| Newtonian Gravity Explains It All:
Action At a Distance.
So far, Newtonian mechanics had only speculated about the motions of particles acted upon by forces which you can easily tell are at work. For instance, if I whirl a stone about my head on a string, it is clear that the reason the stone does not fly off in a straight line (as predicted by the First Law) is that it feels the restraining force applied by my hand through the string (and the tug I feel on my hand is understood in terms of Newton's Third Law). Of course, if I let go of the string, the stone does fly off in whatever direction it is moving when I release it: thus David is said to have slain Goliath with a slingshot. But what of something like an apple falling to the ground? What force is making it do so? There is no obvious connection -- no string -- pulling it downward.
Newton reasoned that there must be some force -- obviously an attractive one, since everything falls down in the same direction -- which could reach across the space between objects. He realized that it must be a weak force, since our puny muscles can withstand the force of the entire planet acting on us. Moreover, he was able to work out that the effect of the gravitational force must fall off like the square of the distance. In other words, a body ten times as far away from the sun as the Earth is would feel only one one-hundredth of the gravitational force the Earth feels from the sun.
Why inverse square? That is in fact a very deep question, not easily answered right now. I will return to it later when we talk about modern theories of gravitation, which differ in some profound ways from Newton's formulation. (Since the time of Einstein, our understanding has changed in complex ways.) But I can tell you how to remember that it is an inverse-square law: simply consider the analogy of a paint sprayer being used to paint a wall (or the analogous behaviour of light, as shown on page 523 of your text). If you stand a certain distance from the wall, so that your sprayer nicely covers 1 square meter of the wall, then your sprayer will cover 4 square metres if you back up to twice that distance. The paint is thinned out by a factor of four, as the diagram makes clear, and the wall gets a thinner coat. This analogy may help you remember the form of the law and why gravity weakens with distance as it does.
By the way, the story of Newton sitting under the apple tree and speculating about why an apple fell onto or near him may be apocryphal, but seems to have had at least a little basis in fact. A lot of profound science stems from someone asking a deep question about an everyday phenomenon.
Testing the Law.The equation describing Newton's "law of universal gravitation" is discussed on pages 138-139 of your text. Notice that he assumed that it applied right across the whole observable heavens, which is why it is called "universal." That is, Newton hypothesised that it should explain not just the falling of an apple but also the motion of the stars and planets, an extrapolation which was very bold. We can only speculate about whether or not Newton considered a geometric analogy like that of the paint-sprayer in developing the law of gravitation (or rationalizing it to himself). What we do know is that he tried to test the dependence, to see if it really was inverse-square and far-reaching. How do you imagine he did that? One obvious possibility is to drop an apple and see how quickly it accelerates downward under the pull of the Earth's gravity, then to go up on top of a high mountain so that we are somewhat farther away from the bulk of the material constituting the Earth. A dropped apple in such a location should feel somewhat less gravity and accelerate less rapidly. This is indeed true, but the problem is that such a change of position is really too small to matter much, even if we climb a very tall mountain. As we will see below, it is our distance from the center of the Earth that matters, and even the top of the tallest mountain is not much farther away than we are at sea level when compared to the 6000 kilometer radius of the planet. It would be better to get very far from the Earth, and drop something from a great height. By Newton's reasoning, it should accelerate only slowly downwards. How can we do this experiment? In modern times, you can imagine going off in a rocket to deep space and doing a controlled experiment, but the amazing thing is that Newton thought a way in which Nature is already doing it for us! He realized that the moon is in fact continuously `falling' as it orbits the Earth. Its sideways motion would carry it off in a straight line except for the Earth's gravity, which causes it to fall toward the Earth by about 1.4 mm for every kilometer of sideways motion. The effect of this continuous behaviour is that the moon orbits the Earth in a near-circular orbit. In fact, Newton's calculations of the rate of the moon's downward fall yielded exactly what would be expected by analogy to the falling apple, except that it is much reduced by the hypothesised (and now confirmed) inverse-square weakening of the gravitational force.
