1 ASTR 324 Longitude, Time, and Navigation Lecture Notes Tom Bensky Physics Department Cal Poly, San Luis Obispo San Luis Obispo, CA 93407 [email protected] Spring 2008 edition 2 Cover Figure If you were a ship’s captain in the 1700s, the scene on the cover would be a big problem for you. Just the sky and the sea, with no sight of land. In fact, a scene like this would be your biggest problem, and is the topic of this book: WHERE ARE YOU? Acknowledgements Special thanks to M. Moelter for his careful and professional proofreading of an early draft of this work (I love that green pen!). Thanks also to D. Doty, J. Keller, and R. Echols for helpful discussions about astronomy. Contents 1 Longitude, Time, and Navigation 6 1.1 Introduction .................................... 6 1.2 The world way back then ............................. 7 1.3 Latitude and Longitude ............................. 8 1.4 Perils of the ignorance of longitude as sea . 12 1.4.1 The Galleon: a death trap ........................ 13 1.4.2 Sir Clowdisley Shovell .......................... 15 1.4.3 George Anson ............................... 15 1.4.4 La Salle .................................. 16 1.5 But still they set sail ............................... 17 1.5.1 Best guesses at one’s location ...................... 17 1.5.2 Deduced Reckoning or DR ........................ 18 1.5.3 Milk your latitude ............................ 19 1.6 So how does one find longitude at sea? ..................... 19 1.6.1 Time and Longitude ........................... 19 1.6.2 Idea #1: The Lunar Method ....................... 21 1.6.3 Idea #2: Keep accurate time as sea ................... 23 1.6.4 The moons of Jupiter ........................... 24 1.7 The Longitude Prize ............................... 25 1.8 The Remarkable John Harrison ......................... 26 1.9 On we go... .................................... 27 2 Excerpts from “The Quest for Longitude” 29 3 The Earth, Moon, Stars, and Navigation 88 3.1 Mapping the Earth ................................ 88 3.1.1 Locating yourself on the earth ...................... 89 3.1.2 Locating points with latitude and longitude . 95 3.2 Angles in space .................................. 98 3.2.1 Use your hand as an “angular ruler” . 98 3.2.2 Angular Measure ............................. 100 3.2.3 Angles and Distance ........................... 104 3.3 Celestial Navigation: Navigating using the Sun, Moon, and Stars . 105 3 CONTENTS 4 3.3.1 The Basic Premise of Celestial Navigation . 105 3.3.2 A basic measurement ........................... 109 3.3.3 The Sky .................................. 114 3.3.4 The Sun .................................. 124 3.3.5 The Moon and Lunar Distances ..................... 126 3.3.6 The Stars ................................. 132 3.3.7 The Moons of Jupiter ........................... 132 3.4 Celestial Navigation: a case study ........................ 142 3.4.1 Let’s find ourselves then ......................... 143 3.4.2 What did we just do? ........................... 158 3.5 How to find latitude ............................... 160 3.5.1 Approximate Easy Latitude: A noon sight of the Sun . 166 3.5.2 Better Easy Latitude: A noon sight of the Sun ............. 169 3.5.3 Easy Latitude again: A sight of Polaris at night . 172 3.5.4 A Guess at History: Using Polaris + the Sun . 173 4 Time and Time Keeping 176 4.1 Introduction .................................... 176 4.2 Measuring Time .................................. 177 4.2.1 Short Times ................................ 178 4.2.2 Longer times ............................... 180 4.2.3 A summary of long and short times ................... 181 4.2.4 Heisenberg: The best we can do ..................... 184 4.2.5 Time in Science .............................. 185 4.3 Time Keeping with a Clock ........................... 186 4.4 Nature’s Clocks .................................. 187 4.4.1 The Pendulum .............................. 187 4.4.2 Mass on a Spring ............................. 192 4.5 Core Elements of Mechanical Clocks ...................... 203 4.5.1 Sustained Energy ............................. 203 4.5.2 Escapements: Controlling the Energy Release . 207 4.6 More Bizarre Clocks ............................... 216 4.6.1 Kepler’s Laws ............................... 216 4.6.2 Atoms ................................... 219 4.6.3 Electronics ................................. 231 4.6.4 Electrical circuits that have a period . 239 4.6.5 Crystals .................................. 242 5 Modern Ideas on Time and Navigation 247 5.1 Introduction .................................... 247 5.2 Albert Einstein .................................. 249 5.2.1 Einstein’s Ideas on Time ......................... 249 5.2.2 Clocks on the move ............................ 