MCC PHS 142 M01 Astronomy Homework Ch.24-25 Adj Prof Astronomy: Sam Wormley <email@example.com> Web: edu-observatory.org Background Material Textbook - Read Chapters 24-25 Textbook - http://highered.mcgraw-hill.com/sites/0073512184/student_view0/chapter24/ Textbook - http://highered.mcgraw-hill.com/sites/0073512184/student_view0/chapter25/ (take the Multiple Choice Quiz for for each chapter) Web - http://edu-observatory.org/eo/cosmology.html Web - http://edu-observatory.org/eo/galaxies.html Web - http://edu-observatory.org/eo/radio_astronomy.html Web - http://www.astro.ucla.edu/~wright/cosmolog.htm Web - http://antwrp.gsfc.nasa.gov/apod/archivepix.html "We shall probably never know whether the first men and women to walk on this planet a few million years ago asked themselves questions about the sun, the moon, the stars. But from the time that the last ice age receded, 10,000 years ago, and human beings began to practice agriculture, we can be sure that they were interested in the seasons and the calendar. There is abundant evidence of astronomical purpose in the most ancient of the stone-age monuments of 3000 B.C.[E.] or so. We know that by 2000 B.C.[E.] the Babylonians were regularly observing Venus, and by the time of the height of the Babylonian civilization, around 600 B.C.[E.], we find sophisticated astronomical knowledge among both the Babylonians and the Chinese. "The intellectual horizon of the human race at any time has always been inextricably bound up with the perceived scale of the universe. Ancient myths and those of surviving primitive peoples in or own times suggest a conception of the heavens as not very far off at all. For example, in the most antique of the Chinese cosmologies, the Kai Thien or hemispherical dome theory, the heavens are seen as a rotating hemispherical cover, the earth as a bowl turned upside down, and the distance between the two as considerably less than the radius of the earth. Stories in which the sun or moon is carried in a chariot or boat suggest a distance scale measured on an even smaller, human scale. And there can be little doubt that a people's perceived scale of the universe must play a fundamental role in its culture and consciousness. "It is tempting to consider the growth of knowledge as a linear progress and to consider ourselves as therefore intellectually superior to the ancients. This fallacy is fueled by a picture of science, surprisingly widely held among scientists today, as data collection and analysis. The accumulation of data does indeed proceed monotonically, but science consists of ideas that give meaning to these data, not the data themselves. When we look at the history of ideas about the scale of the universe, we find that the modern idea of an infinite universe through which countless stars and planets wander is not modern at all, but was held firmly by the Greek atomists and by the Chinese cosmologists of the Hsuan Yeh school. And even Aristotle, whom we have been taught to regard as the obstacle to scientific progress in the Middle Ages, summarized the views of his contemporaries in a way that could easily be adapted to modern cosmological prediction: "All thinkers agree that the world has a beginning, but some maintain that having begun, it is everlasting, others that it is perishable like any other formation of matter, and others again that it alternates, being at one time as it is now, and at another time changing and perishing, and that this process continues unremittingly." -Michael Rowan-Robinson The Cosmological Distance Ladder One of the most important, although difficult, tasks that astronomers have undertaken in the twentieth century has been the determination of the distances to the galaxies. This work is of special importance because galaxies and clusters of galaxies mark the basic structure of the universe. Thus our ability to answer fundamental questions about the origin and fate of the universe depends on how well we know the distances of galaxies. The size of the solar system--a first requirement for the establishment of a cosmic distance scale is the correct measurement of distances within the solar system. The basic step in this procedure is the measurement of the distance to Venus. The most precise way of obtaining this distance is through the use of radar techniques. A radar pulse is sent out in the direction of Venus, and the time between its transmission and reception is measured. Since time measurements can be made with great accuracy, the distance to Venus and the dimensions of its orbit can be established within a kilometer. Once the distance to Venus is known at closest approach a, and most distant separation b, and these measurements are repeated over a number of years, the diameter and eccentricity of both the earth's and Venus s orbit can be computed. The mean distance from the earth to the sun is then directly available as the mean value of (a + b)/2. This distance is called the astronomical unit. A check on the Earth-Venus distance is obtained from trajectories of space vehicles sent to Venus. When observations are made from opposite extremes in the earth's orbit about the sun, a nearby star will appear displaced relative to more distant stars in the same part of the sky. The parallax, p, is defined as half the apparent angular displacement measured in this way. Look closely at this diagram: The diagram may seem a little confusing, after all, I've been telling you for weeks that one astronomical unit (1 AU) is the mean distance between the Earth and Sun. And it is, but astronomers can determine 1 au more precisely as the mean