| The Doppler Effect:
In most astronomy textbooks, including your own, the Doppler effect is introduced as a phenomenon which relies on the wave nature of light (or the wave nature of sound, where it is also applicable). In due course, I will follow that approach, but first I want to present this topic from a slightly unconventional point of view, one which visualises particles in motion and their effects. We will draw a simple parallel between bullets and photons and see where this leads.
In class, I asked you to consider the situation of a fighter pilot flying at some high speed, say 700 miles per hour. (This is approximately the speed of sound in air, although that has nothing to do with the discussion that follows. I settled on this number merely because I tend to think of jet fighters as being supersonic aircraft. The actual speed we consider does not matter.) Now, suppose that a gun on the ground fires a bullet at the aircraft, one which travels at a speed of 701 miles per hour through the air. In real life, air resistance would slow down the bullet fairly quickly, despite its streamlining, but let us ignore that complication here and assume that it keeps up a steady speed of 701 mph. How hard will the bullet hit the fighter? How much damage is it likely to do?
You can see that there is no simple answer to this question: it depends on whether the fighter is moving towards or away from the gun. If it is approaching the gun, it will meet the bullet head-on at high velocity, at an effective "closing speed" of 1401 miles per hour (= 701 + 700). Such an impact will do a lot of damage! On the other hand, if it is flying away from the gun, the bullet will barely catch up to it, and will arrive with the puny relative velocity of 1 mile per hour (= 701-700). It is unlikely that the bullet will even scratch the paint! In short, the relative velocity with which the fighter and the bullet meet determines the energy of the impact, the amount of damage which can be done, and so forth.
If you doubt this reasoning, think of the everyday example of driving a car on the highway. If you are driving at 100 km per hour, which would be more catastrophic: running into (rear-ending) a car which is travelling ahead of you in the same direction but at the slower speed of 98 kph; or running into a car coming the other direction at 98 kph? The answer is obvious.
Now think about light from the corpuscular point of view, as a photon or small "packet" of energy. Suppose, just to be specific, that it has a frequency and energy which corresponds to that of yellow light: it is one of the photons given off in such great abundance by the sun, in the middle of the visible spectrum. By analogy to the jet fighter, the following hypotheses seem reasonable:
If you are travelling at high speed towards the source of the light, the photon will hit you with a little extra `punch,' a bit more energetically than if you were at rest. It is still a photon, but (when it enters your eye) it has enough extra energy, because of your motion towards it, to carry out the chemical change in the cells of your retina which usually only a more energetic photon of blue light can accomplish, just as the head-on impact with the bullet can lead to severe damage to the jet fighter. In short, the photon stimulates the blue colour receptors in the eye, and will look blue! I emphasise that this is not merely a case of semantics: a lamp emitting only yellow photons actually would look blue if we could travel toward it at the colossally high speed necessary. Nor does it depend on human perception for its validity: the photon would have enough energy to accomplish physical effects in any number of impersonal experiments which, in its yellow form, it could not. For instance, yellow photons might not have the energy required to dislodge electrons from a metal plate via the
CONTENT SECTION; but if the metal plate is moving at high speed towards the light source, electrons will be produced in plenty. Indeed, if we raced towards the light source at very nearly the speed of light, the arriving photons would look as energetic as X-rays to us, and penetrate our bodies completely!
Conversely, if you are travelling at high speed away from the source of light, the photon reaches you with just a small fraction of its initial "punch", and is perceived (both by your eye and by any physics experiment you care to suggest) as a red or infra-red photon of relatively low energy.
An important point: you may wonder why I use the vague word "punch" in describing the arrival of the light. Surely the important point is the speed? After all, in the head-on car smash alluded to earlier (you travel at 100 kph and hit a car coming the other way at 98 kph), the two cars come together at an effective speed of 198 kph. If, for example, you are travelling towards the light source at half the usual speed of light, doesn't this mean that you will see the yellow photon arrive at one-and-a-half times the speed of light, whereas it will appear to be travelling at only half the speed of light if you are racing away from the source at that speed? Unfortunately, it is not that simple. At speeds approaching the speed of light, the universe behaves in ways which seems to violate common sense, ways which we will explore later in the term. It is sufficient now for me to tell you that, against all expectation, we would always see a beam of light arrive at exactly the same speed, 300 000 km/sec, regardless of our own state of motion. This is a behavioural property of Space and Time which is described by Einstein's Theory of Relativity.