Gravity: A Modern Perspective.There is one more point I should make about gravitation. I have remarked on its weakness and its inverse-square dependence on distance, but it is also important for you to realize that it is not really special in quite the way Newton thought. He saw it as different from other forces in that it could reach out across empty space "without strings" and provide what he called `action at a distance.' He saw this as completely distinct from everyday forces such as are provided by strings pulling, hands pushing, and the like. In fact, this distinction is irrelevant. The forces provided by strings and hands are macroscopic (i.e. large-scale) manifestations of a microscopic equivalent of the same `action at a distance', except that in this case the force at work is that of electromagnetism. Fundamentally what is happening is that there are repulsive forces between charged particles in your hand and the book you are pushing. The laws that describe these repulsive forces are themselves inverse-square and describe influences which act across the space between the constituent particles: strictly speaking, the electrons and other atomic constituents never really "touch" at all. (Of course, this is a matter of semantics, complicated by the fact that we tend to visualise electrons and other sub-atomic particles as tiny billiard balls, when in fact they are something rather different, as we will learn). In any event, the modern physics understanding of forces has changed considerably, so these kinds of discussions and distinctions are in a sense irrelevant now. In fact, it is currently theorised that gravitation and all other forces, including the electromagnetic force, are simply different manifestations of one force. The underlying mathematics of this is intimidatingly complex, but I will say a bit more about it conversationally later in the next term, in Phys 016.
Calculus, and a Remarkable Simplification.One early problem which Newton faced was to figure out the cumulative effect of a large distribution of matter. To see what I mean, visualise a person standing on top of the spherical Earth, right at the North Pole. Every single atom in the person is attracted by the gravity of every single atom in the Earth, but what is the net effect? For instance, right below the person's feet there are some atoms which should have a relatively strong effect owing to their proximity; but there are many atoms which are farther away, and perhaps by sheer weight of numbers they can dominate, even though individually they are rather far off. Which atoms contribute more to the total downward tug? What is the cumulative effect? To solve such problems, Newton invented what we now call calculus! (Those of you who know some calculus will see the germ of the idea here. How do we add up the grand total effect of a bunch of little pieces, each in a slightly different location? This is really the basis of the integral calculus, as you may have recognized.) The remarkable thing he found was this: a body which has spherical symmetry -- a term I will explain in a moment -- has a total gravitation outside it which is just as strong as though all the matter were concentrated at the centre. In other words, if we were orbiting the Earth in the Space Shuttle and the Earth were suddenly and mysteriously compressed to the size of a golf ball, or a billiard ball, or half its present radius, its gravitational effect on us would be unchanged (provided all the mass were still present, merely squashed together to much higher density). We would continue to orbit just as before! On the other hand, suppose we were to expand the Earth so that its material was reconfigured into a perfectly uniform shell (like a basketball, with a hollow center). If we were now to position ourselves inside it, we would feel none of its gravitation, no matter where we were located! Why not? It is the same reasoning: the material located above us pulls us upward, and the material below us pulls us down. Just as in Newton's original problem, it doesn't matter where we are in the hollow Earth -- we don't have to be precisely located right at the center. The forces balance precisely as long as there is a spherical symmetry to the distribution of material. In calculating the net gravitational force exerted, what is important is the total amount of matter closer to the center of the spherical distribution than you are yourself, and the distance of interest is how far you are from that center of symmetry. Notice that there is no requirement that the spherical body be uniform throughout: it doesn't, for instance, have to have the same composition or density everywhere. It merely needs to be symmetric, which means quite simply that if you drill down to the center from somewhere on the surface you will progress through the same kinds of material and conditions, in the same progression, as you would if you drilled in from anywhere else. The object could be layered, like the Earth itself is (it has a crust made of rocks of one sort, underneath which there are a mantle and a core made of yet different materials), or of uniform composition but perhaps more densely packed in the central parts. The important issue is the total mass!
Digression: A First Thought About Black Holes.In the second term, in Phys 016, we will consider what happens to stars when they die. Some of them become black holes, objects in which all the constituent matter has been squashed down into a vanishingly small volume. Not even light can escape from such objects; hence the name. Suppose, therefore, that the sun suddenly became a black hole (which it never will, as we will learn). That is, suppose all the material in the sun were to be squashed down into an infinitely dense ball, of sub-microscopic size. The Earth would then .... what? Would we be sucked into it, as if by a vacuum cleaner? Or would we fly off into empty space? Neither of these! The Earth would still feel the same gravitational force as before, and would continue to orbit a now-invisible sun. Of course we would be cold and in the dark, but the sun would not act like a cosmic vacuum cleaner and suck in all the planets. Of course, if such a shrinkage did happen, a space traveller would no longer be bothered by the terrible brightness and heat of the sun, so would be able to travel very close to it -- something we dare not do now, since we would be burnt up. But if you did approach very close to the much-compressed sun, you would experience an enormously strong gravitational pull. If you released a small object at rest near the shrunken sun, it would be so strongly attracted by gravity that it would accelerate towards the sun at blinding speed. Why? To understand this, look at the equation on page 138, or simply remember the inverse-square law. Imagine the sun the size of a golf ball, with all its atoms squashed into this tiny volume, and yourself standing just one metre from it. You would now be just one metre from every single one of those atoms, and the cumulative gravitational force would be colossal - strong enough, in fact, to prevent light from escaping (hence `black hole'). If, by contrast, you found yourself exactly 1 metre from the surface of the present sun, which is much more spread out (it is over a million kilometers in diameter), you would be quite near some of the atoms but a very long way from most of them, so the net gravitational pull would be moderate in strength. In other words, things like neutron stars and black holes (about which we will say much more later) have gravitational fields which are very strong locally, but if you are a long way away they are no more dangerous or effective than if the material was distributed as it is in a commonplace star.
