Science A-17 Lecture Notes:
The Astronomical Perspective
[ Index of Lectures]
Welcome to the Lecture Notes. Originally this page contained summaries for the lectures as given during the Fall semester of 1996. They were prepared by students in the course and were not "official". Some of the notes have been revised to reflect the way the lectures were given in 1998 or 1999, and are so identified. If you have comments and suggestions, or find errors or ommissions, please send email to Dave Latham. If there is a student who would like to work with Dave to revise some of the lecture notes that are out of date, please contact him.
Sun Signs and Cycles (revised 1998 DWL)
Friday 17 September 1999
How old is Astrology? This is a relevant question, because in early astrology we see some of the first attempts to make sense out of the motions of the planets. Around 150 AD Ptolemy codified many of the astrological interpretations still used by modern astrologers, but evidence for the Zodiacal Signs can be traced back still earlier to 419 BC, based on a cuneiform tablet now found in the Berlin State Museum. Early astrology developed in Mesopotamia, where the irregular flooding of the Tigris and Euphrates made predictions of great importance. This contrasts with Egypt, where there was little need to develop astrology, as the regular flooding of the Nile gave them cultural stability.
How long is the year? At first glance this question may seem unrelated to the origins of astrology, but there is a connection. The seasons reflect the motion of the Sun along the ecliptic. A solar year might be defined as the time it takes for the Sun to get back to the same point on the ecliptic. Ptolemy describes how an equatorial ring can be used to determine when the Sun returns to the same latitude (north or south) as in previous years. This was demonstrated in lecture, but it is only one of several possible techniques that were used by the ancients. Another way to define a year would be to use the stars. How long does it take a star to return to the same position in the sky compared to the Sun? One technique might be to watch for the dawn when a certain star is first visible, or the dusk when it is last visible.
By watching the Sun and stars for many decades (and keeping good records!) the ancients were able to beat down the errors in the determination of the length of the year. By the time of Hipparchus, about 150 BC, it was realized that the solar and sidereal (stellar) years had slightly different lengths, amounting to 14 days over a millenium:
- solar year = 365.242 days,
- sidereal year = 365.256 days.
Now comes the interesting part, and the connection back to the origins of astrology. The astrological signs are, by convention, fixed on the ecliptic relative to the equinoxes. But, the zodiacal constellations gradually slide along the ecliptic, by 14 days per millenium. For example, the constellation of Taurus is now slightly more than 1/12 of a circle past the position assigned to the astrological sign of Taurus. By figuring out how long ago the constellations were on top of the corresponding signs, we can deduce when the system was established. That was about 2450 years ago. By the way, if you believe that the positions of the planets among the stars at your birth have some influence on your life, you should ponder the fact that the stars have moved out of the signs that they occupied when the system of astrological interpretation was originally set up.
How old are the constellations? Some of the constellations are much older than astrology. One example is the Big Dipper, which is connected to the mythology of the bear in many northern cultures, where the constellation is circumpolar, and to the adze in ancient Egypt, where several constellations appear on tomb ceilings from almost 4000 years ago.
Stonehenge and Eclipses (revised 1998 DWL)
Monday September 27 1999
The armillary sphere is a useful model for visualizing the motion of the Sun along the ecliptic and for seeing how its maximum height above the horizon (at noon) varies with the seasons. A related phenomenon, and one that would have been obvious to ancient observers with a clear horizon, is the gradual march back and forth with the seasons of the Sun along the horizon when it rises (or sets). This effect gets larger as one moves away from the equator.
For example, at the latitude of Stonehenge, on the Salisbury plain in Britain, the sunrise moves almost 90 degrees along the horizon from winter to summer. The motion of the Sun north or south, or along the horizon at sunrise, is most rapid near the time of an equinox, but slows and then reverses itself at the time of a soltice. Several ancient structures are aligned to the point on the horizon where the Sun rises or sets at the summer or winter solstice. Examples are Newgrange in Ireland, with its special window over the entrance to the main corridor, Hovenweep in Utah with its angled windows, and Stonehenge itself with the heel stone marking the direction to sunrise at the summer solstice. Stonehenge is by far the most famous of the Megalithic Monuments. Stonehenge's ring is about a third of the size of a football field, and the largest stones weigh more than 20 tons. While Stonehenge is unlikely to have been an ancient observatory, it is certainly a monument to the discovery that the Sun behaves so regularly.
The Moon takes 29.530 days, on average, to return to the same phase. The phases of the Moon must have been a natural marker of the passage of time for the ancients. But, the number of moonths in a solar year does not come out even, it falls short by 10.88 days. Some calendars still follow the lunar motion religiously (Ramadan gradually works its way through the seasons in about 34 years), while others follow the Metonic cycle, a pattern of 12 years with 12 months and 7 years with 13 months. This explains the pattern of dates for Yom Kippur and also Easter (which has the additional constraint that it must be on a Sunday).
The Moon stays near the ecliptic, but travels on a great circle that is inclined by about 5 degrees to the great circle of the ecliptic. Eclipses can occur at the nodes where these two circles cross. The orbit of the Moon graduallly precesses around, so the nodes slide along the ecliptic. This explains why eclipses don't always come at the same time of year, but gradually move through the seasons with a period of 18 years, the time it takes for the nodes to slide all the way around the ecliptic. Notice that there are two nodes separated by 180 degrees, so eclipses can occur nominally six months apart.
We are at a special time in the history of the solar system, when the orbit of the Moon is just the right size so that the Moon has almost exactly the same apparent size as the Sun. A billion years ago the day was shorter and the Moon was closer, and in the distant future the day will grow longer and the Moon will recede to a larger orbit (we may remember to explain why this happens later in the course). The Moon's orbit is not exactly a circle, so some times it is close enough to completely cover the Sun during an eclipse, while other times the eclipse is only annular. The difference is spectacular, and the last few seconds entering totality are among the most dramatic moments in nature. One moment it is dusky, and the next it is as dark as a night with a full Moon, and there is a black, perfectly round hole where the Sun was. An annular eclipse is a little like kissing your sister. A total eclipse is the real thing, and the first time is something you will never forget.
The Aegean Birth of Science (revised 1999 DWL)
Wednesday 29 September 1999
Although the Egyptians must have been marvelous engineers, to build their pyramids and great tombs, there is hardly any evidence for science in their culture. The famous Rhind Papyrus promises to reveal all secrets, but in fact it contains little more than simple geometry. But, there must have been at least some astronomy, because we see the Sun, Moon, and constellations represented on several tomb ceilings, and the Great Pyramid is lined up north-south to a very impressive accuracy. That feat must have required astronomical observations.
In Mesapotamia we find plenty of evidence that the motions of the Sun and planets were being followed carefully, with planetary positions preserved to this day in a rich collection of cuneiform tablets. For example, a sequence of elongations of Venus are recorded on a tablet that also refers to the reign of King Ammisaduqa, and it has been possible to connect that pattern of planetary positions to a date around 1550 BC.
It is in the mercantile trading centers rimming the Aegean Sea that we find the first attempts to make rational explanations of natural phenomena, both with theoretical principles and mechanical models. The ancient Greeks were remarkably successful in laying out the scale of the cosmos. Not only did they determine a decent value for the radius of the Earth, but they were able to go further and to derive a surprisingly good value for the distance to the Moon, 60 times the radius of the Earth. They even had a way to derive the distance to the Sun that was fine in principle, but difficult to apply in practice.
Thales of Miletus in Asia Minor (ca. 600 BC) has the reputation for being the father of science, but we know almost nothing about his accomplishments. He taught that the world could be understood in terms of a single element, water. The important idea here is not whether water in its various forms can do the job, but rather that there must be some simple underlying principle that can be used to explain nature.
Pythagoras of Samos (ca. 500 BC), discovered that simple integers govern the tones that are harmonious from vibrating strings. For example, identical strings with lengths in the ratio 2:1 sing an octave apart. What a powerful idea this is, that natural phenomena obey simple integers! Pythagoras is also the source of a principle that dominates astronomy for almost two thousand years: motions in the heavens are eternal and therefore must be uniform and circular, or at least must be composed of uniform motion in circles. Of course, most people remember Pythagoras for his theorem about the sides of a right triangle: the sum of the squares of the two legs equals the square of the hypotenuse. The modern proof that we present here cheats, because it uses algebra:
The area of the large outer square can be found in two ways, first from the product of the two sides, and second by adding up the area of the inner square and the four triangles:
Expanding the (a+b)² term,
Empedocles (ca 450 BC) championed the idea that the world could be understood in terms of four fundamental elements: earth, water, air, and fire. Earth was the heaviest, and would naturally seek its rightful place at the center of the universe, followed by water, air, and fire.
Plato (ca 380 BC) was prominent in establishing an intellectual tradition in Athens. He promoted the Pythagorean ideal that only uniform motion in circles could be used to explain celestial motions, and he also believed that integers were fundamenatl to understanding the universe. For example, he speculated that the integers 2, 3, 4, 9, 8, and 27 (2, 3, 2x2, 3x3, 2x2x2, and 3x3x3) could be used to explain the spacings of the planets (the Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn), although he left the proof as an exercise for the reader. He challenged his students to "save the phenomenon", to find rational ways to explain nature. >
One such phenomenon that needed saving was the retrograde motion observed for the planets, e.g. Mars and Mercury. Plato's student Eudoxos (ca 360 BC) took up this challenge, and was able to devise a scheme of nested spheres that could reproduce the observed retrograde motion for Mercury reasonably well, not in exact detail, but at least schematically. Eudoxos was unable to get his model to work for Mars. Nevertheless, we can admire the cleverness of this mechanical model for planetary motions, even if we can't use it to predict planetary positions.
Mars has a more dramatic retrograde motion than any of the other planets, but it still requires careful attention to track the phenomenon as it unfolds over a period of a few months. Most of the time Mars gradually slides to the east through the zodiacal constellations, keeping close to the ecliptic. But, as the season approaches when Mars will be crossing the meridian at midnight, the planet first slows and then reverses its eastward drift, moving back to the west through the stars for two or three months. By the time Mars is prominent in the evening sky, it has once again returned to its normal eastward course. Curiously, in the middle of its retrogression, Mars brightens to its maximum.
Aristarchos of Samos (ca 275 BC) speculated that some of the phenomena could be saved by rotating the Earth once a day instead of rotating the sky, or by putting the Sun at rest and having the Earth move in a yearly circle around the Sun. Of course, these unorthodox ideas violated many common-sense notions about the solidity of the Earth, and what would happen to birds when they took to flight, if the Earth were really in motion. Aristarchos worked out a clever way to deduce the distance of the sun relative to the moon. When the Moon is exactly half illuminated, it makes a right angle to the Earth and Sun. If the angle between the Moon and Sun could be measured at that moment, then the long skinny triangle to the Sun could be solved. Aristarchos concluded that the Sun is about 19 times farther away than the Moon. The trouble is that the angle between the Moon and Sun is very close to a right angle, and it is difficult to measure it accurately enough to get a good distance. Aristarchos' distance to the Sun was too small by about a factor of 20. More successful was his determination of the distance to the Moon, based on the observation that it takes the Moon about three hours to move across the Earth's shadow during a lunar eclipse. This allowed Aristarchos to figure out what fraction of its monthly orbit the Moon had moved during the eclipse, and to compare that to the size of the Earth via its shadow. The geometry is a bit complicated, and Aristarchos only worked out the numbers approximately. Later on Hipparchos did the numbers more carefully and came up with a suprisingly good value of 60 Earth radii for the distance to the Moon.
Eratosthenes (ca 225 BC) was librarian at Alexandria in Egypt during the so-called Hellenistic period that followed the conquests of Alexander the Great. He is famous for figuring out the size of the Earth in real linear units, using Greek stades as his measure. Learn all about the details of this accomplishment by attending your first section. Every educated person knew that the world was round, even in the time of Columbus, although there was some confusion about the size of the stade. Columbus argued that the world was much smaller than the traditional value inherited from the time of Eratosthenes. That was why he thought he had a chance to reach the orient by sailing west. He was wrong, but he got lucky.
Biological evolution is slow. It has taken 3 million years for the volume of the human brain case to double from that of our ancestors, Australopithecus. Sure, we have easy access to mountains of information now, and those of us fortunate enough to live in an affluent society are blessed by abundant nutrition and modern medicine, but it is hard to support the idea that we are any brainier than the Greeks. Just contemplate the magnitude of their accomplishment of laying out the scale of the cosmos.
The Puzzle of the Planets (Revised 1998 DWL
Friday 1 October 1999
Aristotle (ca 350 BC) was another one of Plato's disciples in Athens, and his grand summary of how the cosmos must be put together dominated philosophy for almost two thousand years. At the outer fringe there was the sphere of the sky, carrying the fixed stars with it as it rotated once a day. Ultimately, all celestial motions could be traced back to this prime mover, but with the planets lagging more and more behind as one worked inwards towards the Earth.
To produce retrograde motions for the planets showing that phenomenon (Saturn, Jupiter, Mars, Venus, and Mercury), Aristotle built on the Eudoxan scheme of nested spheres. Each sphere had a uniform rotation about its axis, with the rates of rotation and pivot points adjusted to save the phenomena. Saturn was the outermost planet and the closest to the sphere of the stars, so it lagged the least behind the motion of the stars and had the smallest retrograde loop. Before moving inward to set up the spheres for the motions of the next planet, Jupiter, Aristotle first introduced additional spheres to undo the motions for Saturn so that he could start fresh with Jupiter. This approach doubled the number of spheres needed for each planet, and partly explains why it took Aristotle 55 spheres in total to get the job done for the whole system.
The heaviest element, earth, naturally fell to the center of Aristotle's cosmos, where it remained firmly at rest, surrounded first by the next heaviest element, water, followed by the lighter element of air. This brief outline does little justice to the full impact of the Aristotelian world view, which was breath-taking in the breadth of its argument. To be sure, this was still a philosophical model, to be used to explain how the cosmos works, but not useful for predicting planetary positions with any accuracy.
It is not until five centuries after Aristotle that we see mathematical models good enough to predict accurate positions for the planets. These models are brilliantly reported by Ptolemy (ca 150 AD) in a book known by its Arabic name, the Almagest, the Greatest. Ptolemy tried hard to obey the rules laid down by Plato that celestial motions must be uniform in a circle, or at least be composed of a combination of uniform circular motions, but in the end he found that he had to stretch the rules in order to get his models to save the phenomena accurately.
