Author(s): Neil Degrasse Tyson
Here is the essential companion to Welcome to the Universe, a New York Times bestseller that was inspired by the enormously popular introductory astronomy course for non-science majors that Neil deGrasse Tyson, Michael A. Strauss, and J. Richard Gott taught together at Princeton. This problem book features more than one hundred problems and exercises used in the original course--ideal for anyone who wants to deepen their understanding of the original material and to learn to think like an astrophysicist. Whether you're a student or teacher, citizen scientist or science enthusiast, your guided tour of the cosmos just got even more hands-on with Welcome to the Universe: The Problem Book. * The essential companion book to the acclaimed bestseller* Features the problems used in the original introductory astronomy course for non-science majors at Princeton University* Organized according to the structure of Welcome to the Universe, empowering readers to explore real astrophysical problems that are conceptually introduced in each chapter* Problems are designed to stimulate physical insight into the frontier of astrophysics* Problems develop quantitative skills, yet use math no more advanced than high school algebra* Problems are often multipart, building critical thinking and quantitative skills and developing readers' insight into what astrophysicists do* Ideal for course use--either in tandem with Welcome to the Universe or as a supplement to courses using standard astronomy textbooks--or self-study* Tested in the classroom over numerous semesters for more than a decade* Prefaced with a review of relevant concepts and equations* Full solutions and explanations are provided, allowing students and other readers to check their own understanding
Neil deGrasse Tyson is director of the Hayden Planetarium at the American Museum of Natural History. He is the author of many books, including Space Chronicles: Facing the Ultimate Frontier, and the host of the Emmy-winning documentary Cosmos: A Spacetime Odyssey. Michael A. Strauss is professor of astrophysics at Princeton University. J. Richard Gott is professor emeritus of astrophysics at Princeton University. His other books include The Cosmic Web: Mysterious Architecture of the Universe (Princeton).
Preface xvii Math Tips xxi PART I. STARS, PLANETS, AND LIFE 1 1 | THE SIZE AND SCALE OF THE UNIVERSE 3 1 Scientific notation review 3 Writing numbers in scientific notation. 2 How long is a year? 3 Calculating the number of seconds in a year. 3 How fast does light travel? 3 Calculating the number of kilometers in a light-year. 4 Arcseconds in a radian 3 Calculating the number of arcseconds in a radian, a number used whenever applying the small-angle formula. 5 How far is a parsec? 3 Converting from parsecs to light-years and astronomical units. 6 Looking out in space and back in time 4 Exploring the relationship between distance and time when traveling at the speed of light. 7 Looking at Neptune 4 The time for light to travel from Earth to the planet Neptune depends on where it and we are in our respective orbits. 8 Far, far away; long, long ago 5 There is an intrinsic time delay in communicating with spacecraft elsewhere in the solar system or elsewhere in the Milky Way galaxy. 9 Interstellar travel 6 Calculating how long it takes to travel various distances at various speeds. 10 Traveling to the stars 6 Calculating how long it would take to travel to the nearest stars. 11 Earth's atmosphere 7 Calculating the mass of the air in Earth's atmosphere, and comparing it with the mass of the oceans. 2 | FROM THE DAY AND NIGHT SKY TO PLANETARY ORBITS 8 12 Movements of the Sun, Moon, and stars 8 Exploring when and where one can see various celestial bodies. 13 Looking at the Moon 8 There is a lot you can infer by just looking at the Moon! 14 Rising and setting 9 Questions about when various celestial bodies rise and set. 15 Objects in the sky 9 More questions about what you can learn by looking at objects in the sky. 16 Aristarchus and the Moon 10 Determining the relative distance to the Moon and the Sun using high-school geometry. 17 The distance to Mars 11 Using parallax to determine how far away Mars is. 18 The distance to the Moon 11 Using parallax to determine how far away the Moon is. 19 Masses and densities in the solar system 11 Calculating the density of the Sun and of the solar system. 