Point Masses.As we have seen, Newton's invention of calculus led to the remarkable discovery that spherically symmetric bodies can be treated as though all their mass is right at the very center. This allows for a great simplification in understanding the motions of the planets, sun, and stars. We can treat them as point masses. That is, the orbits of the Earth and planets moving about the sun are just as they would be if you were to replace them by golf balls of the same mass -- there is no need to be concerned over the fact that their material is actually distributed over large volumes (by human standards). This is a tremendous mathematical simplification, although of course we could have handled, albeit with great inconvenience, the need to allow for the effects of every single particle of matter in calculating the total gravitational pull. I should stress that the simplification comes from two things. The first is the mathematical form of the law, the fact that gravity obeys an inverse-square law. (If the law had another form, we would almost certainly not have enjoyed such a pleasant simplification, no matter how the atoms might be distributed.) But we would never have experienced the benefit of this in practice unless the matter in the universe were obliging enough to be distributed in spherical lumps -- as of course it is, thanks to the physics of gravity and its ability to overwhelm other forces. It is the which makes the stars and planets spherical in the first place! If stars and planets could be shaped in other ways, like pretzels or bricks, we might have great difficulty in working out their mutual gravitational effects even though the inverse-square law is exceptionally simple. As a matter of fact, rapidly-rotating stars are not spherically symmetric: they get flattened somewhat by the rotation, like a pizza dough whirled into the air. A planet orbiting such a star would experience forces and engage in motions which would be complicated by this fact. Similarly, the rotating Earth is so too is Jupiter. Indeed, Jupiter rotates faster than the Earth and (being made mostly of hydrogen gas) has less inherent structural rigidity, so the flattening of Jupiter is much more noticeable. Even for the barely-non-spherical Earth, the complicating modifications of the gravitational field cannot be ignored in doing extremely detailed calculations about how artificial satellites will orbit for years or decades to come. But in general this is a negligible problem.
Ballistic Motions.Before going on to consider the motions of planets orbiting the sun, or objects falling to ground (in situations where we can ignore air resistance), let us recognize that they are merely coasting -- that is, they have no rockets squirting out of them, or ropes pulling on them. Such objects are said to be in ballistic motion, following trajectories which are determined only by gravity. It is objects of this sort which we will have to consider when we reproduce Newton's speculations about the way in which fast-moving bodies move when influenced by the gravitational field of the Earth. You may have heard the word ``ballistic'' before in the context of ICBMs -- InterContinental Ballistic Missiles. These constitute part of the weaponry which was so threatening in the deepest part of the Cold War. The danger was that the Russians (or conceivably some other nation) might launch missiles which would be rapidly accelerated using rockets, which would then be shut off to allow the missiles to coast ballistically on a pre-determined path, responding only to gravity (and any air resistance as they come back down towards the surface), until they land on a remote target.
Kepler Explained -- and Expanded!The real beauty of Newton's theory of gravitation, coupled with his three laws of mechanics, is that it showed why the planets behave as they do in every respect. All three of Kepler's laws `popped out' of his analysis, although -- the one describing the shapes of the orbits -- became modified a little. We'll return to the specifically in the next sub-section, but for the moment consider The speed of motion of a planet in its orbit, and thus its orbital period, must clearly depend on the planet's distance from the sun because of the falloff of the strength of the sun's gravitational force. (It is this force which pulls on the planets, and makes them change direction in accordance with Newton's Second Law.) For instance, if Mercury, which is close to the Sun, travelled as slowly as Saturn does, it would not have enough speed to maintain a near-circular orbit, and would fall in precipitously towards the Sun. If Saturn travelled as fast as Mercury does, it would move tangentially outwards, away from the Sun, rather than move in its near-circular path. To clarify your qualitative understanding of this dependence, you might try to visualise yourself riding a bicycle around an enormous funnel, like the `Wall of Death' you can see sometimes at the fair. If you are out where the surface is flat, far from the steep center of the funnel, you can ride very slowly; but if you venture into the throat of the funnel you have to pedal like fury to keep up a rapid pace if you are not to fall right to the bottom. (Of course, the cyclist has the extra problem of needing to combat friction between the tires and the surface, plus wind resistance; the planets merely coast ballistically, but must do so at a speed which maintains their orbits.) Anyway, Newton