How Big is Big and How Small is Small Read online

  HOW BIG IS BIG AND HOW SMALL IS SMALL

  The Sizes of Everything and Why

  by

  Timothy Paul Smith

  Great Clarendon Street, Oxford, OX2 6DP,

  United Kingdom

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  © Timothy Paul Smith 2013

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  First Edition published in 2013

  Impression: 1

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  ISBN 978–0–19–968119–8

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  Contents

  List of Figures

  List of Tables

  1. From Quarks to the Cosmos: An Introduction

  2. Scales of the Living World

  3. Big Numbers; Avogadro’s Number

  4. Scales of Nature

  5. Little Numbers; Boltzmann’s and Planck’s Constants

  6. The Sand Reckoner

  7. Energy

  8. Fleeting Moments of Time

  9. Deep and Epic Time

  10. Down to Atoms

  11. How Small Is Small

  12. Stepping Into Space: the Scales of the Solar System

  13. From the Stars to the Edge of the Universe

  14. A Little Chapter about Truly Big Numbers

  15. Forces That Sculpture Nature and Shape Destiny

  Index

  List of Figures

  1.1 The original definition and measurement of the meter.

  1.2 Towns embedded in counties, embedded in states, embedded in nations.

  2.1 The relationship between surface and volume.

  2.2 As towns grow into cities the traffic for supplies must increase.

  2.3 A collection of Littorina saxatilis (periwinkle) collected in Scotland.

  2.4 Savanna elephant Loxodonta african, Loxodonta catherdralus and Notre Dame de Paris.

  3.1 Loschmidt’s method for determining the size of a molecule.

  4.1 The Beaufort and Saffir–Simpson scales and windspeed.

  4.2 The Richter scale versus the Mercalli scale.

  4.3 The number of hurricanes in the North Atlantic (2003–2012) as a function of windspeed.

  4.4 The Mohs scale versus the absolute hardness of minerals.

  4.5 The stars of Orion as reported in the Almagest.

  4.6 The frets and related frequencies of a guitar.

  5.1 Reversible and nonreversible collisions on a pool table.

  5.2 Order as shown with a die and pennies.

  5.3 The distribution of light described by Planck’s law.

  5.4 The Planck length.

  6.1 Eratosthenes’ method for measuring the size of the Earth.

  6.2 Aristarchus’s method for measuring the size of the Moon.

  6.3 Surveying the distance to unreachable points with angles and baselines.

  6.4 Aristarchus’s method for measuring the size of and distance to the Sun.

  6.5 The size of a geocentric and a heliocentric universe.

  7.1 Energy densities for chemical and nuclear fuels.

  7.2 The energy that is released when hydrogen burns, at the molecular level.

  7.3 The octane molecule.

  7.4 Energy from nuclear fission.

  8.1 The analemma, as viewed from New York City.

  8.2 The discovery of the omega particle.

  9.1 Solar versus sidereal time.

  9.2 The decay of carbon-14 used for dating.

  9.3 The layers or strata of the Grand Canyon.

  9.4 The supercontinent Pangaea about 200–300 million years ago.

  10.1 The cross section of a standard meter bar from the 1880s.

  10.2 A marching band performs a wheel and demonstrates interference.

  10.3 Laser light on a single hair shows an interference pattern.

  10.4 Trees in an orchard demonstrate the facets in a crystal.

  10.5 The interference pattern of X-rays from a crystal and DNA.

  10.6 Wavefunctions of the hydrogen atom.

  10.7 The periodic table of the elements.

  11.1 Detail from a small part of the table of nuclides.

  11.2 The table of nuclides.

  11.3 The nuclear force.

  11.4 An image of a nucleus made of protons and neutrons with quarks inside of them.

  11.5 A collection of different types of particles made of quarks.

  11.6 The interaction between a proton and neutron in terms of pions and quarks.

  11.7 The interaction between quarks in terms of gluons and color charge.

  11.8 The electric charge distribution of a neutron.

  12.1 Kepler’s third law demonstrated by Jupiter’s moons and the planets.

  12.2 The transit of Venus as seen by two separate observers.

  12.3 Some of the moons in our solar system.

  12.4 The rings of Saturn.

  12.5 The Titus–Bode law.

  13.1 Parallax of a near star.

  13.2 The H–R diagram of star color versus luminosity.

  13.3 A map of the Milky Way.

  13.4 The Andromeda galaxy.

  13.5 The Doppler effect in water waves.

  13.6 Comparison of the size of galaxies, group, superclusters and the universe.

  13.7 The large-scale structure of the universe.

  14.1 Everything is small compared to infinity.

  15.1 Different forces of nature dominate different scales.

  15.2 Proton–proton fusion.

  List of Tables

  3.1 Names of numbers.

  6.1 The symbols and values of numerals in the Roman number system.

  6.2 The symbols, values and names of numerals in the Greek number system.

  6.3 A comparison of the astronomical measurements of Aristarchus, of Archimedes and of modern astronomical measurements.

  7.1 Comparison of nuclear and chemical forces and bonds.

  7.2 Comparison of the four fundamental forces in nature.

  8.1 Exotic subnuclear particles; lifetimes, forces and decay.

  9.1 Comparison of decimal and standard time.

  9.2 Days and years of the planets.

  9.3 Present geologic time.

  9.4 Supercontinents through the history of the Earth.

  12.1 Radius of the orbits of the planets: Coper
nicus vs modern measurements.

  12.2 Some properties of the planets.

  12.3 Some properties of the dwarf planets.

  13.1 List of nearby stars.

