The Solar System. I. Introduction. S. Gavin and S. P. Karrer Physics Department, Wayne State University, Detroit, MI,

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1 The Solar System S. Gavin and S. P. Karrer Physics Department, Wayne State University, Detroit, MI, I. Introduction As children we all learned that the Earth is one of many planets circling our Sun. This idea is so central to our understanding of our place in the Universe our cosmology that it s difficult to imagine that somebody once had to conceive this idea and prove it correct. This case study explores where our understanding of the solar system came from. The idea of a Sun-centered solar system followed from the first realization in ancient times that the movement of stars in the sky was linked with cosmology. Origin myths had existed in every culture, but the association of cosmology with something you could observe was truly revolutionary. Things to bear in mind when studying this case: 1. Every generation has had its own cosmology, from Aristotle to Copernicus. We have always been completely convinced that our model is the correct one. Today, the bigbang model explains the observed expansion of the universe along with many other observations. But has anybody heard of dark energy? 2. Forget about gravity! Gravity allowed Newton to calculate planetary orbits with a formula that could be tested in laboratory s on the Earth. This is proof positive of the heliocentric system, and makes Aristotle and Ptolemy seem insane to contemporary students. But that is not the way any of this was discovered. In fact, most details of the solar system had been discovered earlier, and informed Newton s work. 3. Cosmological models have implications for the general public that reach far beyond science. Religion and spirituality has had a profound role in the development and acceptance of these models. The role of the scientific method in the development of the heliocentric model is discussed in The Copernican Revolution by Thomas Kuhn [1]. While understanding the scientific details is not vital to following this case, ref. [2] is a useful web resource for the related astronomy and some of the history. We have also used a number of pictures from the Astronomy Picture of the Day site [3] and other web resources such as the Science U cite, which contains an explanation of retrograde motion [4]. [1] Thomas Kuhn, The Copernican Revolution, (Harvard University Press, Cambridge, 1957). [2] Nick Strobel, Astronomy Notes, [3] Astronomy Picture of the Day, [4] Retrograde Motion, [5] Galileo Galilei, Sidereus Nuncius (the Starry Messenger), trans. and annotated by A. Van Helden, (University of Chicago Press, Chicago, 1989).

2 II. Ancient Greece and the Cosmos: The Sun appears to rise in the east and set in the west. At night, the stars also seem to work their way east-to-west across the heavens. Some stars rise and set as the Sun does, while others travel in circular paths, as shown in fig. 1. The stars don t move relative to one another, as constellations appear move in fixed formation. This lead the ancient Greeks to imagine that the stars were fixed to a celestial sphere that rotated daily. Even today, astronomers describe the position of stars using coordinates that essentially give the longitude and latitude on this celestial sphere. Figure 1: Time-lapse image of star trails the graceful concentric arcs traced by the stars during the night. Josch Hambsch produced this composite of star trails around the South Celestial Pole with an effective "all night" exposure time of almost 11 hours, from [3]. The Greeks believed that the crystalline celestial sphere rotated around a stationary, spherical Earth. Careful observation of the way the stars move at different latitudes suggest that the Earth is spherical. In the northern hemisphere, the number of stars that travel in complete circles (as opposed to rising and setting) decreases the further south you go. Other observations support the picture of a spherical Earth, such as the `sinking of the masts of ships as they sail into the horizon. Eratosthenes was even able to measure the radius of the Earth by observing the shadow of the sun at different latitudes. The Sun and the planets move roughly as the stars do, but with highly important differences. The daily motion of the Sun is practically indistinguishable from that of any star near its position on the celestial sphere. The ancients realized that the stars do not disappear during the day, but that the Sun outshines them when it is above the horizon. However, the Sun s arc across the sky changes seasonally, e.g., the position of the noonday Sun is furthest north in the summer, and the days are longer. This is not so for the stars. This can be explained if the Sun is fixed to its own transparent sphere inside the celestial sphere. The seasonal variation is explained if the north pole of the Sun s crystalline sphere is tilted by 23.5 degrees with respect to the celestial sphere. The motion of the planets is even more interesting, as we