252 CONTENTS 5 5.2.3 It moves so its life will slow ....................... 258 5.2.4 Einstein and Harrison .......................... 259 5.3 The Crown Jewel in Navigation: The Global Positioning System . 259 5.3.1 Introduction ................................ 259 5.3.2 The GPS System ............................. 261 5.3.3 Finding your location ........................... 262 5.3.4 Why Atomic Clocks? ........................... 266 5.3.5 Closing Remarks ............................. 269 6 Could you do what Harrison did? 270 6.1 Introduction .................................... 270 6.2 Problems ...................................... 270 6.2.1 Xprize.org ................................. 271 6.2.2 The Copenhagen Consensus ....................... 271 6.2.3 The Bill and Melinda Gates Foundation . 272 6.2.4 Obvious problems ............................. 272 6.3 The Assignment .................................. 272 7 Map Worksheets 274 Chapter 1 Longitude, Time, and Navigation Dad: Longitude is so boring. -Will Bensky (8 years old) 1.1 Introduction Suppose you were asked you to list some of the biggest problems facing the world today. What would you say? Global warming? Terrorism? Something environmental? Hunger? Energy? Next, suppose $12 million was offered if you could solve one of these problems. Would you pick one and get started working on it? Or would you turn away shaking your head saying “How could I solve such a problem?” Would you be willing to dedicate your entire life to solving one of these problems? In this class, we are going to take a whirlwind look at a huge problem facing the world in the 1700s. In fact until about 1780, this problem had been unsolved for more than 300 years, even though some of the greatest thinkers, such as Isaac Newton (inventor of calculus), Galileo Galilei (first to use a telescope), Edmund Halley (the comet), Huygens (calculus, optics, found Saturn), Euler (calculus, physics), and others worked on it. In the end, an uneducated, unknown, but extremely dedicated and brilliant carpenter, named John Harrison solved the problem after a lifetime of work. In fact, he only lived 3 more years (into his eighties) after winning the prize that would today be worth $12 million. He worked for stretches as long as 15 years straight on various attempts at a solution. Would you do this? Could you? So what was this world-wide problem? Finding longitude at sea. What you are thinking? • Huh? • What’s that? • What is longitude anyway? • Why would this be an important problem? 6 CHAPTER 1. LONGITUDE, TIME, AND NAVIGATION 7 • Why “finding longitude?” • Why “at sea?” 1.2 The world way back then The world in the 16th-18th Century (1500s-1700s) was obviously quite different than today. Societies were well developed in Europe, but the “new world,” consisting primarily of the North American continent was largely unclaimed and unexplored. Portugal kicked off what has come to be known as the “Age of Discovery,” in the mid-1400s. The westernmost country in Europe, Portugal was the first to significantly probe the Atlantic Ocean, colonizing the Azores and other nearby islands, then exploring the west coast of Africa. In 1488, Portuguese explorer Bartolomeu Dias was the first to sail around the southern tip of Africa, and in 1498 his countryman Vasco da Gama repeated the experiment, making it as far as India. Portugal would establish ports as far west as Brazil, as far east as Japan, and along the coasts of Africa, India and China[Smithsoninan, Sept. 2007]. For additional reference dates, think of Christopher Columbus setting sail from Spain in 1492. The 13 original English colonies were established between 1607 (Virginia) and 1732 (Georgia). The American Revolution culminated with a battlefield victory in 1781. The great westward exploration of Lewis and Clark took place in 1804. European societies (Portugal, France, England, etc.) knew of great riches in the new world, and the political gain that could be made by finding and staking claim therein. They just had to get there, and getting there was a problem. At this time, the dominant ocean faring vessel was a “galleon,” which looked something like that shown in Figure 1.1. The look of these ships is probably somewhat familiar to you, as the first “long haul” sailing ship. They were large multi-decked sailing ships used primarily by the nations of Europe from the 16th to 18th centuries (1500s - 1700s). Here was “the problem:” All of the great explorers you may recognize, who used these Galleons, from Columbus to de Gama to Magellan to Drake were essentially “sailing blind.” They had virtually no idea where they actually were as soon as they lost sight of land. Really? Yes, really. Any “great discovery” they made was at a great human cost and more-or-less