Cosmos #9: Lives of the Stars

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Large Numbers and Infinity

Large Numbers

Large numbers are so common in science that a special notation, scientific notation, is used to deal with them.

Our familiar place notation is based on the powers of numbers. For example, 3421 = 3x1000 + 4x100 + 2x10 + 1. Each digit represents a multiple of something ten times as large as the place to the right. We don't have to use 10 as a base. In binary notation, used in computing, powers of two are used. In this notation, 53 is written 110101; 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 1x1.

For really large numbers, it gets cumbersome to write them out. A trillion dollars looks like this: $1,000,000,000,000. In a typical room, there are 10,000,000,000,000,000,000,000,000 air molecules. It's more convenient to write these as 1012 and 1025, respectively. The 12 and 25 are called the exponent of the number. 104 means 10x10x10x10. Obviously 101 simply equals ten. Any number to the first power is itself.

The virtue of scientific notation, apart from compactness, is that it simplifies math. For example, 1000 x 100,000 = 100,000,000 = 103x105 = 108. In other words, to multiply numbers in scientific notation, just add their exponents.

Likewise, 10,000,000/10,000 = 1000 = 107/104 = 103. To divive numbers in scientific notation, just subtract the lower exponent from the upper.

This rule applies even if the lower number is equal or bigger. 1000/1000 = 1 = 103/103 = 100. This baffles a lot of people, but it is perfectly logical: any number to the zero power is one!. Negative exponents mean numbers less than one; 10-3 = 1/1000 or .001.

Sometimes the exponents themselves are so large they are cumbersome. In that case we can nest exponents. Sagan mentions a "googol" - 10100, and a "googolplex" - 10googol. Writing out a googolplex even in exponential form is pretty daunting, and we could never write it entirely; it has more zeroes than there are atomic particles in the Universe. However, we can write a googolplex like this:

                       
                                      100
                                    10
                                  10                       

Who in the world ever needs numbers that big? There are only about 1080 atomic particles in the Universe. But there's a branch of physics called statistical physics that uses such huge numbers. For example, the air molecules in a room have an average energy, but individual molecules have energies ranging from very low to very high. To account for the actual distribution of energies, we have to calculate, among other things, the total number of possible energy states the atoms can have. The exponent of that number is comparable to the total number of atoms; in a typical room that's about 1025.

Infinity

Infinity defies a lot of our common sense notions but there is a lot we can say about infinity. In fact, there are different degrees of infinity.

You can't use ordinary mathematics to deal with infinity. One plus infinity is still infinity; ten times infinity is still infinity, and so on. One of the first modern thinkers to deal rigorously with infinity was Galileo. He pondered whether the collection [set, in mathematical parlance] of even numbers was smaller than the set of all integers. Obviously, every even number can be obtained by multiplying an integer by two. So for every integer, there is an even number. In some strange way, there are as many even integers as integers in general, even though there are only half as many even integers!

Pairing up every even integer with an integer is called putting two sets into one-to-one correspondence. We could do the same thing with all multiples of three, or ten, or a million. Every number in those sets can be paired up with a corresponding integer, and vice versa. Infinite sets are said to be equivalent if you can pair their members up one-to-one.

Mathematician Georg Cantor went further. You can't pair integers up with the points on a line. Regardless how closely you space the integers, there are an infinity of points in between. The points on a line are "more" infinite than the set of integers. Cantor used the Hebrew letter Aleph to denote the different degrees of infinity (or transfinite numbers, to use the correct term). The number of integers is called Aleph-zero, the number of points on a line is Aleph-one.

It is possible to draw an infinitely crinkly line that passes through every point in a plane, so the points in a plane can be put into one-to-one correspondence with the points in a line. Thus the points in the plane also have Aleph-one points. But the number of possible mathematical curves you can draw is greater yet. This set has Aleph-two members. And that's it so far; that's the highest degree of infinity we have been ever able to imagine.

Atoms

How do we know there are atoms? For most of history, the idea was basically an intuitive one that we couldn't simply keep cutting things into tinier and tinier pieces. There had to be a limit.

The first solid clues came from chemistry. Once chemists began to get a handle on what the really fundamental materials of nature were, they found that they didn't mix in arbitrary proportions. You couldn't mix oxygen and hydrogen in weight proportions 5:1 or 3.6: 1, only 8:1 would get water. Any other proportion would yield water plus something left over. (For the chemists in the crowd, 16:1 will also work; that gives H2O2 or hydrogen peroxide.) The fact that materials always reacted in fixed proportions hinted that matter came in discrete units.

Another clue came from certain minerals. When crushed, they always break into regular fragments. Salt, for example comes in tiny cubes because that's how salt breaks when it's crushed. Chemists began to suspect that the cubic shape reflected some fundamental unit that makes up salt, as indeed it does.

How big are atoms? One crude early guess noted that a cubic centimeter of oil could create a film on water at least 10 meters in diameter. One cubic centimeter of volume could spread out to cover an area of a million square centimeters. Thus the film of oil was a millionth of a centimeter thick, and atoms had to be smaller than that. Actually the method overestimates the size of an atom by roughly 100 times.