explained Kepler, but as encouraging as these successes were, Newton did even better. He was able to generalize the laws in an important way: Newton realized that one body moving near or past another and feeling its gravitational tug could travel along a path which is not an ellipse, depending on its initial speed and direction of motion. Let us consider his reasoning. Newton introduced an instructive "thought experiment." Imagine climbing a very tall mountain (the point of this is to get yourself high up above local obstructions and most of the Earth's atmosphere, to minimize air resistance and so forth). Now imagine using a gun to shoot a bullet sideways. What will happen? Clearly, it depends on the speed with which the bullet is launched, so let us consider various possibilities. A Slow-Moving Bullet: Suppose the propellant (gunpowder) in the bullet is damp and scarcely ignites, so that the bullet comes out rather slowly. Clearly, it will fall almost straight to the ground. If the bullet moves with the sort of speed that a real pistol would produce, it will fly off sideways but also curve down toward the ground and eventually hit the surface of the Earth, which inconveniently gets in the way. In the solar system, which is mostly empty space, orbiting objects do not generally run into other objects or surfaces which impede their progress, so we would like to eliminate this unwanted interference. Fortunately, we can call to our aid the great simplification discussed earlier! In carrying out Newton's thought experiment, we really only need the Earth to exist in its present form long enough to allow us to climb the mountain and fire the gun. Once the bullet is on its way, the Earth could instantaneously be shrunk to the size of a golf ball without changing its gravitational effect, so let us suppose that it conveniently does exactly that. The result, as Newton was able to show, is that the bullet would simply move in an ellipse about the compact Earth and eventually return to the starting point, after which it would keep on orbiting. The bullet, of course, is moving in response to the gravitational influence of the Earth itself, which is turns out to be at one of the foci of the ellipse. This is simply a manifestation of Kepler's first law! The bullet which I described, with modest initial speed, has so little sideways motion that gravity makes it fall in rapidly towards the (golf-ball-sized) Earth. The farthest it ever gets from the center of the Earth is when it comes back to the point from which it was launched, a point to which it returns on every orbit thereafter. Our launch point is said to mark the apogee of the orbit, the point of farthest excursion from the Earth. (A planet orbiting the sun is said to be at aphelion when it is farthest from the sun, and at perihelion when closest to it. Remember that 'helios' is the Greek word for the Sun. The equivalent terms for satellites orbiting the Earth are apogee and perigee. ) A Speedier Bullet: If we gave the bullet just the right amount of sideways motion, it would move in a perfect circle around the Earth. (No real gun can do this! A speed of about 18000 miles per hour is needed). That is the general objective of NASA when they launch the Space Shuttle: it is launched to a height of about 100 km above the surface of the Earth and then its speed is adjusted with its rockets until it is moving in essentially a perfectly circular orbit, after which the rockets are turned off. The Shuttle spends almost all of its time moving ballistically, coasting under the influence of the Earth's gravity. A Yet Faster Bullet: If the bullet were launched somewhat more rapidly still, it would move tangentially away from the Earth, climbing in altitude to some extent before falling back in a long looping orbit. The orbit would again be an ellipse, but this time the launch point is the perigee, rather than the apogee (as was the case for the slow-moving bullet). An Ultra-Speedy Bullet: Suppose, finally, that the bullet could be given a lot of speed. Newton was able to show that it could in fact escape the Earth completely: it could move off and never come back, although gradually losing a bit of its speed because of the retarding force of Earth's gravitational attraction. There are now two possibilities, but before exploring those we first have to consider an important qualification. A real particle (a bullet, a rocket, ...) launched from the Earth moves out into regions in which its trajectory is determined not just by the Earth's gravity (which retards its motion) but also by the gravity of other objects, including the sun, other planets, and the stars. As presented here, Newton's arguments are in fact very much oversimplified in the sense that they treat the Earth as though it is alone in space. In reality, of course, a projectile fired at high speed into the depths of space would not merely coast forever in a straight line. Its actual path would be determined by the complex gravitational field of everything else out there, not just that of the Earth. NASA has to consider all that when sending a deep probe out into the Solar System! Having said that, let us visualise a universe in which we have only the Earth and the experimenter who is firing the gun. Consider two distinct cases: Critical Velocity, or Escape Speed: If the bullet is launched with only just enough speed to make an eventual escape, it winds up moving more and more slowly as it