Archimedes was well versed in this method and followed it closely, but then offered one original measurement. He devised an instrument made of cylinders and a measuring rod with which he determined the angular size of the Sun and Moon. He found that they were a thousandth of a great circle, or about 0.36°, close to the 0.5° we now measure. This information would have helped Aristarchus’s calculation. However, Archimedes was already trying to round things up and get the greatest distances he could, ending up with a distance to the Sun of 100 myriad-myriad stade (1010 stradia), or about ten times our modern measurements.
For Archimedes there was one more step. How big is the cosmos, or how far away are the stars? In the end we do not really know how far away Archimedes thought the stars were; we just know he was looking for a huge dimension, and to do that he did something curious. He had several cosmological models available to him: the geocentric model where everything orbits the Earth, the heliocentric model where everything orbits the Sun, and the semi-heliocentric model where Mercury and Venus orbit the Sun and the Sun orbits the Earth. He picked the heliocentric model of Aristarchus, not because it was popular (most astronomers at that time did not like it) or because he personally believed in it (he does not tell us what he thought), but rather because it gave him the largest cosmos. In a heliocentric cosmology the Earth is in motion around the Sun, tracing an orbit of tens of millions of stade in diameter, yet the stars are so distant that we do not see them shift (see Figure 6.5). They must truly be very far away.
Figure 6.5 The size of a geocentric and a heliocentric universe. (Left) In a geocentric universe the stars do not need to be much beyond the Sun. (Right) In a heliocentric universe the stars must be far beyond the Sun, since we do not see them shift as the Earth moves.
What Archimedes was going to do next was to look for parallax. The idea of parallax is that nearby objects seem to shift compared to a background when the observer moves. For instance, hold up a finger at arm’s length and view it with one eye, then note its position compared to a distant wall or landscape. Now close that one eye and open the other. Your finger may not have moved, but it now shows up against a different part of the background. The effect just described is parallax, a type of surveying where the distance between your two eyes forms the baseline of the measurement. When measuring with parallax the longer the baseline, the greater the shift.
What Archimedes does with parallax is curious. First he notes that no astronomer has observed solar parallax. This would be the shifting of the Sun in the sky due to the observer’s motions. Every day our position changes by a distance equal to the diameter of the Earth as the Earth rotates and moves the observer from dawn to dusk. Also, no stellar parallax is observed as we move around the Sun over a six-month period. Therefore, Archimedes concludes that the ratio of the Sun-to-stars to the Earth-to-Sun distances is the same as the ratio of the Earth-to-Sun distance to the Earth-diameter. This gave him a magnificent universe with a radius of about 1014 stade or 100 quadrillion m! That is an amazing leap of the imagination. The distance also happens to be close to about 1 lightyear, or a third of the distance to the nearest star by modern measurements (see Table 6.3).
***
The distance to the fixed stars was not what Archimedes was looking for. He really wanted a big number and would have liked to count the number of grains of sand that would fill all of space. He said that 10,000 grains of sand—one myriad—was equal to a poppy seed and that 40 poppy seeds side by side was a finger’s breadth. From this we know that the sand he was referring to was about 2 × 10−5 m(20μm) across. This is the type of sand used in 500 grade, “superfine” finishing sandpaper. In fact you will rarely find sandpaper this fine. It is more like the grit on emery cloth. Still, if you were to place 7 × 1020 of these superfine grains of sand side by side they would reach Archimedes’ fixed stars. Or, if we were to fill the whole cosmological sphere with sand we would need 1063 grains. That is the large number Archimedes was striving for.
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The normal Greek number system can only count up to a myriad-myriad. What Archimedes did to deal with this limitation was to classify all numbers up to this point as the “first order.” Bigger numbers were of higher orders. So, for example, the population of the Earth today is about seven billion, or
7,000,000,000 = 70,0000,0000
=70 × 108 =70 × myriad-myriad
= 70 of the second order.
