| The Importance of Scale: A First Conservation Law.
Scales Influence Perception.
In the previous section of notes, I showed the facility with which we can handle large numbers when needed. While the magnitude of the number, big or small, is thus not really a concern, there is a natural human preference to use smaller numbers which are more readily visualized. For convenience, therefore, we prefer to think on scales appropriate to the objects we are studying. In astronomy, for instance, big numbers could obviously arise in the context of large distances, since the stars are separated by many trillions of miles, but such numbers can be made small through the simple expedient of adopting a more sensible unit, like the light-year. (Note that this is a unit of distance rather than time -- namely, the distance that light travels in a year, about 10 trillion kilometers. In this new unit, the nearest star is a mere four light years away.) Similarly, when we analyse the properties of the various stars in the sky, we express their masses in solar units -- ten times as massive as the sun, say, or one-half as massive -- rather than in kilograms. But when we buy butter, we ask for lumps of it in kilograms, not in chunks which are said to be one million-trillion-trillionth the mass of the sun!
Thinking about scales can lead to some interesting speculation. For example, almost everyone would agree that ``ants move quickly.'' They seem to be absolutely whizzing about as they race up and down the sidewalk. But you have no trouble outrunning, or even outwalking, an ant. In what sense, then, can they be said to move quickly? The reason for their apparently speedy motion is that they move quickly relative to their own sizes. An ant covers many times its own body length per second, so seems to be moving very rapidly although in fact in an absolute sense (in kilometers per hour) it is moving really rather slowly, at no more than a slow strolling pace for a human.
One counter-example of this is the behaviour of a jumbo jet. When you see it coming in to land at the airport, it seems almost to be hovering in the air as it comes gliding down at a deceptively slow apparent speed. In fact, of course, it is approaching the runway at quite high speed. Why the difference in perception? Really it is just the same as for the ant: when you see the aircraft in the air with nothing but blue sky behind it, the only way you can judge its speed is to see how long it takes to move through its own length (so that its tail moves forward to where the nose was a moment ago). Since the jumbo jet is so large, it takes a fairly long time to accomplish this, even at high speed, and the perception is that the plane is moving slowly. A small plane, like a single-seater Cessna, seems to buzz in to land much faster, although in fact its landing speed is much slower than that of the jumbo. Similar contrasts are seen when you compare the apparent motion of a huge ocean liner to that of a small motorboat.
There are astronomical examples of this as well. The Milky Way galaxy is a big, flattened spinning pinwheel of stars within which our sun is to be found about two-thirds of the way out to edge. You can, if you like, visualise our sun as analogous to a tiny speck of dust somewhere near the outer edge of a spinning Frisbee. As we will learn later, the sun moves around the Milky Way galaxy at a speed of about 200 kilometers per second - almost 800,000 km per hour! This sounds colossally fast, but for the sun to go around the galaxy even once takes over 200 million years, since the galaxy itself is so enormous. If you consider a different analogy, one in which you visualise the Milky Way as a thin vinyl record on a turntable, you can see that it turns very slowly.
There is yet another interesting example. The sun itself is a large object, with a diameter of almost one-and-a-half million kilometers. As it moves within the Milky Way galaxy, travelling along in the direction in which the great pinwheel is turning, it takes almost two hours to move through a distance equal to its own diameter. In this sense, the sun moves rather ponderously through space, not like a bullet absolutely whizzing along.