recedes. Eventually it finds itself far from the Earth, scarcely moving at all. Had it been launched just a little slower, it would come to a halt and fall back; had it been launched somewhat faster, it would still have some significant speed as it continues on its way. Objects which are launched with exactly the critical velocity move off along a path which in fact has the geometric form of a parabola, at least in the idealised case of there being no other gravitational influences of any significance out there once the particle gets far away from the Earth. In the real universe, of course, the particle will change trajectories in response to the gravity of other objects. But any time there is a single dominant source of gravitational force, Newton's thought experiment provides a perfect description. Let's turn the argument around, and think about an incoming object. (See the figure on page 141.) Imagine a comet sitting essentially motionless in space, a very long way out from the sun. Suppose that over billions of years it slowly inches its way ever so slightly closer to the sun. It now feels a slightly stronger gravitational tug, and the imperceptible pace speeds up a little. Eventually, this process accelerates, and the comet will wind up, after a long inward fall, racing past the sun in a parabolic trajectory. After that, it moves on a long outward trajectory, slowly losing speed, until it comes to an effective standstill, very far out. We see such comets from time to time, one-time visitors from the depths of space. But these rare visitors don't always get away! Such a comet can instead make an abrupt change of direction -- something not anticipated by Kepler. Why? Again, it is a question of the influence of other objects which complicate the issue. If the orbital path of the visitor should, for instance, bring it too close to Jupiter, the gravity of that planet will pull it off-course, and the predicted parabolic orbit is not followed. (See the figure on page 145.) The lesson is that you always have to be aware of the influence of other massive objects in the vicinity! Excess Velocity: Particles which are launched with speeds in excess of the critical velocity will, of course, escape the Earth entirely, and will still be moving with some significant speed even when encountered much later, way out in the depths of space. Particles of this sort follow paths which are hyperbolic.
Conic Sections.What do these various possible orbits have in common? They sound very different! Ellipses, circles, parabolas, hyperbolas..... In what sense is a parabola anything like a circle? Well, here is the generalization I referred to earlier. Newton was able to show that a particle moving past a body under the influence of its gravity will follow a path which is in the form of a conic section. All of the aforementioned curves are conic sections of one kind or another. The name 'conic section' comes from the fact that if you take a slice through a cone, it will form a curve which is a circle, an ellipse, a parabola, or a hyperbola, depending on the angle at which you make the cut. (Visualise building an ice-cream-cone out of solid wood -- without the lump of ice cream on top! -- and then sawing through it at various angles. Look at the curved outline made by the newly cut edge of the wood.) Although this has a readily visualised geometrical meaning, it is their mathematical properties that makes them relevant to Newton's reformulation of Kepler's Laws. In short, Newton realized that Kepler's empirical laws did not tell the whole story. They describe only a restricted class of objects, those which happen to be moving around the sun in elliptical orbits. Of course, Kepler cannot be faulted for this! He was able to analyse the orbits of objects for which he had data -- namely, the planets! For them, the orbits are ellipses. (Indeed, they are nearly circular.) Kepler simply did not have the observational data to allow him to study something really eccentric, like Halley's comet (although I emphasise that Kepler's laws apply to that too). Nor did he know about objects like the comet described in the previous section, things which pass through the solar system only once. Objects which have too little speed to escape the Sun are said to be bound to it. They move in elliptical (or possibly circular) orbits. Something which is not bound, like a chunk of rock which enters the solar system from a remote part of the galaxy, will speed up under the Sun's gravitational influence, and will have enough speed as it passes by to eventually make its way out to very large distance again. In other words, infalling bodies cannot simply get "captured" by the gravity of the sun (despite the common occurence of this sort of theme in science fiction movies). For an infalling body to take up an orbit around the sun, it would have to lose speed, which could be accomplished if, by chance, the incoming object ran into something else. You will recognize from this discussion that space probes that are sent to other planets need to be slowed down, by a firing of their rockets, if they are to be put into orbit around the target planet. This is a real inconvenience, since it means that you have to carry a lot of fuel with you the whole way, and pay the penalty of having to get it moving along with everything else, so that you have it when you need it later. Unfortunately that really limits the payload of other more important stuff you can carry.