Still, as long as we stay away from the perplexingly invariable speed of the photons, the analogy works quite straightforwardly. By considering photons as particles, we have reached the correct conclusion that the perceived colour (or, equivalently, the frequency, wavelength, or energy ) of light depends on the relative state of motion of the source of the light and the observer (whether that observer be a human, using his or her eyes, or an impersonal piece of experimental equipment). But now let us reconsider the Doppler effect in terms of the wave nature of light, and see if that is equally profitable.
The Luggage Carousel as a `Wave Generator'.We have just considered some plausible analogies to argue that light, when thought of as photons, would appear to be more energetic (and thus of higher frequency, bluer colour, and shorter wavelength) if you and the source of the light were rushing together than if you were at rest with respect to it. These conclusions are correct, but the arguments are very qualitative and may be unconvincing, especially considering that they referred in such a direct way to the speed of the particle itself (the bullet approaching the jet fighter, for instance). Since the speed of light is a universal constant, we would like to find another description which eliminates the need to think in these terms. By the way, you can see that the speed of the source matters in our 'light-as-particles' discussion, but only because it has such an immediate effect on the final speed of the particle itself. To see why, think of a couple of everyday examples! If you throw a stone at someone, for instance, it will hurt them more if you are running rapidly towards them at the moment of release than if you are running the other way. The high speed attained by bowlers in cricket is attributable in part to the fact that they are sprinting towards the batsmen when they release the ball, not standing still as in baseball. If the jet fighter in the earlier discussion is flying towards a moving gun, such as one mounted in another fighter, the impact of the bullet will depend on how rapidly the other plane is flying. Clearly, in these examples, the speed of the source matters because the speeds add up (your running motion adds to the speed of the stone; the moving enemy fighter releases a bullet which may have a very high speed relative to us). Since this kind of simple addition of velocities is not allowed when we deal with light, we have to find another way to understand why the motion of the source of the light has a bearing on the colour (wavelength) we will perceive. A discussion which considers the wave nature of light (the fact that it produces periodic effects, the crests of the electromagnetic wave passing by) allows us to do this. You may not be surprised to discover that I would like to begin with yet another a helpful analogy. Visualise yourself at the airport, waiting patiently for your luggage to appear on a conveyor belt which carries bags to the waiting passengers. Let us suppose that this belt is fed by a machine which spits out bags with some frequency which I assume to be fixed and unchangeable. In other words, the source of the bags - the spitting machine - is assumed to have some `fundamental physics' about it which determines the rate at which it produces bags. (I will have more to say about this important assumption in a bit.) To be specific, let us suppose it produces a bag every second and drops it onto the conveyor belt. Let us furthermore assume that the belt moves horizontally past the spitting machine at some constant speed, which might be fast or slow but which does not change, and cannot be changed, during the exercise . To be specific, let us assume for now that it moves at a speed of one meter per second. You can see that the frequency with which the spitting machine produces bags, coupled with the speed of the belt, determines the space between bags. In this example, successive bags will be dropped onto the conveyor belt exactly a meter apart. Then, somewhere towards the other end of the room, the luggage will be carried past you at a speed of one meter per second. As a result, you will be able to examine the bags passing by, one after another, at a frequency of one a second - exactly the same frequency with which the machine spits them out. The analogy, of course, is to a wave phenomenon. The regular pattern of bags on the belt represents the crests in the wave, a series of peaks which will be brought past you at some steady speed (that of the conveyor belt) to produce an apparent frequency of arrival. Remember the fundamental equation which describes this: (the true speed at which the pattern passes) = (the frequency with which we see the bags) x (the spacing between successive bags). Quite obviously, a series of bags a meter apart, travelling at a speed of one meter per second, will pass by us at a frequency of one every second. Now, imagine yourself standing idly by the empty conveyor belt, wondering when the planeful of luggage will finally be offloaded and distributed. Suppose the first bag passes by you at exactly three o'clock, catching your eye and alerting you to the fact that the distribution exercise is at last underway. If your bag is the three hundredth to be processed, it will not reach you until three hundred seconds later - a delay of five minutes - since you will necessarily see only one bag a second as each in turn is carried past you for your inspection. But is this really necessary? Are there ways to speed up (or slow down) the rate at which bags are stuck in front of your nose? The answer is ``Yes,'' and developing an understanding of this will provide you with a clear comprehension of the Doppler effect. The essential statement is that the frequency with which you will see bags (or waves) pass by depends not only on the rate at which the machine spits them out but also on your state of motion relative to the source of bags (or waves,) a contention I will now justify.
