Oceans of Truth Chapter 1 The Foundations of Natural Philosophy Chapter Theme How can you measure something you can’t reach? Chapter Goal Use your measurements to build a scale model of the observable universe. 1.1 Early Astronomical Observations Stone-Age Astronomers ..................................................................................................1-2 Astronomy in the Americas ............................................................................................1-4 Review Questions ............................................................................................................1-4 1.2 The Origins of Our Science Mesopotamian Astronomy ..............................................................................................1-5 Egyptian Astronomy ......................................................................................................1-6 Ancient Chinese Astronomy ...........................................................................................1-7 Review Questions ............................................................................................................1-8 1.3 Early Models of the Universe The School of Miletus (630 – 528 BC)..........................................................................1-9 The Eleatic School (570 -444 BC) ...............................................................................1-10 The Atomists (~450 BC) ..............................................................................................1-11 Aristotle (384-322 BC) .................................................................................................1-11 Review Questions ............................................................................................................1-12 1.4 Measurement and Fine-Tuning Aristarchus (310-230 BC) and the Distance to the Heavens .........................................1-13 Eratosthenes (276-194 BC) and the Size of the Earth ...................................................1-16 Ptolemy (100 – 170 AD) and a More Perfect Model .....................................................1-17 A Scientific Model of the Universe ..................................................................................1-19 Review Questions ............................................................................................................1-20 1.5 Our Modern View of the Cosmos An Important Shift in Perspective ...................................................................................1-21 Looking Forward ............................................................................................................................... 1-22 Applications ........................................................................................................................................ 1-23 “If I have seen further, it is by standing on the shoulders of giants.” Sir Isaac Newton in a letter to Robert Hooke, 1676 The Foundations of Natural Philosophy 1-2 1.1 Early Astronomical Observations It is hard to imagine what life was like before there were clocks to tell us what time it was, electricity to light up the night, and libraries full of books with the accumulation of thousands of years of learning. Even though we spend half of our lives in nighttime, many of us almost never see stars in the sky. From today’s cities with bright lights and the inevitable haze of pollution, the night sky is generally a washed-out glow. Since we know from our calendars and wristwatches what season we are in and what the time of day is, the motion of the Sun in the sky is irrelevant. Few of us are ever aware of the current phase of the Moon, and whether or not it will be up in the sky that night, brightening the darkness. We read in the newspaper of upcoming meteor showers and eclipses of the Sun and the Moon, and we take it for granted that we know what they are and when they will occur. We have a sense of control over our environment that is unprecedented in the history of our species. We can’t really understand the perspective of early herders and farmers, who every night faced a sky filled with mysterious lights that slowly swept out subtle and complex patterns of change. For a farmer in the Nile Valley in 5000 BC, being able to anticipate seasonal changes was a matter of survival. There must have been a sense that the motions of the unreachable, mysterious objects in the sky were connected to changes on Earth. If one could read the motion of celestial objects to predict events on Earth, then one had some control over his or her destiny. If one could influence the forces at work behind those motions, that would be even better. The first effort led to the development of astronomy and astrology. The second effort was a goal of religion. Everywhere in the world that ancient buildings have survived, there are traces of an interest and need to measure and seek alignment with the heavens. Stone-Age Astronomers While there is no recorded history of stone-age culture in any part of the world, artifacts remain that allow us to speculate on the level of technical knowledge attained. One of the most spectacular examples of this is at Stonehenge (Figure 1.1), near Salisbury, England. Stonehenge is a circular structure, about 100 [m] across, with large freestanding stone structures arranged in concentric zones at the center. Radiocarbon dating has allowed archaeologists to determine that it was built in three stages: the outermost ring of 56 holes, the large Figure 1.1 Stonehenge III (http://www.megalithic.co.uk) The Foundations of Natural Philosophy 1-3 heelstone, and a rectangular array of 4 stones make up Stonehenge I, probably built around 2700 BC. There was a ring of stones begun but not completed, which is identified as Stonehenge II, and the innermost arrangement of stones, Stonehenge III, was built around 1600 BC (Figure 1.2). It is clear that Stonehenge I was built-in to alignments for observing several key astronomical events. Various stones align with the setting positions of the Sun at the solstices and equinoxes, and other alignments mark the extreme rising and setting positions of the Moon. The site is too large and too precise to be a simple calendar, though, and the ring of 56 holes has no discernible sighting function. In 1966, the astronomer Fred Hoyle discovered that the site could be used to successfully predict eclipses of the Sun and the Moon, a very difficult task. By using the stone alignments as an observatory, and the 56 holes as an elaborate counting system, Hoyle was able to demonstrate a system that suggests that the Neolithic creators of Stonehenge I had a sophisticated understanding of the causes of eclipses, and the exact motion of the Sun and the Moon through the sky. 1 The most interesting feature of the site is that Stonehenge II and Stonehenge III lack the functionality of Stonehenge I. It is as if in the thousand years after the first construction, a significant body of knowledge was lost to the people who lived there. Perhaps the new focus on Bronze-Age metallurgy led to this loss, and the newer construction was a failed attempt to re-create the skills of old.2 1 Hoyle, F. From Stonehenge to Modern Cosmology. W.H. Freeman and Company, San Francisco. 1972 2 ibid. Figure 1.2 Map of Stonehenge Stonehenge I in red, Stonehenge II in green, and Stonehenge III in blue. (adapted from Hoyle, From Stonehenge to Modern Cosmology. W.H. Freeman and Company, San Francisco. 1972.) The Foundations of Natural Philosophy 1-4 Astronomy in the Americas The art of using the Sun, Moon, and stars for keeping calendars and navigation was apparent in very old Native American cultures. At the Anasazi site of Chaco canyon, for example, we find that buildings were aligned with celestial axes, and windows appear to be arranged as calendrical observatories.3 The ancient civilizations of Central America (the Mesoamericans) made much more detailed Astronomical observations. Evidence of their knowledge of Astronomical events, such as eclipses and solstices, comes from stone monuments, architectural arrangements and four Mayan codices (hieroglyphic writings). Evidence of lunar dates such as solar eclipses (from inscriptions on stone slabs or stelae) begin at 357 A.D. One of the four codices, the Dresden codex, (Figure 1.3) contains detailed tables on the cycles of Venus. The latest date recorded in the Dresden codex is 1210 A.D. Review Questions 1. How do you define a day? 2. How do you define a year? 3. What happens to the length of the day during different seasons? 