Although retrograde motion may be the most obvious nonuniformity in the motion of a planet, one that will clearly require combinations of circles or spheres, there is an additional problem that must also be solved. This problem may be easiest to see for the Sun, where there is no retrograde motion. Nevertheless, the Sun does not move uniformly along the ecliptic, but instead moves just a bit slower in summer and faster in winter. This can be seen by adding up the number of days that pass between the equinoxes, which must occur 180 degrees apart on the sky, because they are defined by the intersections of two great circles. It takes 186.5 days for the Sun to move through the summer half of the sky, from March 20 to September 23, but only 178.7 days to cross the winter half. Ptolemy was able to model this nonuniform motion by offsetting the circle that carries the Sun just a little from the Earth. The Sun still moves uniformly in its circle, but to an observer on Earth it appears to move more slowly when it is seen on the side towards the offset, when it is farther away. If you think the explanation for summer is that the Sun is closer to the Earth in that season, then you should think again.
To produce retrograde motion Ptolemy invoked an epicycle, a small circle whose center is carried along the main circle for the planet, the deferent. The planet itself moves around uniformly on the epicycle as the center of the epicycle moves uniformly along the deferrent. The size of the epicycle and its speed of rotation can be adjusted to get the best match to the timing and size observed for the retrograde loops. In addition to producing retrograde motion, the epicycle has another major success; it explains why the outer planets, like Mars and Jupiter, are brightest when they are in the middle of their retrograde loop and therfore are the closest to Earth.
But, it turns out that an eccentric deferent plus an epicycle are not enough to reproduce accurately the variety of sizes and shapes actually observed for the retrograde loops of Mars. To solve this problem Ptolemy invented yet another device, the equant. He discovered that if instead of the center of the deferent he used a different point for the center of uniform motion, then he could reproduce the retrograde loops remarkably well. Curiously, he found that he got the best results when the location of the equant point was on the other side of the Earth, equal and opposite to the center of the deferent.
Ptolemy must have worried that his equant was a cheat and did not follow the full spirit of the rules laid down by Plato 500 years earlier. But, how could you argue with the remarkable success that it allowed? Ptolemy's model was so good at predicting planetary positions that it lasted more than 1000 years.
Copernicus (Revised 1998 DWL)
Monday 4 October 1999
Our guest lecturer, Johannes Cracoviensis, discussed how Copernicus would have been viewed by his colleagues, soon after his death in 1543. Remember, these were heady times, with many exciting changes in the world. Perhaps the biggest impact came from the invention of printing, which made it possible for books to reach a much wider audience. Without printing it is unlikely that we would even know about Martin Luther or Nicolaus Copernicus.
The woodcut of the curious medieval traveller poking his head through the sky, to see heaven and how the celestial machinery works, is out of tune with the prevailing attitudes of the 16th century. It is a fake, first published in 1888 by Flammarion. A woodcut from the so-called Nuremberg Chronicle of 1493 gives a more authentic view, with the spheres of earth, water, air, and fire fixed at the center, surrounded by the crystalline spheres of the planets, with God enthoned above all and attended by his angelic host. This was an orderly universe, and curiosity was unecessary. A brazen attempt to pierce the crystal heavens would have been blasphemous.
Holbein's Ambassadors, painted in 1533, further illustrates the pholosophy of the day. The painting is full of symbolism. The Quadrivium, the advanced part of the standard curriculum after the Trivium (grammar, logic, rhetoric), is symbolized by a globe and Sun dial for astronomy, a lute and song book for music, an arithmetic book, and several examples of perspective for geometry. But, these forms of human knowledge are vanities, fleeting notions compared to the Truth of God, which is symbolized by the crucifix half hidden by the curtains in the background.
Copernicus was born in Torun, Poland, in 1473, and did his undergraduate work at the University of Cracow, which boasted not one, but two astronomy professors. For his graduate work he studied canon law and medicine in Italy. Returning to Poland at age 30, he took up a lifetime post as a canon of the Cathedral of Frombork, which allowed him some time to pursue his astronomical researches. Copernicus first announced his work on the idea that the Earth moves and the Sun stands still in a small manuscript, the Commentariolus, written about 1512 (only three copies survive to this day). Over the next 30 years he continued to refine his ideas and to work out the mathematical details for the planetary motions. He organized the results of his long labors into a magnificent manuscript. The original, in his own hand, survives in the Jagellonian Library.
It is possible that Copernicus' manuscript never would have been published, except for the intervention of a young astronomy professor from Wittenberg, Georg Joachim Rheticus. In 1539 the young Lutheran set out to visit the Catholic Copernicus, to learn more about his heliocentric system. Swept along by the enthusiasm of his young disciple, Copernicus allowed Rheticus to publish the Narratio Prima in 1540, a brief report on the new system and its advantages.
For example, in the Copernican system the retrograde motion of Mars no longer requires an epicycle, but is instead the natural result of the Earth overtaking Mars, as Earth moves past in the next orbit closer to the Sun from Mars. In the Ptolemaic system there was no particular reason why Mars and Jupiter and Saturn always retrograded only when they were near the meridian at midnight, directly opposite the Sun. That's just the way it was. In the Copernican system there is a reason why Mars retrogrades at opposition and is brightest then, because that is when the Earth is bypassing Mars and the distance between the two planets is the smallest.
Another feature of the Copernican system is that the order of the planets in distance from the Sun is the same as the order of increasing period. Mercury, with the shortest period, is closest to the Sun, while Saturn, the most distant, has the longest period. Furthermore, the sizes of the planetary orbits are fixed relative to the Earth's orbit as a common measure, because this is the only way that the retrograde loops come out the right size. In the Ptolemaic system there was no observed feature of Mars' orbit that set the the size of the orbit. What mattered for getting the size of the retrograde loop right was only the relative size of the epicycle and the deferent, not the absolute sizes. Ptolemy chose to pack his planetary circles as close together as the room required for the epicycles allowed, so as not to waste space. But, this choice was arbitrary, as far as saving the phenomena was concerned.
Rheticus convinced Copernicus to have his book printed in Nuremberg, and he took on the job of supervising the task. This responsibility was taken over by a local scholar and theologian, Andreas Osiander, when Rheticus moved on to a new job in Leipzig. In order to disarm possible criticism of the book because of its unorthodox cosmology, which might be seen as contradicting Holy Scripture, Osiander added an unsigned introduction on the nature of hypotheses:
"... For it is the duty of an astronomer to record celestial motions through careful observation. Then, turning to the causes of these motions, he must conceive and devise hypotheses about them, since he cannot in any way attain the true cause. But, from whatever assumptions he adopts, the motions can be correctly calculated from the principles of geometry for the past as well as the future. The present author has performed both these duties excellently. For these hypotheses need not be true or even probable; if they provide a calculus consistent with the observations, that alone is sufficient. ... So far as hypotheses are concered, let no one expect anything certain from astronomy, which cannot furnish it, lest he accept as true ideas conceived for another purpose, and depart from this study a greater fool than when he entered it."
Osiander may also have been involved in changing the title of the book by adding two words Orbium Coelestium to Copernicus' title of De Revolutionibus (On the Revolutions). Copernicus lived barely long enough to see the final printed pages of his book on the day he died.
Why did Johannes Cracoviensis like Copernicus' new system? From the way that the book was annotated by readers in the decades after Copernicus' death, we can see that they were most impressed by the way Copernicus had been able to eliminate the equant, which didn't really obey the axiom of astronomy, that celestial motions must be uniform and circular, or composed of uniform motion in circles. In a way Copernicus was a conservative, and not a revolutionary. He wanted to return astronomy to a purer and more pristine state.
For most early readers the motion of the Earth nowhere offended the principles of mathematics and therefore could be used as a hypothesis for predicting planetary positions. But, the idea that the Earth actually did move and the Sun actually did stand still was in serious trouble on several fronts. The most serious problem must have been that the Holy Scriptures clearly state in several passages that the Earth, and not the Sun, is at rest. Another difficulty was that no annual reflex motion was observed for the stars. Just as the most distant planet, Saturn, shows a small retrograde loop as the Earth bypasses it, so the stars should also appear to follow a tiny loop reflecting the annual motion of the Earth in its orbit. The fact that no such reflex motion was observed for the stars implied that they must be at impossibly large distances, making the cosmos unimaginably huge.
Why did Copernicus feel compelled to introduce his new system? Was it because the old Ptolemaic system had grown too cumbersome, as more and more epicycles had to be added to keep it working accurately? No, this is a modern myth, that the Ptolemaic system was about to crumble under the weight of a jungle of epicycles. In fact, Copernicus ends up using more epicycles than Ptolemy. Was it because the Ptolemaic system was not accurate enough? Copernicus does notice some of the inaccuracies of the old predictions, but he does not mention this as a problem in his book. In fact, the Copernican system is not particularly more accurate than the Ptolemaic system. For a while, the Prutenic Tables, calculated by Erasmus Rheinhold soon after Copernicus' death, are somewhat better than the old Alphonsine Tables, calculated in the 13th century from the Ptolemaic system. But, as time passes the Prutenic Tables gradually drift out of synch too, much like a watch that runs slightly fast or slow drifts out of synch as time passes after it is reset.
Copernicus seems to have been motivated more by aesthetics than by practical problems with the Ptolemaic system. He abhored the equant, which he viewed as a cheat, and he admired the way that his own system ordered the planets by distance together with period, with the actual distance to each planet linked to the size of the Earth's orbit. As he says in De Revolutionibus:
"In this arrangment, therefore, we discover a marvelous symmetry of the universe, and an established harmonius linkage between the motion of the spheres and their size, as can be found in no other way. Thus we perceive why the direct and retorgrade arcs appear greater in Jupiter than in Saturn and smaller than in Mars, and why this reversal in direction appears more frequently in Saturn than in Jupiter, and more rarely in Mars and Venus than in Mercury. All these phenomena proceed from the same cause, which is the Earth's motion. Yet, none of these phenomena appear in the fixed stars. This proves their immense height, which makes the annual parallax vanish from before our eyes. ... So vast without any question is this divine handiwork of the Divine Creator."
Tycho Brahe (Revised 1999 DWL)
Wednesday 6 October 1998
Serious readers of Copernicus' book in the 16th century were not so much impressed by the heliocentric cosmology as they were by the elimination of the equant and the return to a purer form of the old Pythagorean ideal that celestial motions must be uniform and circular, or composed of uniform motion in circles. Evidence for this comes from the notes made in the margins by knowledgeable readers. The pace of discovery in modern astronomy has been breath taking, driven mostly by breakthroughs with new instruments and new technology. In contrast, it took almost a century for widespread acceptance of the revolutionary idea that the sun must be at the center and the earth must move. When the revolution finally came, it was driven by discoveries made with two new technologies, Galileo's telescope on the one hand, and Tycho Brahe's remarkable instruments for measuring positions of stars and planets on the other.
Tycho was born into the Danish nobility but ended up marrying a commoner, which meant that under Danish law his children could not inherit any of his considerable wealth. This may have been an important factor in his decision, late in his career, to move on to Prague and the court of Rudolph II. That was a fateful move, because it allowed the brilliant young German mathematician Johannes Kepler to come to Prague to work with Tycho. It was the access to Tycho's treasure trove of accurate observations of Mars that led to Kepler's reform of astronomy.
The young Tycho was quite a prodigy, and he entered the University of Wittenberg at the unusually early age of 13. Going off to Germany for an advanced education was rather adventuresome compared to the norm for Danish nobility at that time. Tycho was supposed to study law, but he secretly yearned for a career in astronomy. He spent all his spare allowance on astronomy books and snuck out late at night to go observing. In 1563 he followed the progress of Jupiter as it moved towards one of its Great Conjunctions with Saturn, checking to see how well the ephemeris he had bought predicted the correct day for the closest approach of the two planets. He was apalled when the tables, based on the new Copernican theory, missed by more than a day, and he decided then and there at the tender age of 17 that he would devote his life to the reform of astronomy.
Copernicus said that he would be more than happy if he could predict planetary positions to an accuracy of 10 arc minutes, about one third the diameter of the moon. This is rather crude compared to the best detail that the human eye can resolve, which is about 1 arc minute. It was Tycho's goal to improve astronomy to the limit set by the eye. To get a scale big enough to read to 1 arc minute, Tycho reckoned that his instrument would have to be enormous, and he had one built which was 5 meters across. This proved to be too unwieldy for effective use, so Tycho turned his attention to improving the design of both the sighting mechanism and the way to read out the scales.
Before Tycho was able to get his new instruments into operation, a bright new star appeared in the constellation of Cassiopeia. Tycho was able to collect together observations of the nova (it was actually a supernova in modern terms) from all over Europe and to show that everyone saw the star at exactly the same position in the same constellation. This meant that it could not be relatively nearby, but instead must lie at least as far away as the orbit of the moon.
The moon is close enough to the earth that different observers at different geographic locations see it at slightly different positions compared to the background stars, even when they observe at exactly the same time. The angular difference amounts to about 1 degree for observers separated by one earth radius, because the moon is at a distance of 60 earth radii, as known from Greek times. This angle is called the diurnal parallax, as opposed to the annual parallax, because it is related to the radius of the earth as it rotates once a day instead of to the distance from the earth to the sun and the orbital motion of the earth around the sun during the year. In other words, the baseline for the annual parallax is much bigger, because it is the earth's orbit.
Showing that the nova must lie beyond the orbit of the moon was a big deal, because everyone accepted the Aristotelian doctrine that everything in the heavens should be eternal and incorruptible. Okay, maybe the moon could be slightly tainted by the corruptibility of the earth, because it was so near by, and that might explain the obscure markings barely discernible on the face of the moon. But, a new star springing to light in the pure celestial domain? That was unthinkable. Tycho earned considerable attention by lecturing on the topic and publishing a book, De Stella Nova. When he threatened to move back to Germany so that he could pursue his ambitions to build large instruments, the king of Denmark granted him the island of Hven as a fiefdom and money to build an observatory with state-of-the-art instruments.
Even as Tycho's new observatory was under construction, a spectacular comet appeared in the evening sky in 1577. Tycho immediately set out to determine how far away the comet must be, just as he had worked to get the distance to the nova. However, this was a more difficult problem. The new star had held firm to its position in the constellation of Cassiopeia without moving compared to the fixed stars. In contrast, the comet moved compared to the fixed stars, crossing from one constellation to the next in a few days. This meant that Tycho had to be careful to make sure the comet had been observed at the same time when he compared the positions from different observers around Europe. Better yet, he could compare the apparent trajectories observed from different locations. Once again he found that there was no hint of a diurnal parallax, and therefore the comet must also be more distant than the moon. This posed a serious problem for the Aristotelian model of the cosmos, where the planets were supposed to be carried along by crystalline spheres, all carefully nested within each other. Where should the sphere to carry the comet be placed, and how could it avoid interfering with the other spheres?
