How Big is Big and How Small is Small Read online

  Now nothing is ever as simple as it seems when proposed, and the survey across France took seven years to complete (1792–1799). The measurement was hard, the Earth is far from being uniform and the French Revolution was going on. But ideally anyone can establish a meter stick. All you need is to measure a quarter circumference of the Earth and divide by 10,000,000.

  ***

  Why divide by 10,000,000? Why not divide the equator–pole distance by a simple number like 1? Or why not use a more common metric number like a thousand or a million? The answer is that by using ten million we end up with a meter that is of a useful human scale. The meter is about 2 cubits, and it is just over one yard long. It is a very human-sized length. It is interesting to look at another definition of a meter that the French Academy of Science considered. They toyed with the idea of defining a meter as the length of a pendulum with a period of oscillation of 2 sec. However, it turns out that the gravitational attraction of the Earth changes slightly at different latitudes. That would mean that a “meter” established near the equator would be different from one established near the poles, which is not a good way of defining a standard. But in either the Earth-size meter or the pendulum-based meter there were always the curious factors, the 10,000,000 and the 2 sec. Why? The academy already knew approximately the size of the Earth and the length the pendulum would be, and both of them gave a useful, “human-size” meter. A pendulum with a period of 2 sec is 99.4 cm long. The pendulum definition would have led to a very similar meter to the one we have today.

  Figure 1.1 The relationship between surface and volume. From the equator to the pole is 10,000,000 m. Jean Baptiste Joseph Delambre and André Méchain Pierre François measured the distance across France, between Barcelona and Dunkirk, to establish the original meter.

  Throughout this book distance will usually be described in meters and time in seconds. These are the working units of modern science. But for all practical purposes, when you see “meter” you can think “the size of a human” and when you see “second” you can think “a heart beat.”

  ***

  Many disciplines within science have their own specialized units that we will bump into. In atomic physics a measuring stick the size of an atom would be useful. So we have the length unit called the angstrom (symbol Å). The angstrom is roughly the size of a hydrogen atom. It is exactly 10−10 m, or one ten billionth of a meter. Why ten billion and not a billion? Because it is very nearly the size of an atom and, as we will see later, it is very useful. In nuclear physics we use the fermi (symbol fm), which is about the size of a proton or a neutron. It is defined not by the proton or the neutron, but by the meter. A fermi is a quadrillionth of a meter (10−15 m). It is incredibly small, but it is still defined in terms of the human-scale meter.

  Astronomy uses three non-metric units: an astronomical unit (AU): a light-year (ly) and a parsec (pc). An astronomical unit is the distance between the Sun and the Earth. A parsec is defined by the technique for surveying the stars, which we will consider in Chapter 13. A parsec is also roughly the distance between neighboring stars. Finally, the light year reminds us that even if light is very fast, space is that much bigger. But even in astronomy we can, and in this book, will, write stellar, galactic and even cosmic distances in terms of meters.

  ***

  One of the keys to the success of science has been the way scientists organize their observations and data into systems. Carolus Linnaeus (1707–1778), the Swedish naturalist, said that the first step to understanding nature is to uniquely and permanently name the species. But what may be more important is that he then went on and organized animals and plants into his famous binomial system, where a species belongs to a genus, and a genus to a family, an order and so forth. No longer was the world of the naturalist just a jumble of species, but rather there is an organization, a branching system that tells us about connections and relationships between species.

  In this book’s presentation of the size of things we will also need a system, for if we just pour out interesting but randomly organized facts

  it will be a confusing jumble. If I sandwich the description of the lifetime of an exotic particle called the muon between the size of Jupiter and the energy required for an ant to do a push-up, this would not help us understand how big is big or why things are of a certain size. So in this book we will generally work from scales we know, such as the meter or the second, to the scales we are not used to, such as the size of galaxies or the lifetime of an exotic subatomic particle. This does have the problem of making it seem like things are arranged physically next to each other, which is not really right. Phenomena of different scales can be embedded in each other; atoms in molecules, molecules in cells, planets in galaxies. And this is useful to us because this embedding helps us recognize when we have taken a true step in nature. Let us consider one more example. Even if the Earth is much larger than Mercury, it is on the same organizational level because Mercury cannot be contained within the Earth. These planets are embedded within the solar system, and so are on a different level from the solar system. This technique of looking to see what objects can be embedded in other objects will not always help us organize things, but in many cases it can guide us. This may sound a bit too abstract, so let me draw an example from the world of humans.

  Humans created types of governments that are embedded in each other and act differently at different levels because of the way they solve the problem of how to govern. I live in a small town in New England, nestled between the Green and the White Mountains. The town’s job is to maintain roads, especially those that only connect different parts of the town. The town also has school, police, fire and recreation departments. Once a year we gather at a town meeting and debate issues, vote on budgets and impose taxes on ourselves to pay for what we think the town should do. We have commercial centers and a civic life with churches, clubs, schools, sports teams and places to gather to work or learn. These social facets may in fact be a better definition of the town than the government, but that there is a close parallel is not surprising since New England towns grew out of parishes and markets and not the other way around. At the time our town was settled, most people could walk to the town center in about an hour, which is a reasonable amount of time to invest in getting to the market or the church. This limit of an hour’s walk means that the size of the town is about 10 km or half a dozen miles on a side. This size in fact became the standard across much of the United States and Canada in the form of the township and range, or Public Land Survey System, which was endorsed by Thomas Jefferson. This system fitted the size of the town to the needs of its citizens.