Scales Determine Form and Function.This discussion of perceptions may seem trivial, but scale can be a matter of life and death. In general, big things are not merely scaled-up versions of small things, which is why we see no elephant-sized ants or ant-sized elephants. Why is this so? Imagine scaling up an ant to ten metres in length, as in the excerpt I showed you in class from the classical science fiction movie `Them!' (which dates from the early '50s). Unlike its dangerous manifestation in the movie, such a large ant could not function and indeed would not even survive! Why not? The principal point I want to make is that of the limited strength of material, a theme we will come back to in astronomical contexts. Let us see what this means for the ant. As in the movie, let us assume that the new, bigger ant is made of the same material (`ant stuff') as before. It is not, for instance, allowed to have stainless steel bones buried deep within it. Its total weight then depends on its new volume. Let us assume that a typical ant is about one centimeter in length. The giant ant has been scaled up by a factor of one thousand, to ten metres in length, but the same proportions have been preserved. Thus it is also one thousand times wider and one thousand times taller than it was. Its total volume, and total weight, is thus one billion times (1000 x 1000 x 1000) what it was. To understand this, let us suppose that you plan to build a small backyard swimming pool which is intended to be six (6) metres long, three (3) metres wide, and one (1) metre deep. Suppose that you can fill such a pool with one tanker truck full of water. Unexpectedly, you win the lottery and decide to put in the elegant larger pool you have always dreamed of owning, one which is twice as long as your first design, but which has the same pleasing proportions. It will be twelve metres long instead of six, six metres wide instead of three, and two metres deep instead of one. By scaling up the size of the pool by a factor of two in each dimension, you have now designed a pool which will hold eight times (2 x 2 x 2) as much water. (In doing this calculation, it does not matter that the original dimensions of the pool - the length, width, and depth - were not all the same. Since you doubled each of the three dimensions so that the exact proportions would be preserved, the factor by which the volume increases is obtained merely by multiplying the scale factor together three times, once for each dimension.) So, too, with the ant. A thousand-fold increase in each of its dimensions gives it one billion times the volume and, since it still made of `ant stuff', a billion times the total weight. The strength of a leg, however, depends roughly on its cross-sectional area. (This is a general rule in engineering, and applies to girders, timbers, and other building elements, as well as to scaled-up ant legs. One girder twice the size of another can support about four times the load, if it is made of the same material and preserves the same proportions.) Since the ant's leg is 1000 times wider and 1000 times thicker than it was, the leg has one million times (1000 x 1000) the cross-sectional area it had originally, and it is undeniably much stronger than it was - but the load it has to bear (the ant's weight) is disproportionately larger still, and the leg simply collapses under a billion times the original weight. The ant is immobilized. At the other end of the weight scale, we find the elephant, which is very large -- up to about five tons mass. To support this enormous bulk, the elephant has very thick legs, not the spindly equivalent of the ant's limbs, and disproportionately thick bones. A famous twentieth-century version of this discussion appears in an essay entitled "On the Importance of Being the Right Size", by the British biologist J.B.S. Haldane, but the line of thought is certainly not new. It appears in the writings of Galileo, for instance, in his ``Dialogues Concerning Two New Sciences,'' first published in 1638. There, Galileo presented the following figure contrasting the very different proportions of bones from small and large creatures: Considerations of this sort explain any number of everyday phenomena, such as: why ants can lift such big loads, and why grasshoppers can jump so high relative to their body sizes (their muscles are disproportionately strong, and the weight to be lifted is light) why gymnasts are of small stature but very muscular build, rather than looking like NFL linemen. (Can you imagine an NFL lineman, who is undoubtedly very strong in an absolute sense, lifting his own body weight in the sort of manoeuvre you see gymnasts carry out on the rings or high bar?) External factors can matter, of course. The Blue Whale, for instance, is the largest animal ever to live on the Earth, but can be so much larger than land mammals only because its body weight is supported by the buoyancy of water. Once stranded onshore, however, a whale quickly dies because its own body weight compresses its heart and lungs so that they cannot function. There is an optimal size to everything, depending on its environment. In class, I presented to you the example of the tallest person who ever lived, Robert Wadlow, who was almost nine feet in height and still growing at the time of his early death at the age of 22. His death followed an unsuccessful fitting of braces to his feet, with subsequent infection. The braces were needed largely because his legs could not support his own enormous body weight. It is not much exaggeration, therefore, to say that he died tragically young simply because he was too big. (There is a well-illustrated website which tells you more about this young man.)