3 | NEWTON'S LAWS 13 20 Forces on a book 13 Using Newton's laws to understand the forces on a book resting on a table. 21 Going ballistic 13 Calculating the speed of a satellite in low Earth orbit. 22 Escaping Earth's gravity? 14 Calculating the distance at which the gravitational force from Earth and the Moon are equal. 23 Geosynchronous orbits 14 Calculating the radius of the orbit around Earth that is synchronized with Earth's rotation. 24 Centripetal acceleration and kinetic energy in Earth orbit 14 Calculating the damage done by a collision with space debris. 25 Centripetal acceleration of the Moon and the law of universal gravitation 15 Comparing the acceleration of the Moon in its orbit to that of a dropped apple at Earth's surface. 26 Kepler at Jupiter 16 Applying Kepler's laws to the orbits of Jupiter's moons. 27 Neptune and Pluto 17 Calculating the relationship of the orbits of Neptune and Pluto. 28 Is there an asteroid with our name on it? 17 How to deflect an asteroid that is on a collision course with Earth. 29 Halley's comet and the limits of Kepler's third law 18 Applying Kepler's third law to the orbit of Halley's comet. 30 You cannot touch without being touched 19 The motion of the Sun due to the gravitational pull of Jupiter. 31 Aristotle and Copernicus 19 An essay about ancient and modern views of the heavens. 4-6 | HOW STARS RADIATE ENERGY 20 32 Distant supernovae 20 Using the inverse square law relating brightness and luminosity. 33 Spacecraft solar power 20 Calculating how much power solar panels on a spacecraft can generate. 34 You glow! 21 Calculating how much blackbody radiation our bodies give off. 35 Tiny angles 21 Understanding the relationship between motions in space and in the plane of the sky. 36 Thinking about parallax 22 How nearby stars appear to move in the sky relative to more distant stars, due to the Earth's motion around the Sun. 37 Really small angles and distant stars 22 The Gaia spacecraft's ability to measure parallax of distant stars. 38 Brightness, distance, and luminosity 23 Exploring the relationship between brightness and luminosity of various stars. 39 Comparing stars 23 Relating the luminosity, radius, surface temperature, and distance of stars. 40 Hot and radiant 24 Exploring the relation between the properties of stars radiating as blackbodies. 41 A white dwarf star 24 Calculating the distance and size of a white dwarf star. 42 Orbiting a white dwarf 24 Using Kepler's third law to determine the orbit around a white dwarf star. 43 Hydrogen absorbs 25 Using the spectrum of an F star to understand the energy levels of a hydrogen atom. A challenge problem. 7-8 | THE LIVES AND DEATHS OF STARS 27 44 The shining Sun 27 Calculating the rate at which hydrogen fuses to helium in the core of the Sun. 45 Thermonuclear fusion and the Heisenberg uncertainty principle 27 Using quantum mechanics to determine the conditions under which thermonuclear fusion can take place in the core of a star. A challenge problem. 46 Properties of white dwarfs 29 Using direct observations of a white dwarf to determine its radius and density. A challenge problem. 47 Squeezing into a white dwarf 30 Determining how far apart the nuclei in a white dwarf star are. 48 Flashing in the night 30 Determining whether the gravity of a pulsar is adequate to hold it together as it spins. 49 Life on a neutron star 31 Calculating the effects of the extreme gravity of a neutron star. 50 Distance to a supernova 31 Watching a supernova remnant expand, and using this to determine how far away it is. 51 Supernovae are energetic! 32 Putting the luminosity of a supernova in context. 52 Supernovae are dangerous! 33 What would happen if a supernova were to explode within a few hundred light-years of Earth? 53 Neutrinos coursing through us 33 Calculating the flux and detectability of neutrinos emitted during a supernova explosion. 54 A really big explosion 34 Calculating the energy associated with a gamma-ray burst. 55 Kaboom! 36 Calculating the properties of one of the most powerful gamma-ray bursts ever seen. 56 Compact star 36 Calculating the distance between nuclei in a neutron star. 57 Orbiting a neutron star 37 Applying Kepler's third law for an orbit around a neutron star. 58 The Hertzsprung-Russell diagram 37 An essay about the relationship between surface temperature and luminosity of stars. 9 | WHY PLUTO IS NOT A PLANET 38 59 A rival to Pluto? 