One of Tycho's favorite instruments for measuring the angle between two stars (or between a planet and a star) was a sextant. Typically the sextant was mounted on a large universal ball joint so that it could be swung around easily to line up with the two stars to be measured. Tycho invented some ingenious devices for sighting his instruments accurately, and he developed a kind of vernier scheme for his scales so that they could be read out to 1 arc minute. But, the sextant wasn't always the best instrument for a specific project. For example, Tycho built a large mural quadrant on the wall of his living room. One of the primary jobs for this instrument was to measure the maximum altitude above the horizon reached by the sun, so the wall was carefully lined up with the meridian.
A specific application of Tycho's mural quadrant was to determine the orientation of the equator at Hven so that the latitude of the observatory could be calculated. Tycho watched carefully to see the maximum altitude reached by the sun at the summer and winter solstices, so that he could take the average of the two values to find the altitude of the equator. Tycho also observed the pole star carefully, to get its average altitude above the horizon, because that would give him an independent determination of the latitude of Hven. Curiously, the two different ways to get the latitude disagreed by a few arc minutes. Eventually Tycho decided that the problem must be refraction by the earth's atmosphere, which acts like a very weak prism. Near the horizon the light of a star must pass along a much longer path through the atmosphere, and its apparent altitude is displaced upward slightly by the refraction effect. At higher altitudes the effect grows smaller, and it disappears for objects straight overhead. Knowing how to correct for the lifting effect of atmospheric refraction on observations made near the horizon proved to be crucial for Tycho's remarkable campaign to measure the distance to Mars.
In the Ptolemaic system the epicycle of Mars is located on the far side of the sun, and even at its closest approach to earth Mars is always more distant than the sun. In the Copernican system the Earth gets much closer to Mars, when it is on the same side of the sun as Mars and is by-passing Mars. Tycho had the brilliant idea of using this difference to distinguish between the two systems. If he could measure the distance to Mars and show that it gets closer than the sun, then he would have a proof that the Ptolemaic system must be wrong. Conversely, if Mars never gets closer than the sun, then the Copernican system must be wrong.
How good would Tycho's observations have to be, to allow him to get the distance to Mars? The scale of the problem is set by the distance to the sun, which was known already from the work of Aristarchus and Hipparchus to be 19 times the distance to the moon. Since the diurnal parallax for the moon is about a degree, the diurnal parallax for the sun would have to be 19 times smaller, or about 3 arc minutes. In the Copernican system Mars gets about twice as close as the sun, so it's diurnal parallax should be twice as big, about 6 arc minutes. If he could achieve an accuracy of 1 arc minute, a diurnal parallax this big should be within reach. Of course, this would require observing Mars near the horizon as well as high in the sky, to see how its position shifted during the night compared to the fixed stars.
Tycho's first attempts to measure the diurnal parallax of Mars gave inconclusive results. He was frustrated by instabilities in his instruments, which were mounted up on towers. He decided that the Mars parallax project would require a new subterranean observatory with a new generation of more stable instruments. But, in the 1582-3 conjunction with Mars he still failed to detect a parallax, and in the 1585 conjunction the result came out slightly negative. A negative parallax would imply that Mars was more distant than the comparison stars used for the measurents, and that obviously couldn't be right. Apparently the parallax was smaller than the measurement errors, which happened to conspire to give the negative number.
Despite these failures, Tycho was not ready to give up his Mars parallax campaign. He decided that he needed to correct the observations near the horizon for the effects of atmospheric refraction, and he turned his attention to determining how to do this correction more accurately. In the 1587 conjunction he finally managed to get a value for the parallax of Mars, and he sent out a preliminary announcement of this great triumph in some private letters to colleagues.
But, Tycho was not able to confirm the 1587 result in any subsequent conjunctions, and he never formally published a result for the diurnal parallax of Mars. We know now, of course, that the Mars parallax project was doomed to failure from the beginning, because the Sun is actually 20 times farther away than the old value of 19 times the distance to the moon. But, one can hardly blame Tycho for accepting this value for the distance to the sun, because both Ptolemy and Copernicus had used it without question. And, it turns out that this incorrect distance to the sun introduced insidious errors into the way that Tycho was handling the corrections for atmospheric refraction. In the end, the Mars campaign led to a glorious success, because Kepler was able to use Tycho's remarkable treasure trove of observations to show that Mars moves in an elliptical orbit.
Tycho must have admired the Copernican system for the way that it handled retrograde motion so naturally, but he was deeply bothered by the motion of the earth that it required. Copernicus nowhere offended the principles of mathematics, but he threw the earth, "a lazy sluggish body unfit for motion" into such swift movement that the birds would surely be left behind. To solve this problem, Tycho devised a system which returned the earth to the center, where it could be fixed solidly at rest, with the sun in orbit around the earth, but with the planets all in orbit around the sun. This embodied some of the advantages of the Copernican system but avoided the moving earth problem.
Notice that in the Tychonic system the orbit of Mars crosses the orbit of the sun, and Mars can get closer to the Earth than the Sun. How did Tycho reconcile this with his failure to measure a diurnal parallax for Mars? We don't know the answer to this question. Perhaps Tycho was so enamored with the idea of having his own cosmological system and the way it combined the best of the Copernican mathematics with the Aristotelian physics that he was able to live with his nagging failure to measure the distance to Mars.
Kepler and The New Astronomy
Wednesday, October 2
Kepler, born in 1571, was one of the great figures in Astronomy and science in general. He even wrote a forerunner to the modern Science Fiction story--based on a trip to the moon.
His father was a mercenary soldier and he left to fight for the Catholics when Kepler was young (even though the Kepler's had been Lutherans for generations). Kepler went off to college with spectacularly good grades, except for astronomy. He studied under Michael Maestlin, a very distinguished astronomer of the day. Kepler began in a theological program, but eventually ended up teaching astronomy in Graz, where he really began thinking about astronomy, beginning to embrace the Copernican System.
Envisioning God as a "Great Geometer," Kepler constructs a solar system based on five platonic (perfect) solids spacing out the planets. After all, only six planets were as of yet discovered. Kepler published these ideas in his Mysterium Cosmographicum. Maestlin saw to the book's publication, and Kepler sends a copy to Tycho. With Paul Wittich gone, Tycho is in need of a talented mathematician, and so invites Kepler to Hveen.
Kepler doesn't go to Hveen, though. The new king of Denmark, Christian IV, decides not to patronize Tycho, who moves to Prague where Kepler meets him. Kepler wants Tycho's data, and Tycho wants a good mathematician and astronomer. Kepler's assignment as an astronomer is Mars. In 1601, Tycho drinks too much, gets a urinary infection and dies. It is not until 1609 that Kepler is able to publish his Astronomia Nova, in which his Law of Elipses and Law of Equal Areas in Equal Times is laid out--this at a time when he was supposed to be working on the "Rudolphine Tables."
The Warfare on Mars
Friday, October 4
Before Kepler's time, there was little relation between astronomy and physics. After all, hadn't Aristotle said that the Heavens were made out of a special, extraterrestrial ether? But Kepler, somewhat unsuccessfully, tries to introduce physics into the study of astronomy. Kepler begins thinking about the speeds of planets, and treats the Sun as a "governor of planets," with an emanation from the Sun driving those planets nearest it the fastest.
Mars, with its large eccentricty, makes many things about planetary motion more apparent to Kepler. To Kepler, it is more sensible that the sun be the universe's center, rather than some empty point as in Ptolemy's universe. Kepler is a realist: he discovers things and then builds models which make physical sense to him. Kepler is able to achieve very accurate predictions--far more accurate then Copernicus, with his physical worldview and Tycho's data.
Kepler wants to explain the planets' orbits with physics, and he wants the sun to be opposite the equant he has added for prediction purposes. This serves as an intermediate solution. In pursuing the Equant concept, Kepler discovers his Law of Areas.
The "aphelion" is planet's most distant point from the sun and the "perihelion" is the closest to the sun. Kepler, using Tycho's highly precise observations, will eventually develop his famous elliptical model, with the sun at one focus. What is the physcical reason? Kepler assumes a sort of "magnetism" emanating out from the sun--a sort of precusor to Newtonian gravitation.
The last work of Kepler's life is his "Rudolphine Tables," in which Kepler corrects many errors etc. It is one of the first books to use logarithms, a new mathematical tool. The frontpiece, too, is very interesting...depicting five great astronomers, with Kepler lurking below.
Monday, October 7
Galileo Galilei (1564-1642) is one of science's greatest "characters," being very dramatic and very argumentative. He had no fear of questioning authority. He began his training as a "pre-med," but ends up studying mathematics and physics. He gets an assistant professorship, eventually moving on to Padua in Venice, home of the finest glass-makers in the world at the time.
Galileo did some work in optics, developing a telescope. He began manufacturing telescopes and gave them to the military to gain favor with the local nobles. He turned one to the sky at some point. Sensibly, he first looked to the moon, where he saw earth-like features and irregularities. He published his discoveries made with the telescope in Sidereus Nuncius. Observing Jupiter, Galileo saw a "miniature" Copernican system--Jupiter has four moons, much like Earth has one. Galileo had discovered that Greek knowledge was incomplete. Galileo named these moons the "Medicean Stars." (1610) Not surprisingly, Galileo found a place in Florence under the patronage of Cosimo de Medici.
It is important to realize that Galieo's observations were far from perfect. Not only did he "imagine" some things, such as a crater on the moon "roughly the size of Bohemia," he was working with a telscope with an optical defect which Newton would explain and correct later. But Galilieo got the "big picture." Kepler, hearing of Galieo's telescope, worked out a theory of optics very advanced for his day.
Galileo, turning his telescope (which he called a "perspicillium") to the sun, discovered sunspots. People first imagined that these were the silhouettes of planets very close to the sun, but it was quickly realized that the sun had imperfections. So, Galileo discovered quite clearly that Greek knowledge was imperfect, just as the sun and the moon were imperfect.
Galileo also began observing Venus, keeping careful note of its phases. He found that Venus was imitating the moon, possesing a crescent phase. This was a very strong argument for the Copernican system, because, as Galileo said: "Venus has a full phase, so the Ptolemaic system must be wrong." Of course, this is not definitive proof for the Copernican system, for the Tychonic system explains Venus' phases as well.
The Galileo Affair
Monday, October 9
Galileo's protege, Castelli, gets into a discussion with the Grand Duchess Christina, and tells her that Galileo's predictions were true. Christina wonders how you reconcile this with Scripture. The answer given was that the Bible talked in language man could understand--God is "explaining" what he does. Besides, "The Bible tells how to go to heaven, not how the heavens go." Needless to say, Galileo is no longer speaking hypothetically about his theories, angering the Church a great deal.
Before Galileo, the concept of a fixed sun and a moving earth was merely a hypothesis, or a mathematical nicety. Galileo does injury to faith by calling Scripture false. In this, he comes close to making the rhetorical error known as Affirming the Consequent, where he assumes that because Venus shows phases that the Copernican system must be true. This is not a strong statement, because there could be an infinite number of explanations, including the Tychonic system. This makes important members of the Church, such as Bellarmine, very angry. Galileo begins to fear that the Copernican system will be outlawed, and so tries to lobby for it.
In 1616, he goes to Rome to plea for the heliocentric system. Bellarmine warns him to be more hypothetical, and he cools down. The rumors say that Galileo had been forced to recant or repent. Galileo again gets in a fight with the Church in a discussion of comets, this time making enemies in the Jesuits. In 1624, a friend of Galileo (Urban VIII, from Florence) becomes pope. Galileo dedicates his book to this pope, and visits the Vatican to discuss it. Urban says Galileo can write, but only if he treats his theories as hypothetical. So Galileo writes his Dialogues, but they are not at all neutral. The Pope's voice is represented by a character named "Simplicio"--the Pope feels insulted, especially since Simplicio comes across as a stupid character.
Friday, October 11
Mostly because of the insulting Dialogo, Galileo is called in for "vehement" suspicion of heresy. Of course, Galileo is legally able to outwit the Inquisition, but knows he won't be able to get off "scot free." He tries to plea-bargain but ends up being forced to abjure and say that he no longer believes in the Copernican system. Galileo is allowed to return home, but may not write any more. And so began the legend of Galileo, "martyr of science".
In 1638, Galileo breaks the rules and writes a book called the Discoursi, a book on "Two New Sciences," motion and mechanics. The book is smuggled out and published in Holland. In his book, Galileo not only makes discoveries, but breaks the rules of argument in his day. By using induction instead of deduction, he laid the groundwork for the way modern scientific argument would be made.
Galileo attacks the Aristotlean view of motion, which claims that velocity is proportional to force over resistance. He does his famous experiment with dropping balls and inclined planes, trying to discover if acceleration for falling bodies is constant. Using a scantling and pendulum-time interval methods, Galileo is able to determine this. Galileo, along with people at Merton College, are approaching the idea of inertia. Furthermore, Galileo bases his concept of motion on a decomposition into vertical and horizontal elements.
Newton and His Laws
Friday, October 18
Isaac Newton (1642-1727) was history's foremost physicist. His accomplishments include:
- Work in Optics; Reflecting Telescope
- Invented differential and integral calculus
- Formulated the laws of motion (Encapsulated at the beginning of his Principia)
- A Theory of Universal Gravitation
As a "self-help" student, the talented young Newton went off to Trinity college, which promptly closed for the year 1656 because of the plague. During this time, Newton returned home to Woolsthorp for his Annus Mirabilis, during which time he "minds mathematics and physics." Newton claims that he did his greatest work during these years (perhaps Newton is trying to create a myth here). He did in fact make some important discoveries in optics during this time, developing a "Newtonian Telescope" which used mirrors. Upon return to Trinity, Newton shows the telescope to preofessor Isaac Barrow, who resigns the famous Lucasian Chair in favor of Newton. Thanks mainly to the telscope also, Newton became a member of the Royal Society in 1672.
As a member of the Royal Society, Newton publishes a paper on optics containing his theories of colors and prism color division. Robert Hooke, another member of the Royal Society, criticizes the paper harshly. Newton, who never handled criticism well, leaves the society to sulk, not paying his dues. Newton's proponents in the Royal Society paid for him in his absence.
The rivalry between Hooke and Newton goes on...Hooke tries to "draw out" Newton again by asking questions about motion and falling objects. Newton makes an incorrect drawing in his reply, for which Hooke insults him, forcing Newton to sulk once again.
In 1684, Hooke, Halley and Wren are discussing the inverse square relationship of gravity and wondering if Kepler's ideas, particularly an elliptical orbit, could be derived from this. The problem is beyond all of them...Halley goes to the one man who can solve the problem: Newton. We can imagine the conversation between Newton and Halley:
Halley: "What would orbits look like if the gravitational force was 1 over R squared?"
Halley: "How can you know?"
Newton:"I've already calculated it."
Yes, Newton "the loner" had already solved this tremendously difficult problem, but had not published it, lest Hooke should find a reason to criticize him. Unfortunately, Newton had mislaid the solution, but is able to rework the problem.