The Relevance of External Gravity.The scaling arguments I have been discussing are only really relevant in the context of life found on the surface of the Earth, the gravity of which pulls us down and imposes a load -- our own body weight -- on our joints. If you were an ant floating in empty space, but supplied with food, water, and air, there would of course be no structural reason why you couldn't grow very large indeed! Your legs would not snap, since they would not be under the stress of supporting your body weight. As it happens, however, other factors would matter, and you would be doomed none the less. For instance, ants have no circulatory system to get air to all their living tissues. The oxygen merely diffuses into their very tissues, through holes in their exoskeletons which expose the tissue surfaces. For a very big ant, the air would not diffuse far enough to nourish all the tissues, and the ant could not survive. Naturally this would be a consideration on Earth as well: even if the giant ant in 'Them' could overcome its strength limitations, it would still be unable to breathe and would perish. Larger creatures must and do have ways of distributing nutrients to their tissues: central circulatory systems carrying blood into which sugars and oxygen have been dissolved.
The Importance of Proportions.Being the right size can determine how an object or a living body functions, and determines its proportions. But the proportions determine many other relevant factors in addition to mere structural stability, thanks to a very important fact which we will return to repeatedly in astronomy: big objects have less surface area per unit of volume than small similarly-shaped objects do. In class I gave a series of everyday examples which exemplify this point. Making Apple Pie: If you wanted to make an apple pie, would you prefer to peel one hundred tiny apples the size of grapes or one very large one the size of a pumpkin? Your instinctive (and correct) reaction is to choose the large apple; but why? The answer is another example of a scaling argument. Consider two apples, one of which is ten times the diameter of the other. Since the surface area (or total amount of peel) of an apple depends on the square of the radius, the large apple has one hundred (10 x 10) time as much peel as the small apple. But the volume of the apple (the total amount of apple flesh it contains) depends on the cube of the radius, so the large apple has one thousand (10 x 10 x 10) times as much flesh as the small one. For the same total effort, you could peel one hundred small apples or one big one; but the latter option gives you as much flesh as you would have gotten from one thousand small apples. The choice is clear. Radiators: Have you ever looked at a car radiator? I remind you that this is a device to keep your engine from overheating: hot water circulates through it and radiates some of the engine's heat away to the air. To the eye, it looks like a flattened lump of metal, with the proportions of a thick book, and you might be excused for assuming that it is effectively a simple metal box with solid sides. But close inspection reveals that it contains a maze of tiny holes and pipes which allow the air to pass through and around it. Why? The answer is that the heat is most efficiently radiated away to the air if the radiator has a large surface area, with lots of exposure to the surrounding atmosphere, so that heat can escape in many directions. Putting the small holes and pipes into it has the desired effect. The Paradox of the Porridge: When you were a child, I wonder if you were as perplexed as I was by the paradoxical behaviour of the porridge in the story of Goldilocks and the Three Bears. You will remember that Goldilocks finds Papa Bear's porridge to be too hot, Mama Bear's porridge to be too cold, and the porridge of Baby Bear to be just right. But if the porridge was put out at the same time in three different-sized but similarly-shaped bowls, then the porridge in the small bowl should have cooled off most quickly. Once again, this is because the total heat in the bowlful of porridge depends on the volume of the porridge, but the rate at which it loses heat depends on the surface area. For the small bowl, the surface area is larger relative to its volume, and the cooling should be more efficient. (That is why a spoonful of soup cools off so rapidly that it can be swallowed even if the bowlful stays boiling hot.) But why, then, did Mama Bear's porridge cool off so quickly? I could never accept the illogic of that behaviour. Interestingly, some scientifically-literate tellers of the tale avoid the problem by recounting that Goldilocks finds the three bowls of porridge to be of different saltiness. Anyway, the rest of the story hangs together pretty well in physics terms (except perhaps for the talking bears!). For instance, the story is sensible in the presumption that the smallest bed, with its thinner, weaker legs and frame, would collapse under the Goldilocks's weight. If Papa Bear's bed had collapsed, you would probably found that part of the story implausible! Yet the physics of the porridge is just as faulty. A Campfire: When you are leaving camp and want the fire to die, you spread it out to allow the heat to dissipate; if you want to prolong the fire, you bank it up into a tighter pile, minimizing the surface area and thus the efficiency with which the heat can be radiated away. The Effects of Amputation: If you need to have a leg amputated through some misfortune, you will lose some fraction of your body weight but a rather larger fraction of the surface area of skin with which your body is covered. This can actually matter in the context of regulating your body temperature and so forth. Such factors have to be considered if the continued good health of double or multiple amputees is to be guaranteed. The Human Brain: Our general theme finds applications other than in heating and cooling. Think of the human brain, which looks rather like a grapefruit in size and shape. But rather than being smooth, like a grapefruit, it is amazingly convoluted, with `ins and outs' which make it resemble a cauliflower. For an object its size, it has a very large surface area by virtue of these convolutions, and interestingly this seems to be an important factor in whatever it is that gives humans the edge in intelligence over other animals, the brains of which typically have much smoother surfaces.