38 Calculating the properties of a large Kuiper Belt Object in the outer solar system, working directly from observations. A challenge problem. 60 Another Pluto rival 41 Exploring the properties of another large body in the outer solar system. 61 Effects of a planet on its parent star 43 Using observations of the motion of a star under the gravitational influence of an orbiting planet to infer the properties of that planet. 62 Catastrophic asteroid impacts 44 How the impact of an asteroid on the early Earth may have evaporated the oceans. 63 Tearing up planets 45 Calculating the tidal force of a planet on an orbiting moon. A challenge problem. 10 | THE SEARCH FOR LIFE IN THE GALAXY 47 64 Planetary orbits and temperatures 47 Calculating the orbits and equilibrium temperatures of planets orbiting other stars. 65 Water on other planets? 47 Determining whether liquid water can exist on the surface of planets orbiting other stars. 66 Oceans in the solar system 48 Exploring the properties of oceans on Earth, Mars and Europa. 67 Could photosynthetic life survive in Europa's ocean? 49 Determining how much life the sunlight that impinges on Europa could support. 68 An essay on liquid water 50 An essay describing the conditions under which liquid water, necessary for life as we know it, exists. PART II. GALAXIES 51 11-13 | THE MILKY WAY AND THE UNIVERSE OF GALAXIES 53 69 How many stars are there? 53 Calculating the number of stars in the observable universe. 70 The distance between stars 54 Putting the distance between stars into perspective. 71 The emptiness of space 54 Calculating the density of the Milky Way and of the universe as a whole. 72 Squeezing the Milky Way 54 What would happen if you brought all the stars in the Milky Way into one big ball? 73 A star is born 55 How much interstellar gas do you need to bring together to make a star? 74 A massive black hole in the center of the Milky Way 55 Calculating the mass of the black hole at the center of our Galaxy, working directly from observations. 75 Supernovae and the Galaxy 55 How many supernovae are needed to create the heavy elements in the Milky Way? 76 Dark matter halos 56 Calculating the mass of the Milky Way from its observed rotation. 77 Orbiting Galaxy 57 The orbit of the Large Magellanic Cloud around the Milky Way, and what it says about the mass of our Galaxy. 78 Detecting dark matter 57 Calculating how many dark matter particles there are all around us, and how we plan to detect them. A challenge problem. 79 Rotating galaxies 60 Determining whether we can see the rotation of a galaxy on the sky. 80 Measuring the distance to a rotating galaxy 60 Using the apparent motion of stars in a galaxy in the plane of the sky and along the line of sight to determine its distance. 14 | THE EXPANSION OF THE UNIVERSE 61 81 The Hubble Constant 61 Measuring the expansion rate of the universe from the measured properties of galaxies. 82 Which expands faster: The universe or the Atlantic Ocean? 63 The answer may surprise you. 83 The third dimension in astronomy 63 An essay about how we measure distances in the universe. 84 Will the universe expand forever? 63 The relationship between the density of the universe and its future fate. A challenge problem. 85 The motion of the Local Group through space 64 Calculating the gravitational pull from the Virgo galaxy supercluster on our Local Group of galaxies. A challenge problem. 15-16 | THE EARLY UNIVERSE AND QUASARS 67 86 Neutrinos in the early universe 67 Calculating just how numerous the neutrinos produced soon after the Big Bang are. 87 No center to the universe 68 A brief essay explaining why the expanding universe has no center. 88 Luminous quasars 68 Calculating the properties of quasars, and the supermassive black holes that power them. 89 The origin of the elements 68 An essay describing how different elements are formed in the universe. PART III. EINSTEIN AND THE UNIVERSE 69 17-18 | EINSTEIN'S ROAD TO SPECIAL RELATIVITY 71 90 Lorentz factor 71 Exploring the special relativistic relation between lengths as seen in different reference frames. 91 Speedy muons 72 How special relativity is important in understanding the formation and detection of muons created in the upper atmosphere. 92 Energetic cosmic rays 73 Determining relativistic effects for one of the highest-energy particles ever seen. 93 The Titanic is moving 73 Playing relativistic games with the great ship Titanic. 