In 1687, Newton writes his Philosophiae Naturalis Principia Mathematica, laying down many ground-breaking defnitions and laws:
- Definition 1
Realizing Earth's gravitational effects, Newton defines mass as independent of weight.
The quantity of motion can be measured by mass times velocity
Every body continues in its state of rest, or a uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. (Law of Inertia)
The change in motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. (F=ma)
To every action there is always opposed an equal reaction: or, the natural reactions of the two bodies upon each other are always equal, and contrary in direction.
Monday, October 21
Kepler tried to explain the sun as the central driving force with a sort of special "magnetism" driving the planets around it--this was Kepler's attempt to explain the solar system in physical terms. Kepler also suggested that tides are related to the moon, which Galileo thought was a ludicrous idea, going so far as to call it "occult". This is the scene into which Newton entered.
But before discussing Newton, it is important to consider Rene Descrates (1596-1650) He was born into a noble family, studied law, joined a volunteer army, and finally met Isaac Beakman who aroused his interest in philosophy and metaphysics. Descartes begins his work, on the foundations that (1)He must work alone and (2)assume all other philosophy to be wrong. He imagined a "plenary" universe filled with matter infinitely--as much as we can imagine. This matter, proposed Descartes, must have been set into motion by God. Descartes tried to reconcile the heliocentric view and religion by claiming that plenum centered plenum using "vortexes". This is not a good mathematical system, but a philosophical system. In 1644, Descartes published his ideas in Principia Philosophiae, which Newton would read and enjoy--though finding errors--and even mimic the title for his great Principia.
Around the same time, Christian Huygens explained the mysterious "handles" on Saturn as rings. Newton is thinking about this, as well as a wave theory of light and a law of acceleration or central acceleration. If a circular motion is considered, such as a ball spun about on a string of constant length, speed is constant, but the directional force velocity is constantly changing--the derived equation is a = v²/R. (SEE Abell Capter 3)
Wednesday, October 25 (Jimmy Liu)
At this point, the Newton’s laws of physics have been applied to the immediate terrestrial laboratory. However, a great feat of Newton was his formulation of the law of universal gravitation—F = G mM/R2. [Formula worked out in Abbell Chapter 3, p. 50]. Not only could the earth be understood, but also the sun, moon and cosmos. He also calculated how the force drops as distances increases [also refer to Abbell, chapter 3, p. 51]—one over the square of the radius, found through Kepler’s Third Law of Harmonics and central acceleration.
Even though Newton and Kepler essentially agree upon the sun moving the planets around in an orbit, there is a major difference between the Newtonian model and the Keplerian one. In Kepler’s theory, he believed that the planets were pushed from behind, like someone sweeping the orbs across the heavens with a broomstick. The orbs would assume a circle since the natural path of a celestial object would be circular.
To the experienced Newton, he believed that because of universal gravitation seen on earth and in the celestial, the sun pulled on the planets by gravity. The orbit would be a result of a planet’s natural path of a straight line pulled inwards to form a circular orbit. After Newton devised the workings of the sun and planets, the logical question came up, did the same principle hold for the earth and orbiting satellites such as the moon? In order to answer this question, he had no way to prove his hypothesis; unlike the solar system, with the sun having multiple satellites; the earth only had one satellite. However, Newton soon figured out that an apple on earth (one earth radii)—its fall to the ground— and the moon (at 60 earth radii) would suffice to answer his question. Using his second law of motion (F=MA) and universal gravitation, he found out he could cancel the little masses on both sides, as he proved that the inert mass of the little mass (the apple) equaled its gravitation mass through his pendulum experiments. He found out that indeed, the solution worked out to (1/60) squared. One excerpt describes Newton’s work beautifully.
Long, long he toiled, comparing first the curves traced by the cannon-ball as it soared and fell with that great curving road across the sky traced by the sailing moon. Was earth a loadstone holding them to their paths by that dark force whose mystery men have cloaked beneath a name? Yet, when he came to test and prove, he found that all the great deflections of the moon, her shining cadences from the path direct, were utterly inharmonious with the law of that dark force, at such a distance acting, measured from earth’s own centre… For Newton seized them and, with trembling hands, began to work his problem out anew. Then, then as on the page those figures tuned to hieroglyphs of the heaven, and he beheld the moving moon, with awful cadences falling into the path his law ordained, even to the foot and second, his hand shook and dropped the pencil.
"Work it out for me,"
He cried to those around him; for the weight of that celestial music overwhelmed him; and, on his page, those burning hieroglyphs were Thrones and Principalities and Powers… (Noyes, Watchers of the sky p. 197)
His work on universal gravitation was compiled in his Principia, however he had a nervous breakdown in the process. Later, he also worked as master of the mint, where he devised an ingenious serrated teethed coins that could not be clipped for silver. However, the Newton's discoveries would forever change astronomy …
The Discovery of Neptune
Monday, November 1, 1999 (Rachelle Gould 1999)
The Newtonian world view gained widespread acceptance because of its remarkable success in giving quantitative explanations and predictions for a variety of natural phenomena.
First of all, Newton's Laws of Motion together with the Law of Universal Gravitation lead back to Kepler's three laws, but in revised and more powerful forms (this success is discussed in detail in subsequent lectures).
The second success of Newton's laws regarded comets. Edmund Halley studied the comet that appeared in 1682 in detail, and he found a number of historical records that he interpreted as previous appearances of the same comet. After applying Newton's laws to the comet's path, he predicted that this comet had a period of seventy-six years. He was correct in the prediction, and (surprise!) the comet is now known as Halley's comet. It has been clarified that there are two types of comets: long-period comets and short-period comets. Long-period comets have periods of thousands of years, while short-period comets commonly have periods of three to ten years. Halley's comet is the only visible periodic comet, and it was last seen in 1986.
Newton's laws also made an explanation of the ocean's tides possible. The difference in the gravitational force that the moon exerts on the front and back of the earth causes two slight "bulges." As the earth rotates, these bulges move around the earth, causing the twice-daily cycle of high and low tides.
Newton's laws were also used to justify the claim that the earth is not a perfect sphere, but instead is oblate. The centrifugal force exerted upon the Earth's matter is greater at the equator. This matter has more of a tendency to "leave" the earth than does the matter near the poles; thus the earth has a shape that "bows out" at the middle. The oblateness of the earth offers physical proof that it moves, for the centrifugal force is a result of the rotation of the earth. For those who hesitated to believe that the Earth is oblate only because laws theorized that it should be so, proof of the earth's thicker middle could be found in the periods of pendulums. These periods are slightly less at the equator than at latitudes farther north or south. The reason for this is the lower acceleration due to gravity on the surface of the earth at the equator. This lower acceleration results from the fact the matter at the earth's equator is farther away from its center because the radius of the earth is larger, by about 43 kilometers, at the equator.
The fifth success of Newtonian mechanics was the explanation of the earth's precession, which is the drift in the orientation of the earth's axis of rotation. The tilt of the earth's axis is continually changed ever so slightly by the gravitational force that the moon exerts on the earth. The period for the axis to complete one rotation is 26,000 years.
Perhaps the most notable success of Newtonian mechanics was the discovery of the planet Neptune. In 1781 William Hershel, using a reflecting telescope with a mirror constructed from metal rather than glass, was able to notice that a certain celestial object looked different than the surrounding stars. The object was recognized as the seventh planet, Uranus, but there was a difficulty with its orbital path. The historical and modern observations of the object's motion did not match exactly the path predicted by Newton's laws.
The path of Uranus was, in short, not perfectly elliptical; it was tarnished by perturbations. The scientific community set to work to explain the reason for these perturbations. In 1821, when his data tables of Uranus' motion were published, Bouvard mentioned "extraneous and unknown influences" on the orbital motion of Uranus, but did not hypothesize as to what they might be (49).
The idea that orbital perturbations were caused by the existence of another planet was not a new one. In November 1758, the astronomer Clairaut hypothesized that Halley's comet could be "subject to totally unknown forces, such as the action ... of some planet too far distant from the sun ever to be perceived" (49). One of first documented commitments to the idea that the perturbations in Uranus' orbit were caused by another planet was made in 1834 by Dr. Thomas John Hussey. By 1836 Bouvard definitely supported the hypothesis of an exterior planet. Scientists such as Bessel, Madler, and Somerville also supported the hypothesis, and after 1838 the exterior planet hypothesis was quite widely accepted.
Thus was the scientific atmosphere in which young John Couch Adams found himself as he studied at Cambridge University. In 1841 while browsing in a bookshop, Adams came across Airy's report to the British Association for the Advancement of Science. After reading in the report that Bouvard's tables were seriously in error, Adams wrote a memo to himself expressing his goal to investigate the incorrectness of Bouvard's tables after his graduation. He vowed to try to discover the reason behind the perturbations in Uranus' orbit "in order to find whether they may be attributed to the action of an undiscovered planet beyond it ..." (76).
In 1842, Royal Academy of Sciences at Gottingen proposed as a prize question a full discussion of the motion of Uranus, explaining why Bouvard's tables were so inaccurate. Adams may or may not have been encouraged by the challenge; but for whatever reason, by 1843 Adams was convinced that the perturbations in Uranus' orbit were caused by a planet. He was determined to prove its existence, so he set to work on calculations. By October 1843 Adams had a preliminary solution.
Urbain Leverrier, a French astronomer who had distinguished himself through work on the orbital motions of planets and comets, was also concerned with the perturbations in Uranus' orbit. He and Adams worked independently on precisely the same issue, and neither knew of the work of the other. In June 1845, Leverrier decided to focus his attention on the motions of Uranus. Adams had already been working for two years.
In 1845 Adams left a concise summary of his solution to the problem of Uranus at the Royal Observatory in Britain. This solution claimed the existence of an exterior planet and offered details about it. Airy, Britain's Astronomer Royal, coolly did not accept the solution.
On November 5, 1845 Airy wrote to Adams that he would not accept his solution. Just a few days later, on November 10 in France, Leverrier presented his solution to the French Academy of Sciences, and they made a very favorable impression.
Both Adams and Leverrier had offered placements for the new planet, and when Airy learned of Leverrier's results, he was most pleased. Leverrier's placement agreed with Adams' to nearly 1 degree of arc. Airy wrote: "To this time I had considered that there was still room for doubt of the accuracy of Mr. Adams' investigations; for I think that the results of algebraical and numerical computations ... are liable to many risks of error" (103). Though he had refused to accept Adams' calculations, he accepted Leverrier's almost identical calculations readily.
Challis, a professor at Cambridge who worked with Adams in dealing with the unsupportive Airy, began searching for Uranus by "sweeping the sky" in the predicted area, on July 29, 1846. Despite Leverrier's convincing calculations and his eager insistence that a Frenchmen begin looking for the planet, no French telescopes turned toward the region.
Leverrier, eager to prove his calculations correct, wrote to Galle, an assistant at the Berlin observatory, asking him to search for the planet; on September 23, 1846, Galle began the search and observed a "star" of the eighth magnitude that was not on the newly published celestial map.
Challis, when he heard of the discovery, repented the fact that he had not recognized this "star" as a planet, for he too had observed the star. He wrote to Airy... "after four days of observing, the planet was in my grasp, if only I had examined or mapped the observations" (123).
On October 1, Leverrier wrote that he wanted the planet named Neptune, and he said that the French Bureau of Longitudes wanted this name as well. Just as astronomers were accepting his request, Leverrier suddenly changed his mind, saying he wanted the planet named after him. However, Europe's prominent astronomers were none too eager to abandon the tradition of naming planets in order to glorify one man.
When attempts to name the planet were made in England, people were surprised, for Adams had received little recognition for his simultaneous discovery. The first public credit given to Adams came through a letter written by John Hershel on October 3, 1846, in which he gives credit to Adams for independently arriving at the same conclusions as had Leverrier.
The public's unawareness of Adams was largely due to the fact that Airy gave him basically no credit. Airy fully supported Leverrier; he wrote to the Frenchman on October 14, "You are to be recognized beyond doubt as the real predictor of the planet's place. I may add that the English investigations, as I believe, were not quite so extensive as yours. They were known to me earlier than yours" (129).
This letter first alerted Leverrier that his claims were being contested by a young Englishman, and he was upset; a debate between the French and the British ensued. The debate between two historical rivals was possibly affected by unconnected Anglo-French political relations.
By end of 1846, the eighth planet had been observed by most astronomers in Europe and America, and the astronomical community had fixed its name as Neptune. One important discovery was made by William Lassell; he confirmed that the planet had a satellite in July 1847, and it was named Triton.
Leverrier and Adams met for the first time in June 1847. To the credit of both of them, neither still harbored bitterness resulting from the controversy. In fact, the two men became friends and remained so for the rest of their lives. The controversy of the discovery eventually died down, and Adams and Leverrier are now considered the co-discoverers of Neptune.
Note: All page references are taken from the following source: Grosser, Morton. The Discovery of Neptune. New York: Dover Publications, Inc., 1979.
Weighing the Earth
Monday, October 28 (Jimmy Liu)
After Newton worked out his law of universal gravitation, F= GmM/R2, there were two constants that were yet not derived—G (gravity) and Mearth (Mass of earth). The finding of these values could then yield one or the other. So in 1744, Maskeleyne tried to determine the gravitational constant. His experiment involved using a plumb bob, a weight dangling from a lengthy cord and a rather symmetrical mountain, Mount Shiehallion. Using the deflection of the stars to find A, guessing the mass of the mountain, and surveying the radius of the mountain, Maskeleyne was able to calculate a constant for gravity, which was in the vicinity of the correct value. Conceptually, he had succeeded in calculating G. However, it would take more precise experiment to come up with a practical value for G.
In 1798, the Cavendish experiment found a more accurate version of the G constant. A contraption made up of two heavy balls and two smaller balls, measured the attraction between the different balls. By measuring the time is took for the smallers balls to move toward the larger ones and recording the weight and distance, a value 6.7 X 10-8 cm2/gm/sec2 was produced. Plugging this into F=GmM/r2 gave a mass for the earth—6X1027 grams. By using Newtonian physics and only measuring the gravitational constant yielded a powerful result such as the mass of the earth. Many more applications would appear because of this discovery. With the revision of Kepler’s Third Law, R3/P2 = (G/4 pi2) (m+M), the mass of the sun was found out [worked out in source book]. Masses could then also be compared, for example Mars and Phoebos to the Earth and the moon [also refer to the source book].
Was Copernicus Right?
Wednesday, October 30
The lecture's title seems to be a strange question, but we must remember that even Galileo and Newton were still calling the idea that the earth moved a hypothesis! In the following years, the idea that the Earth moved came to be widely accepted becuase it was a convenient concept, but proving the idea was a different matter altogether.
