the amount of orbital motion it has and does NOT change as the planet orbits the Sun. It equals the (planet mass) × (planet’s transverse speed) × (distance from the Sun). The transverse speed is the amount of the planet’s orbital velocity that is in the direction perpendicular to the line between the planet and the Sun. If the distance decreases, then the speed must increase to compensate; if the distance increases, then the speed decreases (a planet’s mass does not change).

Finally, after several more years of calculations, Kepler found a simple, elegant equation relating the distance of a planet from the Sun to how long it takes to orbit the Sun (the planet’s sidereal period). (One planet’s sidereal period/another planet’s sidereal period)² = (one planet’s average distance from Sun/another planet’s average distance from Sun)³. Recall that the semi-major axis is the average distance from the Sun (average of perihelion and aphelion). If you compare the planets to the Earth (with an orbital period = 1 year and a distance = 1 A.U.), then you get a very simple relation: (a planet’s sidereal period in years)² = (semi-major axis of its orbit in A.U.)³. This is now known as Kepler’s 3rd law. A review of exponents and square roots is available in the mathematics review appendix.

For example, Mars’ orbit has a semi-major axis of 1.52 A.U., so 1.52³ = 3.51 and this equals 1.87². The number 1.87 is the number of years it takes Mars to go around the Sun. This simple mathematical equation explained all of the observations throughout history and proved to Kepler that the heliocentric system is real. Actually, the first two laws were sufficient, but the third law was very important for Isaac Newton and is used today to determine the masses of many different types of celestial objects. Kepler’s third law has many uses in astronomy! Although Kepler derived these laws for the motions of the planets around the Sun, they are found to be true for any object orbiting any other object. The fundamental nature of these rules and their wide applicability is why they are considered “laws” of nature.

Select the image to show an animation of Kepler’s 3 laws. A nice java applet for Kepler’s laws is available on the web (select the link to view it in another window).

The UNL Astronomy Education program’s Planetary Orbit Simulator allows you to manipulate the various parameters in Kepler’s laws to understand their effect on planetary orbits (link will appear in a new window).

Vocabulary

• angular momentum
• aphelion
• eccentricity
• ellipse
• focus
• Kepler’s 1st law
• Kepler’s 2nd law
• Kepler’s 3rd law
• perihelion
• semi-major axis

Review Questions 3

1. What shape are planet orbits and where is the Sun with respect to the orbit?
2. What happens to a planet’s orbital speed as it approaches its farthest point from the Sun and as it approaches its closest point? How is it related to angular momentum?
3. How were Kepler’s laws of planetary motion revolutionary or a radical break from earlier descriptions of planetary motion?
4. A moon’s closest distance from a planet is 300,000 km and its farthest distance is 500,000 km. What is the semi-major axis of its elliptical orbit?
5. How will the semi-minor axis compare with the semi-major axis for an ellipse with eccentricity = 0.1, 0.5, 0.8, 0.99? Find the value of (semi-minor/semi-major) for each of the eccentricities.
6. How will the perihelion compare with the aphelion for an ellipse with eccentricity = 0.1, 0.5, 0.8, 0.99? Find the value of (perihelion/aphelion) for each of the eccentricities. [Hint: using the relation that the perihelion + aphelion = 2× semi-major axis and a little algebra, you can find that (perihelion/aphelion) = (1-e)/(1+e).]
7. How is the average distance between a planet and the Sun related to the planet’s orbit period?
8. Which planet has a shorter period—one with a large average distance, or one with a small average distance?
9. What is the semi-major axis of an asteroid orbiting the Sun with a period of 64 years? (Kepler’s third law works for any object orbiting the Sun.)

4.7 Logic of Discovery? Beliefs and Objectivity

Often a mathematical idea or model is discovered with no apparent application to the physical world until many years later. This aspect of pure, basic scientific research is not popular among government officials who want practical applications NOW! How are scientific discoveries made? There are several views about how we make discoveries and why humans are able to do this.

Kepler believed that there is a creationary resonance between the human mind and the laws of nature. In this view God creates humans with the gift of reading the mathematical harmonies of God’s mind. It is only a matter of time for someone to discover God’s plan. A more modern view held by some says that there is an evolutionary resonance between the human mind and the laws of nature. Given the infinite variety of paths of evolution, it is inevitable that creatures will eventually evolve capable of reading the laws of nature. In this view, scientific progress is inevitable.

Is creativity actually a logical process in disguise? It is a common belief today that one’s religious/philosophical beliefs are merely along for the inevitable revolutionary ride and are not necessary to make revolutionary scientific advances. Some believe that there are many technically-capable paths by which the universe can be modeled. Kepler’s neoplatonism was not logically necessary for the discovery of the planetary laws of motion, but, historically, it may have been absolutely necessary for his time and place.

Every age has its paradigms. Though scientists try to be objective, philosophical considerations do intrude on the scientific, creative process. That is not a bad thing because these beliefs are crucial in providing direction to their inquiries and fuel for the creativity mill. Scientists have faith that there is some order in the universe and this faith keeps them striving to solve the cosmic problems.

Facts have little meaning without ideas to interpret them. Because science is a human discipline, there is no machine-like objectivity. Often crucial facts supporting an idea come after a commitment is made to the idea. So is science then all based on an individual’s whim; relative to the scientist’s time and place? The self-corrective enterprise of science is messier than most science textbooks would have you believe. Besides the inevitable cultural prejudices, scientists have, in principle, an infinite number of conceivable ideas to choose from. How do you separate reasonable ideas from the infinite number of merely conceivable ideas?

Sure, there are cultural biases, but science does make us confront the real world — reality kicks back. You can ignore the discrepancies between nature’s truth (observations) and your theories of what should happen only for so long. Experiments are the sole judge of scientific truth — nature eventually wins. The ideas are crucial to understanding the world but they eventually yield to the facts. Science makes us confront the world.

References

For further reading, here are some of my references that cover the development of science using historical records. They also cover science’s philosophical underpinnings using the tools of philosophy.

1. Owen Barfield Saving the Appearances pp. 46–54.
2. Paul K. Feyerabend Galileo and the Tyranny of Truth in The Galileo Affair: A Meeting of Faith and Science ed. Coyne, Heller, Zycinski (Vatican City: 1985), pp. 155-166 and other papers from that symposium held at the Vatican.
3. George S. Johnston 1995, The Galileo Affair (Scepter Press: Princeton, NJ). HTML version version available here.
4. Thomas S. Kuhn The Copernican Revolution: Planetary Astronomy in the Development of Western Thought (Cambridge, Mass: Harvard Univ. Press, 1957).
5. Ronald Pine Science and the Human Prospect (Wadsworth Publ. Co: Belmont, CA, 1989) esp. ch. 5: pp. 130-162.
6. A prelude to Copernicus is Owen Gingerich’s Scientific American article Astronomy in the Age of Columbus Nov. 1992, pp. 100-105.

Is this page a copy of Strobel’s Astronomy Notes?