Essential Astrophysics 2013 Undergraduate Lecture Notes PDF

Undergraduate Lecture Notes in Physics Kenneth R. Lang Essential Astrophysics Undergraduate Lecture Notes in Physics Series Editors Neil Ashby William Brantley Michael Fowler Michael Inglis Elena Sassi Helmy S. Sherif Heinz Klose For further volumes: http://www.springer.com/series/8917 Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: An exceptionally clear and concise treatment of a standard undergraduate subject. A solid undergraduate-level introduction to a graduate, advanced, or non-stan- dard subject. A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career. Kenneth R. Lang Essential Astrophysics 123 Kenneth R. Lang Department of Physics and Astronomy Tufts University Medford, MA USA ISSN 2192-4791 ISSN 2192-4805 (electronic) ISBN 978-3-642-35962-0 ISBN 978-3-642-35963-7 (eBook) DOI 10.1007/978-3-642-35963-7 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012955653 Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science?Business Media (www.springer.com) Preface Essential Astrophysics is a book to learn or teach from, as well as a fundamental reference for anyone interested in astronomy and astrophysics. This unique volume can be used as a textbook, teaching guide, or reference source for just about anyone interested in astronomy and astrophysics. It serves as a comprehensive, introductory text, which takes the student through the field of astrophysics in lecture-sized chapters of basic physical principles applied to the cosmos. Undergraduate students with an interest in the physical sciences, such as astronomy, chemistry, engineering, or physics, will enjoy this one-semester overview. The text is of sufficient breadth and depth to prepare the interested student for more advanced, specialized courses in the future. The clarity and comprehensive nature of Essential Astrophysics make it a significant resource for the curious reader that is unfamiliar with astrophysics or for professional astronomers who may have forgotten the basics. Astronomical examples are provided throughout the text, to reinforce the basic concepts and physics, and to demonstrate the use of the relevant formulae. In this way, the student learns to apply the fundamental equations and principles to cosmic objects and situations. All of the example problems are solved with the rough accuracy needed to portray the basic result. Such order-of-magnitude esti- mates are commonly used in astronomy and astrophysics, where large numbers are involved, and an understanding of the underlying physics does not require engi- neering accuracy. Essential Astrophysics is a serious introduction to astrophysics complete with the necessary formulae. These equations sometimes include the calculus of inte- gration, or adding up, and differentiation, that are found in the author’s classic Astrophysical Formulae and more advanced textbooks. Nevertheless, the end result in Essential Astrophysics is always a simple algebraic relationship that can be applied to cosmic objects. These fundamental equations are given in the text and collected at the end of the book in Appendix III, for future reference and use. Therefore, only elementary algebra is required to solve any of the example problems or other numerical conclusions in this book. v vi Preface There are two types of intended readers. One type will be interested in broad, general conclusions, without use of calculus. This reader will be content with the existing text with no further elaboration. The more mathematically competent reader will want to use Essential Astrophysics as a foundation for more advanced considerations, with the guidance of the references, an instructor, or an advanced textbook, using the formulae found in the text or within set aside Focus Elements of Essential Astrophysics as a starting point. The modern SI (International System) units are used in the equations and example problems, which is another unique aspect of this book when compared to most previous texts of astrophysics. A conversion table between the SI and c.g.s. units is provided in the first chapter, to help the reader follow the details of many papers and textbooks that use the older c.g.s. system. Astronomical and physical constants, units, and fundamental equations are provided in appendices, for quick reference. Essential Astrophysics goes beyond the typical textbook by providing com- prehensive access to astrophysical discoveries, concepts, and facts that are not available in any other way. It gives us access to that long-forgotten formula, idea, or reference, while also providing the material needed to introduce anyone to a new area of astrophysics. Here, the reader can obtain the background required for a general understanding and find guidance to the relevant literature including sem- inal discoveries, original research, and comprehensive up-to-date reviews that will enable the curious reader to delve deeper into a particular topic. A more extensive reference compilation of developments in astrophysics, from then to now, can be found in Astrophysical Formulae. We are the benefactors of 300 years of cumulative discovery in astronomy and astrophysics, and Essential Astrophysics helps pass on these fundamental insights to the next generation. It also reveals both the exciting moments of the past and relatively recent discoveries. Historical aspects are illuminated through a pro- gressive flow of chapter topics and by guidance to the earliest ideas, with reference to the original sources as well as contemporary reviews. Perhaps because of the rapid pace of modern research, contemporary texts often focus on specialized topics and overlook these broader perspectives that Essential Astrophysics provides. There are 50 set-aside focus elements that enhance and amplify the discussion with fascinating details. They include the intriguing development of particular themes, which is missing in most astrophysics textbooks, or provide further astrophysics or equations for use in examples, problems or further investigations. In Essential Astrophysics we can rediscover basic physical concepts such as space, time, radiation, mass, gravity, motion, heat, atoms, radioactivity, and cos- mic rays, which are required to understand the observable universe. These fun- damental topics are discussed in the first seven chapters, beginning with the introductory chapter that describes how astronomers observe the contents of the universe and how astrophysicists interpret them. The SI units of distance, mass, time, energy, and luminosity are introduced, together with their astronomical units Preface vii such as the Ångström, light-year, parsec, and the Sun’s mass, luminosity, and radius. The magnitude unit is also defined, but used sparingly in examples. Chapter 2 describes radiation, of both the visible and invisible sort, which carries messages from the cosmos and tells us much of what we know about it. Chapter 3 discusses gravity, together with mass that helps determine its strength, and related tidal phenomena and space curvature. Chapter 4 discusses cosmic motion, and its balanced equilibrium with gravitation. Chapter 5 discusses the motion of particles in a gas, together with the related concepts of speed distri- bution, heat, temperature, and pressure. The inside of the atom is explored in Chap. 6, where the reader learns about atomic spectral lines and their use in determining the composition of stars and the measurement of motions and mag- netic fields. The transformation of elements in both radioactivity and by sub- atomic bombardment is presented in Chap. 7. The fundamental concepts described in these first seven chapters provide a necessary prelude to the rest of the book. It includes the discoveries that the universe is predominantly hydrogen; that the stars shine by nuclear fusion; that the stars live and die while new ones continue to be formed; that the interstellar spaces are not empty but filled with dust, atoms, and molecules; and that the observable universe is expanding and has a history. The last half of Essential Astrophysics also includes relatively recent discoveries, such as pulsars, black holes, the three- degree cosmic microwave background, the formation of stars and galaxies, invisible dark matter, and the dark energy that is now accelerating the expansion of the universe. Chapter 8 provides an account of the nuclear fusion reactions that make the Sun shine. This is followed in Chap. 9 by modern discoveries of the Sun’s expanding atmosphere, the solar winds, explosions on the Sun, the solar flares and coronal mass ejections, and their space–weather threats to spacecraft and humans in space. Chapter 10 presents an overview of the stars, telling us how far away, bright, luminous, hot, big, and massive they are. It also includes discussions of stellar spectra, as well as the evolution of stars and their role in the origin of the chemical elements. The space between the stars is discussed in Chap. 11, beginning with bright stars that illuminate nearby space and continuing with the dust, gas, radio emis- sion, and molecules within interstellar space. This is naturally followed in Chap. 12 by the ongoing formation of stars and their planets; recent discoveries of protoplanetary disks and planets around nearby stars can also be found in this chapter. The final destiny of stars, when they have depleted their nuclear resources, is presented in Chap. 13. It includes planetary nebulae, white dwarf stars, degenerate pressure, novae, supernovae, neutron stars, pulsars, and stellar black holes. Our last two chapters discuss the observable universe in its entirety, including the Milky Way, the receding galaxies, the big bang with its background radiation, the first atoms, stars, and galaxies, the evolution of galaxies, dark matter and dark energy, and the ultimate destiny of the universe. viii Preface A total of 69 tables provide vital facts and physical information for the main types of cosmic objects; students, teachers, and researchers may also consult this information throughout their careers. In alphabetical order, they include the physical properties of atmospheres, clusters of galaxies, the cosmic microwave background radiation, the Earth, emission nebulae, galaxies, our Galaxy, giant molecular clouds, H I regions, H II regions, interstellar molecules, the Milky Way, our Moon, neutron stars, novae, planetary nebulae, planets, pulsars, radioactive isotopes, the Sun, stars, star clusters, supernova explosions, and supernova remnants. Our tables also include information about cosmic magnetic fields, cosmic rays, cosmological parameters, and nuclear fusion processes, as well as the range of cosmic pressures, cosmic temperatures and stellar luminosity, and the spectral lines of active galaxies, emission nebulae, stars, the Sun’s corona, and the Sun’s photosphere. There are also excellent line drawings, prepared by Kacha Bradonjich, and several images of astronomical objects from the ground and space that help cement our newfound knowledge together. They help crystallize a new concept with a visual excitement that adds another dimension to our understanding. The author also writes another sort of popular book, filled with personal anecdotes, the lives of contributors to the field, and human metaphors, without an equation or reference in sight. For this complementary approach, the reader is referred to the author’s two books The Life and Death of Stars and Parting the Cosmic Veil, which deal with many of the same general topics as Essential Astrophysics in a different, lighter perspective. I am indebted to Gayle Grant for help in assembling this book, and to the Tufts Faculty Research Committee for modest support for typing some equations in it. And last, but not least, the author thanks Ramon Khanna for his skillful editorial suggestions that have made Essential Astrophysics a better book. Medford, November 2012 Kenneth R. Lang Contents 1 Observing the Universe................................ 