CHAPTER 2

The discussion of astrology in an Advancing the Model box provides a useful way to examine if a model or worldview has a scientific basis. Astrology, unlike a scientifically based model, has made few adjustments as the evidence has been changing. A key example is the inability or unwillingness of most of astrology to adjust to precession of the equinoxes. Asking your group of friends a simple question such as "How many of you are Leos?" can then lead to an understanding that the sun signs that are currently assigned do not take into account precession. The objective here is not to show that the astrological worldview is wrong but that it is nonscientific and pseudoscientific in its use of computers, orbits, and action-at-a-distance concepts.

While studying the work of Aristarchus and Eratosthenes, it is important to keep in mind that they were able to accomplish such important work with nothing but simple logical arguments and naked-eye observations. It is not important that their results at the end were not very close to what we know now, especially in the case of Aristarchus. What is important is the logical arguments they used.

1 It is not easy to define what science is. However, any effort to define it must include its methods, its historical development, its social context, and a clear understanding of its language.

2 It is especially important to recognize the different meanings of the words "fact" and "theory" as they are used in science compared to their everyday meanings.

3 The scientific method is a never-ending cycle of hypothesis, prediction, data gathering, and verification.

4 A scientific hypothesis is an educated guess made in describing the results of an experiment or observation. A hypothesis can be wildly speculative but must be testable.

5 A scientific fact is generally a close agreement by competent observers of a series of observations of the same phenomenon.

6 A scientific law or principle is a scientific hypothesis that has been repeatedly tested and has not been contradicted.

7 A scientific model is a description of a phenomenon, based on observation, experiment, and theoretical considerations. It is not necessarily the truth or reality, but a description that allows prediction of future events.

8 A scientific theory is a synthesis of a large body of information that encompasses well-tested (by repeatable experiments) and verified hypotheses about certain aspects of the natural world. No theory can be proven to be true, but data can prove a theory to be false.

9 In science a scheme is not usually called a theory until its ideas are shown to fit observed data successfully. In every-day language, however, the word "theory" is often used to refer to ideas that are much more fanciful and less secure.

Criteria for Scientific Models

1 The model must fit the data.

2 The model must make predictions that can be tested and be of such nature that it would be possible to disprove the model.

3 The model should be as simple as possible. The principle that the best explanation is the one that requires the fewest unverifiable assumptions is called Occam’s razor.

1 We examine these theories to answer the question of where the Earth fits into the scheme of things, to see how well they match the criteria for a good scientific theory.

2 To understand how astronomy—and science in general—works, we must look at how it progresses with time.

1 There is a fundamental difference between the contributions to astronomy made by the ancient Greeks and those made by other ancient civilizations. They were interested in astronomy because of a pure philosophical desire to understand how the universe works. They believed in, and looked for, a sense of symmetry, order, and unity in the cosmos. They took the first steps in creating a unified model of the universe.

2 Aristotle argued that the absence of parallax for the stars in the sky implied that the Earth must be at the center of the solar system. This is a valid scientific argument.

3 Parallax is the apparent shifting of nearby objects with respect to distant ones as the position of the observer changes.

4 Stellar parallax was not observed until 1838; the greatest annual shift observed for any star is only 1.5 arcseconds.

5 Even though Aristotle used a correct logical argument, the conclusion was wrong because it was based on incomplete data. Parallax is hard to observe because stars are at great distances from us.

6 Aristotle used very good arguments to conclude that the Moon and Earth are spherical and that the Sun is farther away from earth than the Moon is.

7 Aristotle saw a difference in the "natural" behavior of Earthly objects compared to heavenly objects. He believed that two different sets of rules existed, one for Earthly objects and one for celestial objects.

8 The Greeks’ love of geometry led them to construct a model of the heavens based on spheres, with the Earth at the center. In order to account for the Sun’s apparent motion in the sky, the Sun was located on a sphere around the Earth, inside the celestial sphere of the stars. The axes of the two spheres were tilted with respect to one another.

9 Ptolemy (150 AD) presented the most comprehensive geocentric model, called the Ptolemaic model. Presented in his book called the Almagest, it held sway for more than 1,300 years.

10 Because the heavens were viewed as perfect, the use in the Ptolemaic model of the symmetrical circle to model the motions of celestial objects was thought to be the most reasonable choice.

11 Five planets are visible to the naked eye: Mercury, Venus, Mars, Jupiter, Saturn.

12 The planets lack the simple, uniform motion of the Sun and Moon. They sometimes stop their eastward motion among the stars and move westward for a while. This is called retrograde motion.

13 The planets always stay near the ecliptic. In addition, Mercury and Venus never appear very far from the position of the Sun in the sky. Thus their elongation (the angle in the sky from an object to the Sun) is small.

14 Any model for the planets must explain these observations.

1 Ptolemy’s geocentric model was able to explain the planetary motions using epicycles. An epicycle is the circular orbit of a planet, the center of which revolves around the Earth in another circle.