Self-Gravity Can Determine Form.So far my discussion of scales and function has focussed on things like ants and elephants, which are affected by the gravity of some nearby immense object like the Earth, or on radiators, where the Earth's gravity is irrelevant (except in the mundane sense that it determines that everything will `stay put'). Moreover, the gravitational force of nearby human-scale objects, even fairly large ones, can safely be ignored. For instance, when you walk past a city bus, you don't have to worry about its gravitational attraction pulling you off the sidewalk and into the road. (Such a force does exist, as we will see, but it is feeble almost beyond telling.) Nor do you have to worry, when you reach out for your morning cup of coffee, that you are moving your hand away from yourself, and that your own body's gravitational pull on your hand is resisting that action, causing you to work extra hard to overcome the resistance. In astronomy, by contrast, we are dealing with very large bodies, like planets and stars. They are dominated by self-gravity, by which I mean that the shape and proportions of any object this big are determined by the way in which the various parts of it feel and respond to the gravitational forces exerted by its own other parts. This consideration explains, among other things, why the Earth and all sufficiently large objects must be spherical (or nearly so). Why should this be? Why couldn't the Earth take on any shape into which we might choose to sculpt it? After all, you are used to the idea that the rocks of the Earth are very rigid and inflexible, at least the ones immediately beneath our feet. Couldn't you mould the Earth into any shape you desired, given sufficiently powerful earth-movers and other equipment? The answer is no. Suppose, for instance, you tried to use enormous bulldozers to raise mountains to a height of even just a few tens of kilometers. You would not succeed: the sheer weight of material pressing down on the lower layers would overwhelm the structural strength provided by the rocks, and they would slump and flow sideways. The new-built mountain would collapse. To understand this, imagine putting a slab of soft clay onto a tabletop and leaning on it. The clay will flow out to either side. If you replace the clay with a slab of steel, you could not press down hard enough on your own to have a similar effect, but hydraulic presses can and do overcome the steel's strength and make it flow. (The presses are very much stronger than you are!) What I am saying is that for objects of a sufficiently large size, no ordinary material can resist the downward pressure of gravity, which acts in a way analogous to the behaviour of an enormous press. This is made clearer if you visualise a large mountain as something like a tiered wedding cake, with a bottom layer which is an immense slab of rock, a few kilometers thick. (See the figure, below.) The rest of the mountain -- the huge heap of rock which sticks up into the air above this base -- is pressing down with all its enormous weight on the bottom layer, thanks to the downward gravitational pull exerted on it by all the atoms contained within the huge volume of the Earth itself. If the mountain is made too big, the base will be unable to support the weight of so much rock above it. The rocks of which the base is made will quite literally flow out to the sides, just as the steel could be made to flow in a hydraulic press, and the mountain will slump and subside. From the known strengths of materials, in fact, you can calculate just about how big a mountain could ever be, here on the Earth: it works out to be roughly the size of Mount Everest. On Mars, which is a smaller planet, the gravity is less, and mountains can be about three times larger. Remarkably, this was known before any spacecraft ever visited Mars, and Carl Sagan, an astronomer at Cornell, predicted that we might find such a huge mountain on Mars - which we did! It is called Olympus Mons (Mount Olympus); it is the largest mountain in the Solar System. In astronomy, therefore, any body of sufficiently large size, bigger than a few hundred kilometers in diameter, MUST be spherical (although to be absolutely correct I should point out that if it is spinning very rapidly, it can be somewhat flattened out, although still possessing a very smooth and regular surface). The important thing to remember is that the apparently very irregular surface of the Earth, with features like hills, mountains, and valleys, is quite misleading: these features are big only compared to us. Relative to the size of the Earth itself, these features are tiny and easily understood in terms of the limited strengths of rocks and minerals, as I have noted. Bigger features simply could not survive. By the way, if you were to build a scale model of the Earth about a meter in diameter, like a very large beach ball, even the very highest mountains on it would be represented by little bumps about half a millimeter in height, scarcely detectable to the touch. (You may be familiar with globes which have been made with a rough surface to represent the mountains and so on. Such features are typically not made to scale, but are instead much exaggerated, to make them readily detectable.)