4. What might change in the sky over the course of a year? 5. How might you tell how long a year is? 6. Why might this have been important in early civilizations? 7. Why do you think Astronomy has been referred to as “The First Science”? 3 http://www.ncafe.com/4corners/astronomy.html Figure 1.3 An image of the Dresden Codex (http://pages.prodigy.net/gbonline/dresden.jpg) The Foundations of Natural Philosophy 1-5 Figure 1.4 The MUL.APIN, an Assyrian text which lays out the rules for keeping the Mesopotamian lunar calendar (~1000 BC). (http://doormann.tripod.com/asssky.htm) 1.2 The Origins of Our Science The technical achievements of ancient cultures in Britain and the Americas have had no impact at all on our modern scientific theories; there was no path by which that knowledge could reach the centers of learning in which our modern version of science took shape. In fact, reconstructing the meaning and purpose of the archaeological artifacts available from those cultures is a challenging process, resulting in conflicting interpretations. Whatever knowledge of the world existed in those ancient civilizations is largely lost to us now. The origins of our modern science lie in the areas with which Ancient Greece, India, and later the Islamic Empire, had contact: Mesopotamia, Egypt, and China. We will take a brief look at those three oldest centers of scientific inquiry. Mesopotamian Astronomy The first records we have of systematic studies of the natural world come from Sumerian, Assyrian, and Babylonian archaeological remains (Figure 1.4). In ancient Sumer, the first great civilization on the Tigris and Euphrates rivers, a lunar calendar was developed that used months based on the cycles of the moon. This was the prototype of the Jewish and Islamic lunar calendars that are still in use today. While it was convenient, since the progress of the months was tied to a clearly visible physical cycle, it didn’t do a good job of keeping track of seasons. The Sumerian year consisted of twelve months which added up to 360 days; not quite as long as our modern year. Every once in a while, when astrologers noticed that the seasons had slipped out of alignment with the calendar, an extra month was thrown it to resynchronize things. Accurate measures of the solar year were clearly not that important at the time. In addition to studying the Moon, the Sumerians created catalogs of the brightest stars, defined the constellations of the zodiac, and observed and recorded the motions of the 5 planets visible to the naked eye (Mercury, Venus, Mars, Jupiter, and Saturn). The first day of a month was defined by the first appearance of the new moon. In order to be able to predict when that would occur required detailed observations of the behavior of the Sun and the Moon over long periods of time. It also required the use of mathematics. Babylonian mathematics was motivated primarily by the need to predict celestial events, both for time-keeping and divination. Remnants of that mathematical system can be found in our modern mathematical practices. For example, our use of 360 degrees to represent one full rotation can be traced directly back to the number of days in the Sumerian and Babylonian year. The Astronomy of the Babylonians developed in stages: the first stage dated back to the second millennium B.C., where thousands of omens were inscribed on vast clay tablets, referring to appearances of the Sun, Moon and planets.4 4 These read something like the astrological predictions in the popular press, for example “If Jupiter [rises] in the path of the [Enlil] stars: the king of Akkad will become strong and [overthrow] his enemies in all lands in battle”(The Babylonian Theory of the Planets, p.94). The Foundations of Natural Philosophy 1-6 Detailed record-keeping of Solar and Lunar motions, the motions of Venus, and eclipses of the Sun and the Moon characterized the next phase, and they were kept for a continuous, 800 year period. The observers used a set of thirty stars as reference, as they noticed that the heavenly bodies moved around only a narrow belt of stars (later known as the Zodiac by the Greeks). They noticed that the Sun and Moon always cross the Zodiac from west to east, but the planets, which generally also move eastwards across the Zodiac, move westwards during certain periods of time. Detailed observations of lunar and solar eclipses were also made. 5 All of this observation and record-keeping provided the data needed to identify the patterns that were so important for astrology and calendar reckoning. Clay tablets known as ephemeredes, dating from the third century BC, contain calculated positions of the Sun, Moon, and planets. Tablets called procedure texts gave rules for computing the tables (similar to a modern computer program).6 Despite the fact that the Babylonians did not attempt to build theoretical models to explain the Universe as we know it, they made a great contribution to the field of astronomy. Perhaps most significant was the effect that they had on neighboring cultures. Both Indian and Greek astronomers and philosophers had access to Babylonian astronomical information, and it shaped their ability to construct their own understanding of the nature of the Universe. In the words of N.M. Swerdlow: “They have left no record of their theoretical analyses and discussions, but to judge from the works they have left us, the Diaries and ephemeredes…contained more rigorous science than the speculations of twenty philosophers speaking Greek, not even Aristotle excepted”.7 Egyptian Astronomy Ancient Egyptians never developed the kind of detailed, precise knowledge of astronomy that the Babylonians did, but they used astronomy to create time-keeping practices that have shaped many later systems, including the one we use today. Unlike that of the Mesopotamian civilizations, the Egyptian civil calendar had nothing to do with lunar cycles. Perhaps because predicting when the annual flood of the Nile would occur was of supreme importance to agriculture, an accurate seasonal calendar was a high priority. The start of the year was marked by the first visible dawn rising of the star Sirius, the brightest star in the night sky. It was known that the year was approximately 365 days long, and this was broken up into 12 “months”. What is interesting is that these months were chosen to be 30 days long each, which doesn’t appear to correspond to any natural cycle. It must have simply been a matter of convenience. There were 5 days left over, but rather than break up the consistent pattern of 30 day months, there was a 5 day festival period at the end of the year that “didn’t count”. 5 The prediction of eclipses was helpful because they were so frightening to many people. In 1504, it is said that Christopher Columbus used tables to predict a total eclipse of the moon on February 29th, which allowed him and his crew to escape during the panic caused by the eclipse among the Native Americans. 6 It has also been suggested that bad weather (resulting in poor visibility) was responsible for the need to develop a mathematical theory. 