value of (a + b)/2. I'm telling you this just so you know. It's harder to make a direct and accurate measurement from the Earth to the Sun. For one reason, the Sun doesn't have a definite edge! What you care about is the relationship d, the distance to a neighbor star, p, the parallax angle, and the radius of the Earth's orbit, 1 AU. Here's the equation: d = 1 astronomical unit / tan(p) Look in Appendix 3 (Conversion Factors) in your textbook. Among those factors, it says that one parsec (which is about 3.26 light years) is equal to 206,265 AU. So you can see how much further stars are apart compared to the planets around our Sun! The nearest star, Proxima Centauri, is 4.24 light years away. Once we know the distance to the nearest stars, by parallax methods, we need other "yard sticks" to build our cosmological distance ladder. Among the objects uses to determine cosmic distances, each kind of object or technique can be used for a range of distances from the nearest object of that type to the greatest distance at which that kind of object can be seen. For instance, Cepheid variable stars, that are discussed in chapters 19 and 24 of your textbook, can be used as indicators for distances between a few hundred parsecs and a few tens of megaparsecs. Your textbook has a better diagram than this one on page 566 showing which indicators are primary, secondary (based on primary) and tertiary (based on secondary). If there are errors in the primary indicators, then the secondary and tertiary indicators just exaggerate the error. The cosmological distance ladder is painstakingly built up, from the solar system up to the largest cosmological scale. The figure on page 545 gives an approximate summary of the cosmological distance ladder in graphic form. Many of these methods can be pushed go greater distances with improved technology and existing ground-based techniques. The parallax an proper-motion satellite Hipparcos, launched in the late 1980's extended the parallax method to about 1000 light years. This new precision, in turn, leads to a more fundamental calibration of Cepheid, RR Lyrae, and nova methods. You saw in the "Mysteries of Deep Space" work by Saul Perlmutter and Alexei V. Filippenko, who are trying to measure the distance and recession velocity of Type Ia Supernovas more than half way across the observable universe. [Did you know that if you put the first four letters of the last names of Perlmutter, Filippenko and Wormley in the American Astronomical Society's on-line directory at http://directory.aas.org/ you get Saul, Alexei, and me!] GRAVITATION AND INERTIA by Ignazio Ciufolini and John Archibald Wheeler Princeton University Press, Princeton, NJ 1995 QC173.59.G44C58 1995 530.1 1--dc20 94-29874 CIP ISBN 0-691-03323-4 In the first chapter Ciufolini and Wheeler introduce concepts and ideas of Einstein's General Theory of Relativity which are developed in the book. This is not an easy book to read for leisure due to the mathematics that is typical in graduate level texts. However, the elegance of conceptual explanation in the first chapter and throughout the text makes this a book well worth reading. Part of the introductory material is reproduced here, because it represents a very simple (yet correct) overview of what General Relativity is and its relationship to inertia. Think of a basketball sitting at the front end of a little read wagon. When you start to pull the wagon forward, the ball tends to stay put (looks like it's rolling to the back of the wagon). That is inertia--the tendency of objects to resist a change of motion (acceleration). "Gravity is not a foreign and physical force transmitted through space and time. It is a manifestation of the curvature of spacetime." That, in a nutshell, is Einstein's theory. What this theory is and what it means, we grasp more fully by looking at its intellectual antecedents. First, there was the idea of Riemann that space, telling mass how to move, must itself--by the principle of action and reaction--be affected by mass. It cannot be an ideal Euclidean perfection, standing in high mightiness above the battles of matter and energy. Space geometry must be a participant in the world of physics. Second, there was the contention of Ernst Mach that the "acceleration relative to absolute space" of Newton is only properly understood when it is viewed as acceleration relative to the sole significant mass there really is, the distant stars. According to this "Mach principle," inertia here arises from mass there. Third was that great insight of Einstein that we summarize in the phrase "free fall is free float": the equivalence principle, one of the best-tested principles in physics, from the inclined tables of Galilei and the pendulum experiments of Galilei, Huygens, and Newton to the highly accurate torsion balance measurements of the twentieth century, and the Lunar Laser Ranging experiment. With those three clues vibrating in his head, the magic of the mind opened to Einstein what remains one of mankind's most precious insights: gravity in manifestation of spacetime curvature. Euclid's (active around 300 B.C.) fifth postulate states that, given any straight line and any point not on it, we can draw through that point one and only one straight line parallel to the given line, that is, a line that will never meet the given one (this alternative formulation of the fifth postulate is essentially due to Proclos). This is the parallel postulate. In the early 1800s the discussion grew lively about whether the properties of parallel lines as presupposed in Euclidean geometry could be derived