Or consider Avogadro’s number
6.022 ×1023 = 602,200,000,000,000,000,000,000
= 6022,0000,0000,0000,0000,0000 = 6022,0000 0000,0000 0000,0000
= six-thousand twenty-two myriad of the third order
This system could easily “name a number” up to a myriad-myriad orders, or in modern notation, 10,000,00010,000,000.
Table 6.3 A comparison of the astronomical measurements of Aristarchus, of Archimedes and of modern astronomical measurements.
However, Archimedes did not stop there. All the numbers up to a myriad-myriad order are called the first period, which of course means that there is a second period, a third period and so on up to a myriad-myriad period. We could write that number as
10, 000, 00010,000,00010,000,000
a number for which even Archimedes did not have a use.
There is actually a parallel between Archimedes’ system and the number system that we presently use, especially when we think about the naming of numbers. Earlier on in this chapter I said that learning a second language or number system helps us understand our first one better. If we were to write out Avogadro’s number as words we would write “six hundred two sextillion, two hundred quintillion.” We could read sextillion and quintillion as “of the sixth -illion” and “of the fifth -illion.” This actually makes more sense in the long-scale number system where million, billion, trillion and quadrillion are powers of a million. That is, in the long scale, billion = million2, trillion = million3 and so forth. That means that “-illion” mimics Archimedes’ “order.”
This observation actually brings us to the limit of Archimedes’ system. His language would not support anything larger. When we write a trillion, we mean the third -illon. Here ‘third’ is an adjective that we have made out of ‘three’, which is a noun. So the largest normal name for a number is vigintillion, which is 1063; coincidentally Archimedes’ sand count. Viginti is Latin for twenty, so vigintillion means 3 × (1 + 20). Beyond twenty the names of numbers are not unique. However, in English we can take any number and make it into an adjective, something Archimedes could not do. Still, his number system, even just the first period, is not only sufficiently large to count all the sand that could be crammed into a sphere with a radius of 1 lightyear, it could also count all the quarks in the modern observable universe, or all the photons. In fact, if we filled the observable universe with particles each a Planck length in diameter, Archimedes could still name that number.
7
Energy
Energy is one of the most common and confusing topics we will consider in this book. It is ubiquitous, touching upon almost every aspect of our lives. We see it in transportation and food, in warmth and lighting and a hundred other areas. But precisely because it is omnipresent we almost fail to see it. Or we do see it but as a hundred different things. It is the forest that is lost in the trees. There are so many ways to describe energy, so many ways to meter it out. A cord of wood contains 15–30 million BTUs of energy. A cookie might contain a few dozen Calories, while the energy needed to boil water for a cup of tea to go with that cookie is about 2000 calories. You may have noticed in the last sentence that the food Calories are not written the same way as the heat calories. You pay an electric bill in kilowatt hours. All of these various units are everyday and human scale.
The destructive energy of a large bomb is measured in megatons. The energy released when a neutron decays into a proton is just under an MeV. Astronomers report that a quasar can spew forth about 1045 ergs each second. Each of these quantities is reported in their own un
its, each is measured in its own way. But those units, the calories or BTUs or ergs, actually tell us something about how that energy is measured or the history of that type of fuel or the expected outcome of releasing that much energy. But we can, and will, sort it all out and find the Rosetta stone of energy. This sorting and cataloging of apparently unrelated facts is what science does well when it is working at its best. Taking a list of chemical elements and their properties, Dmitri Mendeleev formed them into his periodic table of the elements. Taking a list of plants and their anatomy, Carl Linnaeus organized them into his binomial system. But the true genius and value of these systems is that they reflected a deeper aspect of nature. The rows and columns in Mendeleev’s table foretold the structure of the atom and Linnaeus’s system reflected evolutionary history.