7 The Babylonian Theory of the Planets, pp. 181-182 The Foundations of Natural Philosophy 1-7 This abstract measure of time is similar to our modern way of thinking about and measuring the passage of time. The rationality of the Egyptian calendar made it very simple to use for prediction, and even as late as the 1500’s Copernicus used it for his astronomical calculations. Greeks and then Romans adapted the Egyptian calendar for their own use, and the calendar that most of the world uses today retains its 12 months, but has lost some of the original rationality of that scheme. Egypt was apparently the first place that the motion of celestial objects was used to measure time throughout the day. A series of 36 “time-keeping” constellations were identified and used to keep track of time throughout the night. On any given night, 12 of these constellations would rise, one by one, marking out roughly equal measures of time. For daylight time measurement, the passage of Sun across the sky was divided into 12 different “stations”. This is the origin of our 24 hour day. Apart from the important task of keeping an accurate seasonal calendar and measuring time throughout the day, astronomy wasn’t a high-precision endeavor in Egypt. The mathematical tools that were developed in Egypt, early versions of algebra and trigonometry, were developed primarily to support architecture. There are a few copies of ancient Egyptian texts, such as the Ahmes Papyrus (Figure 1.5), which were used to teach useful mathematical techniques. The techniques represented in the document date from about 2000 BC, and include algebraic and geometric problems related to commerce and architecture. Ancient Chinese Astronomy Careful observation of the night sky was an important activity in Ancient China, and Chinese astronomers were among the keenest celestial observers in the world. Although the prime motive was divination, it was tied to the functioning of the government and the meaning of the emperor’s authority. It was, therefore, a highly controlled and well-supported endeavor. Every omen in the sky had the potential to change lives, and observations were diligently recorded and preserved (Figure 1.6). There are astronomical records in China dating back to about 3000 BC. Carved into a cliff at Jiangjunya, in Jiangsu province, is the oldest Chinese star chart known, from about 2000 BC (Figure 1.7). It clearly depicts the Moon and Sun in their various Figure 1.6 Drawings of Comets used as oracles, 200BC. (http://3.bp.blogspot.com) Figure 1.5 A portion of the Ahmes Papyrus (also known as the Rhind Papyrus), currently in the British Museum. It is a portion of a mathematics textbook from about 1700 BC. The Foundations of Natural Philosophy 1-8 seasonal locations, and has a detailed representation of the Milky Way. The Milky Way is shown in such detail that the dark bands and patches that appear in the carving match the features that we see in modern surveys. Temporary changes to the arrangement of the heavens were particularly portentous events, and there are very early records of eclipses and comets. In 1054 AD, Chinese astronomers provided the first documentation of a supernova event, an exploding star that glows very brightly for a few days and then fades away. "In the 1st year of the period Chih-ho, the 5th Moon, the day chi-ch'ou, a guest star appeared... After more than a year it gradually became invisible...”8 The remnants of that supernova are visible today as the Crab Nebula. Review Questions 8. Why do you think the details of many early scientific discoveries were “lost” over time? 9. Why might predicting an annual seasonal calendar have been important in Egypt? 10. What is the origin of our twenty-four hour day? 8 http://www.spacetoday.org/China/ChinaAstronomy.html Figure 1.7 A Chinese rock carving depicting the Milky Way and the Moon in various positions, from about 2000 BC. (from Astronomy Before the Telescope, Christopher Walker, ed. 1996) The Foundations of Natural Philosophy 1-9 1.3 Early Models of the Universe The only records we have of explanations for astronomical phenomena from the three cultures we have looked at so far are religious and mythological. Astronomy played a very practical role, for time-keeping and divination. The patterns observed in celestial motion allowed astronomers to make predictions, but there was no attempt to explain why the patterns existed. A fundamental shift in thinking took place when Greek philosophers were exposed to the knowledge and techniques accumulated in Egypt and Mesopotamia. The foundation of Greek philosophy was pure thought. They were selective observers, and only looked for evidence to confirm the results of their reasoning. It was the synthesis of that reasoning and the information acquired from other cultures that laid the groundwork for modern scientific thinking. The School of Miletus (630 – 528 BC) The first records that we have of a philosopher breaking with religious tradition and offering what we would call “scientific” explanations of why the world is the way it is, are references to Thales of Miletus. Thales founded the Milesian school of natural philosophy around 600 BC, and though none of his original work remains, his accomplishments and ideas were preserved by the writings of Aristotle several hundred years later. Miletus, the site of which is now in Turkey, was a prosperous port with market outposts throughout the known world. Thales had the means to travel to Egypt and possibly elsewhere in the Middle East, as he collected information about Egyptian and Babylonian astronomy. There is speculation9 that the prosperity of the city of Miletus created an environment that celebrated human achievement, instead of attributing all success to the will of the gods. This attitude would have facilitated Thales’ efforts to seek naturalistic explanations of the origin of the world and the causes of everyday phenomena. It was not Thales’ ideas but his approach that defined a new type of philosophy. He sought evidence for each hypothesis, and the conceptual models that he developed were extensions of the observations that he made. The main goal of his school was to identify the origin of all things and describe the nature of matter. Based on the observation that all living things are moist, and water is required for life to continue, Thales proposed that water was the essential element from which life was generated. Further noticing that in the mouths of rivers, islands form and build up, he speculated that water could also create earth itself. Metal workers routinely melted metals into a “wet” state by heating them up, so perhaps earth could be returned to its 9 www.utm.edu/research/iep/t/thales.htm Figure 1.8 Reconstruction of the Greek conception of the Universe at Homer’s time (~900BC) The Earth was a flat disk, surrounded by ocean. The heavens formed a ceiling, upon which the Sun, Moon, and stars moved. This model, with minor variations, was widely assumed to be accurate until about 350 BC. (http://www.henry-davis.com/MAPS/Ancient%20Web%20Pages/AncientL.html) The Foundations of Natural Philosophy 1-10 watery nature. Water could be seen to evaporate and create air, as well, and Thales concluded that water was the one essential element from which all else resulted. Thales envisioned the Earth floating on a sea of water. Since all things were made from water, and eventually returned to water, this made sense. There were well-known examples of floating islands, and these served as his model for this idea. An advantage of this model was that it gave a natural explanation for a terrifying phenomenon: earthquakes. If the Earth floated on a bed of water, then surely earthquakes were caused by waves on that water; no angry gods required. One of the interesting features of the Milesian school was that Thales’ successors didn’t necessarily agree with his models. The highest priority was to seek the truth using whatever evidence was available, rather than to remain faithful to a particular idea. This was an essential element of their style of inquiry, and it remains the most important aspect of scientific investigation today. Anaximander of Miletus had more abstract notions than Thales of the fundamental principle that gave rise to the world. He did make some practical observations, however, concerning the structure of the Universe as a whole. From watching stars in the northern sky that turn in circles but never dip below the horizon, he suggested that all stars, in fact, travel in full circles around the Earth. He