3 discuss shortly. In this picture, each planet requires a crystalline sphere of its own. What really delighted the ancient Greeks was the symmetry of spheres within spheres. In Aristotle's view the outer celestial sphere was the driving force for the many crystalline spheres that the Sun, Moon, and the planets rode upon. All of this was centered about the Earth. But there were many competing views to this `Aristotelian cosmology. Atomists believed that the Earth was just another planet and that no particular place in the universe was much different from another. The Pythagoreans suggested that the Earth was not the center of the universe but, rather, an unseen Altar of Zeus was. Heraclides put the Earth back in the center, but thought that the Earth rotated. He also imagined that Mercury and Venus orbited the sun. Aristarchus came closest to our modern view. He said that the sun was the center about which the Earth and the other planets rotated. This system still contained the sphere of stars, but much farther away. Many of these `alternative cosmologies were dismissed and ridiculed, especially in the Middle Ages. Q1. How does this picture fit in to the scientific method? What testable predictions does it suggest? Q2. What are the similarities and differences from the case of free fall studied earlier? Q3. The sphere-within-sphere model is very useful for astronomy and navigation. Good calendars are really useful in farming and other seasonal activities. Of what use are the distinctions between the ideas of Aristotle and Aristarchus? Q4. What is the role may spirituality and religious considerations have had in the invention of this celestial sphere model? Fig. 2. Venus traces out this S shape on the celestial sphere. Astronomer Tunc Tezel constructed this composite illustrating Venus' retrograde motion against the background stars. This picture from [3] combines a series of 29 images recorded from April 3rd through August 7, 2004.

4 The motion of the planets presented a challenge for the sphere-within-sphere cosmology. Like the stars, the Sun and the planets move daily across the night sky from east to west. Viewed against the background of the fixed stars, careful observation shows that the Sun slowly drifts from west to east. The planets also drift toward the east most of the time. Sometimes, however, they seem to double back and move from east to west on the celestial sphere. These periods of retrograde motion, shown in fig. 2, were a challenge to understand or predict. III. The Standard Model Claudius Ptolemaeus (90 to 168 AD), or Ptolemy, created the first standard model of the cosmos, a model that lasted 1500 years. He developed a system describing the motion and placement of all the known major objects in our solar system. Following Aristotle, he placed the Earth at the center, unmoving, with everything else revolving around it. This worked well for the Sun, Moon, and stars, but not as well for the wandering planets. To describe retrograde motion in a Geocentric System, Ptolemy used an idea known as epicycles. Epicycles are extra orbits that circle an imaginary point called the `eccentric. It is the eccentric that follows a circular orbit around the Earth. Ptolemy s system is shown in fig. 3. Prior to Ptolemy, others had used epicycles to describe planetary motion. Ptolemy took the best of these models and consolidated them. He then used observations and added refinements of the model to improve agreement. See [1] and [2] for a discussion of some of the details. What emerged was a messy but effective way to predict the motion of planets and other astronomical events. Importantly, Ptolemy published his work and made it widely available this may have been the secret of his lasting impact! To be sure, this model conflicted with the fundamental ideas of the Aristotelian sphere-withinsphere cosmology, since there is no way for a planet to follow an epicycle without crashing through its crystal sphere. Ptolemy s view was that the model was only a mathematical tool for predicting planetary motion, the sole justification of which is its agreement of predictions with future observations. In contemporary physics, we call this a phenomenological approach. Such an approach is typically useful in developing future experiments in research areas where calculations from first principles are not practical. It s worth mentioning that interest in Astrology was one of the major driving forces behind Ptolemy s investigations. While he was skeptical of the extent of Astrology s predictive powers (he felt other factors such as upbringing also mattered), he wrote the most influential and popular work on Astrology from the time. His work was very influential in the Islamic world and parts of Europe.