2 The model retained the idea of perfect heavenly circles and uniform speeds. (In an effort to match the observations, Ptolemy actually did try models with varying speeds for the planets and moved the Earth off-center among the planets’ paths.) The model explained why the planets never move far from the ecliptic, but treated Mercury and Venus as special cases in order to explain their small elongations.

3 Ptolemy’s model meets the first two criteria fairly well but it is much less successful with the third.

4 The Ptolemaic model did fit the data, so we must judge it as an acceptable model even though it lacked that certain neatness we would like.

1 400 years before Ptolemy, around 280 BC, the Greek philosopher Aristarchus proposed a moving-Earth solution to explain celestial motions. He introduced the concept of a spinning Earth and the first heliocentric model, 1800 years before Copernicus.

2 Even though Aristarchus could not explain the lack of observable parallax at his time (Aristotle’s argument), he believed that the Sun was at the center of the solar system because it was much bigger in size than the Earth.

3 With powerful and simple arguments based on observations he concluded that the Sun was about 20 times farther from the Earth than the Moon is. He showed that the Earth is 3 times larger than the Moon in diameter, and that the Sun is about 20 times larger than the Moon in diameter. This implies that the Sun is about 7 times larger than the Earth in diameter.

4 Aristarchus was the first to create a map of the solar system. He simply did not have the scale for it.

1 Eratosthenes (276–195 BC) was the first person to clearly understand the shape and approximate size of the Earth.

2 By comparing shadows at noon during summer solstice at two different locations he understood that the Sun must be directly overhead (at the zenith) in Syene but that the Sun’s direction was off the vertical by 7° in Alexandria.

3 He realized that the 7° difference was due to the Earth’s curvature and therefore the Earth’s circumference was about 360/7 C 50 times the distance between the two cities. Knowing this distance he was able to find the Earth’s diameter. His calculation was very close to the correct value.

4 Combining the calculations of Aristarchus and Eratosthenes, the ancient Greeks had for the first time measurements of the radii of Earth, Moon, and Sun and their relative distances. We had to wait until 1769 AD to observe the actual value of the astronomical unit and thus the true dimensions of the solar system.

5 The important point here is not the accuracy of the measurements but the power of simple logical arguments that allowed the ancient Greeks to have a very good sense of the solar system more than 2000 years ago.

1 In the 13th century St. Thomas Aquinas blended the natural philosophy of Aristotle and Ptolemy’s work with Christian beliefs.

2 A central, unmoving Earth fit perfectly with Christian thinking and a literal interpretation of the Bible.

3 People during the Middle Ages placed a great reliance on authority, especially authorities of the past.

1 Copernicus, a contemporary of Columbus, worked for 40 years on a heliocentric—Sun-centered—model for two reasons:

(a) Ptolemy’s predicted positions for celestial objects had become less accurate over time.

(b) The Ptolemaic model was not aesthetically pleasing enough.

1 Copernicus’ system revived many of the ideas of Aristarchus. An Earth that rotates from west to east under a stationary sky produces the same observations as a rotating celestial sphere from east to west around a stationary Earth.

2 Copernicus’ system is heliocentric with the Earth being just another one of the planets, all of them revolving around the Sun.

3 As seen from high above the Earth’s North Pole, all planets move in a counterclockwise direction, with the planets closer to the Sun moving faster than those farther away.

4 In order to explain the apparent motion of the Sun in the sky, Copernicus’ model had the plane of the Earth’s equator tilted with respect to the plane of its orbit around the Sun.

5 Copernicus’ model explains the generally west to east motion of the planets, as does Ptolemy’s. However, the observed retrograde motion of planets such as Mars is explained more simply in the Copernican system. Retrograde motion is a natural result in a heliocentric system.

6 Copernicus had the Moon revolving around the Earth and all the planets circling the Sun.

1 Accuracy in Fitting the Data

(a) Copernicus’ model was not accurate enough to account for all observed planetary motions. Because of his assumption of uniform motion (like Ptolemy), Copernicus was forced to add small epicycles of his own to improve accuracy.

(b) Copernicus did not abandon the circle as the preferred planetary orbit because he thought circles are the best representative of the repetitive motions of the heavens.

(c) Once parallax was observed (in 1838), it provided obvious evidence that the heliocentric model is the better one. Stellar parallaxes prove the Earth moves. Parallax also provided evidence that stars are not all at the same distance from Earth, which was assumed in both the Copernican model and the Ptolemaic model.

(d) Using the evidence available in the 1500s, both models had about the same errors.

2 Predictive Power

(a) A good theory (or model) must make testable predictions that might allow the theory (or model) to be disproved.

(b) Both the Copernican and Prolemaic models made predictions about parallax. When parallax was finally observed, it proved that the Ptolemaic model was wrong.

(c) The Copernican model also made predictions about relative distances of the then known planets from the Sun; these predictions were (much) later confirmed.

3 Simplicity: Mercury and Venus

(a) Copernicus liked his model because it was aesthetically more pleasing than the Ptolemaic model. A good model is nearly always simple and elegant in its power to explain and predict.