Kepler's Second Law Revisited.You will remember that Kepler described his Second Law in terms of equal areas being swept out in equal times. In fact, Newton realised that this was merely another way of describing the conservation of angular momentum. To understand this, consider an object (like an asteroid) moving through the inner solar system. Its motion can be divided into two different components: one directed towards (or away from) the central body; and one component which is "sideways." It is important to note that any part of the motion which is directed straight towards (or away from) the sun doesn't enter the calculation of the angular momentum: only the sideways motion matters. (See the figure on page 137.) An object falling straight in at high speed has no angular momentum whatsoever! The amount of angular momentum the asteroid has is given by "(its mass ) x (its distance from the sun ) x (the amount of sideways [ or transverse] motion it has)." Now since the mass of the asteroid doesn't change as it orbits (it always contains the same number of atoms), ignore that factor, and note that the angular momentum depends upon the product of its distance from the sun (call that R) and the amount of transverse motion it has (call that V). The makes the important statement that this calculated quantity does not change as the asteroid moves. Think of an experiment. If the Earth were mysteriously brought completely to rest, it would have no sideways motion and thus no angular momentum about the sun. Of course, the stalled Earth would still feel the sun's gravitational force, so it would start to fall towards (and eventually straight into) the sun. In doing so, however, it would never build up any sideways motion. This is an obvious consequence of the fact that the Sun's gravitational force is pulling it straight in, pushing it neither to one side or the other. But it is also a good example of the conservation of angular momentum! Once brought to a stop, the Earth has zero angular momentum, a circumstance which is true at every moment of the subsequent death plunge, even though the radial component (the inward speed) builds up enormously. Contrast this to the behaviour of a body like Halley's comet. Visualize it in the outer part of its orbit, just starting to move back in so that it has some very small motion towards the sun. It will also some small transverse motion because it is just rounding the slow outer curve of its orbit, moving in a long lazy arc. Naturally, as it subsequently moves inward, its radial speed will build up. (It is, after all, falling towards the sun.) But the conservation law now tells us that its transverse motion will speed up too, in accordance with the requirement that the product of V and R must equal some fixed quantity. As R becomes smaller, you must have a larger V: the speed of transverse motion must increase. In short, the object speeds up in both senses of its motion. Later on, when the comet moves away from the sun, similar considerations hold in reverse: it slows down in both components of its motion. This explains why a body like Halley's comet, moving on a very eccentric orbit, absolutely whizzes past the sun but then spends the vast majority of its time moving very slowly in the remote outer parts of its orbit. In more generality, one can show that this sort of thinking provides a complete explanation of Kepler's second law, which turns out to be nothing more or less than a statement that Kepler did not think in these terms -- no one did, in his era -- but spoke instead in terms of `areas swept out,' a geometrical visualisation. Part of Newton's legacy is that he allowed us to see broader physical principles in more generally applicable form. In class, I described the classical example of the conservation of angular momentum: the spinning figure skater who brings his or her arms in closer to the rotation axis and thereby speeds up the spin. In this case, of course, the atoms in the skater's arms are not moving inwards under the influence of the gravity of the skater's body, but rather in response to muscle power! Nor is there a central lump, like the Sun, which governs the motion of all the other atoms! Instead, you have a cohesive body which is moving as a unit in some rotatory sense. But the conservation of angular momentum is true in general, regardless of what kinds of forces are acting, and is equally applicable here. Thus, when the atoms in the arms are pulled in, they get closer to the axis around which you are spinning, and must speed up in their sideways motion so that angular momentum will be conserved. Since the whole body is an intact object, the pulling in of the arms leads to a speeding up of the rotation of the whole skater.
Dumbbells in Space.There is yet another important implication, this time one which follows from Newton's third law. This is most easily understood if we consider a somewhat simplified situation. Suppose, for instance, that the solar system contained only the Earth and the sun. (This allows us to ignore the extra tugs we feel from Jupiter and the other planets.) If there were no sun at all, the Earth would sail along in a straight line through empty space, as we know from the First Law. The presence of the large lump of matter we call the Sun changes that: the Earth does not follow a straight line, but rather a curved orbit, continuously `falling' toward the sun in response to its gravitational pull. (Indeed, only the Earth's sideways motion prevents it from actually falling right into the sun. As we noted earlier, this is exactly what would happen if we could `put on the brakes' and bring the Earth to a complete halt! -- it would start to fall straight towards the sun and would eventually run into it. This death plunge would take a couple of months, by the way.) So the Earth moves in response to the force the sun applies to it. But surely the third law tells us that the sun must feel an equal and opposite force towards the Earth. Why doesn't the sun fall towards us? In particular, if we brought the Earth to a halt, so that it and the Sun were both at rest, wouldn't they fall towards each other? The sun is more massive, admittedly, but surely it would move a little in response to the Earth's gravitational pull? How can the Sun be completely at rest in the middle of the Solar System? Even if we think about the Earth in motion, the problem persists. After all, in June we are on one side of the Sun; in January we are on the other side. Doesn't this mean that the Sun feels a yearly cycle of