94 Aging astronaut 74 Understanding how the relativistic effects of moving an astronaut at close to the speed of light. 95 Reunions 74 How two friends can differ on the passage of time. 96 Traveling to another star 74 Calculating how time ticks slower for an astronaut traveling at close to the speed of light. 97 Clocks on Earth are slow 74 Calculating the difference between a clock in orbit around the Sun and one standing still. 98 Antimatter! 74 Should you run if trucks made of matter and antimatter collide with one another? 99 Energy in a glass of water 75 Calculating how much energy could be extracted from the fusion of the hydrogen in a glass of water. 100 Motion through spacetime 75 Drawing the path of the Earth's orbit around the Sun in spacetime. 101 Can you go faster than the speed of light? 75 Why the postulates of special relativity do not allow travel faster than the speed of light. 102 Short questions in special relativity 76 Quick questions which can be answered in a few sentences. 19 | EINSTEIN'S GENERAL THEORY OF RELATIVITY 77 103 Tin Can Land 77 Exploring the nature of geodesics on a familiar two-dimensional surface. 104 Negative mass 78 Would a dropped ball of negative mass fall down? 105 Aging in orbit 79 Exploring special and general relativistic effects on your clock while in orbit. A challenge problem. 106 Short questions in general relativity 80 Quick questions that can be answered in a few sentences. 20 | BLACK HOLES 82 107 A black hole at the center of the Milky Way 82 Calculating the properties of the supermassive black hole at the center of our Galaxy. 108 Quick questions about black holes 82 Short questions that can be answered in a few sentences. 109 Big black holes 83 Exploring the properties of the biggest black holes in the universe. 110 A Hitchhiker's challenge 83 The Hitchhiker's Guide to the Galaxy inspires a problem on black holes. A challenge problem. 111 Colliding black holes! 84 Measurements of gravitational waves from a pair of merging black holes allows us to determine their properties. 112 Extracting energy from a pair of black holes 85 Using ideas from Stephen Hawking to determine how much energy can be released when black holes collide. 21 | COSMIC STRINGS, WORMHOLES, AND TIME TRAVEL 87 113 Quick questions about time travel 87 Short questions that can be answered in a few sentences. 114 Time travel tennis 87 Playing a tennis game with yourself with the help of time travel. A challenge problem. 115 Science fiction 89 Writing a science fiction story that uses concepts from astrophysics: the challenge is to make it as scientifically realistic as possible. 22 | THE SHAPE OF THE UNIVERSE AND THE BIG BANG 91 116 Mapping the universe 91 Ranking the distance of various astronomical objects from the Earth. 117 Gnomonic projections 91 Exploring the geometry of an unusual mapping of the night sky onto a flat piece of paper. 118 Doctor Who in Flatland 95 Using concepts from general relativity to understand the nature of Dr. Who's Tardis. 119 Quick questions about the shape of the universe 96 Short questions that can be answered in a few sentences. 23 | INFLATION AND RECENT DEVELOPMENTS IN COSMOLOGY 97 120 The earliest possible time 97 Calculating it using both general relativity and quantum mechanics. 121 The worst approximation in all of physics 98 Can the Planck density give us a reasonable estimate for the density of dark energy? Hint: no. 122 Not a blunder after all? 99 Describing the relationship between Einstein's desire for a static universe and the accelerated expansion we now observe. 123 The Big Bang 99 An essay describing the empirical evidence that the universe started in a Big Bang. 24 | OUR FUTURE IN THE UNIVERSE 100 124 Getting to Mars 100 Calculating the most efficient orbit to get from Earth to Mars. 125 Interstellar travel: Solar sails 101 Using the pressure of light from the Sun to propel a spacecraft for interstellar travel. 126 Copernican arguments 102 Applied to time. 127 Copernicus in action 102 An essay about Copernican arguments in our understanding of the structure of the universe and our place in it. 128 Quick questions for our future in the universe 103 Short questions that can be answered in a few sentences. 129 Directed panspermia 103 Exploring how humankind could colonize the Milky Way with robotic probes. Useful Numbers and Equations 107 Solutions 113