  14.1 Examples of sets that have an infinite number of members.

  1

  From Quarks to the Cosmos:An Introduction

  If the meter is the measure of humans, then we are closer to quarks than we are to quasars. However, if we take the second as the heartbeat of our lives, then we are closer to the age of the universe than to the lifetime of elementary particles. There are well over forty-five orders of magnitude between the largest things we have ever measured—the grand breadth of the universe itself—and our smallest measurement, the probing of those iotas of matter, quarks, electrons and gluons. There are also over forty orders of magnitude between the fastest events clocked and the slowest events, which we continue to watch evolve and unfold. Somewhere in the middle of these ranges are the scales where we humans spend our lives, the scales measured in meters and seconds. The breadth of these scales is truly astonishing and it is a credit to modern science and to our intellectual capacity that we can write down forty-digit numbers with certainty and a fair amount of precision.

  This book is a guide to understanding and appreciating those numbers, both the large distances and the small. A concert or an art exhibit may be beautiful and awe-inspiring, and quarks and the cosmos are awe-inspiring and, I would argue, beautiful too. But the concert or exhibit may be better appreciated with program notes or a guide. Likewise, with diligence and a good guide, we can learn what these vast and infinitesimal numbers mean. We can internalize and appreciate their beauty.

  The universe is about 1027 macross. That is a 1 followed by twenty-seven zeros: 1,000,000,000,000,000,000,000,000,000 m. At the other extreme, we have probed neutrons and protons. We know that they contain quarks and we know that the quarks are smaller than 10−18 m. That is a 1 preceded by seventeen zeros: 0.000,000,000,000,000,001 m. But these are numbers with which we have no connection in our everyday experiences. These numbers are bigger than the national debt, or even the gross national product of all the nations on Earth expressed in pennies. How are we ever going to understand these monstrous numbers? We will do it with analogies and by developing an appreciation for what scales mean. We will also look at systems and slices of nature the scales of which have a smaller range. We will build up, with small steps, to these massive scales. But you should not be disappointed that we do not start out with the ultimate of scales, for it is a grand tour to get there. It is a journey through atoms, sequoias, the Sloan Great Wall, pianos, whales, quarks and rock concerts. On the way we will encounter geniuses and madmen, surveyors and seismographers, horses and hurricanes.

  Humans can measure things that are 10−18 m across. If we were to count how many of these tiny things could be placed side by side across the universe we would end up with a number with forty-five digits! But that is only one of the endpoints, one of the brackets that hold all of nature. It is not the starting point. We start, curiously enough, with what we understand and use every day. We start with the measuring stick of man.

  ***

  Humans have developed a number of different standards to measure our world, and in general these standards have reflected our own stature, or our labors or our lives. The second is about the time between heartbeats. The traditional acre is related to a day’s labor. The day itself may be astronomical in origin, but its importance lies in how the Earth’s rotation affects our lives by telling us when to sleep or have breakfast. The pound is a handful of dirt and the cup is a good size for a drink. The traditional units of bushels, pecks, gills and gallons are all useful units for measuring things we use every day.

  However, it is the length scale that is of primary interest when measuring the universe, and we humans have been prolific when devising standards by which we can measure the length of things. The Romans measured long distances with the mile, which literally means “one thousand paces” (mille passus), where the pace is two steps or the distance from the right footstep to the right footstep. This is a practical unit for marching off distances with an army.

  More ancient than the mile is the stade. The stade has an important distinction in athletics: it is the distance of the ancient Olympic footrace. We know the Olympic stade is 192.8 m, since we can still go to Olympia and measure where the races were held. It is from the stade that we derive the name of the place where races, and now other athletic events, are held: the stadium. In the history of astronomy, the stade also has an important role, for Eratosthenes (276–194 BC), the Greek mathematician and astronomer calculated the size of the Earth and obtained a circumference of 250,000 stades. Unfortunately for us, it is not clear which stade he used. Was it the Olympic stade, the Roman stade of 185 m or another unit: the itinerary stade of 157 m? In any case, the distance he calculated of 39,000–46,000 km is amazingly close to the 40,100 km modern measurements yield. Still, the mile or the stade may be good standards when measuring the length of an army’s march or the breadth of an empire, or even the girth of our home planet, but they are not the units we use when we measure ourselves. When we measure ourselves and our homes we use the foot, the yard or the cubit. Every single one of these units starts out with some part of the human body, as well as a story which comes out of the misty past.