The future Astronomer Royal James Bradley decided to renew the search for the "stellar parallax" that would prove that the earth was moving. Along the way, Bradley discovered a phenomenon known as the "Aberration of starlight" (1744) which could be explained in terms of the earth's moving. Since light is coming through a telescope which is positioned on a moving earth, the telescope in some instances must be tilted slightly in the direction of the motion--much like a person tilting their umbrella forward in order to keep their feet dry as the walk through the rain. Robert Hooke observed the same phenomenon, and incorrectly claimed that he had discovered the "stellar parallax. Bradley realized that the shift was in the wrong direction and developed his idea of aberration.
Astronomers continued to search for the "stellar parallax." Earlier Friedrich Bessel chose the star 61-Cygni to observe in hopes of discovering a parallactic effect, finally measuring the tiny parallax with an ingenious split-lens telescope which allowed him, through comparison of his observed star with other stars rather than simple angular position, to eliminate intereference caused by atmospheric disturbances in his telescope's accuracy. Bessel's discovery allows him to find the distance to the star, developing the unit called "parsecs", or "parallax seconds." 1 parsec equals about 3.26 light years or 206,000 astronomical units (AU).
The Russian Astronomer Wilhelm Struve also supposedly discovered the stellar parallax at the same time, using the star Vega. He tries to measure something very tiny--too tiny in fact for his instruments, especially factoring in the atmosphere. As soon as Bessel's more conclusive results came out, Struve's observations seemed to gain a "confidence coefficient" and they soon began to give a sine-wave type of pattern which "proved" the stellar parallax.
Another important question is how does one prove the rotation of the earth? Would you be thrown off? If we were to go to the equator, where the acceleration a person would experience due to the rotation would be greatest, the force of gravity would still far overpower the acceleration of the earth, as can be proven mathematically.
Another way to answer the question is using "Foucault's Pendulum"--a large pendulum will rotate around due to the rotation of the earth. At the North or South Pole the Foucault's pendulum would cycle through every 24 hours, at middle lattitudes somewhat longer, and at the equator the pendulum will would not precess at all. This elaborate and rather theatrical experiment can be used as another proof that the earth rotates.
But we are still reminded of Pope Urban the Eighth's comment to Galileo, essentially that God might have done things any way he pleased and that these explanations do not conclusively prove anything. However, we can look to yet other evidence for the Earth's rotation, such as the Coriolus Effect. This effect is apparent on a large scale: in cloud swirls and artillery shells which are fired for long distances. So some of the evidence further pointing to the correctness of Copernicus includes: (1)the Oblateness of the Earth (2) Foucault Pendulum and (3)Coriolus Forces.
To start building for the next lecture, the concept of Angular Momentum (Angular Momentum = mvR) is indeed commensurate with Kepler's Law of Areas. A good example of the conservation of angular momentum is the spinning ice skater who speeds their rotation by pulling their arms in.
The Energy Alternative
Friday, November 1
Newton, in his Principia, was very interested in directed motion (mv). For instance, consider a straight-line motion at a constant velocity, as shown by the line in the diagram to the right. The triangles created by drawing lines from equal time intervals along the motion line to any point create triangles of equal areas, for their bases and altitudes are all the same. The triangles are all of equal area, and Newton would explain this with his "Conservation of Angular Momentum." And this conservation of angular momentum holds true for centrally directed blows--a way to interpret gravity's force on an orbiting body. Here, Kepler's Law of Equal Areas applies to more than planetary orbits--Newton has found it applies to any centrally oriented motion.
Some new definitions, particularly concerning energy, were to come about. Non-directed motion is considered energy. Work is the product of force and distance: force alone does not constitute physical "work." Potential Energy, the Energy of Position, is equal to mass times the acceleration of gravity times the altitude. The pendulum is a good way to observe potential energy and velocity, as potential energy is converted into motion through the swing of the pendulum.Kinetic energy is defined as half of m times v squared. There is also a third type of energy, heat.
We can easily apply this knowledge to astronomy, developing the concept of "escape velocity," useful in firing rockets into orbit, considering the escape of atmospheric elements, and contemplating the size of a black hole's radius.
Einstein and Relativity (Prof. Gerald Holton)
Monday, November 4
Einstein is the most famous twentieth century physicist. He was tuly great, delving into many topics--he found time to write nearly 45,000 letters, asked questions about the nature of the world and God, etc. Einstein was first and foremost a unifier: he broke barriers.
When Einstein set out to write his first scientific papers, Einstein was something of an enigma. It was a time when physics "didn't need an Einstein." Many other physicists were making great advances. Theoretical physics was progressing nicely, and many master researchers and experimenters were working at this time. Names like J. J. Thompson, Maxwell and Michelson were making discoveries--x-rays, electrons, radiation--at a dizzying pace.
But even then, there were slight discrepancies between the experimental facts and the theory. As new experimental discoveries were made which contradicted theory, changes would be made to the theories--in a sense, "adding an epicycle." At this time, physicists were considering light as travelling through ether. Maxwell, for instance, sought to discover the effects of ether on light.
The physicist H. A. Lorentz was a highly respected contemporary of Einstein's. He was studying an "interference" problem, discussing the nature of ether as it "conspired" to change physicists' results--experamentalists were all getting a "negative answer," and the theory was failing in light of the data.
Around 1900, there was a great debate in physics: if all of nature's mysteries were known--that is, all physics had been discovered--would everything be explained in mechanical terms (Newtonian) or in electromagnetic terms?
In the midst of this debate, Einstein notices an "assymetry"-- he is worried about aesthetic principles at a time when people can only think about theory and experiments. The assymetry was between the way physics described the result of a magnetic object moving through a coil compared to the coil moving around the magnet. Electromagnetics, optics and mechanics all "work together" in the same situations. Why must they be considered seperate ideas? Why not unify them?
And so Einstein did. He began to think about the nature of time, realizing that it is no distinct "breathing in and out" of the universe--it is just sychronized, simultaneous events. And so space must be considered along with time. With his new relativistic worldview, the dichotomy between mechanics and electrophysics begins to break down. Again, we see Einstein as a barrier-breaker.
He makes great leaps--not crawling from experiment to experiment. He is not afraid to simply throw out ideas that don't make sense, no matter how well established they are--he throws away the concept of ether. He refuses to give up his ideas in the face of a challenge, and eventually he is proven to be more right than the people who challenged him with the supposed ultimate proof: experimental results. Basically, Einstein's intellectual philosophy can be summed up in three points:
- It is possible to build a complete world view (Grand Unification Theory)
- Don't be interested in small problems: ask the biggest wquestions (such as "Did God have a choice in the creation of the universe?")
- Do we really have the ability to do and understand things? What is most incomprehensible is that we find parts of nature comprehensible at all
Conservation Laws and Creativity
Wednesday, November 6
As Bronowski posited in his essay on creativity in art and science, there are three levels of achievement: discovery, invention and creation. Beethoven's 9th Symphony is a good example of a creation--if Beethoven had never lived, the work would never have been written. On the other hand, a scientist--say Galileo--only discovers, right? Though we might casually talk about his "discovery" of the moons of Jupiter, what he really discovered was some points of light. Galileo "invented" the idea that these points of light were orbiting moons. Christian Huygens, similiarly, can be said to have "invented" the rings of Saturn.
If Beethoven had never been born, we wouldn't have his 9th Symphony. But if Newton had died of smallpox at a very young age, would his laws and physics still have been developed? Essentially, are the laws there, waiting to be discovered? Or does the genius explain nature with laws which are creative--that is, one of a number of ways to explain a phenomena? Was gravitation waiting to be discovered, or did Newton "invent" it?
If there is another way to explain Newton's ideas without using his terms--inertia, mass, force, etc--than it is fair to say that at least to some extent Newton was an inventor, or a creator. And so, we are searching for a sort of "alternative physics". One that can explain Newton's ideas in a nonmechanical way. One possible way to solve this problem would be to use "conservation laws." In our hypothetical history, some Newton-like genius could explain the universe in terms of momentum, angular momentum, and energy. Force, inertia, and vectors have little or nothing to do with this system, but these rules can be used to work back to Kepler's laws. Yes, we have succeeded in discovering an alternative physics, and many common problems are actually easier to solve in this system than they would be using classical physics. Arguably, Newton was a "creator" in the same sense as Beethoven. The line between scientific and artistic processes is perhaps not as distinct as we generally suppose.
Einstein and His Gravitation
Friday, November 8
At this point in the course, two basic themes are beginnning to come together. First, the conception of gravitation and planetary motion and second, that science is a creative endeavor.
Einstein's Gravitation is also known as "General Relativity," a property of "space-time", with light and gravity following paths that are curved in space-time. This is opposed to "Special Relativity," which deals with reference frames that are not accelerating (1905 paper). Of course, this is experimentally inconvenient, since it is impossible to determine absolute rest. Theoretical implications of Special Relativity include that you can never observe something travelling faster than the speed of light, and that an object moving fast enough appears to be shortened.
General Relativity allows acceleration and gravity, and in fact, assumes that upward acceleration and gravity's force are indistinguishable--the "Equivalence Principle." This can be demonstrated, at least in discussion, in a simple elevator experiment in which the elevator accelerates upwards, making the laser appear to "bend" downwards. This same result will occur for an elevator sitting still on theground with gravity working on it. Thus, gravity bends light. Essentially, inertial mass equals gravitational mass.
This Equivalence Principle could be used as a basis for a theory of gravitation. Einstein's General relativity will give you the same answers as Newtonian physics, except under extreme velocity or gravitation. The mathematics for this is very challenging, and is generally solved by making small "cheats." Shwarzchild was the first to solve the problem under a specific set of circumstances, in effect "predicting" black holes.
Some of the classical tests of General Relativity include:
- Starlight is bent by the sun's gravity, as viewing stars whose light passes near the sun during an eclipse proves (Edington in 1919). Einstein made a factor of two mistake in his prediction, but luckily corrected it before the expedition went out to discover if he was right. Indeed, this "bending of starlight" was observed. This discovery really captured the imagination of the public.
- Professional astronomers were more interested in its explanation of the advance of the perihelion of Mercury's orbit. The axis was projected to move 531 arcseconds per century, but the observed motion is actually 574. General Relativity explains this.
Testing Relativity: The Pulsars
Wednesday, November 13 (Jimmy Liu)
Now that Einstein had formulated his theory of general relativity, many tests were concocted to test out his postulations. Tests of general relativty were:
- The Bending of Light and Radio Waves: As mentioned in the last lecture, the first of these experiments were carried out by Edington in his 1919 eclipse expedition. Later, the Viking orbiter further proved the bending of light when, as it was orbiting Mars, it was directly opposite of Earth with the Sun blocking Mars. Scientists measured the time of the radio transimissions, and indeed, they correlated with Einsteinian relativity. Furthermore, a similar experiment with the Mariner missions to Mercury when at its perihelion confirmed this. Relativity is accurate and off by less than 1%.
- Gravitation Redshift: Longer wavelengths were observed when light and other forms of radiation escaped a gravitational field.
- Pound-Rebka Experiment: An experiment, performed here at Harvard University (in Jefferson Hall), sought to confirm the theory of relativity, however, through the minute gravitational difference between the floor and the top of a building. Through measuring the potent emission of radioactive nuclei breakdown, Pound was able to figure out a redshift that was predicted by relativity.
The dawning of radioastronomy further created more possible testing of the Einstein principle. Radio astronomy was pioneered by the finding that the universe continuous produced radio "hiss" (Carl Janski) and that cool hydrogen emitted these tranmissions (Ed Purcell). Many devices were created to measure these radio transmissions through radio antennaes.
Not long after, the discovery of stars that emitted periodic and clear radio signal were observed. Many theorems sought to explain this phenomenon, including an advanced civilization of "little green men" and pulsation. However, the most possible explanation is the constant rotation of a star, much faster than that of the Sun. If a star did indeed spin rapidly, it must conform to angular momentum, thus the star must be small and compact; it could possibly be a neutron star (which is around 20 kilometers in diameter).
The binary pulsar, because of its periodicity and properties, was found to be the best way to measure relativity. Since a binary pulsar distorts the time-space continuum in a periodic manner, different shifts attributed to differing gravtitational fields were found to be easily measurable. And the results of the shifts of wavelength confirmed Einstein's theory of relativity.
Collapse to Black Holes
Friday, November 15
What holds things up? Or how can things sometimes not be "held up" and collapse into black holes? And what keeps things from collapsing against gravitation? Well the moons of Mars (Deimos and Phoebus) are small enough and strong enough that their gravitation can't even pull them into a spherical shape.
But why don't galaxies and solar systems collapse? Rotation and angular momentum have something to do with this. The sun itself is rotating (though not fast enough that we really can see oblateness as in the earth) as its very large mass pulls it into a spherical shape. And perhaps the thermal pressure due to the motion of atoms plays a role. But the question remains: what keeps the sun from collapsing under its own gravity?
Some of the pioneering work done concerning the structure of stars was done by Sir Arthur Eddington, who supposed that stars, despite their great density, are gaseous. This is of course very counterintuitive, since we don't normally associate gaseous objects with great density. He thought that perhaps a collapsing of the sun might account for its heat and light energy, but that was not too reasonable a solution. Much of the sun's energy is really from a nuclear reaction which in fact helps to "hold up" the sun from collapsing.
Eddington also found a direct correlation between mass and luminosity, with a crucial exception: the binary companion star of Sirius. Sirius' companion is a "white dwarf" which has collapsed rather more than the typical star, and is becoming degenerate. At some point, the star would collapse even farther, down into a "Neutron Star" where the atoms themselves were compressed. The Swiss physicist Fritz Zwickey was one of the first to consider the possibility of Neutron Stars, particularly those left over from a Supernova explosions or evidenced by pulsars.
It is even possible to reach a greater density with higher gravitational force, where the escape velocity would be faster than the speed of light. This of course, is a black hole, as predicted by Karl Schwarzchild's solution to Einstein's General Relativity. The "Schwarzchild Radius" takes the form R=2GM/C². So you could have very small black holes, with small masses--the radius would just have to be incredibly tiny. You could, for instance, theoretically squeeze an apple to a small enough radius that it would become a sort of "mini-black hole." But the only force that we imagine to be able to achieve the conditions necessary for a black hole are found in the gravitational forces of very dense, very massive objects.
The Astrophysicist Chandrasekhar determined the upper limit for the mass of a star that can be supported against gravity by degenerate pressure, giving us a group of "Chandra Limits": After a size of about 1.4 solar masses, a white dwarf will collapse into a neutron star. In the collapse, a supernova or nova may occur.