1 1.1 What Do Astronomers and Astrophysicists Do?........... 1 1.2 Our Place on Earth............................... 2 1.3 Location in the Sky.............................. 4 1.4 Measuring Angle and Size.......................... 9 1.5 The Locations of the Stars are Slowly Changing.......... 10 1.6 What Time is It?................................ 15 1.7 Telling Time by the Stars.......................... 17 1.8 Optical Telescopes Observe Visible Light............... 19 1.9 Telescopes that Detect Invisible Radiation............... 23 1.10 Units Used by Astronomers and Astrophysicists........... 27 1.11 Physical Constants............................... 30 2 Radiation.......................................... 33 2.1 Electromagnetic Waves............................ 33 2.2 The Electromagnetic Spectrum....................... 37 2.3 Moving Perspectives.............................. 40 2.4 Thermal (Blackbody) Radiation...................... 44 2.5 How Far Away is the Sun, and How Bright, Big and Hot is it?................................ 50 2.5.1 Distance of the Sun........................ 50 2.5.2 How Big is the Sun?........................ 54 2.5.3 The Unit of Energy......................... 54 2.5.4 The Sun’s Luminosity....................... 55 2.5.5 Taking the Sun’s Temperature................. 55 2.5.6 How Hot are the Planets?.................... 56 2.6 The Energy of Light.............................. 59 2.7 Radiation Scattering and Transfer..................... 61 2.7.1 Why is the Sky Blue and the Sunsets Red?........ 61 2.7.2 Rayleigh Scattering......................... 62 ix x Contents 2.7.3 Thomson and Compton Scattering.............. 63 2.7.4 Radiation Transfer......................... 65 3 Gravity........................................... 69 3.1 Ceaseless, Repetitive Paths Across the Sky.............. 69 3.2 Universal Gravitational Attraction.................... 73 3.3 Mass of the Sun................................. 80 3.4 Tidal Effects................................... 81 3.4.1 The Ocean Tides.......................... 81 3.4.2 Tidal Locking into Synchronous Rotation......... 85 3.4.3 The Days are Getting Longer.................. 85 3.4.4 The Moon is Moving Away from the Earth........ 87 3.4.5 A Planet’s Differential Gravitational Attraction Accounts for Planetary Rings.................. 90 3.5 What Causes Gravity?............................. 93 4 Cosmic Motion...................................... 99 4.1 Motion Opposes Gravity........................... 99 4.1.1 Everything Moves......................... 99 4.1.2 Escape Speed............................. 99 4.2 Orbital Motion.................................. 101 4.3 The Moving Stars................................ 105 4.3.1 Are the Stars Moving?...................... 105 4.3.2 Components of Stellar Velocity................ 105 4.3.3 Proper Motion............................ 107 4.3.4 Radial Velocity........................... 107 4.3.5 Observed Proper Motions of Stars.............. 109 4.3.6 Motions in Star Clusters..................... 111 4.3.7 Runaway Stars............................ 114 4.4 Cosmic Rotation................................. 116 4.4.1 Unexpected Planetary Rotation................. 116 4.4.2 The Sun’s Differential Rotation................ 120 4.4.3 Stellar Rotation and Age..................... 124 5 Moving Particles.................................... 125 5.1 Elementary Constituents of Matter.................... 125 5.2 Heat, Temperature, and Speed....................... 130 5.2.1 Where Does Heat Come From?................ 130 5.2.2 Thermal Velocity.......................... 132 5.2.3 Collisions............................... 134 5.2.4 The Distribution of Speeds................... 135 5.3 Molecules in Planetary Atmospheres................... 138 Contents xi 5.4 Gas Pressure................................... 141 5.4.1 What Keeps Our Atmosphere Up?.............. 141 5.4.2 The Ideal Gas Law......................... 142 5.4.3 The Earth’s Sun-Layered Atmosphere............ 144 5.4.4 Pressure, Temperature, and Density Inside the Sun... 148 5.5 Plasma....................................... 149 5.5.1 Ionized Gas.............................. 149 5.5.2 Plasma Oscillations and the Plasma Frequency..... 152 5.5.3 Atoms are Torn Apart into Plasma Within the Sun... 153 5.6 Sound Waves and Magnetic Waves................... 154 5.6.1 Sound Waves............................. 154 5.6.2 Magnetic Waves........................... 156 6 Detecting Atoms in Stars.............................. 159 6.1 What is the Sun Made Out Of?...................... 159 6.2 Quantization of Atomic Systems..................... 165 6.3 Some Atoms are Excited Out of Their Lowest-Energy Ground State................................... 173 6.4 Ionization and Element Abundance in the Sun and Other Stars................................. 176 6.5 Wavelengths and Shapes of Spectral Lines.............. 180 6.5.1 Radial Motion Produces a Wavelength Shift....... 180 6.5.2 Gravitational Redshift....................... 181 6.5.3 Thermal Motion Broadens Spectral Lines......... 183 6.5.4 Rotation or Expansion of the Radiating Source can Broaden Spectral Lines................... 184 6.5.5 Curve of Growth.......................... 185 6.5.6 Magnetic Fields Split Spectral Lines............. 186 7 Transmutation of the Elements.......................... 191 7.1 The Electron, X-rays and Radium..................... 191 7.2 Radioactivity................................... 193 7.3 Tunneling Out of the Atomic Nucleus.................. 196 7.4 The Electron and the Neutrino....................... 199 7.5 Cosmic Rays................................... 202 7.6 Nuclear Transformation by Bombardment............... 209 8 What Makes the Sun Shine?............................ 215 8.1 Can Gravitational Contraction Supply the Sun’s Luminosity?............................. 215 8.2 How Hot is the Center of the Sun?.................... 217 8.3 Nuclear Fusion Reactions in the Sun’s Core............. 219 8.3.1 Mass Lost is Energy Gained.................. 219 8.3.2 Understanding Thermonuclear Reactions.......... 225 xii Contents 8.3.3 Hydrogen Burning......................... 231 8.3.4 Why Doesn’t the Sun Blow Up?............... 237 8.4 The Mystery of Solar Neutrinos...................... 237 8.4.1 The Elusive Neutrino....................... 237 8.4.2 Solar Neutrino Detectors Buried Deep Underground......................... 239 8.4.3 Solving the Solar Neutrino Problem............. 242 8.5 How the Energy Gets Out.......................... 244 8.6 The Faint-Young-Sun Paradox....................... 252 8.7 The Sun’s Destiny............................... 253 9 The Extended Solar Atmosphere......................... 255 9.1 Hot, Volatile, Magnetized Gas....................... 255 9.1.1 The Million-Degree Solar Corona............... 255 9.1.2 Varying Sunspots and Ever-Changing Magnetic Fields........................... 258 9.1.3 Coronal Loops............................ 261 9.1.4 What Heats the Corona?..................... 266 9.1.5 Coronal Holes............................ 268 9.2 The Sun’s Varying Winds.......................... 268 9.2.1 The Expanding Sun Envelops the Earth.......... 268 9.2.2 Properties of the Solar Wind.................. 271 9.2.3 Where Do the Two Solar Winds Come From?...... 274 9.2.4 Where Does the Solar Wind End?.............. 275 9.3 Explosions on the Sun............................. 276 9.3.1 Solar Flares.............................. 276 9.3.2 Coronal Mass Ejections...................... 281 9.4 Space Weather.................................. 283 9.4.1 Earth’s Protective Magnetosphere............... 283 9.4.2 Trapped Particles.......................... 287 9.4.3 Earth’s Magnetic Storms..................... 288 9.4.4 Solar Explosions Threaten Humans in Outer Space............................ 289 9.4.5 Disrupting Communication................... 290 9.4.6 Satellites in Danger........................ 291 9.4.7 Forecasting Space Weather................... 292 10 The Sun Amongst the Stars............................ 293 10.1 Comparisons of the Sun with Other Stars............... 293 10.1.1 How Far Away are the Stars?................. 293 10.1.2 How Bright are the Stars?.................... 296 10.1.3 How Luminous are the Stars?................. 298 10.1.4 The Temperatures of Stars.................... 303 10.1.5 The Colors of Stars......................... 304 Contents xiii 10.1.6 The Spectral Sequence...................... 305 10.1.7 Radius of the Stars......................... 306 10.1.8 How Massive are the Stars?................... 310 10.2 Main-Sequence and Giant Stars...................... 318 10.2.1 The Hertzsprung–Russell Diagram.............. 318 10.2.2 The Luminosity Class....................... 321 10.2.3 Life on the Main Sequence................... 323 10.2.4 The Red Giants and Supergiants................ 326 10.3 Nuclear Reactions Inside Stars....................... 329 10.3.1 The Internal Constitution of Stars............... 329 10.3.2 Two Ways to Burn Hydrogen in Main-Sequence Stars...................... 