reasoned, therefore, that all of the objects in the sky must travel in full circles. He imagined that there was much more depth to the sky than previously thought, and that some of the objects we see are very far away. It made no sense, then, that the Earth would be floating on water; it had to be suspended in empty space. The problem arises, though, of what supports the obviously heavy Earth in that case. He knew that a large object could be balanced, if its weight was distributed evenly around the point at which it was supported. Extending this idea, he reasoned that the Earth must be somehow “evenly distributed” in space. This meant that Earth must be exactly centered at the mid-point of the Universe. That takes care of the Earth, but what keeps the Sun, the Moon, and the stars from falling? They must also be somehow “balanced” about the center. He envisioned huge wheels, spinning with their axes centered on the center of the Universe. Inside the hollow rim of each wheel was fire, and through an opening in the rim we could see through to that fire. This internal fire, seen through the opening, was what we perceived as the Sun, the Moon, or a star. Eclipses of the Sun occurred when that hole in the “Sun wheel” closed temporarily. The phases of the Moon were due to a cycle of closing and opening of the hole in the Moon wheel. Anaximenes of Miletus used a series of experiments and observations to determine that air was the fundamental element from which all else was made. When pressed together, air cools down and condenses to form earth. When air is forced to expand, it heats up and forms fire. He demonstrated this by instructing people to blow on the palms of their hands. When you purse your lips and blow (pressing the air together), the wind on your hand is cool and “dense”, exerting a noticeable push on your hand. It is closer to being like earth. When you blow on your hand with your mouth open, the air reaching your hand is hot with a very light touch, more like fire. Anaximenes envisioned a flat, disk-shaped Earth floating on a bed of the prime element, air. The stars were fixed to a crystal sphere, and the Sun, Moon and stars were made of fire. According to Heraclitus, change was the essential principle that governed the Universe, and fire was a symbol of perpetual change. It transforms one substance into another, without having any substance itself. He thus identified the primordial element to be fire. To demonstrate that “change” was the fundamental quality of existence, he pointed out that a river is in a constant state of change because it always has different water in it than just a moment before. In fact, it is this constant change that makes it a river, and not a lake. Change is needed, therefore, to give things their identity. The Eleatic School (570 -444 BC) The Eleatic School was founded by Xenophanes, whose main teaching was that the Universe is eternal and changeless. All apparent change was an illusion, even motion. He suggested that earth was the primordial element. The Foundations of Natural Philosophy 1-11 Zeno, also of the Eleatic school, made many arguments against the reality of motion. The most famous of these is that an arrow flying through the air is at any instant in exactly one place, occupying a space exactly as long as its natural length and is therefore identical to a stationary arrow during that instant. If it is stationary at all instants, it cannot, in fact, be moving. Many of Zeno’s paradoxes of motion can’t be fully resolved without the theory of Special Relativity, developed by Albert Einstein almost two and a half thousand years later! It is interesting to note that the colony of Elea had a violent and insecure history in Xenophanes’ lifetime, forced by war to relocate from Ionia to what is now Corsica, and again from Corsica to what is now Sicily. Xenophanes himself lived much of his life as a refugee in exile, before joining the colony. We can speculate that the turmoil of his life must have colored Xenophanes’ philosophy, and that his intellectual obsession with stability and consistency was the result.10 The Atomists (~450 BC) Leucippus and Democritus disagreed with the Eleatics, that change is an illusion. They proposed that nature was entirely composed of just two things: space and matter. Space was the vessel and matter occupied it. Matter was made of small, indivisible particles called “atoms”, and atoms were indivisible because they contained no empty space. There were different kinds of atoms, and their size and shape distinguished them from each other. Atoms were constantly in random motion, like particles of dust in a beam of light. When two atoms collide, the outcome of the collision depends on their shapes. If they have complementary, interlocking shapes, they might stick together to form a cluster. Otherwise, they will bounce off of each other like billiard balls. All of the objects that we see are made up of bonded clusters of atoms, and the properties of materials depend on which types of atoms they are made up of. The atomist view was very unpopular, most likely because it reduced all interactions to the collisions of randomly moving atoms. There was no room for a human soul, or a sense of purpose in such a model, except as a consequence of unpredictable atomic collisions. Despite eerie similarities to our modern models for the nature of matter, the ideas of the Atomists went ignored for over two thousand years. Following the Atomists, Greek philosophy took a turn towards ethics and social philosophy. The Atomists were the last of the true natural philosophers whose top priority was to understand the nature, composition, and behavior of the physical world. Aristotle (384-322 BC) Aristotle was a student of Plato’s, the tutor of Alexander the Great, and the founder of a school at the Lyceum in Athens. Aristotle set out to systematize existing knowledge about all subjects, including natural philosophy, and he sorted through and organized 250 years worth of ideas in the process. Aristotle sought, through reason and logic, to develop a self-consistent, simple, and practical model for what the Universe looked like and why it was the way it was. By surveying the ideas and arguments of his predecessors, he was able to choose from the myriad of hypotheses the ones that best withstood the test of his logical process. It is through his writings, which were hidden away by his successors for safekeeping and re-discovered 200 years later, that we know most of the details of the various schools of philosophy that preceded him. In Aristotle’s book “On the Heavens”, he discusses the ideas of the Eleatic, Milesian, and Atomist schools, and constructs from them the most sensible picture of what the Universe looks like. He describes four fundamental elements (all of the single primordial elements of his predecessors): earth, water, air, and fire11. Of these, earth and water are “heavy”, while air and fire are “light”. Objects made from these elements have a natural inclination to move. Heavy elements naturally move towards the center of the Universe. The Earth, being made of heavy things, naturally finds itself at the center. Standing on the Earth, we see that objects made of earth and water fall “down”. The heavier the object, the more quickly it moves down. Also, the closer it gets to the center of the Earth, the faster it moves to complete its journey. The light elements have a 10 http://www.mathpages.com/rr/s3-07/3-07.htm 11 This was actually first suggested by Empedocles around 450 BC The Foundations of Natural Philosophy 1-12 natural inclination to move away from the center, or “up.” Fire is the lightest of the elements, and earth is the heaviest. The Earth must be spherical, since as heavy objects accumulate around the center point of the Universe, they will pack down around that point to the same depth in all directions. Aristotle also provided some useful evidence that the Earth was a sphere, based on a series of simple observations. First of all, when one travels north, new stars appear at the northern horizon, while