If black holes do occur under even higher gravitational forces, then where are they? Finding a black hole is very tricky, and "visible-light" observation is certianly not an option. One famous candidate for a black hole is known as Cygnus X-1, discovered with X-ray astronomy. We see only one star, but it is behaving as a binary star. It's black hole companion should be in the neighborhood of about 8 to 16 solar masses.
In 1991, there was a nova in the constellation Muscae, and we see evidence of a gravitational force of about 5 solar masses--a very good candidate indeed. But even more convincing is the examination of the cores of galaxies, where there cores are spinning at such fast speeds that there must be a very massive object at the center.
At the Center for Astrophysics, radio astronomy has suggested a galaxy (M-106) centered on a huge concentration. Where the milky way is about as dense as 1 solar mass per cubic parsec, there are about 35 billion solar masses per cubic parsec in this region of M-106. This, then, is a very convincing candidate for a huge black hole.
Discovering the Nebulae
Wednesday, November 20
At this point, after the second hour exam, the course begins to shift its focus. Just what is man's place in the universe? We will explore this question through modern problems in astronomy: how big is the universe? Is the universe expanding? How old is the universe? We will begin to discuss what is known as Cosmology.
Returning to the 18th century, we come to French astronomer Charles Messier, who was very interested in comets. To aid in his and other studies, he began to compile a catalog of "noninteresting objects" which looked like comets because of their "fuzziness" but weren't moving. Ironically, these "Messier Objects" are some of the most interesting in astronomy at higher resolutions: M-1, for instance, is the Crab Nebula.
At the sime time, across the channel, discover of Uranus and Astronomer Royal William Herschel gets funding from George III and builds great telescopes that push the limits of technology. Herschel had a relatively young son (for Herschel's age) named John who would become a very important astronomer. William's sister, Caroline Herschel, was also an accomplished astronomer. John catalogues more than 2,000 nebulous objects. He moved to Capetown South Africa and was able to some Southern-Hemisphere observing there. He also was an accomplished chemist--he perhaps invented the word "photograph" and perfected some aspects of photography. He figured out how to get rid of darkening crystals so that pictures sould be saved permanently, instead of slowly darkening. However, this photography doesn't quite work in Astronomy--we need Eastman's dry-exposure film for the long exposure photography that Astronomy demands.
Lord Rosse, in Ireland, builds a truly giant telescope. Despite horrible weather in Ireland, he is able to make some good observations of Messier objects, particularly noticing the spiral structure of M-51. Meanwhile, Dreyer developes his New General Catalog (NGC) which lists about 7,000 Messier objects.
In the American Southwest, many great observatories were being built. Foremost among them was the Lick Observatory near San Jose, which, at the time, had the largest refracting telescope in the world. Near the turn of the century, when Eastman perfected his photography processes, long-exposure photgraphs of the sky became feasible, and some spectacular photographs of Nebula were produced.
The most famous of the giant telscope builders was George Ellery Hale. He convinced Charles Yerkes, a chicgo trolley magnate, to finance the building of the Yerkes Observatory, which was big enough to "lick the Lick." In search of a warmer climate, Hale moved to the Southwest, persuading Carnegie to fund the construction of a 60-inch reflecting telescope at the Mt. Wilson Observatory (1917). Astronomer Edwin Hubble would be a famous user of this telescope. And at this time, long-exposure techniques were becoming more and more refined, and astronomers were beginning to see farther and farther into space, revealing nebulae, galaxies, globular clusters, etc., which before appeared as little more than smudges to Astronomers such as Messier back in the late 1700s.
At Mt. Wilson (using a 60-inch telescope), Harlow Shapley becomes interested in variable stars, and is able to use the period of their oscillations to determine their distance. His work led us to believe that our solar system is located on the outside of the Milky Way.
Hale's last great telescope is a 200-inch one, built on Mt. Palomar, which he did not live to see finished. The telescope was so powerful that a smaller 48-inch Schmidt telescope had to be used to decide find areas worth pointing the larger telescope, which was so powerful that it a "sky" survey would take an unreasonable amount of time.
The Milky Way Galaxy
Friday, November 22
We have talked about the quest to build bigger and bigger telescopes, and the mapping of objects in space. But we have only considered where objects are in the sky--not how far away they are. We must consider distance as well angular position...we must even consider the time aspect, particularly for stars farther and farther away.
Astronomer Thomas Wright, from Durham, England, explains the Milky Way as as a "plane of stars" in which the sun lies. His explanation was largely championed by German philosopher Immanuel Kant, despite the fact that many of Wright's ideas were quite absurd.
William Herschel, an original thinker, tries to discern the "depth" of the Milky Way, using the idea that "Faintness Means Farness." Herschel would through his work prove that this is not always true, (using "double stars" which are actually orbiting each other that other factors can influence luminosity besides distance--all stars are not intrinsically the same) but would confirm that it is a good general rule of thumb.
Christian Huygens had said that "Faintness Means Farness," as he studied the way that light dims over distance. The intensity of light is found to equal lumonosity divided by the distance it travels squared (I=L/D²). Huygens was able to find this ratio even using naked-eye observation, estimating a distance to Sirius compared to the sun. Kapteyn counts the stars and tries to make some guesses about the shape of the Milky Way, seeing us at its center. Huygens, trying to get a relative scale, takes a "statstical" approach and assumes that for most stars, faintness will indeed mean farness.
Astronomers, looking for an average distribution of stars, expect the number of stars of certain magnitudes to increase according to the volume of space surveyed. Taking an certain "block" of sky, we see 16 first magnitude stars, 47 second magnitude stars, and only 162 third magnitude stars. What is happening? Why are we missing so many stars? Is the density of stars changing farther away from the solar system? This led some astronomers to believe that we are at the center of the galaxy, though in actuality it was mostly interstellar dust that was obstructing far off starlight.
Dutch astronomer Kapteyn worked on the nature of the Milky Way--finding that we are "in the middle" of a very large disk. He does this by noting that the density of stars seems to decreases further from our solar system.
At the Mt. Wilson observatory, Harlow Shapley was studying cephied variable stars and saw a concentration of globular clusters to one side of the "Aitoff Projection" map of the sky. Using cepheid variables and other known distances he was able to fashion a "distance ladder" and eventually develop his period/luminosity scale for the distance of cepheids--uses these cepheids as "landmarks in space" to determine distances. He is able to postulate that we are not in the central "hub" of the universe, but towards the edge of the galaxy. Soon Shapley would get into great debates over the scale of the universe.
Wednesday 1 December 1999 (Rachelle Gould 1999)
In 1900 the standard view of the universe was that the sun is located at the center of a vast assemblage of stars shaped like wheel, with a diameter of something like 10,000 light years. This picture was based on star counts, an approach that was pioneered in the 1780s by William Hershel. It is often called the Kapteyn Universe, after the Dutch astronomer Jacobus Kapteyn, who worked hard to establish the actual size of the system.
The emergence of great observatories in the United States led to a flowering of observational astronomy in America. In 1890, the Lick Observatory in California was built. It housed a thirty-six inch refractor, which was at the time the largest in the world. In 1897, as a result of the entrepreneurship of George Ellery Hale, this telescope was surpassed by the Yerkes refractor of the University of Chicago, which measured forty inches and was located in Williams Bay, Wisconsin. Hale tired of the bitter winters in Wisconsin, and moved on to build a new observatory on Mt. Wilson, outside Los Angeles. For his new Observatory Hale build giant reflecting telescopes, the first one with a 60-inch diameter mirror, followed by the 100-inch Hooker telescope. These instruments were technological marvels, and excelled at taking deep photographs of faint objects. It was the 60-inch telescope that Harlow Shapley used to observe the variable stars in globular clusters in order to derive a new and much larger size for the sidereal universe.
After studying astronomy as an undergraduate at the University of Missouri, Shapley moved to Princeton for his PhD thesis research. There he worked with eclipsing binary stars and increased the number with solutions from ten to one hundred. He was then hired by Hale to work at the Mt. Wilson observatory, where he set to work on the globular clusters.
As Shapley studied the globular clusters, he was struck by the fact that a third of all of these clusters in the Milky Way were concentrated in an area covering five percent of the sky in the direction of the constellation of Sagittarius. He concluded that this concentration marked the center of the sideral universe.
Shapley wanted to know how far away the clusters, and thus the center of the universe, were located. His experience working with binary stars at Princeton was good preparation for his work with the variable stars in globular clusters. He recognized that the brightness variations he observed in the globular cluster variables were not caused by the orbits of binary stars; rather, these variable stars were individually pulsating with a regular period. The crucial characteristic of this type of star is that there is an observed relationship between these measurable periods and the instrinsic luminosities of the stars.
That period-luminosity relation was discovered by Henrietta Leavitt, working at Harvard. In 1912, Leavitt created detailed plots comparing the magnitudes and periods of the Cepheid variable stars in the Magellanic Clouds. She used the photographic plates from Harvard's southern observatory in Arequipa, Peru, obtained over the period from 1893 through 1906. The use of the Magellanic Clouds allowed Leavitt to make accurate comparisons of the stars in the Clouds, which are located at a relatively large distance. This means that the difference in distance between the fronts and backs of the clouds is small; therefore within each of the two clouds the stars' apparent brightness will be proportional to their inherent brightness. In other words, two stars that appear to have the same apparent brightness must have the same intrinsic luminosity. However, since the actual distances to the Magellenic clouds were not known, the intrinsic luminosities of the Cepheid variables in the clouds were also unknown. Thus, in order to allow Leavitt's period-luminosity diagram to be used to determine actual distances, it needed to be calibrated.
In 1918 Shapley determined the necessary calibration by finding the distance to a small number of nearby Cepheid variables in the Milky Way using the method of statistical parallax, in which proper motions (the rate at which stars' apparent positions in the sky change with respect to very distant stars or galaxies) and radial velocities are analyzed statistically. A detailed explanation of statistical parallax can be found on page 375 of the textbook, and a summary can be found on page 175 of the sourcebook).
Once the distance to a few nearby Cepheids was found through the use of statistical parallax, it was possible for Shapley to apply the principle of faintness means farness to discover their intrinsic luminosities, because he now had both their observed magnitudes and their distances. The basic idea of the principle is that the greater the distance of a light source from the earth, the fainter it will appear. The principle is based upon characteristics of the propagation of light first demonstrated by Newton. Its basic mathematical premise is the inverse square law, which states that the amount of light passing through a specific area will decrease with the square of the distance from the light source. For example, if two stars have the same luminosity, one that is twice as far away will appear to be one fourth as bright as the closer. When two stars are compared, the formula for distance calculation is: l/L = (D/d)2, where l and d are the luminosity and distance for one star, and L and D are the luminosity and distance for the other. When a star's actual intrinsic luminosity, L, and apparent brightness, b, are being compared, the following formula can be used: b = L/d2.
Thus Shapley was able to calibrate Leavitt's period-brightness diagram and convert it into a period-luminosity diagram. Then the lumniosity of a variable star in a globular cluster could be determined by deriving a period for its light variation and reading off the luminosity for that period from the period-luminosity diagram. Shapley used this approach to lay out the distances to the globular clusters. He found that the center of the distribution of clusters lay at a distance of about 50,000 light years in the direction of the concentration of clusters towards Sagittarius.
Shapley's study of the Cepheids thus lead him to realize that the sun was not at the center of the sidereal universe, but was located well off to one side. He also believed that the spiral nebulae were "local whirlpools of gas" found within the Milky Way, for he thought that the Milky Way was the entire universe. His ideas of an off-center sun and an enormous size for our sidereal system were extremely radical at the time at which he proposed them.
Heber Curtis, who was working at the Lick Observatory, was a representative of the traditional view of a much smaller universe centered on the sun. Curtis also believed that each of the spiral nebulae was its own universe; he called them "island universes." Obviously, this view directly contradicted Shapley's view that the Milky Way was so vast that it encompassed these nebulae. On April 26, 1920, Shapley and Curtis faced one another in a debate on "The Scale of the Universe" before the National Academy of Sciences. Shapley argued that the universe, which was in his opinion the Milky Way galaxy, was 300,000 light years across, a distance that greatly exceeded contemporary ideas of its size. Curtis, however, agreed with most other contemporary ideas of the size of the universe, saying that practically all of those who have investigated the subject had deduced diameters of 7,000 to 30,000 light-years. Shapley argued that the variable stars in globular clusters were intrinsically very bright. Since they also had relatively bright apparent magnitudes, Shapley deduced that, by the principle "faintness means farness," these stars must be extremely distant. Curtis, however, disagreed with this interpretation. He argued not only that Shapley, who studied only eleven stars, was basing his distance calibration on too small a sample, but also that the variable stars in the globular clusters were not intrinsically any brighter than the sun, and thus that their distance from the sun did not have to be nearly as great as Shapley claimed.
In their debate, Shapley argued that the sun was off-center in the galaxy and that the galaxy was larger than existing estimates. He argued that the globular clusters were very distant and yet that they were still contained in the Milky Way; hence the size of the Milky Way was necessarily very great. Curtis argued that the Cepheids were not as distant as Shapley claimed them to be and thus that the Milky Way was much smaller than he claimed it to be. Curtis also argued that the spiral nebulae were "island universes" outside the Milky Way, while Shapley asserted that observations did not support the idea that the nebulae were galaxies. In just a few years, new work would show that Shapley was basically right about the large size of the Milky Way, but Curtis was right about the island universes.
Shapley was not the only astronomer studying Cepheid stars; Edward Hubble, while studying novae in the Andromeda galaxy, noticed a number of Cepheids in M31 and commenced to study them. In 1924, he calculated the distance to these stars to be 900,000 light years. With Hubble's calculation, it was evident that M31 was a full-fledged galaxy well beyond the outskirts of the Milky Way. It was a weighty discovery, for now astronomers knew that the small smears they observed in their photographs could possibly be entire galaxies like Andromeda, 100,000 light-years across.
After Hubble demonstrated that the spiral nebulae are island universe, astronomers began the process of mapping out their distribution in space. Eventually, it was determined that the "local group" consists of three spiral galaxies (the Milky Way, Andromeda, and M33), the Large and Small Magellenic Clouds, and more than a dozen smaller galaxies. Located at an enormous distance, about fifty million light years, from this local group is the Virgo cluster, which contains a concentration of over 1,000 - possibly as many as 5,000 - galaxies, most of them elliptical. At greater and greater distances from the sun, we continue to encounter rich clusters of galaxies. For instance, the Coma Cluster, located at 300 million light years, contains more than ten thousand galaxies. The extremely faint Hydra cluster is three billion light years away. The universe is a very big place!.
Friday, December 3, 1999 (Rachelle Gould 1999)
As of 1920, there were basically two views of the world system. That held by Shapley was that the universe was the Milky Way, a flattened disk with a nucleus. In this "galactocentric" view, the sun was not at the center of the universe but rather revolved around that nucleus. The opposing traditional view, represented in the 1920s by Kapteyn, its leading expert, placed the sun at the center of the Milky Way, which was a cloud of stars.