335 10.3.3 Helium Burning in Giant Stars................. 340 10.4 Using Star Clusters to Watch How Stars Evolve.......... 343 10.5 Where did the Chemical Elements Come From?.......... 348 10.5.1 Advanced Nuclear Burning Stages in Massive Supergiant Stars................... 348 10.5.2 Origin of the Material World.................. 349 10.5.3 The Observed Abundance of the Elements........ 350 10.5.4 Synthesis of the Elements Inside Stars........... 351 10.5.5 Big-Bang Nucleosynthesis.................... 353 10.5.6 The First and Second Generation of Stars......... 354 10.5.7 Cosmic Implications of the Origin of the Elements... 355 11 The Material Between the Stars......................... 357 11.1 Gaseous Emission Nebulae......................... 357 11.2 Solid Dust Particles in Interstellar Space................ 366 11.3 Radio Emission from the Milky Way.................. 369 11.4 Interstellar Hydrogen Atoms........................ 375 11.5 Interstellar Molecules............................. 378 12 Formation of the Stars and Their Planets.................. 381 12.1 How the Solar System Came into Being................ 381 12.1.1 The Nebular Hypothesis..................... 381 12.1.2 Composition of the Planets................... 382 12.1.3 Mass and Angular Momentum in the Solar System......................... 385 12.2 Star Formation.................................. 388 12.2.1 Giant Molecular Clouds..................... 388 12.2.2 Gravitational Collapse....................... 389 12.2.3 Triggering Gravitational Collapse............... 392 12.2.4 Protostars................................ 395 12.2.5 Losing Mass and Spin....................... 398 xiv Contents 12.3 Planet-Forming Disks and Planets Around Nearby Stars..... 400 12.3.1 The Plurality of Worlds...................... 400 12.3.2 Proto-Planetary Disks....................... 400 12.3.3 The First Discoveries of Exoplanets............. 403 12.3.4 Hundreds of New Worlds Circling Nearby Stars.... 408 12.3.5 Searching for Habitable Planets................ 409 13 Stellar End States.................................... 411 13.1 A Range of Destinies............................. 411 13.2 Planetary Nebulae................................ 412 13.3 Stars the Size of the Earth.......................... 418 13.3.1 The Discovery of White Dwarf Stars............ 418 13.3.2 Unveiling White Dwarf Stars.................. 419 13.3.3 The High Mass Density of White Dwarf Stars...... 420 13.4 The Degenerate Electron Gas........................ 423 13.4.1 Nuclei Pull a White Dwarf Together as Electrons Support It...................... 423 13.4.2 Radius and Mass of a White Dwarf............. 427 13.5 Exploding Stars................................. 429 13.5.1 Guest Stars, the Novae...................... 429 13.5.2 What Makes a Nova Happen?................. 430 13.5.3 A Rare and Violent End, the Supernovae......... 433 13.5.4 Why do Supernova Explosions Occur?........... 436 13.5.5 When a Nearby Star Detonates Its Companion...... 437 13.5.6 Stars that Blow Themselves Up................ 438 13.5.7 Light of a Billion Suns, SN 1987A.............. 439 13.5.8 Will the Sun Explode?...................... 443 13.6 Expanding Stellar Remnants........................ 443 13.7 Neutron Stars and Pulsars.......................... 450 13.7.1 Neutron Stars............................. 450 13.7.2 Radio Pulsars from Isolated Neutron Stars......... 453 13.7.3 X-ray Pulsars from Neutron Stars in Binary Star Systems............................. 460 13.8 Stellar Black Holes............................... 465 13.8.1 Imagining Black Holes...................... 465 13.8.2 Observing Stellar Black Holes................. 466 13.8.3 Describing Black Holes...................... 467 14 A Larger, Expanding Universe.......................... 471 14.1 The Milky Way................................. 471 14.1.1 A Fathomless Disk of Stars................... 471 14.1.2 The Sun is Not at the Center of Our Stellar System............................ 473 14.1.3 The Rotating Galactic Disk................... 479 Contents xv 14.1.4 Whirling Coils of the Milky Way............... 482 14.1.5 A Central Super-Massive Black Hole............ 484 14.1.6 Dark Matter Envelops the Milky Way............ 486 14.2 The Discovery of Galaxies......................... 487 14.3 The Galaxies are Moving Away from us and from Each Other............................. 491 14.4 Galaxies Gather and Stream Together.................. 500 14.4.1 Clusters of Galaxies........................ 500 14.4.2 Dark Matter in Clusters of Galaxies............. 502 14.4.3 Cosmic Streams........................... 508 14.4.4 Galaxy Walls and Voids..................... 510 14.5 Looking Back into Time........................... 512 14.6 Using Einstein’s General Theory of Relativity to Explain the Expansion........................... 517 15 Origin, Evolution, and Destiny of the Observable Universe..... 523 15.1 Hotter Than Anything Else......................... 523 15.2 Three Degrees Above Absolute Zero.................. 526 15.2.1 An Unexpected Source of Noise................ 526 15.2.2 Blackbody Spectrum........................ 527 15.2.3 As Smooth as Silk......................... 529 15.2.4 Cosmic Ripples........................... 529 15.3 The Beginning of the Material Universe................ 532 15.3.1 The First Three Minutes..................... 532 15.3.2 Formation of the First Atoms, and the Amount of Invisible Dark Matter..................... 535 15.3.3 History of the Expanding Universe.............. 537 15.4 The First Stars and Galaxies........................ 541 15.4.1 Pulling Primordial Material Together............ 541 15.4.2 When Stars Began to Shine................... 542 15.5 The Evolution of Galaxies.......................... 545 15.5.1 Active Galactic Nuclei...................... 545 15.5.2 Super-Massive Black Holes................... 550 15.5.3 Gamma-Ray Bursts......................... 552 15.6 Dark Energy, the Cosmological Constant, and How it All Ends..................................... 554 15.6.1 Discovery of Dark Energy.................... 554 15.6.2 Using the Cosmological Constant to Describe Dark Energy............................. 555 15.6.3 When Stars Cease to Shine................... 560 16 References......................................... 561 xvi Contents Appendix I: Constants.................................... 607 Appendix II: Units...................................... 609 Appendix III: Fundamental Equations........................ 611 Author Index.......................................... 615 Subject Index.......................................... 619 Focus Elements Focus 1.1 Astronomical catalogues.......................... 6 Focus 1.2 The elongated shape of the Earth.................... 11 Focus 1.3 Stellar aberration............................... 14 Focus 1.4 The great observatories........................... 25 Focus 2.1 Plane waves of electromagnetic radiation.............. 34 Focus 2.2 Light, the fastest thing around...................... 36 Focus 2.3 The Michelson-Morley experiment................... 41 Focus 2.4 The solar parallax and the Sun’s distance.............. 52 Focus 2.5 Global warming by the greenhouse effect.............. 58 Focus 3.1 Moving along an elliptical trajectory................. 70 Focus 3.2 The Earth’s gravity............................. 79 Focus 3.3 Tidal friction slows the rotation of the Earth............ 85 Focus 3.4 Conservation of angular momentum in the Earth-Moon system......................... 87 Focus 3.5 The Roche limit................................ 91 Focus 3.6 Testing relativity with the binary pulsar............... 96 Focus 4.1 How fast can a planet or star rotate?................. 119 Focus 5.1 The Earth’s ionosphere........................... 146 Focus 6.1 Hydrogen, the most abundant element in the Sun and most stars......................... 179 Focus 7.1 Nuclear nomenclatures........................... 195 Focus 7.2 The age of the solar system........................ 198 Focus 8.1 The temperatures necessary for thermonuclear reactions.... 221 Focus 8.2 Non-resonant thermonuclear reaction rates............. 229 Focus 8.3 Secondary nuclear fusion reactions in the Sun........... 235 Focus 8.4 Trillions upon trillions of neutrinos.................. 238 Focus 8.5 Leptons...................................... 242 Focus 8.6 Convection................................... 248 Focus 8.7 Helioseismology............................... 251 Focus 9.1 Magnetic pressure and gas pressure.................. 264 xvii xviii Focus Elements Focus 9.2 Discovery of the solar wind....................... 268 Focus 9.3 Mass loss from the Sun.......................... 273 Focus 9.4 Physical properties of coronal mass ejections........... 282 Focus 9.5 Planetary magnetospheres......................... 285 Focus 10.1 The upper mass limit for a star..................... 312 Focus 10.2 Determining the stellar mass in a spectroscopic binary system.................... 316 Focus 10.3 The equations of stellar structure.................... 332 Focus 10.4 The proton–proton chain.......................... 335 Focus 10.5 The CNO cycle................................ 336 Focus 11.1 Charged particles gyrate around magnetic fields......... 371 Focus 12.1 How fast was the young Sun rotating?................ 385 Focus 12.2 Magnetic energy............................... 393 Focus 12.3 Determining the mass and orbital distance of an exoplanet................................ 406 Focus 13.1 Radius and mass density of a white dwarf star.......... 420 Focus 13.2 Neutrinos generated during a supernova............... 441 Focus 13.3 Luminosity, rotational energy, and magnetic field strength of a radio pulsar......................... 457 Focus 13.4 Accretion luminosity and the Eddington limit........... 463 Focus 14.1 Cepheid variable stars............................ 474 Focus 14.2 Differential rotation of the Milky Way................ 480 Focus 14.3 Density and total number of galaxies................. 498 Focus 14.4 How old is the observable universe?................. 513 Focus 15.1 Before the Big Bang............................. 523 Tables Table 1.1 Celestial positions of the equinoxes and solstices........ 8 Table 1.2 Principal SI units and their conversion to corresponding c.g.s. units....................... 30 Table 2.1 Approximate wavelengths of colors................. 38 Table 2.2 The electromagnetic spectrum..................... 39 Table 2.3 Radiation constants............................. 