other stars slip below the southern horizon. If the Earth were flat, the stars should look the same from anywhere on Earth. Similarly, as a ship sails away, the hull of the ship dips below the horizon before the sails do. This can only happen if the surface of the ocean curves downward. Since this happens in any direction, the Earth must be a sphere. It was known from Babylonian astronomers that a lunar eclipse was caused when the Earth’s shadow covers the Moon, blocking the Sunlight that normally illuminates it. The shape of the Earth’s shadow can be observed, then, and it is always round. The only shape that always casts a round shadow is a sphere. There are only two directions of natural motion, up and down, and these are opposites of each other. In the heavens we see objects moving along circular paths. This must be a new kind of natural motion, one that has no opposite. This must be a new element, therefore, and since there is no opposite of its natural motion, and the circular path has no beginning and no end, it must be eternal and unchanging. He named this element “aether”. Thus there is a sharp distinction between the celestial realm, changeless and made of aether, and the terrestrial realm, made of the four standard elements. The celestial realm is made of nested spheres that surround the Earth, the outermost being the sphere of the stars. Since he assumes that the Earth is at rest, the sphere of the stars turns around the Earth. It was clear that the Moon is between the Earth and the planets since it can be seen to block our view of the planets. The sphere of the stars has perfect motion, and each of the planets, the Sun, and the Moon travel on their own sphere, with more complex motion. Although he doesn’t claim to explain the cause of this complex motion, it is satisfying that the sphere most distant from the Earth is the most ideal, the most heavenly. It is interesting that Aristotle denies the concept of empty space put forth by the atomists; there are only objects. The spherical Earth fills the nested set of celestial spheres. Also, motion is not simply something that happens to objects, motion as discussed by Aristotle is an inherent part of an object’s elemental structure. A rock is always striving to fall. Aristotle’s whole picture of the Universe, description of matter, and explanation of motion rely on the fact that the Earth is at the very center. Review Questions 11. What were the differences between the study of Astronomy in early civilizations and the approach of Greek philosophers? 12. What was the difference between the explanations for the origins of the world provided by philosophers such as Thales and the explanations of Egyptian and Babylonian astronomers? 13. What observations led to the idea that the Earth is round and not flat? 14. What shadow would a flat cylinder cast when viewed from side-on? 15. What main idea from the Milesian school forms the basis of modern scientific investigation? 16 (a) Why did Anaximander believe that stars travel in full circles around the Earth? (b) What might be an alternative explanation for this phenomenon? 17. Why was the view of the atomists unpopular? 18. If you were Aristotle, how would you explain the observation that a stone fell to the ground after it was dropped? 19. Did Aristotle believe that the motion of objects near the surface of the Earth behaved in a similar way to the motion of objects in the celestial realm. If not, how were they different? The Foundations of Natural Philosophy 1-13 1.4 Measurement and Fine-Tuning Aristarchus (310-230 BC) and the Distance to the Heavens Aristarchus was most likely in Alexandria, which was a part of Greece at the time he lived. He described a different model for the structure of the Universe than Aristotle did, but the origins of his ideas on the subject are not known. According to Aristarchus, the Moon orbits the Earth, and the Moon-Earth system orbits the Sun. The stars and planets presumably also orbit the Sun. There were a few problems with this idea. First of all, there was no indication that the Earth was in motion. If the Earth were flying through space, surely we would feel it. Also, as we moved around the Sun, we should see the apparent positions of the stars shifting due to our changing perspective. Aristarchus argued that the stars must therefore be unimaginably far away. These notions of a moving Earth and an enormous Universe must have seemed bizarre in comparison to Aristotle’s deliberate, sensible reasoning. Although his model for the structure of the Universe was essentially ignored, Aristarchus made some ingenious measurements of the sizes and distances of the Sun and the Moon. This was the first time anyone had successfully attempted to measure celestial distances, and the first indication of the size of the Universe. Despite some of the strange theories about the causes of eclipses proposed by philosophers of the Miletus School, Aristarchus knew that they are caused by special alignments of the Sun, the Earth and the Moon. The Moon is bright because it reflects light from the Sun. During a “full Moon”, the Moon is on one side of us, and the Sun on the other. Thus we see a full Moon at night (when the Sun is below the horizon), and the whole side of the Moon facing us is illuminated (Figures 1.9 and 1.10). A “new Moon” occurs when the Moon is on the same side of us as the Sun. The illuminated side of the Moon faces away from us, and the Moon appears dark. Furthermore, it is in the sky at daytime, since it appears in the same part of the sky as the Sun. When we see a very slim crescent Moon, it is near Sunrise or Sunset, and the Moon is very close to being “new”. Figure 1.9 The conditions for a full Moon Figure 1.10 The conditions for a new Moon The Foundations of Natural Philosophy 1-14 If the motion of the Moon and the motion of the Earth (or the Sun, in Aristotle’s theory) were all in the same plane, then every new Moon would appear exactly in front of the Sun, blocking it from view. This is what we call a “solar eclipse”. Similarly, during every full Moon, the Earth would cast a shadow on the Moon, and it would no longer be illuminated. This is a “lunar eclipse” (Figures 1.11 and 1.12). These eclipses would only last about an hour at a time, but we would see one of each every month, as the Moon gets to the right part of its cycle. It turns out that the orbit of the Moon around the Earth is slightly tilted (11°) with respect to the plane of the Earth/Sun motion, so the Sun, Moon, and Earth rarely line up precisely enough for this to occur. Nevertheless, eclipses do occasionally happen, and Babylonian astronomers had been recording and predicting these events for hundreds of years. A solar eclipse occurs during a new Moon, when the path of the Moon intersects the line between the Sun and the Earth. When this happens, the Moon's shadow falls on the Earth and blocks the Sunlight. If you are standing where this shadow falls, you see the Sun "covered" by the Moon. It was well known even in antiquity that a total solar eclipse, in which the Sun appears completely covered, is visible from a very limited area on Earth. This must mean that the shadow of the Moon on the Earth is very small. So, either the Moon itself is small, or the shadow “tapers” down to almost nothing (about 100 miles across) by the time it reaches the Earth. This tapering effect would be possible if the Sun was much bigger than the Moon (see diagram). A lunar eclipse would occur during a full Moon, if the Moon’s path brought it exactly in line with the Sun and Earth. In this situation, the Earth's shadow would fall across the Moon, blocking the Sunlight that normally illuminates it. The Earth’s shadow on the Moon had long been observed to be round (this was one of Thales’ Figure 1.11 The tapering shadow of the Moon on the Earth during a solar eclipse Figure 1.12 The taper of the Earth’s shadow during a lunar eclipse The Foundations of Natural Philosophy 1-15 arguments for a spherical Earth). By studying exactly how the curvature of the Earth's shadow compared