Speeding Up a Slow Process: Raise the Apparent Frequency.Let us suppose that you are particularly anxious to collect your bag and get out of the airport. Is there anything you could do to speed up the process, to quickly scrutinize more bags in a shorter span of time? The answer is ``yes,'' but there are subtleties which need exploration. More importantly, there is one surprising option which would really assist you if it could be implemented, an option which is absolutely central to your understanding of the Doppler effect. To start with, let us imagine that your bag is unmistakably distinct in some way - the only one which is Day-Glo pink, for instance. All it takes is a glance from you to see whether the bag passing by belongs to you or not - you don't have to mull the matter over. Given that, you can afford to examine bags with very high frequency, perhaps considering dozens per second if only they could be presented to you at that rate. What are the various possibilities? First, here is something that would work in real life, but which I will disallow: we could stand on tip-toe and peer towards the spitting machine, looking back along the length of the conveyor belt in the hope of seeing our fluorescent pink bag among the ones which have not yet reached us. In this way, we could examine quite a lot of bags at once! We could then `jump the queue' and go and collect the bag rather than wait for it to be carried to where we are standing. I am going to disallow such behaviour, however, because I am going to apply the reasoning to the passage of an electromagnetic wave , and you can detect a light signal only when it actually reaches your eye - you can't ``see it coming'' before it gets here! To preserve the analogy, therefore, let us imagine ourselves as being particularly nearsighted, unable to see the bags until they are right in front of our noses. (For this reason, you would not even know that any bags are on their way until the first one actually reaches you, just as an astronomer doesn't know that a new source of light is shining in her direction until the first glimmer of light itself actually arrives.) After that, you watch carefully until your bag arrives, five minutes later. Here is something that, perhaps surprisingly, would not help: t he use of a much faster c onveyor belt would make no difference: your bag will still arrive five full minutes after the first one passes you by. At first this seems illogical, but it is easily demonstrable. Do you remember my use of `limiting cases' in earlier physical arguments? Try that here. Let us suppose that the conveyor belt moves furiously fast, so that every bag being dropped onto the belt hurtles rapidly down the full length of the room past every waiting passenger. Of course, this would make grabbing your bag, when it finally goes by, very difficult, but that is not my point. The point is that you will still see only one bag every second because that is the frequency with which the spitting machine drops them onto the belt.. After the first bag goes by, alerting you that the distribution operation is underway, you still have to wait five minutes for your bag to arrive, in three-hundredth position. The reason there is no advantage is that the rapid pace of the conveyor belt makes the separation between each bag and the next one - the `wavelength' - very large (the first one gets carried quite a long way before the second one falls onto the belt). As a result, there is no net gain. The bags still reach you at the frequency with which they are produced, and you must wait just as long as before for the three-hundredth bag to arrive. Equally, there would be no penalty if the belt were to be run at a slower speed: the bags would be closer together, but would still pass you by at one-second intervals, just as before. There would be a real benefit in speeding up the conveyor belt once the baggage is on board, but of course this is not allowed. You can see why it would help by thinking again of a limiting case. Start with the conveyor belt just barely moving, and dump all the bags in a closely packed group. Now send the belt off at high speed. The passengers learn (from the arrival of the first of the bags) that the process has started; and they find their own bags almost immediately therafter because the bags are all closely packed together and travelling rapidly. Why is this useful technique ``of course not allowed''? It is because in Nature the `conveyor belt' simply cannot be sped up or slowed down on a whim: a beam of light travelling through space travels at a fixed, immutable velocity (300 000 km/sec). Let's see how far we have gotten: We have disallowed looking down the length of the conveyor belt to anticipate the arrival of our bag. To preserve the analogy to light, we have ruled out speeding up the conveyor belt in mid-operation. We have recognized that the actual running speed is irrelevant in determining the frequency of delivery of the bags for examination. They will simply appear with the frequency at which they were dropped onto the belt by the spitting machine. This is sounding more and more like there is nothing we can do to hurry up the discovery of our bag somewhere in the pattern, but that is wrong. Before reading further, ask yourself what you might do in real life. Here is the situation: you are standing beside a conveyor belt along which you now know (from the arrival of a first bag) that a whole platoon of bags are approaching at a steady pace. Unfortunately you are too nearsighted to look off into the distance in the hopes of seeing your bag: you will only recognize it when it is very close to you. You want to be especially expeditious in finding your luggage. How can you inspect bags at a higher rate than the sedate once-a-second pace at which they are presented to you by the conveyor belt? The answer is simplicity itself, and I expect that most of you leapt to it right away. You merely need to start walking briskly `against the flow', towards the source of the bags (the spitting machine). If we walk just as quickly as the luggage is being transported, for instance, we will see two bags pass by every second, rather than one. If you run like the wind, you might be able to look at them all in just a few seconds. The frequency with which you encounter bags is raised thanks to the simple expedient of moving towards the source (the machine which spits the bags out).