49 Table 2.4 Distances, visual albedos, effective temperatures, and mean temperatures of the planets................ 57 Table 3.1 Earth’s orbital and physical properties................ 77 Table 3.2 Orbital and physical properties of the Moon........... 82 Table 4.1 Mass, radius, and escape speed of some cosmic objects... 102 Table 4.2 Stars with the highest proper motion................. 111 Table 4.3 Physical properties of star clusters.................. 111 Table 4.4 Oblateness of the giant planets and the Earth........... 119 Table 4.5 Rotation periods and rotation velocities of some planets and stars......................... 121 Table 4.6 Differential rotation of the Sun..................... 121 Table 5.1 Physical properties of electrons, protons, neutrons, and atoms................................... 129 Table 5.2 Range of cosmic temperatures..................... 131 Table 5.3 Atmospheres of Venus, Mars, and Earth.............. 138 Table 5.4 Atmospheres of the giant planets and the Sun.......... 138 Table 5.5 Range of cosmic pressures........................ 142 Table 6.1 Prominent absorption lines and elements detected in sunlight............................. 163 Table 6.2 The twenty most abundant elements in the Sun......... 164 Table 6.3 Wavelengths of the m to n transitions of hydrogen for n = 1 to n = 5 and m = 2 to m = 10................................... 171 xix xx Tables Table 6.4 Atomic number Z, atomic mass MA, and atomic Rydberg constant RA for the most abundant atoms in the cosmos... 173 Table 6.5 Ionization potentials v for different stages of ionization the most abundant atoms in the cosmos............... 178 Table 6.6 Cosmic magnetic fields.......................... 189 Table 7.1 Long-lived radioactive isotopes used for dating......... 198 Table 7.2 Average fluxes of primary cosmic rays at the top of the atmosphere.............................. 204 Table 7.3 Particle speeds at different particle energies, expressed as fractions of the speed of light, c.................. 205 Table 7.4 Nobel Prizes related to experimental investigations of subatomic matter............................ 208 Table 8.1 Physical properties of the Sun..................... 219 Table 8.2 Binding energy, EB, and binding energy per nucleon, f = EB=A, for some nuclei of atomic mass number A..... 228 Table 9.1 Strong forbidden emission lines in the visible light of the Sun’s low corona......................... 257 Table 9.2 Prominent soft x-ray and extreme ultraviolet emission lines from the Sun’s low corona and transition region..... 258 Table 9.3 Mean values of solar-wind parameters at the Earth’s orbit.................................. 272 Table 10.1 The ten brightest stars as seen from Earth............. 297 Table 10.2 Apparent visual magnitudes, mV, of some astronomical objects............................ 298 Table 10.3 The range in stellar luminosity..................... 299 Table 10.4 The spectral classification of stars.................. 305 Table 10.5 Some well-known large stars...................... 306 Table 10.6 The Morgan–Keenan (M–K) luminosity classes......... 321 Table 10.7 The main-sequence stars......................... 324 Table 10.8 Nuclear fusion processes in a supergiant star of 25 solar masses............................. 349 Table 11.1 Bright named emission nebulae.................... 359 Table 11.2 Intense spectral lines of emission nebulae............. 360 Table 11.3 Physical properties of emission nebulae (H II regions).... 361 Table 11.4 Physical properties of atomic hydrogen (H I) regions..... 378 Table 11.5 Abundant interstellar molecules.................... 380 Table 12.1 Physical properties of giant molecular clouds........... 388 Table 12.2 Stars with an excess of infrared radiation detected from the IRAS satellite.......................... 401 Table 13.1 Representative mass, radius, and mean mass density of the stars............................. 412 Table 13.2 Physical properties of planetary nebulae.............. 414 Table 13.3 Bright named planetary nebulae.................... 417 Table 13.4 Physical properties of white dwarf stars.............. 423 Tables xxi Table 13.5 Physical properties of some novae.................. 430 Table 13.6 Historical supernovae visible with the unaided eye....... 434 Table 13.7 Characteristics of supernova types.................. 436 Table 13.8 Supernova SN 1987A........................... 441 Table 13.9 Physical properties of the Crab Nebula supernova remnant............................. 447 Table 13.10 Physical properties of neutron stars.................. 451 Table 13.11 Physical properties of radio pulsars.................. 454 Table 13.12 Physical properties of binary x-ray pulsars............. 461 Table 14.1 Physical properties of the Milky Way disk............ 478 Table 14.2 Physical properties of the globular cluster spheroid...... 478 Table 14.3 Physical properties of galaxies..................... 492 Table 14.4 Physical properties of rich clusters of galaxies.......... 502 Table 15.1 Physical properties of the cosmic microwave background radiation............................ 531 Table 15.2 Cosmological parameters inferred from WMAP observations.................................. 532 Table 15.3 Crucial times during the expansion of the universe....... 540 Table 15.4 Intense emission lines found in Seyfert galaxies......... 546 Chapter 1 Observing the Universe 1.1 What Do Astronomers and Astrophysicists Do? Astronomy is an ongoing, cumulative science in which astronomers either discover previously unseen constituents of the observable universe or determine physical properties of known ones. They measure the mass, luminosity, distance, size, chemical composition, motion, and magnetic fields of planets, stars, galaxies, and their surroundings. Astrophysicists apply the laws of physics to celestial objects and events, thereby interpreting and explaining the astronomical observations. They assume that the physical laws that apply on Earth are valid throughout the Cosmos, but often under extreme conditions that cannot be achieved on our planet. The diverse aspects of physics used in astrophysics include radiation processes and universal gravitation, cosmic and particle motion, atomic and nuclear physics, and special and general relativity. Astronomers and astrophysicists together investigate how everything in the universe originates, changes, interacts, moves, and radiates. Theoretical studies, analytical models, and numerical simulations with computers are also employed to help understand these processes. Astronomy, and therefore astrophysics, is an instrument-driven science. Many of the seminal discoveries in astronomy have been accidental and unanticipated, often made when using unique telescopes, new technology, and novel detection equipment (Lang 2009). These instruments extend our vision to places that are not accessible to direct observation, enabling us to ‘‘see’’ the invisible and permitting us to look at the universe in new ways. Without a telescope, for example, the vast majority of stars cannot be seen, and all but a very few of the billions of galaxies and most of the expanding universe are invisible to the unaided eye. Observations provide the crucial data for our celestial science. Without them, astrophysicists would have nothing to describe. Fortunately, an astronomical object can be observed over and over again, in different ways, once it has been K. R. Lang, Essential Astrophysics, Undergraduate Lecture Notes in Physics, 1 DOI: 10.1007/978-3-642-35963-7_1, Ó Springer-Verlag Berlin Heidelberg 2013 2 1 Observing the Universe discovered. These observations require knowledge of our location on the Earth, the location of the object in the sky, and an understanding of both angular measure and passing time. 1.2 Our Place on Earth In order to observe cosmic objects with any accuracy, we must first establish our bearings here on Earth. In arguments used by Pythagoras (572-479 BC), and subsequently recorded by Aristotle (384-322 BC), it was shown that the Earth is a sphere. During a lunar eclipse, when the Moon’s motion carries it through the Earth’s shadow, observers at different locations invariably saw a curved shadow on the Moon. Only a spherical body can cast a round shape in all orientations. The curved surface of the ocean was also inferred by watching a ship disappear over the horizon; first the hull and then the mast disappear from view. So we can, to first approximation, assume the Earth is a sphere, and locate ourselves within a grid of great circles on it. A great circle divides the sphere in half; the name derives from the fact that no greater circles can be drawn on a sphere. A great circle halfway between the North and South Poles is called the Equator because it is equally distant between both poles. Circles of longitude are great circles that pass around the Earth from pole to pole, perpendicular to the Equator. Each circle of longitude intersects the equator in two points that are 180° apart. We halve the great circles of longitude into semicircles, called meridians. Long ago, in 1884, it was decided that the half-circle Fig. 1.1 Latitude and longitude Great circles through the North and South Poles of the Earth create circles of longitude. They are perpendicular to the Equator where they intersect it. The circle of longitude that passes through Greenwich England is called the Prime Meridian. The longitude of any point, P, is the angle lambda, k, measured westward along the Equator from the intersection of the Prime Meridian with the Equator to the equatorial intersection of the circle of longitude that passes through the point. The latitude is the angle phi, /, measured northward (positive) or southward (negative) along the circle of longitude from the Equator to the point. In this figure, the point P corresponds to San Francisco 1.2 Our Place on Earth 3 of longitude passing through the old Royal Observatory in Greenwich, England, would mark 0° longitude. It was designated as the ‘‘Prime Meridian’’, the starting point of counting longitudes. The longitude, denoted by the Greek letter k, of any point on the Earth’s surface is the angle measured westward from the intersection of the Prime Meridian with the Equator to the equatorial intersection of the circle of longitude that passes through the point (Fig. 1.1). The latitude, designated by the symbol /, is the angle measured northward (positive) or southward (negative) along a circle of longitude from the equator to the point. Sobel (1995) has discussed early determinations of terrestrial