  The cubit is one of those ancient measurements that is very convenient for humans. It is the distance from the elbow to the outstretched fingertips. We all come with a built-in cubit measuring stick. It corresponds to about 18 in or 45 cm. We are told that Noah’s Ark was 300 cubits long, 50 cubits wide and 30 cubits high, and the Ark of the Covenant was one and a half cubits high and wide and two and a half cubits long. But the ancient world did not have the ISO (International Organization for Standardization) watching over it and, much like the stade, there was a multitude of cubits. An important and well-studied cubit is referred to as the “Egyptian Old Royal Cubit.” In this particular case we know from ancient documents that the Great Pyramid of Giza is 280 royal cubits in length and, since the pyramid is still there, we can go out and actually measure it, and so we know that this royal cubit is equal to 52 cm.

  Here we can see the start of a problem. The cubit may be a very convenient unit and any master shipwright or stone cutter can offer his forearm as a standard. But if the stone blocks for the pyramid are to be, let us say, ten cubits on a side, and one quarry uses 45 cm and another quarry uses 52 cm, they will not fit together at the building site. So civilization developed standards. We may like the story of King Henry I of England offering his foot as the royal standard foot (he really was not tall enough to have a foot about 30 cm long), but far more useful was the iron bar mounted in the wall of the marketplace with standard lengths (at least for that market) inscribed on them.

  And then we move into the age of the Enlightenment and the metric system. The idea was that this new standard, the meter, would not depend upon a particular man, for after the French Revolution there was no king left who could offer his royal foot as a standard. For all men are created equal, even if we all have our own unique stature. So the meter was to be based on something outside of a simple human dimension, something universal. It should also be based on something that all humankind had access to, such that anyone could go out and create their own meter stick. Thus 10,000,000 m was defined as the distance between the equator and the poles of the Earth. What could be more noble and permanent, more universally accessible than the planet upon which we stand?

  Establishing an ideal meter is all well and good, but in a practical sense, how does one go about creating a meter stick? How far is it from the equator to the pole and how did we measure this distance in 1792, or at any other time? No one would visit the North Pole until Robert Edwin Peary’s team did so in 1909, and the continent upon which the South Pole is located was still terra incognita. In fact Antarctica (Terra Australis) was not even sighted until the crew of the Russian ship Vostok saw it in 1820. Th
e first expedition to reach the South Pole, led by the Norwegian Roald Amundsen, did not reach it until 1911, and they were not measuring the distance there with meticulous detail.

  So how do we establish the meter? What we would like to do is pace off the distance from the equator to a pole. Imagine that I start in Ecuador and march north across the Andes mountains, then through the jungles and deserts of Central America and Mexico, always keeping Polaris, the north star, in front of me. On across the plains of the United States, the boreal forest of Canada and finally the tundra and ice of the Arctic to the North Pole. After my epic trek I find (let us suppose) that I have taken 12,345,678 steps (left foot to left foot is two steps). Then 12,345,678 steps is 10,000,000 m, and after a bit of division I find that 1.2345678 of my steps is 1 m. I can now mass-produce my meter stick!

  The beauty of this system is that if we are careful, your stride does not have to be the same length as my stride. In fact you could replace the steps with revolutions of a bicycle wheel or a surveyor’s chain and you and I would end up with meter sticks of the same length. If the Earth was smooth and regular you could perform your measurement someplace else, such as through London or Paris or even through Thomas Jefferson’s home at Monticello. You could even stride off to the South Pole and you would still end up with the exact same meter.

  But there is a problem with this technique, and it is not just the mountains, canyons, swamps and rivers that will mess up our uniform stride length—a good surveyor can measure across all of these. The problem was that no one had gone to either pole until over a century after the meter was established. Yet the French Academy of Science, which was the organization that established the meter, understood this limit. So when they sent out their two survey teams, headed by Jean Baptiste Joseph Delambre and André Méchain Pierre François, they were only to measure the distance between Barcelona and Dunkirk and their respective latitudes (see Figure 1.1). The reason that this works is that if you can only measure a fraction of the equatorial–polar trek, but you know what fraction you have measured, you can calculate the length of the whole trek. For example, if you measure the distance from 36° north to 45° north you have measured 9° out of 90°, one tenth of the whole distance, or 1,000,000 m.