to the shape of the Moon as the eclipse progressed, Aristarchus was able to estimate that the Earth's shadow was 2.5 times bigger than the Moon (see diagram). Now let’s consider the possibilities. If the shadows didn’t taper, then the small size of the Moon’s shadow during a solar eclipse would imply that the Moon itself was very small, about 100 miles in diameter. But since the Earth’s shadow during a lunar eclipse is only 2.5 times the size of the Moon, this would mean that the Earth is only 2.5 times larger than the Moon, or about 250 miles. Since the Earth was obviously larger than that, the non-tapering shadows model didn’t make sense. So, Aristarchus assumed that the shadows did taper, and in fact the Moon’s shadow tapered from the full diameter of the Moon down to almost nothing. In other words, during a solar eclipse, in the distance between the Moon and the Earth the Moon’s shadow was reduced by almost the full width of the Moon. He reasoned that during a lunar eclipse, the Earth’s shadow would be subject to the same tapering effect. Since the distance between the Earth and the Moon is the same in both cases, the Earth’s shadow should be reduced by approximately the same amount: the full width of the Moon. This means that if Earth’s shadow upon the Moon was 2.5 times the size of the Moon, the Earth’s shadow must have started out to be 3.5 times the size of the Moon. Therefore the Earth must be 3.5 times as large as the Moon! When we look at an object in the sky, we can’t judge how far away it is until we know its size12. Aristarchus’ measurement of the size of the Moon now allowed him to estimate the distance to the Moon, using some simple geometry. We can define the angular size of an object in the sky by how big of an angle it takes up. Since the sky stretches 180 degrees overhead from east to west, something that takes up half the sky would have an angular size of 90 degrees. Keep in mind that this doesn’t tell us how big it is, just how much of an angle it subtends from our point of view. Not having the benefits of modern trigonometry, Aristarchus used an argument based on similar triangles. One way to measure the angular size of the Moon is to hold up a small object (a coin, for example), and move it back and forth until it looks exactly the same size as the Moon. Use a stick to measure the final distance of the coin from your eye. As seen in Figure 1.13, we can see that the ratio of the diameter of the coin, D, to the length of the stick, L, must be the same as the ratio of the Moon’s diameter to its distance from us. Aristarchus was able to use this technique to express the distance to the Moon in terms of its size, and he knew its size relative to the Earth. So this gave him the size and distance to the Moon in terms of the size of the Earth. You will do this experiment yourself, and calculate the actual distance to the Moon as a laboratory exercise. 12 There is an interesting story related to this that happened in the late 1990’s in San Francisco. A Bay Area news show reported spotting an unidentified flying object in the background of an on-location report from Twin Peaks. Taped footage showed an object apparently over the Bay, streaking across the screen. Experts were brought in to estimate the velocity and acceleration of this unknown craft, and deemed it to be traveling at impossible speeds. In the end it was revealed to have been an out-of-focus fly just a few feet in front of the camera, traveling at a very reasonable speed after all. Figure 1.13 Measuring the angular size of the Moon The Foundations of Natural Philosophy 1-16 One interesting coincidence is that the angular size of the Moon and the angular size of the Sun are the same when viewed from Earth. If they weren’t almost exactly the same, eclipses of the Sun would be much different. All that was left for Aristarchus to do was to try and measure the distance to the Sun. Knowing its angular size would also give him an estimate of its true size. To measure the distance to the Sun, Aristarchus made use of another clever bit of reasoning. Knowing that when the Moon is half-full it must be illuminated directly from the side by the Sun, he tried to measure the angular separation between the two bodies in the sky (Figure 1.14). This gave him a measure of the right triangle formed by the Earth, Moon, and Sun at that moment. This was a tricky measurement to make, and it was very close to a right angle. This indicates that the Sun is extremely far away. Aristarchus erred on the low side for the angle measurement, getting a value for the hypotenuse of 20 times the Earth-Moon leg of the triangle. This also means that the Sun was 20 times bigger than the Moon, since they have the same angular size. The actual values should actually be about 20 times that much. His result was very sensitive to small errors in the angle measurement, so it is not surprising that he wasn’t more accurate. He also may have intentionally underestimated the distances, since the real values would have required an extraordinary shift in perspective for him and his contemporaries. Eratosthenes (276-194 BC) and the Size of the Earth Aristarchus had successfully measured the size of the Moon and Sun relative to the size of the Earth, but the question of the size of the Earth remained. The only way to measure large distances was to pace them out on land. While the known part of the Earth, mapped by the Greeks (Figure 1.15), was growing, no one had traveled all the way around the sphere. Direct measurement was clearly impossible. Figure 1.14 Measuring the angular separation of the Moon and Sun in the sky Figure 1.15 The scope of the known world at the time of Eratosthenes (~220 BC) (http://www.henry-davis.com/MAPS/Ancient%20Web%20Pages/AncientL.html) The Foundations of Natural Philosophy 1-17 Eratosthenes (who was the second chief librarian at the Library of Alexandria) had access to a great deal of information, and came up with a very clever way to make an indirect measurement of the size of the Earth. He knew that at noon on June 22 (the summer solstice), the Sun was as high in the sky as it ever got, and a vertical stick cast the shortest shadow. From library records, he knew that on that same day every year the Sun must have been directly overhead in the town of Syene (which is now Aswan), in southern Egypt. It was observed that Sunlight went straight to the bottom of a deep well, and a vertical stick cast no shadows at all. Alexandria was due north of Syene, and on the same day, vertical sticks cast a distinct, measurable shadow. So the Sun that was directly overhead in Syene was not overhead in Alexandria, which must be due to the curvature of the Earth. If he could use the shadow of the stick to measure the angle of the Sun’s rays in Alexandria, he could calculate the angle subtended by the two cities (Figure 1.16). This is easily accomplished, by comparing the length of the shadow to the length of the stick. Although the trigonometric functions sine, cosine, and tangent were unknown to Eratosthenes13, all he needed to work out was what fraction of 360° that angle represented. This could easily be accomplished using contemporary knowledge of the lengths of chords in circles. He knew that the angle separation of Alexandria from Syene was some fraction of 360 degrees, and the distance between the two cities was the same fraction of the full circumference. The distance between the two cities was carefully measured since it was a well-traveled route, and from this he estimated the circumference of the Earth to be about 40,000 km. This is very close to modern measurements. A remarkable achievement, considering that he did this without leaving Alexandria. Ptolemy (100 – 170 AD) and a More Perfect Model One of the shortcomings of Aristotle’s model of the Universe was that the planets didn’t behave like other stars, but wandered at changing rates across the sky, even going backwards for periods of time (this is known as retrograde motion). This didn’t quite fit in with the vision of the eternal, changeless, celestial