longitude, whereas Carter and Carter (2002) have provided a historical account of latitude variations. Alder (2002) has discussed early measurements of the size of the Earth, and the associated beginning of the metric system. Example: Location and rotation speed on the Earth The length of the day and the rotation period is the same for every place on Earth, but the speed of rotation around its axis depends on the surface location. The surface speed of rotation is greatest at the equator and reduces to lower values at higher latitudes. Using an equatorial radius of about 6,378 km, the Earth would have to be rotating at a speed of about 464 m s-1 to spin about its equatorial circumference once every 24 h. To calculate this speed, just multiply the equatorial radius by 2p to get the equatorial cir- cumference, and divide by 24 h, where there are 86,400 s per hour. The constant p = 3.1416. At higher latitudes, closer to the poles, the circum- ferential distance around the Earth, and perpendicular to a great circle of longitude, is less, so the speed is less. The speed diminishes to almost nothing at the geographic poles, which are pierced by the rotation axis. Every location on the Earth rotates about an axis that pierces the Earth and extends between its North and South Poles. The period of rotation, and the length of the day, is everywhere the same, but the rotation speed is fastest at the equator and systematically lower at higher latitudes. The geographic description of the location of an observatory includes its height, h, in meters above mean sea level. The geodetic coordinates of longitude, latitude and height, designated k, /, and h, are specified online in The Astronomical Almanac for all observatories engaged in professional programs of astronomical observations. The Global Positioning System (GPS) is now used to determine reliable loca- tion and time information. It is a system of about 30 navigation satellites devel- oped, maintained and operated by the U.S. Air Force for military and civilian purposes. Each satellite is constantly beaming radio signals that contain the exact time. These signals take a few milliseconds to travel from a satellite to the GPS receiver, and it has a built-in computer that calculates its precise position on Earth using signal time delays from four or more satellites. The time differences are 4 1 Observing the Universe converted into distance by multiplication with the speed of light, and these dis- tances are translated by triangulation into an exact position accurate to about 100 m. The numbers after the N (north) notation on the GPS receiver indicate its latitude, while the numbers after the W (west) notation indicates its longitude. The GPS devices used in automobiles specify the longitude and latitude of your start and end points, and map the route between them. The GPS was initially developed by the military and is still used by them. Soldiers can use a GPS device to find an enemy objective, even in the dark or unfamiliar territory; weapons systems can use them to track potential ground and air targets. All GPS receivers capable of functioning above 18 km in altitude and moving faster than 515 m s-1 are classified as weapons. 1.3 Location in the Sky You may have watched the stars as they rise at the horizon on one side of the Earth, slowly move overhead, and eventually set on the other side of the planet, only to reappear the next night. This slow coursing of stars was initially attributed to a revolving celestial sphere, which carried its embedded stars about a stationary Earth, but appearances can be deceiving. The Earth is instead spinning under an imaginary celestial sphere concentric with the Earth, on which the stars and other astronomical objects are placed. Such a celestial sphere explains why people located at different places on Earth invariably see just half of all the stellar sky. As the Earth rotates, day turns into night and these stars glide by. Astronomers define points and circles on the celestial sphere (Fig. 1.2). If you extend the Earth’s rotation axis in both directions, it intersects the celestial sphere at the north and south celestial poles. They are the pivotal points of the night sky’s apparent daily rotation. When the plane of the Earth’s Equator is extended outward in all directions, it cuts the celestial sphere in half, at the celestial equator. The point where the Sun crosses the celestial equator going northward in spring is called the Vernal Equinox. The Vernal Equinox is sometimes called the first point of Aries, and is given the symbol c. The projection of the plane of the Earth’s orbit onto the celestial sphere is known as the ecliptic, and the angular separation between the ecliptic and the celestial equator is called the obliquity of the ecliptic, designated e, which is about 23.5°. The obliquity is also the angle between the Earth’s rotational axis and a line perpendicular to its orbital plane. On the standard reference date of January 1.5, or at noon in January 1, in the year 2000.0, the slowly changing obliquity had the exact value of: e ¼ 23 260 21:40600 ; 1.3 Location in the Sky 5 North Celestial Pole Celestial Summer Sphere Solstice Autumnal Equinox Ecliptic Equator Dec.=δ Celestial Equator R.A.=α 23.5° East Vernal Equinox/ Celestial Winter First Point of Aries Sphere Solstice South Celestial Pole Fig. 1.2 Celestial coordinates Stars, galaxies, and other cosmic objects are placed on an imaginary celestial sphere. The celestial equator divides the sphere into northern and southern halves, and the ecliptic is the annual path of the Sun on the celestial sphere. The celestial equator intersects the ecliptic at the Vernal Equinox and the Autumnal Equinox. Every cosmic object has two celestial coordinates. They are the right ascension, designated by the angle alpha, a, or by R.A., and the declination, denoted by the angle delta, d, or Dec. Right ascension is measured eastward along the celestial equator from the Vernal Equinox to the foot of the great circle that passes through the object. Declination is the angular distance from the celestial equator to the object along the great circle that passes through the object, positive to the north and negative to the south. Precession results in a slow motion of the Vernal Equinox, producing a steady change in the celestial coordinates where the symbol ° denotes angular degrees, the 0 symbol designates minutes of arc or angle, and the symbol 00 denotes seconds of arc. Positions on the celestial sphere are defined by angles along great circles. By analogy with terrestrial longitude, right ascension, denoted a, is a celestial object’s longitude, but it is measured eastward along the celestial equator from the Vernal Equinox. The right ascension is expressed in hours and minutes of time, with 24 h in the complete circle of 360 degrees, denoted as 360°. For conversion, 1 h of time is equivalent to 15° of angle, or 1 h = 15°; 1 s of time is equal to 15 s of arc, or 1 s = 1500 ; and 1 min of arc equals 4 s of time, or 10 = 4 s. Just as latitude is a measure of a one’s distance from the Equator of the Earth, declination, denoted d, is a celestial object’s angular distance from the celestial equator. The declination is positive for objects located north of the celestial equator, and they are observed by inhabitants of the northern hemisphere of the Earth. Celestial objects in the southern sky have negative declinations, and people living in the southern half of our planet observe them. 6 1 Observing the Universe Example: What can you see in the night sky from where you are? An observer can see only half of the celestial sphere – the half above the local horizon. The celestial objects that can be seen depend on only two things, the observer’s latitude, denoted by /, and the object’s declination, designated d. At the Earth’s geographic North Pole, and latitude / = 90°, the north celestial pole is directly overhead and located near the star Polaris, and the horizon runs along the celestial equator. This observer can therefore only see the northern half of the celestial sphere, and objects with positive, northern declination; all the objects with negative, southern declinations are forever invisible from this location. For an observer located at different northern latitudes /, stars can be observed with declinations greater than / - 90°, or d [ / - 90°, for 0 B / B 90°; only these stars rise above the horizon. The southern half of the celestial sphere is fully visible from the geographic South Pole, while the northern sky is unseen. At the Equator where / = 0, the complete range of positive and negative declinations are visible. Of course, the observer has to wait for the rotating Earth to bring any potentially observable object above the horizon and into the observable half of the celestial sphere. For centuries, astronomers have used catalogues of right ascension, a, and declination, d, of celestial objects to locate them in the sky (Focus 1.1). Modern celestial positions of the highest accuracy are referred to the center of mass of all bodies in the solar system, known as the solar system barycenter, and specified within an International Celestial Reference System (Kaplan 2005). Focus 1.1 Astronomical catalogues The positions, brightness, spectra, angular size, and other data of different kinds for celestial objects are given in catalogues provided over centuries of meticulous observations by dedicated astronomers. Stars were the first celestial objects to be catalogued. Their accurate positions were compiled and used to discover such things as stellar motions, the planet Uranus, and the first asteroid. The positions and spectral classification of about 235,000 stars were compiled in the famous Henry Draper Catalogue, published between 1918 and 1924. The letters ‘‘HD’’ followed by the listed number in this catalogue often designates a star. Gliese (1969) and Gliese and Jahreiss (1979) have catalogued the closest stars other than the Sun. Planets are now being discovered around such nearby stars designated by ‘‘GJ’’ followed by their number in this catalogue – GJ 581, for example. The French astronomer Charles Messier (1730-1817) compiled one of the most famous catalogues (Messier 1781). His list of just over 100 bright non-stellar objects includes some of the most widely studied objects in the universe, now known as emission nebulae, galaxies, star clusters, and supernova remnants. The letter ‘‘M’’ followed by the number in the Messier 1.3 Location in the Sky 7 Catalogue indicates them. The