In an effort to clarify the true nature of the universe, Curtis studied the spiral nebulae. He observed novae in these nebulae that would flare up and fade away. After observing a nova in 1918 in Aquilae, which is located in our own Milky Way, he had the information necessary to calculate the distance to the novae in the nebulae if he assumed that the novae he observed were similar to the nova in Aquilae. We now know that most of the novae that Curtis studied are actually binary systems in which one member is a white dwarf. A nova explosion occurs as a result of the normal star's transition into a red giant. In these systems, mass is transferred from the star to the white dwarf when small pieces of matter leaving the normal star cross the point between the two stars at which gravitational forces on the matter are equal; when past this point, the matter is drawn to the white dwarf. As the normal star expands, becoming a red giant, its boundaries eventually reach this point. When this occurs, the matter is "unsure" as to where it belongs, and the result is that hydrogen-rich material very slowly transfers from the surface of the red giant to that of the white dwarf. This material collects, for years to centuries, and its weight increases until the temperature of the material causes the hydrogen to ignite like a nuclear bomb. After the explosion, the white dwarf is almost unchanged; in addition, because the red giant continues to leak hydrogen onto it, the process tends to repeat. Thus theoretically all novae repeat with some regular interval. To use Professor Gingerich's analogy, a nova is actually more like a sneeze than an explosion; it's heart-stoppingly violent for a moment, but it actually causes little damage to the person (the star, in this case). And a nova, like a sneeze, can happen repeatedly.
A distinction must be made between novae and supernovae. Within the population of novae, intensities differ; for example, in 1885, a nova that occurred in Andromeda was visible to the naked eye. The novae that were studied by Hubble, in contrast, were difficult to observe even under magnification. Supernovae, different than novae, are huge explosions that are extremely obvious. The last two in our galaxy occurred in 1604 (observed by Kepler) and 1572 (observed by Tycho Brahe); there has not been a supernova in our galaxy for hundreds of years.
Shapley, like Curtis, used information on the novae to make conclusions about the universe. He decided that the nova in Andromeda was normal (not excessively bright), and he thus interpreted that the "nebula" must be close to us; this agrees with his erroneous idea that all observed light sources are contained within the Milky Way because it composes the entire universe. Shapley's faulty interpretation demonstrates the ambiguity and difficulty of trying to determine the distances to stars. This ambiguity played a fundamental role in astronomical development, for there was never complete certainty about distance calculations. The principle of "faintness means farness," with its inverse square law, provides a viable method of calculating distance if the intrinsic luminosities of objects are known. The fundamental challenge in applying "faintness means farness" is to figure out what you're looking at; in this case, Shapley was mistaken in his identification of an object and thus his distance calculations were incorrect.
In the 1920s, astronomer Jan Oort was also working on distance calculations. He used very distant Cepheid variables in our Milky Way, still observable because of their large luminosities, and observed their radial velocities. Radial velocity is an object's velocity as it relates to the line of sight; it is the speed at which an object is moving toward or away from the observer. Because it reveals information about the motion of celestial objects, radial velocity would become an important component of observations of the universe.
Radial velocity was used to support the idea of the differential rotation of the Milky Way. The fact that certain stars seem to be receding from us and that other stars seem to be approaching can be explained by a system in which these stars are orbiting the galaxy's center in orbits both smaller and larger than our own. For some of these stars we observe no radial velocities. When a star is directly between the galactic center and the sun we observe no Doppler shift (caused by radial velocity) because the star is moving neither directly toward nor directly away from us. However, when a star in an orbit smaller than ours is located at "ahead of" us in relation to the galactic center (in the direction of our revolution), it will appear to recede from us by showing a positive radial velocity. This is because its velocity is greater than ours (which agrees with Newton's Law of Gravitation), so the distance between the star and us is actually increasing. A positive radial velocity is also observed for stars in orbits larger than ours that are located "behind" us (in the opposite direction from our revolution). Since we are moving more quickly than these stars, they appear to be receding. A negative Doppler shift can be seen in stars in larger orbits that are located "in front of" us. Again, we are moving more quickly than these stars, and thus as we "gain on them," they appear to approach us. This negative radial velocity is also observed in stars in smaller orbits that are "behind" us; since their velocities are greater than ours, they "gain on us." Oort's examination of radial velocities supported the idea that there is a galactic nucleus at Sagittarius and that the stars and our sun revolve around it in accordance with Newton's Law of Gravitation. Oort's work was extremely significant, because it identified Sagittarius as the center of the galaxy using a method completely independent of Shapley's globular cluster analysis. Thus, this examination of radial velocities convinced many people of the validity of Shapley's view of the galaxy as a flat disk with a nucleus around which our sun revolves.
Joseph von Fraunhofer, who was rescued in 1801 by Maxmillian II, was a pioneer in the study of optics and the spectra of light. By viewing the light of the sun through a slit placed in front of a prism, Fraunhofer discovered the dark lines in the spectrum of the sun's light that would come to bear his name. By 1914, Fraunhofer had plotted more than 500 of these lines. He labeled them with letters, and the line labeled D, found in the yellow area of the spectrum, holds the most historical significance.
In Heidleburg, Robert Bunsen was working to produce pure chemicals and flames that were as pure and thus colorless as possible. He introduced different compounds to his flames and observed the resulting color changes. Kirchhoff was a student of Bunsen, and he has been called Bunsen's greatest discovery. At the student's suggestion that a prism be used to observe the light of the flames Bunsen and Kirchhoff built the first proper spectroscope out of a cigar box. From their work with spectroscopes, they discovered that every element has its own characteristic spectrum.
The work of Fraunhofer combined with that of the Heidelberg pair to allow a major advancement in the study of the universe to take place. Kirchhoff and Bunsen knew that sodium produced yellow-orange spectral lines. They wanted to see if the location of sodium's spectral lines matched the dark lines found in the sun's spectrum. To their joy, in 1860 they discovered that the two lines were found in exactly the same place. What this suggested with fairly high certainty was that the "pure" (undelineated) light from the sun was passing through sodium gas before reaching the earth; thus, the outer layers of the sun must be composed at least partly of sodium gas. We observed the phenomena in lecture; when a sodium flame was placed in front of the light producing a continuous spectrum, definite dark lines in the yellow region of the spectrum became evident. The significance of this discovery was great, for it was now possible to determine the composition of the sun and other celestial objects through spectral analysis.
These characteristic lines found in the spectra of stars also serve another very important purpose: they serve as reference points that allow determination of the shift in an object's spectrum caused by the Doppler effect. If an object is moving, its characteristic dark lines will change position in its spectrum. The lines serve as indicators of the object's motion.
The use of spectroscopy in astronomy began to increase near the end of the nineteenth century, when the application of photography to astronomy made spectroscopy possible. Another source of astronomical discovery was the Lowell Observatory in Arizona. Lowell, the brother of President Lowell of Harvard, was a great believer in a civilization on Mars, and he hired Vesto Slipher mainly to look for canals on Mars. Slipher managed, however, to do some work in astrophysics; he observed the spectra of thirteen of the spiral nebulae, looking for evidence that they were rotating. Instead he found what would become a critical piece used in the solution of the puzzle of the universe: the redshift explained by the Doppler effect.
The Doppler effect is the phenomenon that can be heard when, for instance, a train approaches and then moves away from an observer. As the train approaches, the distances between the sound waves produced by its whistle decrease, causing the whistle to sound more and more high-pitched. As the train moves away, the distances between the whistle's sound waves increase, and the whistle's pitch thus becomes lower and lower. This phenomenon is present in light in addition to sound waves. If a luminous object is moving away from the observer, its light waves are further spaced and its spectrum will be shifted toward the red. If, conversely, the object is moving toward the observer, its light waves are closer together and its spectrum will be shifted toward the blue. The degree of change in the spectrum can yield the object's velocity; the method used can be found on page 187 of the sourcebook.
Slipher's data showed that the spectra of all of the observed nebulae except for that of the Andromeda galaxy were shifted toward the red, and by amounts much larger than those for the stars. This meant that all of these nebulae were moving with fairly high speeds away from the earth. In 1914, he reported his findings to the American Astronomical Society. He continued to collect spectra and calculate velocities for the next ten years. Some of the velocities calculated by Slipher were as great as 500 km/sec, a speed that almost exceeds the escape velocity of the Milky Way. Thus Slipher's observations were one of the first important clues that the spiral nebulae were not part of the Milky Way but were independent objects and were very far away.
Little additional interpretation of Slipher's findings occurred, however, until Hubble in the 1920s was able to calculate the distances to galaxies using Cepheid stars. In 1924, his announcement that the distance to the Andromeda galaxy was 900,000 light years suggested with little doubt that M31 was an independent galaxy and not a nebula found in the Milky Way. Hubble also calculated the distances to a number of additional galaxies. Using Slipher's unpublished spectra and some of his own, he looked at the relationship between his calculated distances to galaxies other than Andromeda and their redshifts. Through this comparison, what has been called "...the prime fact of the twentieth century, the most amazing scientific discovery of all time..." was revealed.
When Hubble plotted the distances of galaxies versus their redshift, he invented (or discovered; see "Creativity in Science" lecture :)) the fact that a galaxy's distance is proportional to its redshift velocity. Basically, the farther away a galaxy is, the more quickly it is receding from the earth. This principle has come to be known as the Hubble law. The motion of the galaxies resembles the motion caused by an explosion; thus the idea that an enormous explosion had occurred at the beginning of the life of the universe was born. The Hubble law is mathematically represented by the equation v = H x d, where v is the velocity calculated from redshift, H is "the Hubble constant," and d is the distance to the galaxy. If it is assumed that the expansion of the universe has been constant, the time since the beginning of the universe can be found by simply dividing any given object's distance by its velocity (by the elementary relationship of distance equals rate multiplied by time, d = r t). When the formulas t = d/v and v = H d were combined, it became evident that the age of the universe could be calculated by inverting the Hubble constant; the age of the universe would be 1/H. However, Hubble's calculation of a billion years was not compatible with the knowledge of geologists; they knew that the earth was much more ancient than a mere billion years. In the early 1900s, geologists had used the sodium levels in the ocean to calculate the age of the earth, believing that erosion would have caused them to rise uniformly since the creation of the earth. However, those investigating the age of the earth had recently abandoned this method and had begun using the method of radiometric dating, which was believed to be much more accurate. In his 1913 book The Age of the Earth, which provided a detailed analysis of the various contemporary methods used to estimate the Earth's age, Arthur Holmes wrote, "The earth must be older than the oldest analyzed radioactive minerals (1,750 million years) ... a minimum estimate of 1,900 to 2,000 million years is indicated for the age of the earth." Since 1,000 million years is a billion, this demonstrates that scientists of the time were fairly certain that the earth's age exceeded a billion years.
No advancements were made to the problem of the age of the universe until 1952, when Walter Baade discovered that the distinction between the two kinds of Cepheid variables. The group of stars that had been labeled as "Cepheids" was composed of two types of stars: the actual Cepheids and the RR Lyrae stars. The Cepheids have longer periods and higher luminosities, while the RR Lyrae have shorter periods and lower luminosities. The Cepheids are massive stars with short lifetimes, and the RR Lyrae are older stars with lower masses. Shapley and Hubble had been unaware of the distinction, and this was the reason for their incorrect calibrations and thus incorrect values for the Hubble constant and the age of the universe.
Milton Humason, an astronomer who was not very active in the formation of theory but was a superb observer, was able to obtain spectra of extremely distant galaxies. These new spectra, faint though they were, allowed the Hubble law to be tested at larger distances. As these spectra were examined, Allan Sandage, a student of Hubble's, began to understand that not all of the distances established by Hubble were reliable. Many of the "stars" Hubble had placed were actually nebulae at much greater distances. However, because the scale used by Hubble was constant, the ratios of galactic distance (even though they were wrong) were constant; thus the idea that the universe is expanding can still be arrived at from his work.
Sandage, using the new information provided from modern observations, was able to recalibrate Hubble's system. Using this recalibration, he calculated an age for the universe that was somewhere between ten and fifteen billion years. This figure was much more agreeable with the knowledge of geologists.
The incorporation of spectroscopy with astronomy thus lead to a virtual revolution in the view of the structure and nature of the universe.
The Expanding Universe
Monday, December 2
The commonly used distance indicators for galaxies include cepheid variables (out to the Virgo Cluster), Globular Clusters (past Virgo), supernovae (distant clusters) and brightest cluster galaxies (also for distant clusters). So, if the Universe is expanding, v=Hd, with H as the "Hubble Constant" with dimension of one over time. (For instance, we can look at the expansion of raisin bread, with the raisins analagous to galaxies. If H equals 5 cm per 10 min per 10 cm, than the age of the raisin bread is one over H or 20 minutes.)
The value of the Hubble Constant has changed greatly over time, as the following table summarizes:
View Year Value (Km/s/MPC) 1/H (Age of Universe) Hubble 1935 535 2 billion years Sandage 1964 50 20 billion years de Vaucouleurs 1980s 1000 10 billion years Modern 1990s 70 15 billion years
Our modern value for the expansion time contradicts data hinting at the age of the universe, such as the ages of the oldest stars. The way that theory doesn't quite mesh with fact is something of a problem currently. Furthermore, it can be supposed that very long ago, expansion was in fact faster, and has been slowing down at some rate--making the expansion time shirter than 1/H. The more massive that we estimate the universe to be, the younger it must be, because the gravitational force would be stronger and hence a greater initial expansion speed would be necessary to give the expansion rate observed now. Scientists have tried to look back to the "slowing down" near the beginning of the universe, but studies have been inconclusive. And of course, perhaps our assumptions are wrong--maybe galaxies have been changing throughout time. If, say, younger galaxies were brighter because of "bursts of star formation" or the merging of old galaxies induced brightness, we would have other reasons why our current view might be wrong--in fact, the evolution of galaxy brightness is quite a morass currently. Or we could look at supernovae. What results do they give? And Virgo and Coma Clusters have galaxies orbiting at velocities which aren't explained by their visible masses--perhaps some sort of "dark matter" exists in these clusters of galaxies. Clearly, there are many questions left to be answered in astronomy.
The Riddle of The Quasars
Wednesday, December 4
Redshift distances can be interpreted as faintness means farness (reciprocal of the distance squared) or smallness means farness. Redshift measurements are one of the great "cosmological" measurements used to show the expansion of the universe by interpreting it as a doppler shift. Is the expansion of the universe "isotropic" (uniform)? Well, there are complicated orbits going on (such as the Milky Way and M31) that complicate the situation.