So we will take Linnaeus and Mendeleev as our inspiration and expect that out of apparent chaos, order will arise. We will take our energy factoids and stir them about. We will walk about them like a sculptor eyeing a block of marble, trying to see the statue inside. We will rearrange them until a pattern emerges at which point we expect to see something deeper and more universally significant.
In the end I would like to plot a diagram, as I have done in most other chapters, which has objects with low energy at one end and high energy at the other end. And in the process we will create order and learn something.
So what is energy? It is one of those things that at first is hard to define but, like beauty, you know it when you see it. So let us start with a event that we will all agree contains a lot of energy.
***
On the front page of the sports section of a newspaper is a color photograph from a soccer match. In the middle of the picture is a soccer ball, frozen in time such that we can see the black pentagons and white hexagons that make up the ball. Also frozen in the air, with his fingers 10 cm from the ball, is the goalkeeper. He is wearing a yellow jersey and is apparently diving towards the ball with outstretched, gloved fingers. Finally, in the foreground, is a striker clad in sky blue. He too is frozen in time, with one foot stretched out in front of him, toe pointed towards the ball. Clearly there is a lot of energy in this scene. Or is there?
What if the whole picture had been staged?
What if the ball and the keeper were suspended from thin wires? What if the striker had ballet-like balance and could stand on the toe of one foot? In that case the scene might in fact be static, with no motion and so no energy. The photograph is crystal clear. Under a magnifying glass we can see the pixels from the printing press, but we cannot see any blurring to tell us about motion.
Actually there is one more piece of information available to us on the page. We know that the event is important, even newsworthy, since it is on the front page of the sports section. Not only was there a shot on goal, but it must have been a pivotal moment in the match. So, did he score or did the keeper defend his net?
If we also had a photograph from a tenth of a second later all would be clear. The keeper crashing into the turf, the striker with a triumphant glow, the ball 3 m forward and well within the net. There indeed was a lot of energy in that photograph.
Three meters in a tenth of a second tells us that the ball is rocketing along at 30 m/s (over 100 km/h or 60 mph). Since the ball must meet FIFA standards it must have a mass of 410–450 g, which also means it has an energy of about 200 J. Finally, we have met the most basic unit of energy: the joule (J).
When we talk about motion, like a soccer ball in flight, energy is clearly involved. However, it might be useful to start out with a motor or engine to get a handle on energy; engines tend to have steadier and more regular paths than soccer balls, and when we talk about engines they tend to be rated in terms of watts or, more traditionally, in terms of horsepower. Now we are back again to all these charmingly and curiously named measurement units, with their own stories. Horsepower was a term that was coined by James Watt (1736–1819) a Scottish engineer who vastly improved the steam engine. Watt was trying to sell his engines to mines to pump out water. Prior to the steam engine, pumping had been done by horses walking on a mill wheel (like a big gerbil wheel). Watt determined that one horse could produce “33,000 foot-pounds per minute,” which is the same as a horse raising 33,000 pounds one foot in one minute, or 55 pounds a distance of foot in a second, or 11 pounds up 5 feet in a second, or many other combinations.
But that is not energy.
James Watt also lent his name to another metric unit, the watt. That too is not energy. The horsepower (hp) and the watt (W) are units that measure power. A horse will produce one horsepower. If it works a minute or a day, it is one horsepower. But clearly if it toils all day it is using a lot more energy than if it only labored a single minute. Energy is power exerted, multiplied by the amount of time involved. In terms of an electric bill, a kilowatt hour is the power of a kilowatt, used for one hour. Conversely, power is the rate at which energy is used.
So that soccer ball with 200 J of energy has the same energy that a 40 W light bulb uses in 5 s. Or, since the striker got the ball moving in a tenth of a second, during that brief contact he exerted about 2000 W.