Crab Nebula supernova remnant, M 1, is the first on the list, and M 31 is the closest spiral galaxy, also known as Andromeda. In astronomical parlance, a nebula is a diffuse, non-stellar object. Some extragalactic nebulae, which reside outside our Milky Way, were eventually designated as galaxies. They contain as many as 100 billion stars, as well as diffuse gaseous nebulae. An emission nebula consists of interstellar gas glowing from the ultraviolet light of a nearby luminous star. William Herschel (1738-1822) dramatically increased the number of known non-stellar objects to 2,500, during 20 years of systematically observing the heavens, from 1783 to 1802. This sweep of the sky’s northern hemisphere was extended to the southern hemisphere by William’s son, Sir John Herschel (1792-1871), who published data for 5,079 objects in his General Catalogue in 1864, the combined result of more than half a century of painstaking observations. Using the Herschel catalogue as a basis, J. L. E. Dreyer (1852-1926) published his New General Catalogue (NGC) of nebulae and star clusters, followed by two Index Catalogues, designated IC. Many galaxies, as well as emission nebulae and star clusters, are still known by their NGC and IC numbers. Later on, the photographic Palomar Sky Survey, using the wide-angle 1.2 m (48 in.) Schmidt telescope on Palomar Mountain, was used to cata- logue tens of thousands of galaxies; an observatory telescope is often des- ignated by a name and its diameter in meters or inches. In 1958, George Abell (1927-1983) used it to create a catalogue of 2,712 rich clusters of galaxies; the designation ‘‘A’’ followed by the number in his catalogue is still used today. Millions of galaxies were catalogued in the mid-twentieth cen- tury using photographs taken using large telescopes, but galaxy catalogues have now become computerized. An important example is the Sloan Digital Sky Survey, created from a dedicated, computer-driven 2.5 m (98 in.) telescope. Bright radio sources are designated by ‘‘3C’’ followed by the number in the Third Cambridge Catalogue of Radio Sources published in 1959, a famous example is 3C 273, the first quasar to be discovered. The most intense radio and x-ray sources have been named after the constellation they appear in. For example, Cygnus A is a bright radio galaxy, Cygnus X-1 is a bright x-ray source and a candidate black hole, and Centaurus X-3 is an x-ray pulsar. Specific types of objects, such as pulsars, supernova remnants, and white dwarf stars, have their own catalogues, and are often designated by letters like ‘‘PSR’’, ‘‘SNR’’, or ‘‘WD’’ followed by their celestial position. 8 1 Observing the Universe The international celestial reference frame is defined by a catalogue of exceedingly accurate positions for extragalactic radio sources observed with Very Long Baseline Interferometry. At optically visible wavelengths, the Tycho-2 catalogue of positions for more than 2.5 million stars, observed from the HIPPARCOS satellite, is used. Accurate positions of major solar system bodies are given as a function of time in an Ephemerides provided by the Jet Propulsion Laboratory. As the Sun moves along the ecliptic, it crosses the celestial equator twice, on its way north at the Vernal Equinox, on about March 20, and then at the Autumnal Equinox on about September 23. On either equinox, the Sun lies in the Earth’s equatorial plane, so the twilight zone that separates night and day then cuts the Earth in equal parts and the days and nights are equally long. The point at which the Sun is farthest north, is the Summer Solstice (on about June 21), and its most southerly point is the Winter Solstice on about December 22. The days in the northern hemisphere are the longest on the Summer Solstice, and shortest on the Winter Solstice. So the crossing of the Sun at the equinoxes and solstices mark the beginning of the seasons in the Earth’s northern hemisphere, and the location of these points on the celestial sphere are given in Table 1.1. Right ascension and declination provide celestial bearings in the equatorial coordinate system. Another celestial coordinate system is the horizon, or hori- zontal, coordinate system that employs great-circle angles measured with respect to the observer’s zenith and horizon. The zenith is located above your head, directly away from the center of the Earth. It is the point of intersection of the celestial sphere with the upward prolongation of the observer’s plumb line, whose bob is drawn to the terrestrial center by gravity. If a plane is extended outward from the observer’s feet, perpendicular to the plumb line, it intersects the celestial sphere in a great circle known as the horizon. Celestial objects are only visible if they are above the horizon. The altitude measures the angular distance from the horizon to the object in question, along a great circle that intersects the object and the zenith. The azimuth, an angle measured along the horizon, provides the second dimension to this coordinate system. Angles are used to designate celestial posi- tions in both types of celestial coordinate systems. Table 1.1 Celestial positions of the equinoxes and solstices Position Right ascension a (2000.0) Declination d (2000.0) h Vernal (Spring) equinox 0 0° Summer solstice 6h 23° 260 21.400 Autumnal (Fall) equinox 12 h 0° Winter solstice 24 h -23° 260 21.400 1.4 Measuring Angle and Size 9 1.4 Measuring Angle and Size Astronomers measure angles in degrees, designated by the superscript °, and there are 360° in a circle. They also use the second of arc, or arc second, denoted by the symbol 00 , and the minute or arc, or arc minute, abbreviated by 0 , as a units of angle. The units mimic a clock with 60 s in a minute, or 6000 = 10. A full degree of angle contains 60 min of arc, so 1° = 600 = 3,60000. Mathematicians use a different unit of angular measurement called a radian. The radian is the ratio between the length of an arc and its radius. The ratio of linear size to distance is expressed in radians, where an angle of one radian, when viewed from the center of a circle, results in an arc on that circle equal to the radius of the circle (Fig. 1.3). That is, the radian unit of angular measure is defined such that an angle of one radian subtended from the center of a unit circle produces an arc length of one. A full circle subtends 2p rad and 360°, where p = 3.141592654, so 1 rad = 360=(2p) = 57.2958°, and the conversion factors between seconds of arc and radians are: 100 ¼ 4:848 106 rad; and 00 1 rad ¼ 2:06265 105 ¼ 57:2958 : Fig. 1.3 The Sun’s angular size and radius The solar radius can be determined from the Sun’s angular size and distance. As long as this angle is small, the physical size is only a small arc of a large circle, denoted by the dashed line, and the angular size is the ratio of the physical size to the distance. Astronomers specify this angle as a partial arc of a full circle of 360°; for the Sun it is about 32 min of arc, in which there are 60 min of arc in 1°. This angle has been enlarged to display it in this illustration. In mathematics, the radian is the standard unit of angular measure. It describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. When the arc length is equal to the arc radius, the angle is 1 rad. We can convert between the two methods of describing angles by noting that the circumference of a circle is 2p times its radius; therefore 1 rad is equal to 360°=(2p), or 57.2958°. For the Sun, the angular size h ¼ 2R =D radians, where R denotes the Sun’s radius and the mean distance of the Sun, D , is 1 AU. The observed angular size of the Sun corresponds to a radius of 695.5 million meters 10 1 Observing the Universe A telescope can be used to measure the angular size, hsize, of a celestial source if the angular resolution of the telescope is smaller than the angular size of the source. If the source distance, D, is known, then one can infer its linear extent, L, perpendicular to the line of sight, or if spherical its radius, R, using the angular size and the relation: L 2R hsize ¼ ¼ rad: D D Example: Measuring the size of the Sun The linear radius of the Sun, denoted by R , can be determined from the Sun’s angular diameter, denoted by h , using 2R h ¼ rad D where the mean distance between the Earth and the Sun has a value of D = 1 AU = 1.496 9 1011 m. As illustrated in Fig. 1.3, this expression uses the mathematician’s radian unit of angular measurement, and it has to be converted to astronomical measurements of angle. The radian is the ratio between the length of an arc and its radius. The mean equatorial angular diameter of the Sun is h = 31.97 min of arc = 31.970 = 1,918.200 , and this is equal to 0.009299 rad, since 1 rad = 206265 s of arc. So the radius of the Sun is given by h D R ¼ ¼ 6:956 108 m; 2 which is 109 times the radius of the Earth. By the way, the angular diameter of the Sun is about the same angle as that subtended by the thumb when viewed at arm’s length. In one of those fascinating coincidences, the angular diameter of the Sun is also about the same as the angular diameter of the Moon, which is much closer to us and smaller in radius than the Sun. Because of this similarity in angular size, the Moon can pass in front of the Sun during a total solar eclipse, blocking out the sunlight. 