gravitational tugs in various back-and-forth directions? Why doesn't it move, even a little, in response? The answer is that it does. In fact the two objects (the Earth and the Sun) both move around their common center-of-mass, like the two ends of a dumbbell, or like two people on a teeter-totter going up and down with a balance point in between. But the sun is enormously more massive than the Earth (by a factor of about 300,000), so its tiny orbital motions are very small by comparison to ours. To understand this, it may help to imagine a teeter-totter with an adult and an infant on opposite sides. To balance it, the infant has to sit way out on the end, with the heavy adult much nearer the pivot point. Likewise, the `pivot point' around which the sun and Earth move is very close to the more massive sun, and indeed is right inside it, so the sun moves back and forth by an amount which is less in size than its own diameter as the Earth goes around its big orbit. I demonstrated the effect in class with some dumbbells which had balls of different masses on the ends. As you throw the dumbbell through the air, the center of mass follows a simple path, with the two balls wobbling around it as the dumbbell turns. Such objects, however, are very simple in comparison to the solar system, where the sun is surrounded by no fewer than nine planets, some of which are quite a bit more massive than the Earth. The gravitational interactions of all the bits and pieces are consequently quite complex. The sun's resultant `wobbling about' is small, because even the biggest planet (Jupiter) is quite a bit less massive than the sun. Its effect is, however, the dominant one: Jupiter and the sun represent the child and adult on the teeter-totter. The other planets, being quite a bit less massive, have smaller influences, rather as though a few flies and mosquitos were to land on the teeter-totter in various locations. While this affects the balance a bit, the effects are not very important in practice. None the less, the wobble from Jupiter, and to a lesser extent from the other planets, is real and measurable -- a direct consequence of the Third Law. We will see later that such wobbles allow us to detect and study binary stars. (Indeed, such observations have now even led to the detection of planets in many other solar systems. See pages 243-248.) This is an opportune moment to consider one way in which ``wobbles'' might lead to the identification of a binary star. Essentially all of the stars nearest the sun are seen to drift slowly across the pattern of more remote background stars as the decades and centuries pass. This tells us that the stars are actually moving through space. But some of them do not move in a straight line: they wobble up and down as they travel. This must be because they are orbiting some other object, with both objects moving around their common center of mass. The bright star Sirius is one example of this, and careful study following this discovery led to the finding of a very faint but fairly massive star of the type now called `white dwarfs'. We will study these later, in Phys 016.
Escaping from a Planet: Rockets.Earlier, we discussed the notion of escape velocity, the speed with which an object must be moving to escape the Earth's gravity. It is about 11 km/sec, or 25000 miles per hour. (To maintain a circular orbit, the Space Shuttle, or indeed any satellite orbiting the Earth at a height of about 100 km above the ground, has to be moving at about 18000 miles per hour.) My discussion might have left you with the impression that an object like a rocket absolutely must be moving at such a speed if it is ever to leave the Earth completely. This is not in fact correct, as I will now explain. Imagine firing a bullet upwards. As it rises, it is constantly being slowed (accelerated downwards) by the gravitational pull of the Earth. The interesting question is whether this will cause it eventually to come to a stop and to fall back to the Earth, as in the old expression "what goes up must come down". (Of course, a real bullet fired in the classroom, say, would hit the ceiling and presumably not escape, even if it had very high speed. Here we are considering the ideal case of unimpeded motion without obstacles and air resistance.) As noted, a speed of about 25,000 miles per hour (or its metric equivalent, about 11 km/sec) is needed for this to happen -- but only if you are considering objects moving ballistically (i.e. coasting, without rockets firing or anything like that). In principle, however, it would be possible to build a powerful rocket which could lift off very slowly, rockets blazing away, and gradually inch its way upward. You can imagine such a machine taking a week, say, to climb up to the level of the clouds, firing its rockets all the time. Or perhaps it would take a decade to slowly inch its way to the moon, travelling at an average speed of about 3 miles an hour (about as fast as a person walks). As I say, this is possible in principle. In practice, however, there are serious problems. The most important of these is that in such an approach most of the energy is being expended in holding the rocket above the ground. Since the rocket contains tons and tons of fuel which is on board for later use, you are wasting a lot of energy making only a little progress with the payload, the part of the device which is intended eventually to escape. It would be much better to get up to speed really fast, getting rid of redundant mass as quickly as possible. Burn up fuel rapidly and cast off empty fuel tanks as you go! Real rockets are designed in this way, and get up to speed quite quickly -- but not in a single instantaneous blast of energy at takeoff, as with a bullet from a gun, because the sudden acceleration would squash the astronauts and equipment on board. With this approach of a short early period of quick acceleration, we do indeed approximate a bullet being fired, and our rockets are quickly brought up to high speed after which they coast ballistically except for subsequent minor course corrections. But please remember that there is nothing in principle which prevents us from having long-firing rockets, or other slow means of acceleration, which could permit us to move at modest speeds and yet still eventually escape the Earth or indeed the whole solar system. (We will return to this theme much later in a discussion of interplanetary and interstellar travel.) So the notion of "escape velocity" is valid and important, but please remember its limitations.