Self-Gravity Determines Temperature.Everyday experience teaches us that when an object falls under the influence of gravity, it picks up speed and hit the ground with some force. In the terminology of physics, it first has some amount of potential energy, that which arises as a result of its location. You can harness this energy to accomplish some work, like driving a nail into a piece of wood, by dropping a heavy stone onto the nail. (The water falling over Niagara Falls is used to drive turbines and create hydro-electricity.) Immediately after its release, the potential energy is converted to kinetic energy, the energy of motion, but shortly thereafter the falling object hits the floor and is brought to a halt. (Think, for instance, of a book falling onto the floor of the lecture room.) The body is again at rest, but in a new location, lower down than it was originally, and thus possessing less potential energy than it had originally. Has some or all of the energy vanished? The answer is no. We will see later in the course that modern physics includes a rich variety of so-called Conservation Laws, one of which, the Law of the Conservation of Energy, means that the energy does not just vanish but must be present in some other form. In the case of the book falling onto the floor, the enery is dissipated in various ways, including the production of sound: the molecules of air are made to jiggle around (and you hear the sound of the impact as the disturbance reaches your ears). Of more interest is the fact that the force of the impact sets the atoms and molecules in the floor and book to jiggling around with increased vigour. It is the random jiggling about of the atoms and molecules that we call heat and quantify with a measurement of temperature. (See page 123 of your text to learn more about what temperature means in terms of the fundamental behaviour of atoms and matter.) So far we've been considering dropping one object onto a second pre-existing one, like a book onto a floor. Instead of that, think now of a great number of atoms, molecules, particles, pebbles, and what-have-you widely distributed throughout empty space, long before the sun itself came into existence. Each particle feels the gravitational tug of all the other particles, and if the conditions are right (something we will come back to later) the particles will start to move towards one another, drawing together in more concentrated fashion. Each particle picks up speed as it falls inward (just as you fall faster and faster if you jump off a cliff) because the combined gravitational force of all the other particles continues to act on it . Moreover, the gravitational forces get even stronger as the particles draw closer together, so the infalling material accelerates quite a lot. As a consequence, the particles are moving very rapidly indeed by the time they have fallen into the central regions, where they meet and collide. The random coming together of all these particles, and their consequent myriad impacts, lead to a general vigorous jiggling about of all the constituent particles. As noted above, however, this random jiggling is what we call heat. In other words, big objects which coalesce under the influence of gravity will be hot.
Putting It All Together.We have seen that gravity makes big things spherical and hot. This immediately explains why the universe is full of very hot round objects: we call them stars! We have also seen that big spherical things have proportionally less surface area than do small spherical things. (Remember the apples!) This consideration alone explains why the Earth is not like the moon. Although both the Earth and moon were probably formed in much the same way at nearly the same time, a few billion years ago, the moon, with a relatively large surface area, has been able efficiently to radiate away its internal heat (like Baby Bear's porridge ought to have done!). But while the moon has turned into an inert dead lump of rock, the Earth, less efficient at getting rid of heat, still has a molten core. As a result, it is undergoing active geological processes like continental drift and so forth. It is all because of size - and gravity. Gravity rules the universe! Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Friday, 28 January, 2022.)