Give Yourself Time to Think: Lower the Apparent Frequency.While some people might be impatient about the delay between the passage of successive bags (as you surely would be if the spitting machine dropped them onto the belt at a rate of, say, once a minute) others might think that the bags arrive all too frequently. Perhaps the bags are so nearly alike in appearance, or you are so jet-lagged, that you have trouble deciding if a particular one is really yours - but before you can persuade yourself one way or the other, another bag has arrived and is demanding your attention. Consequently, it might be important for you to slow down the frequency of arrival, to allow yourself longer intervals of time between successive bags so that you can think coherently. How could you accomplish this? There are several possibilities: In real life, a sure-fire way would be to haul any likely-looking bag off the conveyor belt and to examine it at your leisure as time permits. For instance, perhaps a subsequent bag will pass by which is obviously not yours; you dismiss it instantly and take the opportunity to snatch a quick extra peek at the bag at your feet. If the one you pulled off can eventually also be rejected, it can then be put back onto the conveyor belt. (I am sure that many of you have seen people doing exactly this at airport luggage carousels. I certainly have.) Of course, if many bags look alike, you will soon be surrounded by a heap, all needing later inspection, and a big backlog builds up. In any event, such behaviour is forbidden in our analogy. In essence, you have changed the pace of the conveyor belt, effectively bringing it to a stop at your location. This is not allowed: the bags must be permitted to go on their uninterrupted way. Nor can we look off into the distance at bags which have already passed us by: they can only be examined when passing close to us, neither earlier nor later. For the reasons discussed in the previous section, using a slow conveyor belt brings no benefit. If the spitting machine places a bag onto the belt once every second, that is inevitably the frequency with which they will come past your location. As before, we are also prohibited from changing the speed of the conveyor belt once the operation has begun. Just as before, however, we are not without a solution: we need merely walk in the same direction as the belt is moving, so that the bags appear to overtake you at a slower speed - or equivalently with reduced frequency. (Indeed, if you walked at the same speed as the conveyor belt, you could wind up strolling along directly opposite one bag and never see any of the others. That would provide you a very long time to debate the likelihood that the bag is yours.) You can see, from this simple analogy, that the frequency with which we see bags on the conveyor belt will depend on whether or not we are moving with respect to the source of the bags. But there is another way of looking at this.
Consider a Moving Source.There is one more surprising option, not readily realisable in the airport luggage hall but nonetheless valid in principle, which I now want you to consider. Suppose the spitting machine itself can move - what then? To be specific, imagine that the spitting machine still produces one bag every second, but that it is moving alongside the conveyor belt in your direction, at a speed of half a metre per second (half the speed of the belt itself). The first bag which the machine spits out lands on the conveyor belt and is carried towards you at 1 metre per second. After exactly one second has passed, the bag is a metre closer to you than it was when it first landed on the conveyor belt, but the spitting machine itself is half a metre closer than it was initially, so it is only half a metre behind the first bag it produced. At that instant, it spits out another bag, which of course lands just half a metre behind the first one. Since this process is repeated again and again, bags are put out onto the belt with a uniform separation of just half a metre (rather than one full metre). In this way, the machine produces a platoon of bags which are now more closely spaced than expected (``of shorter wavelength''). The inexorable motion of the conveyor belt carries them past you at a frequency of two per second, even though the machine itself is spitting them out only once a second. The frequency you perceive is higher than that actually produced by the machine because the distance between you and it is decreasing. The result is that successive 'crests in the wave' (bags on the belt) are produced closer to you than earlier ones and get 'scrunched together.' Conversely, suppose the machine itself is moving away from you. Successive bags are dropped onto the belt with a spacing which is greater than one meter, and the bags are eventually carried past your position at a lower frequency than once per second. Of course, matters can get complicated if you and the machine are both moving. For instance, what if you are walking slowly parallel to the conveyor belt and away from the machine, but the machine itself is moving your direction at a somewhat faster pace, with the net effect of lessening the separation? A little thought will reveal that you will see bags pass at a higher frequency than that with which the machine itself spits them out; and so on. These analogies make the fundamental physics of the Doppler effect very clear. The really crucial thing to note is that it does not matter what is moving, you or the source. The effect is the same. It is simply a question of the relative speed, as follows: if the distance between you and source is decreasing, you will see bags more frequently than you would if you were at rest with respect to it; but if the distance between you and the source is increasing, you will see bags less frequently than you would if you were at rest with respect to it. That's all there is to it.