Escaping from a Planet: Gases.The notion of escape velocity has an immediate and practical application when we consider the behaviour of the atmosphere of a planet. Before exploring that, let us ask ourselves how it is that we can breathe, given that the Earth has a gravitational pull that causes things to fall to the surface. Shouldn't all the molecules and atoms of air be lying right down on the ground? Yet if you climb a hill, you can still breathe. The answer lies in the fundamental notion of temperature (or the heat content ) of a material. The temperature of a gas (or a liquid or solid) is basically a measure of how much energy it contains, as manifested by the energy with which the individual atoms jiggle about or race around. In a solid, the atoms are in more-or-less fixed locations with respect to one another. In a liquid, there is some cohesion, but things can slop around. In a gas, individual atoms or molecules move independently. A solid (like ice) melts because as we add heat to it the atoms and molecules jiggle with increasing vigour until the bonds between them are disrupted. In our gaseous atmosphere, the atoms and molecules are moving very quickly and undergoing enormous numbers of collisions with one another. (There are, for instance, many more molecules in a cubic meter of the air in the lecture room than there are stars in the entire observable universe). If you consider a particular molecule high above the ground, gravity acts to cause it to fall towards the ground, but as it does so it is buffeted in collisions with trillions of other atoms, and the net effect is that there is a sustaining pressure which keeps the atmosphere puffed up. Think of an immediate application of this: a hot-air balloon. A propane burner heats air which flows into the balloon. The heat means that the atoms are moving around with extra vigour, and the material of the balloon feels an outward pressure which causes it to expand. This expansion makes the balloon less dense than the surrounding atmosphere, and it is literally buoyed upwards like a cork in water. There is a limit to how high it can get, because the outside air at great altitude is less dense than it is at sea level. (By the way, something like the Goodyear Blimp is not a hot air balloon. It floats because it is filled with helium gas, which is already less dense than the oxygen-and-nitrogen-rich atmosphere.) Considering a corollary of this, by the way, requires me to shoot down the tale of Winnie-the-Pooh and the Honey Tree. In that story, Pooh Bear borrows a balloon (a blue one, so that it looks like a patch of sky) from Christopher Robin, who blows it up. Hanging onto a string tied to the balloon, Pooh floats up into the air to get close to the bees' nest and collect the honey. (He has rolled in dark mud to disguise himself as a small black cloud.) Unfortunately, no balloon inflated in this way will ever rise into the air! Your breath is not hot, and the only reason the balloon expands at all is because it is overfilled with a great many molecules of air which exert an outward pressure on the rubber skin. The balloon is consequently much denser than air, and falls to the ground. Now, what would happen if the Earth were to cool down very drastically (or if you were to cool down some section of it, like a sealed-off room)? The answer is that the atmosphere would not be puffed up to the same height as before, and the air would become more concentrated towards the surface. As the temperature dropped, certain gases would actually liquefy or even solidify. Some molecules would bond together, and their ``jiggling'' would be too tepid to disrupt the bonds. You would find pools of liquid nitrogen, for instance, on the ground. The fact that the Earth has liquid water on the surface and some water vapour in the atmosphere is a reflection of the fact that the temperature is poised at just the right level, thanks to our distance from the sun. This is important for life as we know it, but there may be other planets where the temperature is lower (or higher) but some other gas/liquid plays an equivalent role for a life form. On the other hand, if the Earth were to become very much warmer, the atmospheric constituents might fly around with such vigour that they could literally reach escape velocity - the atmosphere would boil off (or evaporate) into space. There are three factors that determine such effects: the temperature of the planet, which depends on its proximity to the sun. Closer planets are warmer in general, so less likely to hold on to an atmosphere the size (mass and radius) of the planet, since this determines the escape velocity. Small planets, with weaker gravitational fields, are less likely to retain their atmospheres. the gas itself. In the atmosphere, the numerous collisions between particles lead to the energy being shared in such a way that the bigger, heavier atoms move about rather ponderously while the lighter ones whiz about quite rapidly. Thus light gases, like hydrogen, might boil off while the heavier gases, like nitrogen, are retained. When you put all these factors together, it is easy to understand why the moon has no atmosphere (it gets the same amount of sunlight as the Earth, but has less gravity); why the Earth has very little hydrogen and helium in its atmosphere (they are light gases and can easily escape over the millennia that the Earth has been around); why Jupiter has a dense hydrogen-rich atmosphere: it is enormous, so it has a very strong gravitational field. Moreover, it is farther out from the Sun than we are, so receives less heating in the form of sunlight. As a consequence, it has been able to hang onto all the hydrogen and helium it had when it formed, and is very different in composition from the Earth. This is a theme we will return to later. 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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(Tuesday, 20 October, 2020.)