realm, and it brought up two distinct problems. First of all, there was no known mechanism that could result in this 13 These functions were a Hindu development, from about 500 AD. Figure 1.16 Eratosthenes’ measurement of the size of the Earth. Note that the angle subtended by Alexandria and Syene matches the angle that the Sun’s rays make with the top of the stick (opposite interior angles). The Foundations of Natural Philosophy 1-18 behavior, whereas all other motion in the Aristotelian system was clearly explained. Second of all, there was no way to predict the motions of the planets, which was of primary importance for astrologers. Steps toward a solution to this problem were taken by Appolonius, another scholar of the Alexandrian Library at around 200 BC. Appolonius developed the idea of the “epicycle”; a circular path that rides on the rim of another circular path (Figure 1.17). This provided a model for the irregular motion of the planets. Hipparchos of Nicea was a precise observer, who contributed to our understanding of the structure of the heavens in two ways. He built on the work of Aristotle and Apollonius to explain the mechanisms of celestial motion, and he also carried out the most exacting and thorough astronomical measurements that we have a record of. He found ways to improve on Aristarchus’ measurements of the distances to the Moon and the Sun, and he measured the length of the solar year to much greater precision than anyone had before. Hipparchos compiled a catalog of positions of the fixed stars that was used until relatively recent times, and formed the foundation of our modern surveys of the night sky. Hipparchos’ measurements were precise enough to notice a very subtle year-to-year variation in the position of the entire celestial sphere. It turns out that there is a slow “wobble” in the sky. In fact, it is so slow that it takes 26,000 years for it to complete one cycle! It is a testament to Hipparchos’ thoroughness that he was able to identify this cycle in only one lifetime’s observation. This wobble is known as the “precession of the equinoxes”, and there was nothing in Aristotle’s or Appolonius’ models of the heavens that could explain it. Claudius Ptolemy lived in Alexandria in the second century AD. Despite his name, he was no relation to the Greek pharaohs of Egypt who ruled from that city during the Ptolemaic dynasty. He studied many scientific phenomena, but he is most well known for his groundbreaking contributions to astronomy and cartography. His crowning achievement was a book entitled “He Megiste Syntaxis”14 (the Greatest Compilation), which put together all that was known in Greek astronomy with a significant number of original ideas to make it all work. This body of work formed the basis of all of Islamic and European astronomy until the 16th century. 14 This book was widely read by Arab scholars, and was entitled “Al Magesti” in Arabic. When it was rediscovered by Eurpoeans, in Latin translation from the Arabic, it bore the title “Almageste”, which is how it is most commonly known today. Earth Earth Figure 1.17 Appolonius’ explanation of the irregular motion of planets. On the left is Aristotle’s uniform circular motion; on the right is a planet on an epicycle, circular motion around a point undergoing its own circular motion. The planet on the left will be seen to move at a uniform rate across the sky; the planet on the right will have a changing speed, possibly even moving backwards at times. The Foundations of Natural Philosophy 1-19 Figure 1.18 Drawing of a Ptolemaic Sphere (http://www.hps.cam.ac.uk/starry/ptolarmill.html) Ptolemy made use of Hipparchos’ measurements of the stars and the long records of Babylonian observations of the Moon to build a complex mechanical model of the structure of the celestial realm (Figure 1.18). He made corrections to earlier efforts by using epicycles upon epicycles to reproduce the complex motions of planets. He also introduced the “equant”; a shift in the center of celestial motions. This displaced the circular motions of heavenly objects to account for observed irregularities in the lengths of seasons and the “wobble” of the whole sky. The beauty of Ptolemy’s solution is that he based the entire model on the uniform circular motions which were at the heart of Aristotle’s philosophy. The weakness is that the circles were no longer centered directly on the Earth. This compromised the symmetry of Aristotle’s theory. Nevertheless, with further adjustments the rather complicated model proved to be quite accurate, and served the needs of astrologers, scientists, and navigators for over 1400 years. A Scientific Model of the Universe The significance of Ptolemy’s model is that it bridged the gap between the quantitative astronomy of the Babylonians and the philosophical cosmology of the Greeks. For the first time, there was one description of the Universe that could both predict measurable phenomena and explain why they occurred. The model could be tested by carefully comparing its predictions to precise astronomical observations. Erroneous predictions provided clues as to how the model needed to be adjusted; successful predictions increased confidence that the model was accurate. This aspect of the model, its “test-ability”, is the definitive characteristic of all scientific theories. Of course, credit for developing the first scientific model of the Cosmos does not belong to any one person. Ptolemy relied on the results of thousands of years of astronomy and hundreds of years of philosophical debate. This, too, is characteristic of scientific investigation. The Foundations of Natural Philosophy 1-20 Review Questions Aristarchus and the Distance to the Heavens 20. What was the basis for Aristarchus’ argument that the stars must be unimaginably far away? 21. Pick out a distant object and look at it while you move over a small distance. How do your observations support Aristarchus’ argument? 22. What does a “full Moon” look like in the sky, and at what time of day do we see it? 23. What would we “see” every full Moon if the motion of the Earth and the motion of the Moon were in the same plane? 24. Why doesn’t the effect described above happen every month? 25. What causes a solar eclipse? 26. How would Aristarchus have known that the shadow of the Moon on the Earth was very small during a solar eclipse (in comparison to the size of the Earth)? 27. What two possible scenarios could cause the small shadow described above? 28. If the Moon’s shadow on the Earth (during a solar eclipse) did not “taper”, how big would the Moon be? 29. If the Earth’s shadow on the Moon (during a lunar eclipse) did not taper, how big would the Earth be? 30. How did Aristarchus know whether or not this was a reasonable answer? 31. What does this imply about what happens to the shadows cast between the Moon and the Earth? 32. Give an example (different from an example in the book) of an object close to you with the same angular size as something far away from you (give rough distances in your example)… 33. What observation leads to the conclusion that the Earth and the Sun have the same angular size? 34. If you can see the Moon, measure it using the method described in the Chapter. (Show your calculations.) Eratosthenes and the Size of the Earth 35. Refer to figure 1.16. (a) Why should the two angles marked be the same size, and what must be true about the Sun’s rays for this to happen? (b) Why should the ratio of the length of the stick to the length of the shadow be equivalent to the ratio of the radius of the Earth to the distance between Alexandria and Syene (Geometry)? Ptolemy and a More Perfect Model 36. What is meant by “retrograde” motion? 37. At which point in its motion would a planet appear to switch direction in the sky if it were traveling in an “epicycle” (figure 1.17)? 