James Prescott Joule (1818–1889), the person for whom the energy unit is named, was a brewer and an active scientist who lived in Manchester, England. He was interested in the relationship between heat and energy and in the middle of the nineteenth century he quantified their relationship. He performed a series of experiments that involved putting a measured amount of energy into water and seeing how the temperature of the water changed. These included using electrical energy and compression energy. But his most famous experiment involved the vigorous stirring of water—the use of kinetic or mechanical energy—to get its temperature to rise. To do this he placed a small paddle wheel in the water and attached the wheel to a string. The other end of the string had a weight that he could release, thus producing energy in a very controlled way. The amount of heat he produced was minuscule, a fact that left some of his contemporaries skeptical. In fact, recent reconstructions of his experiment have suggested that the only reason that he could measure this temperature change was because of his experience with brewing. Perhaps the most important thing about Joule’s experiments was that three very different experiments got nearly (to about 10%) the same conversion factor between energy and heat. In modern, metric units, one calorie of heat is equal to 4.184 J of energy.
These results, plus those of a number of Joule’s contemporaries, finally pushed us toward the principle of the conservation of energy: energy is never created nor destroyed, but it can change its form. As James Joule wrote:
Believing that the power to destroy belongs to the Creator alone, I entirely coincide with Roget and Faraday in the opinion that any theory which, when carried out, demands the annihilation of force, is necessarily erroneous.
James Joule, 1845
In this case force means heat or energy.
Once it was established that energy was conserved, it was possible to talk about the flow of energy. We can see that the energy of a steam engine can turn a generator that produces electricity that can pass through a resistive wire and get hot, perhaps heating more water and making more steam. We know that it is not perfect. At every stage some of that energy is transformed into entropy, a form of energy that is not useable to us. But we should also ask what it was at the start of the flow that heated the water. Where did the original energy come from? We need the initial heat from a burning lump of coal or chunk of wood. But the piece of raw wood is not hot or in motion to start with. Where is its energy?
***
What we have in a piece of wood, or a bucket of kerosene is potential energy; it is potential as in “unrealized” or “possibly in the future;” it may cause motion or heat. But it really is energy right now too, even if it is not in motion or heat. Potential energy includes things like the weights on a cuckoo clock or on Joule’s water-stirring machine. Energy is put into the clock at the beginning of the week when the clock’s c
aretaker raises the weights. That energy is released a bit at a time when the weights slowly fall and the hands on the face of the clock spin.
There is also energy in a sandwich. This can give us the energy to climb a hill or wind a clock. There is potential in springs and batteries. When Sisyphus, from Greek mythology, rolled his stone up a hill in the underworld, he was putting potential energy into it, energy that became motion when the stone rolled back down. All food and all fuel contains potential energy, including hydrogen, the fuel our Sun burns, a uranium nucleus that might decay, and an asteroid, which may plummet to the Earth.
***
Let us step back a moment and try to see where we are headed. We are looking for an organizing principle for energy, something that will take oil and speeding comets and let us plot them relative to each other and see where energy is fundamentally big and where it is small. We now have one organizing strategy. We can categorize energy as motion (kinetic), heat, or potential. But that does not effect the plot of big energy and little energy in a fundamental way. We can sort into three groups, but we do not have a real basic scheme. So we need to keep hunting and looking at energy and trying to find a pattern.
***
Lets go back to the idea that energy flows and see where it carries us. The source of nearly all energy in our solar system is the Sun. Within the Sun, two protons in hydrogen atoms collide and form a helium atom, which then settles into a deuterium atom (an isotope of hydrogen) and in the process produces about 2 × 10−13 J of energy. That is not a lot of energy, but this sort of reaction happens in the Sun about 1039 times each second.
As a whole, the Sun radiates about 4 × 1026 W or 4 × 1026 joules of energy per second. This energy is carried outward by the motion of particles that have been boiled off the surface of the Sun, which we call the solar wind. It is also carried off as radiation: light, radio waves, X-rays, ultraviolet, infrared and other forms. All this energy streams outward in every direction.
How Big is Big and How Small is Small Page 11