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

  The next size of government is that of the county (see Figure 1.2). Whereas counties have many functions, including sheriff’s departments, roads, parks and social support agencies, it is most likely the county courthouse that you might need to visit; not to see the judge, but rather the county clerk. For this is the place where land transactions and other important events are registered and the records maintained. In Jefferson’s public land survey system the county is six townships from east to west and six townships from north to south, or 36 miles – 60 km – on a side. This means that it is not more than about 20 miles or 30 km for the average citizen to travel to the clerk and courthouse. To travel there on foot or even horse is an all-day event. The place is accessible, but not on a weekly basis.

  In many places the role of the town, the county, or the city are different from what I am describing here, but the point of this exercise is that different systems can embed themselves in each other. Also, we note that the larger the area that the government unit covers, the more remote and abstract is the interaction with the individual.

  When we move to the next level of government—the state—it plays a very different role than towns or counties, and I also interact with it in a very different way. To start with when we ask the question, “What
is the natural size of a state?” we see a great range of solutions, from the small New England states like Rhode Island or New Hampshire, to the mega-states like Texas or Alaska. Their boundaries are often defined by geographic constraints, such as rivers and oceans, or by history. Curiously enough, we will see that galaxies are similar in this respect. Moreover, the mechanisms of interaction are new. I have no reason to go to the state capital. I can send things by mail or email, or if I need to go to the Department of Motor Vehicles, I can go to a satellite office. My input is through my state representative. This means that the size of the state is not simply limited by how far I can travel. The state still deals with serving individuals. The state licenses you to drive or be married; it licenses certain types of business; it charters certain types of organizations and sets standards for schools and restaurants. It also legislates the rules by which towns and counties operate. But it is dealing a bit more with other levels of government, and a bit less with individuals.

  The federal government is even more remote from our lives than the state. It may dominate the news, but it really is less of a factor in our everyday lives. I carry my state driver’s license in my wallet every day, but I only carry my passport on the rarer days that I travel internationally. My method of input to the federal government is even more remote. I am represented by people I have never met and who sit in a legislature a thousand kilometers away. An educational policy set by the federal government affects a few days of school a year, but the state sets minimum standards for all classes, and it is the budget of the town and the policies of the local board that determine when schools start every day and who is teaching what. The federal government deals most directly with other governments at its own or similar levels, with states, other nations and the whole planet.

  Finally there is the United Nations, which is an organization and not truly a government, for its members have not relinquished their sovereignty. It is a meeting place of nations, a forum where the world can meet together and work out solutions to common problems.

  This nested system of governments I have just described—town, county, state, nation and international organization—is not the only way that governments could be organized. Most readers will not live in a place like my town. They will point out that I have missed cities, villages and boroughs, or that the state could be replaced with a province, a district, or a territory. Also, the balance between the responsibilities of different levels of government is continuously in flux. For example, with the recent establishment of a Welsh Parliament in Cardiff and a Scottish one in Edinburgh and the increasing importance of the European Union, will the role of the United Kingdom change? (Will England get its own Parliament?)

  Government levels may be in flux, and particular types of government are not universal, but there are certain general trends. Units of government have a more direct effect on units of government of a similar magnitude. Also, it is very unusual to get a government that is embedded in more than one other government. Although there are school districts that cross state lines this is not common or simple. Nature tends to divide itself with more firmness and distinction. Atomic, planetary and galactic systems are well separated in magnitude. Systems that neighbor each other in size do affect each other. If I want to understand chemistry it would help to understand atomic physics, but it would not help as much to study nuclear physics or cosmology. So in this book, as in governments, we will see the embedding of systems, and the effect that different levels of nature have on each other.

  ***

  At the beginning of this chapter we introduced the meter, and argued that it was a unit we could understand and that we could do this because it is the size of a human, but so far we have not really measured anything. We talked about systems that can be embedded in bigger systems, using as an example the different levels of government, but we have not really talked about the different levels of nature. Describing all the levels of nature is a huge task, which is beyond the scope of this chapter, but it would be a disappointment to stop here without at least sketching out the size of the universe and quarks. Envisioning forty-five orders of magnitude is a great mental challenge, and we will spend the rest of this book learning how to deal with these numbers, but I think we deserve a quick tour of nature right now.

  We start with the meter, the measuring stick of our everyday lives. The average adult human is about 1.7 m tall, and we tend to pass the 1-m mark in height when we are in kindergarten. Chairs are half a meter up and beds are about 2 m long. Meters measure things that fit us.