Quasars are "Quasi-Stellar Raadio Sources". During WWII, radar was developed, and during peace-time, this technology became available for radio astronomy. Quickly people realized that there were many radio-sources in the sky, which could be pinpointed and often found to correspond with optical observations. What could the quasars be attributed to in the visual spectrum? This is what Maarten Schmidt was most interested in. In the spectra of quasars, Schmidt noticed that they have broad, "fuzzy" emission lines. Schmidt eventually realized that these "weird" emission lines were ultraviolet wavelengths being shifted up into the visible spectrum. To explain this redshift, quasars must be extroadinarily far away and moving very, very quickly. Yet, they are very bright--if "faintness means farness" then these objects must be extroardinarily luminous intrinsically. This is "The Riddle of The Quasars."
If we explain the redshift in terms of doppler shift, then the luminosity of the quasars must be prodigous. But, if we assume that they are driven by some as of yet undiscovered physics, as Holton Arp does, we can see that their luminosity is much more normal. Most astronmers accept that the quasars are very far away. To generate their brightness, quasars must have alot of energy--nuclear burning isn't sufficient. We might guess that some great gravitational force, perhaps a black hole, is the source of this gravitational energy.
Arp, as we've said, sees quasars as near. For him, they are associated with galaxies. Using a statistical method to show that the fact that so many quasars are found near bright galaxies is very unlikely given most astronomers explanations. Because it seems so unlikely that quasars are associated with galaxies "by chance", they must be "with" the galxies, that is at the same distance despite differing redshifts, as the redshift, to Arp, is some new redshift which physics has yet to explain.
Quasars & The Violent Universe
Friday, December 6
Quasars, as we understand them, must be prodigous events, implying that it is a violent universe we live in. (This is of course in contrats with Arp who argues that quasar redshifts are thanks to some "discrepant cause" and that they are really associated with near galaxies). Mainline astronomers, though, feel that Arp's "examples" are just accidental alignments, and that redshifts are due to ordinary cosmological expansion. Quasars, in fact due seem to be associated with very distant galaxies that are just now being discovered--perhaps some of these galaxies have quasars at their centers).
Quasar redshifts are often greater than 1. This implies (using the classical doppler shift) that the quasar is receding faster than the speed of light. But using a "relativistic doppler shift", we get a more reasonable answer for this large redshift. The largest quasar redshifts observed are around 4.5, though 2 is about average.
As an aside, it is important to note that sometimes quasi-stellar objects (no radio sources) are discovered. They don'y give radio emissions, but they do have the huge redshift.
Quasars are extreme examples of a family of active galaxies. Even M31 has some kind of bright activity at its core. This is not unusual--there is activity going on in the cores of many galaxies. For quasars we suspect that the "central engine" is a massive black hole.
One of the most powerful arguments for the main stream view of quasars involves the "Lyman Alpha Forest", a series of asorption lines in the spectrum of hydrogen. The "Lyman Alpha Forest", with a large number of these absorption lines, suggests that there are many clouds of gas between the quasar and the observation point, so the quasar must be behind many such clouds and hence at a great distance.
Quasars are all very different. In one case though, we find side-by side quasars that are identical. What is going on here? We discover that this is a case of "gravitational lensing" with the gravity bending the quasar-light in some sort of shifted ringlet, makeing the quasar appear as two images to us. This lensing is caused as the light passes nearby galaxies. Because there are more than a dozen cases gravitational lensing with quasars, they must be at a cosmological distance.
In general, Quasars have redshifts of about 2, and so far none have exceeded 4.5. It seems that quasars are more common in an earlier period of the history of the universe, which agrees with the classical redshift-based notion that quasars are at cosmological distances.
Did the Universe Begin With A Bang?
Monday, December 9
Beginning with the Creation in Genesis, we have long had a conception of a "beginning" in the Judeo-Christian Tradition (i.e. Western Theology). For modern cosmologists, the question of the beginning is coupled with the question of the "end"--will gravity or the kinetic energy of expansion win out?
In the "steady-state' cosmology, there is no concrete beginning--only exponential growth. Stephen Hawking, in A Short History of the Universe, imagines that at some point near the beginning, time and space became confused, There was thus no "beginning", but nonetheless we could have a "History of the Universe."
Interestingly, as we jump up magnitudes, the amount of quasars increase by factors of ten. So, it seems that there is a sort of shell of quasars around us. Are we in some way at the center? (A very anti-copernican idea indeed) No, a better explanation is that going further back into time (at the great distances of the quasars) we encounter a "time of the quasars." And going even further back in time quasars become less dense. The fact that there was a "time of the quasars" speaks against the steady-state model.
To take the entire universe down into a very tiny pinpoint, we need to do more than to compress atoms and molecules. Using qunatum mechanics, all matter could be compressed into a very tiny, very hot, energetic point. Within a small fraction of a second from the explosion, quarks would form and then neutrons, and then protons. The mixture of protons and neutrons would be 7 protons per 7 neutrons per 1 neutron. Deuterium would from, and then helium--by weight, 25 percent would be helium and 75 percent would be hydrogen. Astonishingly, very close to the ratio of the lements in the universe that we see at the present time. Non big-bang models have difficulty explaining this ratio.
Hoyle, working on the steady-state hypothesis, sees stars and supernovae as building up unusual elements. Hydrogen atoms appear mysteriously, but how does Helium form then? Gamov developed a concept of a substance called "Ylem." Frustatingly for him (as well as Hoyle) he could get Helium in his explanation but nothing else.
As for the creation of the elements, it seems that all of this must happen in the "first three minutes" after the explosion. As the explosion happens, the density drops, and the nuclear reaction die down. Many of the heavier elements, thus, must have come about inside the stars.
Interestingly, very old, reshifted photons can be found in the microwave spectrum. This "relic radiation" was discovered at Bell Labs by Penzius and Wilson. It seems to come from all over, as space's curvature allows us to "see the biug bang" in all directions.
The Curvature of Space
Wednesday, December 11
Will the Universe expand forever? This question has something to do with the geometry of relativity (The curvature of space, etc.) COBE, the Cosmic Background Explorer, shows some of the irregularities.
Lobochevsky considered a question that has troubled geometers since Euclid: Euclid postulates that you can draw one and only one parallel line to another line through a particular point. Lobochevsky says that lines can in fact intersect and still both be parallel to another, third line, as long as this is in a sort of curved, saddle-shaped space. In Lobochevsky geometry, traingls can have less than 180 degres in them. Riemann, who would propose a sort of spherical geometry, where lines would be the equivalent of "great circles" (geodesics) on an earth-globe. These various geometries get very different results:
Geometer Euclid Lobochevky Riemann Parallel Lines 1 Infinite Zero Triangles 180 degress less than 180 degrees More than 180 degrees Universe Future (just) expand forever expand forever collapse Density critical lower higher Galaxy Count R cubed greater than R cubed less than R cubed
Galaxies seem to be rushing away from us--is there some sort of "great attractor" out there? And galaxies do seem to "bunch up." We can't assume that galaxy distribution isn't good enough to tell wether the universe is expanding or not.
Will The Universe Collapse?
Friday, December 13
Will the Universe Collapse? As we know from Wednesday's lecture, this largely depends on which geometry is used. A Euclidean universe will result in the Universe just barely expanding forever. A Lobochevsky universe will also expand forever: an "open universe" or "icy whimper." On the other hand, a universe of Riemann space results in a closed universe which will collapse down on itself.
Looking at a redshift diagram, we cannot determine what kind of universe we live in. But looking to another process, we can use the equation for escape velocity and ask our question of wether or not the universe will collapse in this way: is it expanding faster than its own escape velocity? We can calculate the corresponding escape velocity, and find the "critical density" at which a any more density will cause the universe to be closed, collapsing.
How can we find density? We weigh and count galaxies, and use Newton's revision of Kepler's third law. Dividing density by the critical density gives us a value that we will refer to a "Omega." If Omega is greater than 1, then the universe is closed. Using the galaxy method, Omega equals .01 (Not a very satisfying number--where is the rest of the mass if there is more?) Considering that there is some invisible neutral hydrogen surrounding galaxies, we can get a value of .03 for Omega. Looking at clusters of galaxies, we see that there is in fact "more mass than meets the eye." For instance, what holds the great Coma cluster together? The visible galaxies are not sufficient for this. So, using "rich clusters," Omega is about .1. The Virgo cluster is suggestive of even more mass, and pushes the value of Omega up to .15 or .20. But still we are not even close to 1.
A more satisfactory way of studying the question is to look back to the "first three minutes" of the universe. If the universe was not dense (an open universe) alot of deuterium would be produced--and deuterium is very difficult to make in stars. A measurement of deuterium would tell us what Omega should be. Using the Keck telescope, looking at Lyman-Alpha Forests, our analysis of deuterium in the Universe also suggests an open universe. And, after all of this, astronomers still don't know what 90 percent of the mass in the universe is, for Omega seems to actually be very near 1, as we will discuss in the next lecture...
The Nature of Science
Monday, December 16
Deuterium abundance is a very good indication of the density of the universe, at the beginning as well as now, and measurements of it suggest an open universe: an "icy whimper." But there are some nagging reasons why the universe might be closed. Consider the "Horizon Problem," which ackknowledges that if the universe is a certain age, then there is a "horizon distance" determined by the speed of light and the age. No objects that are farther apart than this horizon distance could ever have been in contact. But yet objects horizon distances apart still have the same temperature: how did this happen? And how could the universe be so close to critical density as it is (with Omega near 1)? This is part of the "flatness problem." It seems that, near the beginning, the universe must have been nearly perfectly Euclidean. And at the beginning, all four physical forces might have been unified, and extreme expansion would have caused flatness. This seems to imply that we have nearly Euclidean space, which in turn implies that Omega should be very near 1. If that is so, then we are missing 90 percent of the mass of the universe. This mass might be in the form of "Dark Hot Matter" (neutrinos), "Dark Cold Matter," planetary black holes, etc. But all of this is of course theoretical.
We turn now to the Reading Period assignment: Thomas Kuhn's The Structure of Scientific Revolutions. Why did he write this? Well, we have a theory of scientific method in our culture, with hypotheses, ovservations, deductions, theories, laws, etc. But in actuality, very little of the great work of science has been achieved through aimless observation, for usually, scientists know what they're looking for before they start.
So science is not just a subject (a noun)--it is an action. It is a faith that the universe is rational. Take some of Kepler's more ludicrous ideas for example, even they display this trait. Einstein does much to rehabilitate the science of Kepler. Scientists try to see the stucture, make causal connections, and than test it. What is the strange efficacy that science has? In other word, why the heck does it work? It can be justified with the philosophy of "logical positivism". It asks: what are facts? How do you pin them down? But the logical positivists really had little concept of the way that science really works.
It is important to see how theories replace other theories, and how scientific revolutions seem. Kuhn wrote his great book on the Copernican Revolution, and followed it with The Structure of Scientific Revolutions. Kuhn elaborated on Einstein's views on scientific creativity. In this view, we see a multi-teared conceptual explanation of science: there is a reality level and a theoretical level, or "paradigm." The theoretical structure is tested against the reality level until an anomaly is found, and then ane theoretical structure is devised (paradigm shift) which explains the anamoly and the rest of observation. And it is the anomoly's explanation which is the most powerful in overthrowing old theory.
For instance, take Einstein's Theory of Relativity. It was not the prediction that gravity bends light that made it so popular with scientists: Einstein mught just have goten lucky on that question, which no one had ever really cared about before. What was truly the selling point for relativity was its explanation of the advance of the perihelion of Mercury--that was an old problem which had irked professional astronomers for 50 years.
It is important to note that because each paradigm shift is so different, it is in actuality extremely hard to generalize about the structure of scientific revolutions. Nonetheless, Thomas Kuhn tries...
And finally, in the spirit of Winter Recess, some historical notes on the Star of Bethlehem. Giotto, having seen Halley's comet, paints the Christms Star as a comet. But we have no evidence that was so--we have no astronomical account to map into the historical account. Was the star a supernova then? Kepler seemed to think so, basing the supernova that he saw on the conjunction of some "fiery trigon" in the sky. He traces the conjuction back to Charlemagne, and then back to the birth of Christ.
Index of Lectures
(Click on Lecture to view it)
[Friday 18 September 1998: Sun Signs and Cycles]
[Monday 21 September 1998: Stonehenge and Eclipses]
[Wednesday 23 September 1998: The Aegean Birth of Science]
[Friday 25 September 25 1998: The Puzzle of The Planets]
[Monday 28 September 1998: Copernicus]
[Wednesday 30 September 1998: Tycho Brahe]
[Wednesday, October 2, 1996: Kepler and the New Astronomy]
[Friday, October 4, 1996: The Warfare on Mars]
[Monday,October 7, 1996: Galileo's Astronomy]
[Wednesday, October 9, 1996: The Galileo Affair]
[Friday, October 11, 1996: Galileo's Mechanics]
[Friday, October 18, 1996: Newton and His Laws]
[Monday, October 21, 1996: Central Acceleration]
[Wednesday, October 23, 1996: Universal Gravitation]
[Friday, October 25, 1996: The Discovery of Neptune]
[Monday, October 28, 1996: Weighing the Earth]
[Wednesday, October 30, 1996: Was Copernicus Right?]
[Friday, November 1, 1996: The Energy Alternative]
[Monday, November 4, 1996: Einstein and Relativity]
[Wednesday, November 6, 1996: Conservation Laws and Creativity]
[Friday, November 8, 1996: Einstein's Gravitation]
[Wednesday, November 13, 1996: Testing Relativity: The Pulsars]
[Friday, November 15, 1996: Collapse To Black Holes]
[Wednesday, November 20, 1996: Discovering the Nebulae]
[Friday, November 22, 1996: The Milky Way Galaxy]
[Monday, November 25, 1996: Galaxy Redshifts]
[Wednesday, November 27, 1996: Galaxy Distances]
[Monday, December 2, 1996: The Expanding Universe]
[Wednesday, December 4, 1996: The Riddle of The Quasars]
[Friday, December 6, 1996: Quasars & the Violent Universe]
[Monday, December 9, 1996: Did the Universe Begin with a Bang?]
[Wednesday, December 11, 1996: the Curvature of Space]
[Friday, December 13, 1996: Will the Universe Collapse?]
[Monday, December 16, 1996: The Nature of Science]
Science A-17: the Astronomical Perspective, Harvard University. Professors Owen Gingerich and David Latham. The 1996 lecture notes were created originally by Garrett Moritz and Jimmy Liu, the 1998 revisions can be blamed on Dave Latham. Send questions or comments to email@example.com. If you would like to help with the development of this site, perhaps even as a term project, there are many places where figures or even animated applets would be very helpful. I have lots of ideas and very little time to invest in this, but would be pleased to have your help.