38. What is “scientific” about Ptolemy’s model of the Universe? The Foundations of Natural Philosophy 1-21 1.5 Our Modern View of the Cosmos The Ptolemaic system, with adjustments and refinements, was a remarkably successful tool for making astronomical calculations. For the next 1400 years, it was the most accurate way to predict celestial motions; and was thus indispensable to the any fields that relied on knowledge of those motions. This included natural philosophy, but also politics, medicine, theology, time-keeping, navigation, and astrology. An Important Shift in Perspective 1300 years after Ptolemy, the Aristotelian model of an Earth centered universe was challenged by Nicolaus Copernicus (1473 AD -1543 AD), and Galileo Galilei (1564 AD – 1642 AD), two characters that we will spend more time talking about in Chapter 3. Copernicus and Galileo were both proponents of a Suncenttere model, reminiscent of Aristarchus, although they each had different reasons for believing in this. The Sun-centered universe vastly simplified the calculations of celestial motion, and this was Copernicus’ chief rationale for this shift in perspective. What the Universe now looked like was the Sun, with Mercury, Venus, Earth, Mars, Jupiter and Saturn all in orbit around it. The Moon, however, orbited the Earth. All of the orbits were uniform circles, and the size of the orbits was related to how long it took for the planet to go all of the way around the Sun. Saturn took about 30 years to orbit the Sun, and traveled in the largest circle. Mercury was closest to the Sun, with an orbital period of only about 88 days. Shifting the Sun to the center of the motions eliminated all of the deferents and epicycles of the Ptolemaic system, vastly reducing the complexity of the calculations. Galileo, on the other hand, had made direct observations of the Sun, Moon, Venus, and Jupiter that all suggested that the Aristotelian/Ptolemaic model was inaccurate. Through his telescope Galileo had seen spots on the Sun, craters on the Moon, and “ears” on Saturn evidence that these celestial bodies were not the perfect spheres they had been assumed to be. He saw that although Venus always remained near the Sun in the sky, it nevertheless had distinct phases, like our Moon. This observation could only be explained by supposing that Venus orbits the Sun, not the Earth. Furthermore, he saw 4 small stars that moved back and forth, always next to Jupiter. These had to be moons of Jupiter, orbiting that planet. Clearly, not all heavenly bodies orbited the Earth. These points of view were not well-received, or widely adopted, for a few reasons. First of all, Christian doctrine assumed that since Man was God’s most important creation, Earth must be the center of the Universe. It was an extremely strange suggestion to say that this was not the case. Furthermore, if the Earth is itself orbiting the Sun, then the Earth must be in motion. Not only is it moving through space to accomplish its orbit, it must be spinning around its axis once every day to account for the cycles of day and night. The Foundations of Natural Philosophy 1-22 Chapter 1: Looking Forward Some of the more sophisticated concepts covered later in the book will build on your understanding on a number of the topics covered in this chapter. For a full appreciation of future topics, you should make sure that you have a good understanding of the topics listed below: (Material covered in supplemental material is noted as such under the “Chapter” column.) Chapter 1 Topic Key concept Future Topic Chapter Ray Model of Light A light ‘ray’ travels in a straight line from its source. Ray Optics Wave-particle duality 2 Suppl. Aristotelian Cosmology The Earth is at rest at the center of the universe and a medium known as ‘aether’ fills the space between the Earth and the stars Satellite Motion 3 The Ptolemaic System The planets, the Sun and the Moon orbit the Earth by means of ‘epicycles’ Satellite Motion 3 Solar System Model The Earth is orbiting the Sun – i.e. it is moving at a very high speed (relative to the background stars). Special Relativity – the Michelson-Morley Experiment Suppl. Big Bang Model The Universe started with an explosion from a tiny mass at extremely high density and temperature. The Universe has been expanding ever since. The accelerating Universe Suppl. The Foundations of Natural Philosophy 1-23 Chapter 1: Applications Section 1: What planet are these questions from? (For Questions #1-9) As you slowly wake from suspended animation, you realize that your space-craft has crash-landed on an alien world; you don’t have any idea where you are. You know from the date on your watch that you must still be in our solar system; not enough time has passed for you to have gone too far astray. You quickly inventory the tools at your disposal: a pencil, a tape measure, a mirror, some cardboard, and a Daft Punk CD. You put on your space-suit and open the door… 1. You immediately notice that the sun looks smaller in the sky than you are used to. In an effort to figure out where you are, you set up a pinhole and a screen to project an image of the sun. You find that you get an image that is exactly 1 centimeter in diameter when the pinhole is 572 centimeters from the screen. What is the angular size of the sun as viewed from your location? Draw a diagram and show your calculations. 2. Are you closer to, or farther from, the Sun than the Earth is? Explain. 3. Your watch is still working, and you note that the image of the Sun on your screen is moving. It takes it 1.5 minutes to move one full diameter. At what rate (in degrees per hour) is the Sun moving across the sky? 4. How long will it take the Sun to go around a full 360 degrees? 5. The next day, you notice a truly huge object rising in the sky at the horizon. It must be an enormous planet. As it slowly clears the horizon, you recognize its swirling, stormy atmosphere from photographs: the planet Jupiter. You must be on one of its many moons! To measure its angular size, you run back to the ship and get the CD. Holding the CD (diameter: 12 centimeters) in front of your face, you move it back and forth until it exactly masks the planet. This happens when the CD is 60 centimeters from your eye. What is the angular size of Jupiter in the sky? 6. Sketch the relative sizes of Jupiter and the Sun as they appear in your sky. 7. You notice that only half of Jupiter is illuminated by the Sun from your point of view. Sketch the arrangement of Jupiter, the Sun, and the moon you are standing on that would lead to this observation. 8. If you are stuck on this moon for a while, are you likely to experience total solar eclipses? Explain, using a diagram. 9. Are you likely to experience eclipses of Jupiter? If so, will they be total eclipses? Explain, using a diagram. Section 2: Angular Size, Phases and Eclipses 1. The Moon has a diameter of 3.5 x 103 km and its angular size, as seen from the Earth, is 0.5 o . Calculate the distance from the Earth to the Moon. 2. Aristarchus determined that the diameter of the Earth is 3.5 times the diameter of the Moon. Calculate the angular size of the Earth as viewed from the Moon. 3. Sketch the arrangement of the Earth-Sun-Moon system during a half moon as seen from the Earth. 4. The distance from the Earth to the Sun is 1.5 x 108 km. Using the answer you calculated for the distance from the Earth to the Moon, and your sketch from question 3, calculate the distance from the Moon to the Sun. 5. Given that that the angular size of the Sun is 0.5 o as viewed from the Earth, calculate its diameter. 6. What would you expect the angular size of the Sun to be for an observer standing on the Moon? Explain your reasoning and state any assumptions that you have made. 7. Note: For this problem, the relative sizes of the two objects should be shown approximately to scale. (a) Sketch the appearance of the Earth as seen from the Moon for the arrangement in question 3 (a half moon as seen from Earth). (b) Add a sketch of the appearance of the Sun, as seen from the Moon, to your answer for (a). 8. Sketch the arrangement of the Earth, Sun, and Moon that would result in an eclipse of the Sun, as seen by an observer on the Moon. 9. What would an observer on the Earth see during this same alignment of Earth, Moon, and Sun? Explain.