5 Fig. 3. Ptolemy s Earth-centered system. Planets rotate about certain points, which themselves follow circular paths about the Earth. From ref. [4] Q5. What evidence did Ptolemy have for/against his system? Q6. Introductory physics courses use a number of phenomenological models. Can you name some? One of these models used in early chapters is later given a more fundamental explanation in later chapters. Which one? IV. Copernicus to Galileo Steady progress in observational astronomy was made throughout the Middle Ages. Islamic civilization rose in the Arabic countries, while Europe languished in poverty. The works of the Greek scholars were translated into Arabic. Astronomers thoroughly cataloged the heavens, giving many of the stars their present Arabic names. Work had continued on developing Ptolemy s model to make it more consistent with these improved observations. New cycles were added and the speed of rotation around these cycles modified. Ptolemy s work, presented in his 13 volume series called the Almagest was already complex. The improved models were even more so. Nicolaus Copernicus ( ) brought us the first draft of our current solar system model. He realized that the daily motion of the stars, Sun, and planets could be explained if the Earth was a sphere spinning on its axis. He explained the yearly drifting motion of the Sun from the Earth s perspective as a consequence of the Earth s rotation about the Sun. A 23.5 degree tilt of the Earth s axis relative to the Sun explained the seasons. Importantly, by placing the planets in the order shown in fig. 4, he could explain their retrograde motions in a fashion that its much more straightforward than Ptolemy. Figure 5 shows the Earth racing past an outer planet such as Mars. As the Earth passes that planet, the apparent motion of that planet relative to the Earth seems to reverse direction. Similarly, as an inner planet such as Venus executes retrograde motion when it passes the Earth [4]. Elements of the heliocentric system had been proposed earlier, and Copernicus was aware of this. As mentioned before, Heraclides and Aristarchus placed the sun at the center. Later, Nicholas of Cusa made arguments for the motion of the Earth itself. From the voyages of Columbus and others, it was well known that there was no end to the Earth. Most people

6 were convinced that the Earth was a sphere. Figure 4: Heliocentric system from ref. [4] Figure 5: Retrograde motion: as the Earth passes the outer planet, it image of the planet against the fixed stars appears to form a loop. From ref. [2]. Philosophical and religious considerations guided Copernicus thinking. He believed that God should the center about which everything revolved, so he followed Aristarchus and placed the Sun at the center. He argued that the natural motion of a sphere was to move in a circle, so the orbits of the planets were circular. His universe, too, was spherical, because it contains the most volume and is the most perfect of all shapes. But there was also an element of physics in Copernicus thinking. Those that argued against a spinning Earth claimed that it would fly apart. Copernicus argued that a spinning celestial sphere, being much larger in size, would experience an outward force that was much larger. It was much easier for the Earth to withstand this, being made of solid, earthy stuff. Nevertheless, it is important to realize that Copernicus was firmly rooted in the Ptolemaic tradition, even as he was breaking with it. His own writings were every bit as mathematical and technical as Ptolemy s, and ware intended for the same select audience of learned astronomers. Moreover, since Copernicus believed the planets followed spherical orbits at uniform speeds, there were small discrepancies with detailed observations. To account for these discrepancies, he added small epicycles of his own. But he could achieve a comparable accuracy to Ptolemaic model with far fewer epicycles. (We now know that the orbits are ellipses with speeds following Kepler s laws.) Copernicus felt that the most convincing aspect of his heliocentric picture was its relative mathematical and scientific simplicity. The epicycles and eccentrics of Ptolemy are reduced in complexity if you put the sun in the center. It is also more natural, as the speeds of the planetary orbits decrease as they move farther from the Sun. In contrast, their speeds and other characteristics were less systematic the Ptolemaic system. With Copernicus, planetary distances could be determined in terms of the distance from the Earth to the Sun using trigonometry and observations. In the geocentric system, a change in one epicycle meant nothing to other planets. But despite its simplicity and natural feel, Copernicus model was still only as accurate as the Ptolemaic model. Knowledge of Copernicus work was limited to refined circles because of the difficulty of his