If we cut a pie in half repeatedly, Sagan points out it takes 90 cuts to get from apple pie to atom. That is, an apple pie is 290 times bigger than an apple pie. How big is that? It so happens that 210 is 1024, or approximately 1000, a nice tie-in between binary and decimal math. When computer types talk about kilobytes of memory (thousands of bytes), they are actually talking about 1024 bytes. To do this problem, we need another rule: to raise a number in scientific notation to a higher power, multiply the two exponents. So 290 = 29x10 = (210)9 = 10009 = 1027. That actually overestimates the number of atoms in a pie a bit. Note for math-phobes: there will be no calculations on the exams.

Does it surprise you that there can be more atoms in a pie than in a roomful of air? Why do you suppose that might be so? Atoms can be thought of for many purposes as spheres, but you can't pack spheres together to fill space completely. In solids, about half the volume of the solid is actually interatomic space. In liquids, about two-thirds of the volume is interatomic space - the atoms are moving around a lot more freely. The same is true, by the way, in the human body, so when your friends say you're not all there, they're absolutely right. In air, though, atoms account for only a couple of percent. The atoms in air are widely spaced; a typical air molecule can travel about 500 times its diameter before hitting another one. Matter is indeed, as Sagan notes, "chiefly made of nothing."

When we look at the interior of the atom, the emptiness becomes even more amazing. Once it became clear that atoms could emit smaller particles, researchers began shooting at various targets. In experimenting with gold leaf, which is only a few hundred atoms thick, Ernest Rutherford found to his amazement that once in a great while particles were sharply deflected, even occasionally flung backward. He likened this to seeing an artillery shell bounce off a sheet of tissue paper. Clearly there was some sort of hard object in the center of an atom, which Rutherford christened the nucleus. From the fraction of particles deflected, it was found that the nucleus is about 1/100,000 the size of the atom. In a scale model with the nucleus a centimeter across, the atom is a kilometer across.

It was in the Cavendish Laboratory shown in the video that many of the discoveries about the interior of the atom were made. As Sagan notes, Physics and Chemistry reduce the world to an astonishing simplicity of whole numbers. Chemical compounds have whole numbers of atoms, atoms have whole numbers of atomic particles. Pythagoras and his followers would have been delighted.

Atoms consist of swarms of electrically negative particles, electrons, orbiting around a positively charged nucleus about 1/100,000 the diameter of a typical atom. It's the nucleus that determines what an element is. The nucleus consists of massive positively charged protons and electrically neutral neutrons. To keep everything electrically balanced, the number of electrons equals the number of protons. (To see what happens when the numbers get even slightly out of balance, shuffle across a rug on a cold, dry day and then touch the doorknob.) Each chemical element is determined by the number of protons in the nucleus: hydrogen is 1, silicon is 14, iron is 26, gold is 79, uranium is 92.

The electron cloud, though negligible in mass, is the part of the atom that interacts with other atoms. The electrons orbit in a shell structure and successive shells have similar electron arangements. Thus, if we arrange elements in a table, we can line up elements with similar properties in columns. This is the basis of the familiar Periodic Table. The nucleus is pretty much inaccessible to the electrons of other atoms, and the energy binding particles to the nucleus is thousands or millions of times greater than that binding electrons to the atom. That's why the alchemists never realized their dream of changing base metals into gold. They were using only chemical processes to attack the nucleus, about like attacking a tank with a Nerf Bat. We can do it because we have the technology to accelerate nuclear particles to the necessary energies.

As Isaac Asimov once put it, had the alchemists succeeded, we would by now have a lot of gold around. But they failed, and in the process laid the basis for chemistry, so we have dyes and plastics and synthetic fibers and new metals and antibiotics. Which would you prefer?

Forces in Nature

There are four known fundamental forces in nature:

From time to time there are announcements or speculations about other forces in nature, but so far none have stood up to examination.

Most of us are familiar with the first two; they are long-range forces that drop off slowly with distance (one over distance squared to be exact). Many people wonder what keeps the protons in the nucleus from flying apart. The answer is that there are two short-range forces that are extremely strong within the nucleus but drop off so rapidly with distance that they are undetectable even in a neighboring atom. We study these forces by slamming atomic particles into each other at high speed. The neutrons in an atom supply the strong nuclear force and help bind the nucleus together. But for nuclei with very large numbers of protons, even the strong nuclear force can't hold things together perfectly well. That's why heavy elements are unstable, and the heavier the atom is, the shorter its lifetime.

These four forces, by the way, help explain why most scientists are skeptical of claims of the paranormal. To affect matter by mind power, you'd have to do one of two things: generate one of the known forces, or generate an unknown force. If your mind can generate one of the known forces (say transmit radio waves) it should be easily detectable. If there exist unknown forces, then there are a lot of deep questions that nobody has ever answered satisfactorily. How are the forces created? How do they propagate across space? How do they interact with matter. Do they account for any of the effects we normally ascribe to known forces? If yes, which ones? How will we have to modify our view of the known forces? If no, then what effects do they produce? These are serious issues because the four known forces seem to account for just about every physical process we observe.

The Sun

Information on the Sun

Stars

Everything in the universe exists in a state of balance between gravity and some other force.

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Significant Points

Note: you will have to visit some of the pages on stars and the Sun to answer these:

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Created 23 March 1998, Last Update 9 April 1998

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