  On this quick tour we will now jump to things that are one hundred thousand times bigger than us. What things are 100,000 m (105 m) or 100 km in length? This is the distance you can drive in an hour. It is also the distance across a state like Connecticut or a country like Wales or the Lebanon.

  At 100,000 times the distance across Connecticut (1010 m), we are looking at the world at the planetary level. This is 26 times the distance from the Earth to the Moon, but only 7% of the way to the Sun. All the problems of state versus nation seem insignificant at this scale.

  Another jump of 100,000 times (1015 m) brings us to a scale bigger than our solar system, a chunk of space encompassing all the planets and even part of the Oort cloud, the belt of comets that surround our star system. It takes light 40 days to travel this distance.

  Now a jump of another 100,000 times (1020 m) brings us to galactic scales. 1020 m is 3.3 kpc (kiloparsecs). Our galaxy is about 1021 m across.

  Another jump of 100,000 means that we are no longer looking at individual galaxies, but rather at distributions of galaxies. At 1025 m across is the Sloan Great Wall of galaxies.

  Finally, one more jump brings us to 1030 m. This is bigger than the whole of the observable universe. The edge of everything—and this really is a difficult idea—is 1027 m “out there.” Ten followed by 27 zeros. And beyond that?

  ***

  What about the complementary tour? How small is small?

  One hundred thousand times smaller than a meter (10−5 m) is the world of cells. Cells come in a range of sizes from the ostrich egg, which can be up to 15 cm long, to the bacterium, which is a millionth of a meter in length. But the red blood cell, at 10−5 m, is often cited as typical. In the light of our recent tour of big, we note that a red blood cell is as much smaller than our whole body as a person is smaller than Connecticut. So you can think of that single cell wandering around the body as much like a person wandering in Connecticut. Actually the analogy is not quite right, for a human wandering tends to stay on the surface of the Earth, whereas the cell can circulate through the whole three-dimensional body.

  One hundred thousand times smaller than the red blood cell (10−10 m) is the atomic world. The diameter of the simplest atom, hydrogen, is 1.06 × 10−10 m. Heavier atoms, such as gold or lead, can be 200 times more massive, but only three or four times the diameter.

  One hundred thousand times smaller than an atom (10−15 m) is the world of nuclear physics. The proton is 1.7 × 10−15 m across. At this distance the electromagnetic force, which holds together atoms and molecules, is unimportant. Gravity, the force which holds together galaxies, is even less important than the electromagnetic force. Both of these are dwarfed by the Strong Force, the force which holds together nuclear matter. This is a force we ignore in our everyday lives, even though it is so very powerful and it is embedded in every iota of what we are made of.

  One more step of one hundred thousand and we fall off the other end of the scale. We do not know what the world looks like at 10−20 m. This limit is a bit different than the 1027 m edge of the observable universe. Here we are limited by our biggest experiments. When at the beginning of this chapter I said that the size of a quark is less the 10−18 m, I meant that this is the limit of what we can see with our biggest accelerators. We have good theories that lead us to believe that quarks may be much much smaller. They may be as much smaller than us as we are smaller than the universe, but we really do not know. A jump fro
m our present experimental limit to this ultimate Planck scale is like jumping from the distances between stars to distances that span the whole universe and ignoring galaxies and galactic structure along the way. There could be a lot of interesting stuff between our smallest present measurements and this ultimate Planck scale; we just do not know.

  ***

  So are we closer to quarks than to quasars? That is a hard question to answer, for the answers seem to be full of all sorts of caveats. Things like “What do we mean by closer?” and “What are these experimental limits?” And, by the way, since we are asking tough questions, “How in the world am I supposed to understand a number with forty-five digits?”

  2

  Scales of the Living World

  The voyage down to quarks and up to stars and galaxies is epic. But more interesting than simply “How big are things?” is the question of why are things the size they are. Why does scale matter? Why are there no Lilliputians or giants? Why it is that we cannot have a solar system inside of an atom? As in any voyage of exploration we need to do a lot of preparation in the next few chapters, but first we will embark upon a short cruise around a familiar neighborhood. We will look at the scales of the living world.

  ***

  The living world still has an impressive span, from things that measure a millionth of a meter to those that are a hundred meters in length. Lifetimes range from 20 min for bacterial division to almost 5000 years for the Great Basin bristlecone pine Pinus longaeva. Life covers eight orders of magnitude in both time and space, and someplace in the middle are humans. So this will all feel familiar and we will be very much at home.

  The most important tools in science are our eyes and our brain. We have five senses, but the vast majority of the information we have about the size and shape of our world comes through our eyes. Eyes are our windows on the world; they tell us what is here and what is there. They let us see clouds and rainbows, something no other sense tells us about. We can see hair, which is less than a tenth of a millimeter thick. In fact hair can range from 50–150 µ m(µm—a micrometer—is a millionth of a meter) in thickness, as it ranges from blond to black. Blond hair is only half a dozen cells across, yet it is easily visible.