1.5 The Locations of the Stars are Slowly Changing While most stars sweep by as the Earth rotates, a star that is aligned with our planet’s rotation axis, at the north celestial pole, seems to remain placed in an unchanging location at 90° north declination. The Earth’s northern rotation axis, for example, now points close to Polaris, also known as the North Star or the Pole 1.5 The Locations of the Stars are Slowly Changing 11 Star, which would lie approximately overhead when viewed from the Earth’s geographic North Pole. The latitude of any location in the Earth’s Northern Hemisphere is equal, within about 1°, to the angular altitude of Polaris. The uncertainty is due to the fact that Polaris is not exactly at the north celestial pole, where the north end of the Earth’s rotation axis pierces the night sky. We can locate Polaris by following the line joining the two stars farthest from the handle of the Big Dipper, which accounts for the phrase ‘‘follow the drinking gourd’’ used by southern slaves escaping to the northern parts of the United States. Mariners have also used the North Star for navigation, to find the direction of north and the latitude of their ship. Nevertheless, everything in the universe is in a state of perpetual change, and the locations of the so-called fixed stars on the celestial sphere are no exception. Their change in position is related to the Earth’s elongated shape (Focus 1.2), which has sent the Earth into a wobbling rotation that resembles a spinning top. This causes a very slow change of the celestial positions of the north celestial pole, the Pole Star and all the other stars, called precession. The changing positions of bright stars on the celestial sphere were first observed by Hipparchus, a Greek astronomer who lived in the second century BC (Hipparchus, 125 BC); the tele- scope was not invented until 17 centuries after Hipparchus established the stellar positions using his eyes. Focus 1.2 The elongated shape of the Earth The Earth isn’t precisely spherical in shape. It has a slight bulge around its equatorial middle and is flattened at its poles, with a shape more like an egg than a marble or billiard ball. This elongated, oblate shape is caused by the Earth’s rapid rotation. The outward force of rotation opposes the inward gravitational force, and this reduces the pull of gravity in the direction of spin. Since this effect is most pronounced at the equator, and least at the poles, the solid Earth adjusts into an oblate shape that is elongated along the equator. An ellipse of eccentricity, e, and major axis, ae, which is rotated about the polar axis, defines the Earth’s reference ellipsoid at sea level. The planet’s equatorial radius is ae, and its polar radius, ap. They are given by: ap ¼ ae ð1 f Þ ¼ ae ð1 e2 Þ1=2 ; where the flattening factor f = (ae - ap)=ae is related to the eccentricity, e, by e2 = 2f - f 2. The mean surface radius of the Earth, hai, is given by: 1 hai ¼ a2e ap 3 6:371 106 m; which is the radius of a sphere of volume equal to the Earth ellipsoid. Geophysicists use another definition of mean radius given by (2ae ? ap)=3. 12 1 Observing the Universe The radius, r, of the surface of the Earth geoid at any latitude / is given by r ¼ ae ð1 f sin2 /Þ: Two of the primary constants of the International Astronomical Union are (Kaplan 2005): Equatorial radius of the Earth ¼ ae ¼ 6:3781366 106 m; and Flattening factor for Earth ¼ f ¼ 0:0033528197 ¼ 1=298:25642: These values of ae and f give a polar radius for the Earth of ap = 6.356752 9 106 m, and the difference between the equatorial and polar radius is 21,385 m or about 21 km. The world geodetic system, which is the basis of terrestrial locations obtained from the Global Positioning System, or GPS for short, uses an Earth ellipsoid with ae = 6.378137 9 106 m and f = 1=298.257223563. The changing locations of celestial objects are caused by the gravitational action of the Moon, Sun and planets on the spinning, oblate Earth. As a result of this gravitational torque, the Earth’s rotation axis is constantly changing with respect to a space-fixed reference system. The precessional motion of the Earth’s rotation axis is caused by the tidal action of the Moon and Sun on the spinning Earth. That is, because the Moon and the Sun lie in the ecliptic plane, which is inclined by 23.5° to the plane of the Earth’s Equator, they exert a gravitational force on the Earth’s equatorial bulge. This causes the rotation axis to sweep out a cone in space, centered at the axis of the Earth’s orbital motion and completing one circuit in about 26,000 years (Fig. 1.4). So the Earth is not placed firmly in space; instead it wobbles about causing the identity of the Pole Star to gradually change over time scales of thousands of years. The northern projection of the Earth’s rotation axis is currently within about 0.75° of Polaris and will move slowly toward it in the next century. After that, the north celestial pole will move away from Polaris and, in about 12,000 years, the Earth’s rotation axis will point to within 5° of the bright star Vega. The slow conical motion of precession carries the Earth’s Equator with it; as that Equator moves, the two intersections between the celestial equator and the Sun’s path, or ecliptic, move westward against the background stars. One of these intersections is the Vernal, or Spring, Equinox, from which right ascension is measured. This equinox point moves forward (westward) along the ecliptic at the rate of about 50 s of arc, denoted 5000 , per year, which is equivalent to 3.33 s per year. As the Earth’s rotational axis precesses, declinations also change, through a range of 47°, or twice 23.5°, over 26,000 years. 1.5 The Locations of the Stars are Slowly Changing 13 To Polaris/ 26,000 Year To Vega North Pole Star Precession 23.5° Equator 23.5° Fig. 1.4 Precession The Earth’s rotation axis traces out a circle on the sky once every 26,000 years, sweeping out a cone with an angular radius of about 23.5°. The Greek astronomer Hipparchus (c. 146 BC) discovered this precession in the second century BC. The north celestial pole, which marks the intersection of this rotation axis with the northern half of the celestial sphere, now lies near the bright star Polaris. However, as the result of precession, the rotational axis will point toward another bright north star, Vega, in roughly 13,000 years. This motion of the Earth’s rotational axis also causes a slow change in the celestial coordinates of any cosmic object Because the long period, 26,000 year, conical motion of the Earth’s rotation axis is caused by the gravitational action of the Moon and Sun, it is called lunisolar precession. Simon Newcomb (1835-1909) derived the detailed theory for com- puting the corrections to astronomical coordinates for this precession (Newcomb 1895). In addition to this steady, progressive motion, there are small, periodic varia- tions in both precessional speed and axial tilt caused by the gravitational action of the planets on the Earth’s equatorial bulge. The most important term in this nutation, first observed by James Bradley (1693-1762), induces an 18.6 year periodic wobble in the precessional motion with a size of 1700 in the direction of precession and 900 perpendicular to it (Bradley 1748). Because of positional changes caused by precession and nutation, the equinox, or reference date, must be given when specifying the right ascension or declination of any cosmic object. The standard epoch that is now in used for celestial positions is: J2000:0 ¼ 2000 January 1:5 ¼ JD2451545:0; where JD denotes Julian Date and the prefix J denotes the current system of measuring time in Julian centuries of exactly 36,525 days in 100 years, with each day having a duration of 86,400 s. The combination of lunisolar and planetary precession is called general pre- cession, and the astronomical constants at standard epoch J2000.0 include (Kaplan 2005): General precession in longitude ¼ q ¼ 5,028.79619500 per Julian century: 14 1 Observing the Universe The nutation term, N, for that epoch is Constant of nutation ¼ N ¼ 9:205233100 : Kaplan (2005) provides modern formulas for precession and nutation. They describe the transformation of celestial coordinates from one date to another, as a function of time since a reference epoch. The astronomical constants also include an aberration constant, j, which accounts for the observed position shift of an astronomical object in the direction of the Earth’s motion (Focus 1.3). The aberration constant at the standard epoch J 2000.0 is Constant of aberration ¼ j ¼ 20:4955200 : Focus 1.3 Stellar aberration As the Earth orbits the Sun, the stars all appear to be shifted in the direction of motion, a phenomenon called stellar aberration, described by James Bradley (1693-1767) in 1728. Because the speed of light is finite, the apparent direction of a celestial object detected by a moving observer is not the same as its geometric direction at the time. For stars, the normal practice is to ignore the correction for the motion of the celestial object, and to compute the stellar aberration due to the motion of the observer. This also gave Bradley a means to improve on the accuracy of previous estimates for the speed of light (Bradley 1728). The magnitude Dh of stellar aberration depends on the ratio of the velocity of the observer, V, to the speed of light, c, and the angle, h, between the direction of observation and the direction of motion. The displacement, Dh, in the sense of apparent minus mean place, is given by 3 V 1 V 2 V Dh sinh sin 2h þ sin h cos2 h 0:33 sin3 h þ ::: c 2 c c As the Earth orbits the Sun, it is moving at a velocity of approximately 30 km s-1, and the speed of light c & 300,000 km s-1, so the term of order V=c is 10-4 rad or 20 s of arc, denoted 2000 , and the term (V=c)2 has a maximum value of 0.001 s of arc. Bradley (1728) used aberration obser- vations to determine the speed of light as approximately 183,000 miles per second or 294,500 km s-1. The constant of aberration, j, at standard epoch J2000.0 is given by 2pa 1 V Constant of aberration ¼ j ¼ ð1 e2 Þ2 ¼ ¼ 0:9365 104 rad, Pc c 1.5 The Locations of the Stars are Slowly Changing 15 which is equivalent to: j ¼ 20:4955200 Here the constant p & 3.14592654, the mean distance between the Earth and the Sun is a = 1 AU = 1.49598 9 1011 m, known as the astronomical unit, e = 0.01671 is the mean eccentricity of the Earth’s orbit, P = 3.1558 9 107 s is the length of the sidereal year, c = 2.997925 9 108 m s-1 is the speed of light, and one radian = 2.062648 9 105 00. When the observer is moving directly at the star, h is zero and there is no aberration shift at all. The shift achieves its greatest value of about 20.500 when the observer’s motion is perpendicular to the direction of the star, with h = 90°. 