7 writings. Opponents to the Copernican picture did, however, raise troublesome questions. Why did we not feel the Earth s motion? It would be centuries before the underlying mechanics became clear and the Coriolis effect would be demonstrated. If the Earth is circling the Sun, why didn t the stars move throughout the year? Very refined telescopes were needed to demonstrate the parallax effect, because the stars are so very far away. But the truly profound questions about our place in the universe did not come up until much later. Q7. Occam's razor states that the simplest explanation is best. Is this a valid scientific principle? Can you give an example of a modern day theory where simplicity and elegance are the primary justifications? Q8. It is also a valid scientific principle to stick to tried-but-true models that explain all observations. Nobody wants to waste effort on wrong theories. Can you name a few exciting scientific ideas that turned out to be wrong? Erasmus Reinhold lived at about the same time as Copernicus and was the first to create astronomical tables using Copernican methods. His Prutenic Tables, named for the Prussian sponsor, were more accurate than the formerly used (and very old) tables. The new methods gave higher accuracy on many important fronts (for example the positions of the planets). They were less accurate, however, on determining the length of the year. Despite this shortcoming, the new tables were used by many astronomers. This helped the heliocentric system gain a foothold. Q9. What does the lowered accuracy of the year say about the Copernican system? Q10. What does this show about scientific consensus and community? Tycho Brahé ( ) is remarkable more for his observations than his theories. With only his naked eye and what was essentially a million-dollar protractor, he was able to make extremely accurate measurements of planetary motion (to within one arc-minute for stars and four arc-minutes for planets). This laid the groundwork for Kepler to design his famous laws. He also developed a hybrid geo-heliocentric system in which the Earth was still at the center. The sun revolved around the Earth but all the other planets revolved around the sun. Brahé also made two key observations: a nova and comets. A new star appeared in the constellation Cassiopeia and gave evidence against the perfection of the heavens. The measurement of parallax of comets showed they were part of the heavens and provided more evidence. The comets also showed that the planets did not ride on crystalline spheres. If there were such spheres, the comets would have smashed through them, but they moved unhindered as the stars or planets. Q11. How do Brahé's measurements stand out in that era of science? Johannes Kepler ( ) systematized Brahé's observations in the framework of the heliocentric solar system. His three laws of planetary motion greatly simplified Copernicus model and rid the world of epicycles. Kepler's First Law states that planetary orbits are ellipses and the sun is at one of the foci of this ellipse. In particular, Brahé's observations of the orbit of Mars were described to high accuracy by an elliptical orbit. His Second Law states

8 that in a given time of orbit, the sector created by the arc that the planet has traveled will always have the same area. This law took the function of equant point from Ptolemy's system. Kepler's Third Law states that the square of the period of orbit (the year ) is proportional to the cube of the orbital radius. Q12. Kepler s observation that planetary orbits where ellipses put the final nail through Aristotle s crystalline spheres. But there was no fundamental justification for elliptical orbits. Is this any better than Ptolemy s description? Galileo Galilei ( ) brought the heliocentric solar system to the world stage with striking new discoveries. The key to his success was the telescope. He hand-built one of the first telescopes and used it to investigate the other planets. It is worth noting that his telescope was about as good as a pair of binoculars with modern optics. First, on looking at supposedly blank space in the sky he observed countless (and before, invisible) stars. Though not considered a scientific argument today, for the philosophically oriented renaissance audience, this was evidence that the heavens were not perfect as assumed. Next, he observed the moon and other objects and found them to be imperfect, with surface features similar to the Earth. This put all the planets on equal footing, instead of the other planets being perfect, heavenly spheres. Galileo also discovered that Jupiter had moons. He argued that the moons orbiting Jupiter was a microcosm of the solar system a large body with smaller bodies orbiting around it. He discovered that Venus has a complete set of phases, as does the moon. Copernicus had suggested that an observation of Venus phases would provide direct evidence on the nature of its orbit. In the Ptolemaic system, Venus orbits within the sphere of the Sun. Copernicus argued that in Ptolemy s model, Venus would exhibit only crescent or new phases when facing the Sun as shown in fig. 7. It would therefore not be possible to see a full or gibbous Venus. One might imagine an alternative geocentric model with Venus outside the Sun s sphere, but in that case only gibbous and full phases would be possible. Galileo saw all the phases and sizes of Venus in the sequence predicted using the heliocentric model, see fig. 6. Galileo had a very high profile and wrote for the public in plain language. This brought his revolutionary ideas to the public, and he was criticized from all quarters. In particular, the New Testament states that "the world is firmly established, it cannot be moved," which is at least superficially at odds its rotation and orbiting in the heliocentric model. There were many powerful people who did not like Galileo's new arguments, in particular Cardinal Bellarmine. There were trials and questions, and eventually Galileo was given a life sentence of house arrest. Q13. How is each piece of evidence credible/not credible? Q14. Do any of these arguments specifically prove the Copernican system? Discussion question: If you were an astronomer in Copernicus day, which model would you use? At what point in history would you start using the heliocentric model. Bear in mind that in science the easiest discoveries with the greatest impact are often made first. On the other hand, your colleagues will laugh at you if you work on something they consider crazy.

9 Figure 6. Sequence of Venus sizes and phases according to Copernicus from ref. [5], pg 107. Figure 7. Sequence of Venus sizes and phases according to Ptolemy from ref. [5], pg 108.