(b) The Copernican model was more aesthetically pleasing: it could explain the motions of Mercury and Venus without resorting to special rules needed by the Ptolemaic model. (Recall that Mercury and Venus never appear far from the Sun in the sky.)

(c) Copernicus offered a simpler explanation for retrograde motion that required no use of epicycles. He did use epicycles, however, in order to make his model fit as accurate as possible.

(d) Copernicus, who died in 1543 just as his book De Revolutionibus was published, started such an upheaval in people’s thinking that the word "revolution" took on a second meaning that is so familiar to us today.

1 Tycho (1546–1601) was born 3 years after Copernicus died. After developing an interest in astronomy and learning that both the Ptolemaic and Copernican models were based on inaccurate recorded data, he decided to obtain more accurate observations of planetary positions.

2 Tycho built the largest and most accurate naked-eye instruments yet constructed. (The telescope had not yet been invented.) They allowed him to measure angles to within 0.1°, close to the limit the human eye can observe.

3 He not only made careful measurements, but he recorded the accuracy of each measurement. The inclusion of an error in a measurement is now a common practice in science.

1 The quality of his observations led Tycho to reject the Copernican model because he could not observe parallax for the distant stars. The irony here is that his observations of a "new" star (a supernova) and a bright comet led him to conclude that these objects were far away from the Earth because he could not measure their parallax.

2 Tycho’s model of the heavens was a mix between the Ptolemaic and Copernican models. He positioned the Earth at the center with the Sun revolving around it, but he argued that the other planets were revolving around the Sun.

1 In 1600, a year before Tycho died, he hired Johaness Kepler as his assistant. Tycho’s accurate measurements of planetary positions (especially for Mars) proved invaluable for Kepler.

2 After 4 years and 70 combinations of circles and epicycles, Kepler devised a combination that would predict Mars’ position when compared to Tycho’s data to within 0.13°. Knowing the quality of Tycho’s work, Kepler decided to abandon the circle as the basic motion of the planets.

3 The shape that worked not only for Mars but for every planet Kepler had data was the ellipse.

1 The ellipse is a geometrical shape of which every point is the same total distance from two fixed points (the foci).

2 Eccentricity is the ratio of the distance between the foci divided by the longest distance across the ellipse (major axis).

1 First Law: Each planet’s path around the Sun is an ellipse, with the Sun at one focus of the ellipse (there is nothing at the other focus).

2 Second Law: A planet moves along its elliptical path with a speed that changes in such a way that a line from the planet to the Sun sweeps out equal areas in equal intervals of time.

The second law implies that a planet moves fastest when it is at the nearest point to the Sun (the perihelion) while it moves most slowly at its farthest point from the Sun (the aphelion).

1 Third Law: The ratio of the cube of a planet’s semimajor axis (the average distance a from the Sun) to the square of its sidereal period P is the same for each planet: a 3 / P 2 = constant

2 A planet’s sidereal period is the time the planet takes to complete a full orbit around the Sun, relative to the stars. This differs from the synodic period, which is the time between two successive identical configurations between a planet and the Sun, as seen from Earth.

1 Kepler’s modification to the Copernican model brought it into conformity with the data. Finally, the heliocentric theory worked better than the old geocentric theory.

2 Kepler’s breakthrough choice of ellipses to explain planetary motion was empirical —ellipses worked but he did not know why they worked.

We begin this chapter with a discussion of what science is and science’s "ways of knowing." We describe three criteria in evaluating a scientific theory: fitting the data, the ability to make testable predictions, and being simple ("aesthetically pleasing"). We describe and contrast the Ptolemaic (geocentric) model and the Copernican (heliocentric) model of the heavens and use them as examples to study the question of what makes a good scientific theory. We highlight the retrograde motion of Mars as a key observation that planetary motion models must explain. We also present the first heliocentric model by Aristarchus and describe Eratosthenes’ method to measure the Earth’s radius. We then discuss the observational contributions of Tycho Brahe and Kepler’s laws of planetary motion (as a modification to the Copernican system).

After studying the material in this chapter you should be able to:

1 Give three criteria for scientific models.

2 Describe the fundamental basis of the Greek celestial model.

3 Describe the observed motions of the planets that must be explained by a scientific model explaining planetary motion.

4 Evaluate astrology on the basis of criteria for scientific models.

5 Describe the arguments used by Eratosthenes in finding the Earth’s radius.

6 Describe the arguments used by Aristarchus in finding the relative distances between the Earth, Moon, and Sun, their relative sizes, and the argument in support of a heliocentric system.

7 Describe the cultural mindset at the time of Copernicus that placed a heavy reliance on authority.

8 Describe the heliocentric model, and the kinds of planetary motions it could predict.

9 Describe how the Ptolemaic and Copernican model could be compared to determine how each was rated on the desirable model attributes.

10 Describe why circular orbits were abandoned in favor of elliptical orbits.

11 Draw an ellipse using string and two thumbtacks.

12 Describe Kepler’s three laws of planetary motion.