1.6 What Time is It? There are two ways of keeping time in common use today. One is atomic time, the basis of the Système International, abbreviated SI, second, and the other is based on the rotation of the Earth (Kaplan 2005; Seidelmann 2005). The SI second is the fundamental unit of atomic time, which is specified by atomic clocks that use cesium-beam and other atomic frequency standards to an accuracy of 1.5 9 10-14, or to the fourteenth decimal place (Essen 1969). The frequency standards form a standard timescale known as International Atomic Time, abbreviated TAI for Temps Atomique International. The time distributed by the Global Positioning System, or GPS, remains at a constant offset with International Atomic Time. On 21 November 2010, TAI-GPS = 19 s. The clocks we use in daily life are set to the Earth’s rotation with respect to the Sun. It establishes our daily rhythm, from sunrise to sunset and back to sunrise again. The 24 h solar day is the time it takes for the Sun to make one circuit around the local sky. By definition: 1 solar day ¼ 24 h ¼ 1,440 min ¼ 86,400 s: This is known as the unit of Sun time, or solar time. Also by definition, the Julian century has 36,525 solar days and one day is defined as 86,400 s of International Atomic Time, or TAI. So: 1 Julian year ¼ 365.25 solar days ¼ 8,766 h ¼ 525,960 min ¼ 31,557,600 s Solar time is the basis of Universal Time, abbreviated UT, which has been defined to be as uniform as possible despite variation in the Earth’s rotation. The worldwide system of Civil Time, and the clock on your wall or the watch on your 16 1 Observing the Universe arm, are synchronized to Coordinated Universal Time, denoted UTC, which has been corrected for small variations in the Earth’s rotation, but the atomic clocks in satellites and the time in the Global Positioning System, abbreviated GPS, are not corrected in this way. Because of irregularities in the Earth’s rotation and the lengthening of its rotation period due to Moon-induced tidal friction, this Sun time does not advance at a uniform rate, and it increasingly lags behind the SI-second time scale. UTC is therefore a hybrid time scale using the SI second on the spinning Earth as a fundamental unit, but subject to occasional 1 s adjustments. The difference between atomic time and universal time, or TAL - UTC is an integral number of seconds, which increases by 1 whenever a leap second is introduced into UTC, so the two kinds of clock share the same seconds tick. When necessary, the adjust- ments to UTC are introduced at the end of June or December, by international agreement. Universal Time is equivalent to the standard Civil Time for 0° longitude, which is defined to be the Prime Meridian at Greenwich, England. By specifying the longitude of any other location of the terrestrial globe, we can exactly infer the local Sun time. For example, the Sun will be overhead at noon in Boston about 4 h 44 min later than noon in Greenwich; because the longitude of Boston is 71.06° = 4.733 h, where 360° equals 24 h. And in the same way, noon in New York City will occur about 10 min later than in Boston, because New York City is slightly west of Boston. The world has been divided into standard time zones based on about 1 h, or 15°, increments in longitude, so our watches differ from others in hourly increments and are slightly out of synchronism with the Sun. Standard time is the result of synchronizing clocks in different geological locations within a time zone. Blaise (2000) has described the creation of standard time. Terrestrial Time (TT) is the modern time standard used for time measurements of astronomical observations from the surface of the Earth. The unit of TT is the SI second. The Astronomical Almanac uses TT in the tables of positions, or Ephe- merides, of the Sun, Moon and planets as seen from the Earth. TT is slightly ahead of atomic time, and can be approximated by TT TAI þ 32:184 s; which is equivalent to TT GPS þ 51:184 s With the advent of atomic clocks and the exact targeting of planetary space- craft, measurements of time were further refined with the introduction of the Barycentric Dynamical Time, or TDB for short. It was adopted to take into account the relativistic time dilation (Sect. 2.3) when calculating orbits and astronomical ephemeris, and it applies to the solar-system-barycentric reference frame. The barycenter is the center of mass of two or more orbiting bodies. The 1.6 What Time is It? 17 difference between TDB and yet another Barycentric Coordinate Time, or TCB, is about 16.6 s, and Mean ratio of the TCB second to the TDB second ¼ 1 LB ; where LB = 1.550519767 9 10-8, an exceedingly small number. In every day life we use Sun time, based on the solar day that is exactly 24 h long, in solar time. This solar day is the interval between two successive passages of the Sun across an observer’s meridian. Astronomers also use star time, or sidereal time. A sidereal day is the time between successive passages of a star across the local meridian, and this star day is about 4 min less than a solar day. 1.7 Telling Time by the Stars Astronomers use another sort of time, called sidereal time, to know when and how to point their telescopes to view a particular star or any other cosmic object. The term sidereal is derived from the Latin sidus meaning ‘‘star.’’ As with solar time, this star time is based on the Earth’s rate of rotation, but measured relative to the fixed stars rather than the Sun. The sidereal day is the time it takes for a star – or any other celestial objects – to proceed from its highest point in the sky one day to its highest point the next day. The Earth makes one rotation about its axis in a sidereal day, but during that time it moves a short distance along its orbit around the Sun. At the end of a sidereal day, the Earth therefore needs to rotate a little more before the Sun reaches its highest point (Fig. 1.5). A solar day is therefore about 4 min longer than the sidereal day. To be exact: 1 sidereal day ¼ 23 h 56 m 4:09 s ¼ 23:93447 h ¼ 86,164:1 s, where the hours, minutes and seconds are in solar time. Observatories have two kinds of clocks that tell either the local solar time or the local star time. The two kinds of time can also be determined using clocks or time simulators on the web. At any moment, the Local Sidereal Time equals the right ascension, designated a, of a celestial object on the local meridian, so this time tells an observer when a given celestial object with a particular right ascension can be seen, provided its declination is in the observable range for the observer’s latitude. Example: When is a celestial object visible in your part of the sky? For an object with an observable declination, the time at which it can be observed depends upon the object’s right ascension, a, and the local sidereal time, abbreviated LST. The object crosses the local meridian when a = LST, and may be visible for several hours before and after this time. However, a terrestrial clock or watch is geared to the Sun rather than the 18 1 Observing the Universe stars, so it does not keep local sidereal time. A terrestrial clock is equal to the local sidereal time only at midnight of the Autumnal Equinox, about Sep- tember 23, and thereafter the local sidereal time gains 2 h on the terrestrial clock for each succeeding month. The geographic longitude, denoted k, of the observer relates the Local Sidereal Time (LST) to the Greenwich Sidereal Time, or GST for short, at the Prime Meridian by LST ¼ GSTkðobserverÞ; where the longitude is measured positive westward and can be converted into time using 24 h = 360°. The local meridian is an imaginary half circle stretching from the horizon due north, through the zenith, to the horizon due south. The zenith is directly overhead, an extension of a plumb line from the center of the Earth Fig. 1.5 Sun time and star time The Sun reaches its highest point in the daytime sky, its culmination, at noon, and this happens every 24 h in a solar day. A distant star returns to its highest point in the night sky every sidereal day of 23 h 56 min 04 s, which is the unit of star time. The Earth rotates once around its axis in one sidereal day, but during that time the Earth has moved along its orbit around the Sun (bottom). The star (small circle, top) and the Sun (below star) are at culmination, crossing the local meridian, at just one time (left). After a sidereal day, the Earth has to rotate for another 3 min 56 s before the Sun reaches its highest point (right). A solar day is therefore nearly 4 min longer than a sidereal day 1.7 Telling Time by the Stars 19 through the observer to the celestial sphere. The Prime Meridian is the local meridian for the old Royal Observatory in Greenwich, England. We define the Local Hour Angle (LHA) of any object on the celestial sphere to be the time since it last crossed the meridian, or LHA ðobjectÞ ¼ LST a ðobjectÞ ¼ GST kðobserverÞ aðobjectÞ: The Local Sidereal Time is equal to zero when the Vernal, or Spring, Equinox is on the local meridian. The solar and star times are related by: One mean solar day = 24 h 03 min 56.555 s of mean sidereal time One mean solar day = 1.0027379 mean sidereal days, One mean sidereal day = 23 h 56 min 04.09 s of mean solar time One mean sidereal day = 0.99726957 mean solar days. 1.8 Optical Telescopes Observe Visible Light Telescopes collect and magnify electromagnetic radiation from a cosmic object, and bigger telescopes provide two advantages. They gather more radiation than a smaller telescope, permitting the detection of fainter objects and providing a brighter image of any cosmic object for analysis. Big telescopes also provide greater angular resolution, which is the ability to see the separation between objects that are close together. Better resolution permits observation of finer detail on the object emitting the radiation. Kitchin (2013) provides a thorough discussion of telescopes and the techniques of using them to observe the cosmos. Regardless of what cosmic object a telescope is pointed at, the object’s radi- ation that carries information to the Earth travels in rays that are parallel to one another. A telescope’s lenses and/or mirrors are used to focus and collect visible radiation, or light. They are described by the science of optics; therefore the study of visible light from cosmic objects is called optical astronomy. There are two types of optical telescopes, the refractor and the reflector, which respectively use a lens and a mirror to gather and focus optically visible light (Fig. 1.6). A tele- scope’s lens bends the incoming rays by refraction, focusing them to a point where they meet, called the focal point. A curved mirror reflects the incoming rays, sending them to the focal point. In a refractor, light is bent by refraction at the curved surface of a lens, called an objective, toward a focal point where the different rays of light meet. If we place a detector at the focal point, in the plane parallel to the lens, we can record an image of whatever the telescope is observing. The distance from the lens to the focal point is called the focal length, which determines the overall size of the image. The critical thing is the diameter, or aperture, of the light-gathering lens. 20 1 Observing the Universe Sun Sun Prime Focus Aperture Cassegrain Objective Lens Secondary Newtonian Focus Newtonian Secondary Focal Length To Coudé Focus Coudé

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