from the other postulate and axioms, or whether the parallel postulate had to be assumed independently. More than two thousand years after Euclid, Karl Friedrich Gauss, Jnos Bolyai, and Nikolai Ivanovich Lobacevskij discovered pencil-and paper geometric systems that satisfy all the axioms and postulates of Euclidean geometry except the parallel postulate. These geometries showed not only the parallel postulate must be assumed in order to obtain Euclidean geometry but, more important, that non-Euclidean geometries as mathematical abstractions can and do exist. Consider the two-dimensional surface of a sphere, itself embedded in the three-dimensional space geometry of everyday existence. Euclid s system accurately describes the geometry of ordinary three-dimensional space, but not the geometry on the surface of a sphere. Let us consider two lines locally parallel on the surface of a sphere. They propagate on the surface as straight as any lines could possibly be, they bend in their courses one whit neither to left or right. Yet they meet and cross. Clearly, geodesic lines (on a surface, a geodesic is the shortest line between two nearby points) on the curved surface of a sphere do not obey Euclid's parallel postulate. The thoughts of the great mathematician Karl Friedrich Gauss about curvature stemmed not from theoretical spheres drawn on paper but from concrete, down-to-Earth measurements. Commissioned by the government in 1827 to make a survey map of the region for miles around Göttingen, he found that the sum of the angles in his largest survey triangle was different from 180. The deviation from 180 observed by Gauss--almost 15 seconds of arc--was both inescapable evidence for and a measure of the curvature of the surface of Earth. To recognize that straight and initially parallel lines on the surface of a sphere can meet was the first step in exploring the idea of a curved space. Second came the discovery of Gauss that we do not need to consider a sphere or other two-dimensional surface to be embedded in a three-dimensional space to define its geometry. It is enough to consider measurements made entirely within that two-dimensional geometry, such as, would be made by an ant forever restricted to live on that surface. The ant would know that the surface is curved by measuring that the sum of the internal angles of a large triangle differs from 180, or by measuring that the ratio between a large circumference and its radius R differs from 2 pi. Gauss did not limit himself to thinking of a curved two-dimensional surface floating in a flat three-dimensional universe. In an 1824 letter to Ferdinand Karl Schweikart, he dared to conceive that space itself is curved: "Indeed I have therefore from time to time in jest expressed the desire that Euclidean geometry would not be correct." He also wrote: "Although geometers have given much attention to general investigations of curved surfaces and their results cover a significant portion of the domain of higher geometry, this subject is still so far from being exhausted, that it can well be said that, up to this time, but a small portion of an exceedingly fruitful field has been cultivated" (Royal Society of Göttingen, 8 October 1827). The inspiration of these thoughts, dreams, and hopes passed from Gauss to his student, Bernhard Riemann. Bernhard Riemann went on to generalize the ideas of Gauss so that they could be used to describe curved spaces of three or more dimensions. Gauss had found that the curvature in the neighborhood of a given point of a specified two-dimensional space geometry is given by a single number: the Gaussian curvature. Riemann found that six numbers are needed to describe the curvature of a three-dimensional space at a given point, and that 20 numbers at each point are required for a four-dimensional geometry: the 20 independent components of the so-called Riemann curvature tensor. In a famous lecture he gave 10 June 1854, entitled On the Hypothesis That Lie at the Foundations of Geometry, Riemann emphasized that the truth about space is to be discovered not from perusal of the 2000-year-old books of Euclid but from physical experience. He pointed out that space could be highly irregular at very small distances and yet appear smooth at everyday distances. At very great distances, he also noted, large-scale curvature of space might show up, perhaps even bending the universe into a closed system like a gigantic ball. He wrote: "Space [in the large] if one ascribes to it a constant curvature, is necessarily finite, provided only that this curvature has a positive value, however small.... It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry.... The curvature in the three directions can have arbitrary values if only the entire curvature for every sizable region of space does not differ greatly from zero.... The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience." But as Einstein was later to remark, "Physicists were still far removed from such a way of thinking: space was still, for them, a rigid, homogeneous something, susceptible of no change or conditions". Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible." Even as the 39-year-old Riemann lay dying of tuberculosis at Selasca on Lake Maggiore in the summer of 1866, having already achieved his great mathematical description of space curvature, he was working on a unified description of electromagnetism and gravitation. Why then did he not, half a century before Einstein, arrive at a geometric account of gravity? No obstacle in his way was greater than this: he thought only of space and the curvature of space, whereas Einstein discovered that he had to deal with spacetime and the spacetime curvature. Einstein could not thank Riemann, who ought to have been still alive. A letter of warm thanks he did, however, write to Mach. In it he explained how mass there does indeed influence inertia here, through its influence on the enveloping spacetime geometry. Einstein's geometrodynamics had transmuted Mach's bit of philosophy into a bit of physics, susceptible to calculation, prediction, and test. Let us bring out the main idea in what we may call poor man's language. Inertia here, in the sense of local inertial frames, that is the grip of spacetime here on mass here, is fully defined by the geometry, the curvature, the structure of spacetime here. The geometry here, however, has to fit smoothly to the geometry of the immediate surroundings; those domains, onto their surroundings; and so on, all the way around the great curve of space. Moreover, the geometry in each local region responds in its curvature to the mass in that region. Therefore every bit of momentum-energy, wherever located, makes its influence felt on the geometry of space throughout the whole universe--and felt, thus, on inertia right here. The bumpy surface of a potato is easy to picture. It is the two-dimensional analogue of a bumpy three-sphere, the space geometry of a universe loaded irregularly here and there with concentrations and distributions of momentum-energy. If the spacetime has a Cauchy surface, that three-geometry once known--mathematical solutions as it is of the so-called initial-value problem of geometrodynamics--the future evolution of that geometry follows straightforwardly and deterministically. In other words, inertia (local inertial frames) everywhere and at all times is totally fixed, specified, determined, by the initial distribution of momentum-energy, of mass and mass-in-motion. The mathematics cries out with all the force at its command that mass there does determine inertia here. One exciting experiment to be performed by the turn of the century will be the measurement of frame-dragging by the Earth as it rotates. It is estimated that the gravitational effect of the rotating Earth on the local spacetime nearby is a measurable effect [330 milliarcsec per year]. The mass of the Earth has about 0.698 billionth total voting power as the rest of the universe on our local spacetime! Homework Problems Note the answers to the odd (Conceptual Questions, Problems and Figure-Based Questions) are in the back of your textbook. It is strongly suggested that you do some of those in every chapter so you have immediate feedback as how well you are understanding the material. There are online multiple choice quizzes for each chapter of your textbook. Goto http://www.mhhe.com/fix then click on Your book Student Edition Choose a chapter Multiple Choice Quiz You are expected to do all of your own homework. Statistical patterns showing copying or collaboration will result in no credit for the homework assignment for all participants involved. The Code of Academic Conduct for Iowa Valley Community College District is found in the Student Handbook. Physical Science classes require the use of mathematics. If you don't know algebra, you sould NOT be taking this class. If you need to review, look at Introduction to Algebra http://www.math.armstrong.edu/MathTutorial/ WolframAlpha is way faster than a scientific calculator. http://www.wolframalpha.com There is little excuse for turning homework in late. You have a whole week between classes to read the chapters and do the homework. Homework one week late - half credit. Two or more weeks late - no credit. Do the homework during the week, not in class! You got homework questions, email me 24/7. firstname.lastname@example.org Even if you don't have a homework question, email me anyway! Problem 1: Using you starwheel (planisphere), name three constellations that are circumpolar. Problem 2: Suppose a quasar is observed to increase tenfold in brightness in 4 day. Approximately how large could the quasar be? Problem 3: How do electrons and the energy they carry get into the lobes of radio galaxies? Problem 4: How have the quasars evolved over the last 10 billion years? Problem 5: What evidence do we have that there may be dead quasars in the centers of normal galaxies today? Problem 6: Use Figure 24.3 to find the distance of a galaxy that has a redshift of z = 1.0. Problem 7: Use Figure 24.10 to find the apparent speed of a component of a quasar that is moving at 90% of the speed of light in a direction that makes an angle of 20° to the line of sight to the quasar. Problem 8: If the general picture of active galaxies presented in Figure 24.23 is correct, what would we call a low-luminosity active galaxy that we see at a large viewing angle? Problem 9: What evidence do we have that there is more dark matter than luminous matter in the Local Group of galaxies? Problem 10: What is the best way to detect extremely hot gas in clusters of galaxies? Problem 11: As I mentioned earlier one of the biggest problems in astronomy is measuring distances, especially distances between galaxies. The origin and fate of the universe depends on how well we know the distances of galaxies. We also need to know how much mass there is, the expansion rate, the deceleration rate, and similar cosmological parameters. General Relativity is central to those questions, so I want you to have some understanding of it. "Please answer me this", as Papa Joe would say, "What is or are the most important thing(s) you learned in this astronomy class"?