Drawing Interesting Inferences.Now think of the deductive inferences possible. Let us suppose that you are waiting for your luggage, inside the airport hall. You have a friend outside in a car waiting for you to come out, and she is wondering why you are taking so long. By cell phone, you are keeping her informed of the frustration of waiting for your bag. The conversation goes something like this: ``Let me see - is this my bag? No." (long pause) "Next one? No." (long pause) "How about this one? No." ... (and so on) Suppose your friend is worried about getting away before the start of rush hour and urges you to hurry up the process. After a moment, she hears you say "Okay, how about this bag? No." (very brief pause) "This one? No." (very brief pause) "This one? No." .... (and so on). Clearly something has changed! -- you are now examining bags with increased frequency, but why? If your friend knows that the machine always spits out the bags at a constant fixed frequency (that the `underlying physics' of the source is unchanging) and that the belt speed cannot be adjusted (that the `speed of light' is a constant), then she can correctly infer that you and the spitting machine must now be moving closer together, but that is all she can infer. She does not know if the spitting machine, usually fixed on its base, has suddenly started moving towards you for some reason, so that the bags pass you with increased frequency, or if it is you who has started walking towards the machine (clearly the most likely answer in this situation, but not absolutely guaranteed!). An important warning: this part of the analogy seems to stress change, as though the Doppler effect is noticed only when some observed phenomenon changes in frequency. That is not the case. Suppose your friend understands all the machinery well (the 'physics of the source' and the `speed of light'), so that she knows the frequency with which bags are usually presented. If she hears you say ``Not my bag ... not my bag ... not my bag...'' with high frequency when you first get on the phone, she can rightly conclude that you and the machine must be approaching one another. (She does not know which is moving, but will presumably most likely deduce that you are walking towards the machine.) But there is no need for her to have noticed a change: all she needs to observe is that the frequency with which you are inspecting the bags is higher than that with which the machine itself spits them out.
The Physics of the Source: An Application to Light.Our conclusions about the relative speed of the source and observer are correct only if we are right about the nature of the source. If the spitting machine breaks down and is replaced by another which delivers two bags a second, an unmoving observer will see bags go by at this new, higher frequency. The friend on the telephone will conclude that the observer is walking briskly towards the spitter, but this inference is wrong. There is a completely different source there, throwing out bags according to some completely different prescription. Unless you know for certain what the source is like intrinsically - how many bags a second it really generates - you can conclude nothing about the relative motions on the basis of what frequency you observe. Let us think about this in the context of a light source. Suppose we have a particular lamp which gives off light (photons, waves) of one discrete kind only - yellow in colour; middling in wavelength, energy, and frequency; right in the middle of the visible spectrum. If someone shines this lamp at you from afar, the arriving light should look yellow. But: if the photons look red, then you may conclude that the distance between you and the lamp is increasing. Perhaps it is being carried at high speed away from you. Alternatively, perhaps it is at rest, and you are rushing away from it. Maybe you are both moving; if so, it is in such a way that you are getting farther apart. On the other hand, if the photons look blue, you and the lamp must be drawing closer together, thanks to the motion of one or both of you. But: if someone has tricked you by replacing the lamp with one of another description - say, one that emits only blue light - then you will be quite wrong about the motions. You need to know the physics of the source (exactly what is it emitting?) to draw any valid conclusions about the motions.
Consider a Surfer.The bags on the conveyor belt represent the peaks in a series of waves, in our homespun analogy. But now let us think of a simple example where real waves are present. Imagine yourself as a surfer floating in the ocean near a Hawaiian beach, waiting for the right wave. As you sit there on your board, waves pass by, and you bob up and down in a rhythmic way - once every five seconds, say. Then you see a good wave, and paddle so that you are moving along with the waves - moving away from the source, which is far out at sea. The waves now pass you by with reduced frequency, and indeed the ideal is to match the speed of the waves so that you are riding along a single perfect wave rather than having any pass you by.
Some History: The Doppler Effect in Sound.The Doppler effect was first recognized and predicted by Doppler (and Fizeau) in the early 1800's, but had never been observed to that date. Why not? Well, the obvious way to test it is with sound , which is a periodic effect: a sound wave moves through the air, causing a periodic compression and rarefaction of the density of the particles in the air (like the waves passing through the big `slinky toy' which I used in a classroom demonstration). Before the 1800's, however, motion was too slow to provide any everyday examples of how this would affect our perception of sound. The development of fast-moving trains changed that. Doppler recognized the opportunity this provided, and conducted an experiment using a train moving at about 60 km/h. He had a group of musicians stand in an open rail carriage and play a note on their trumpets - an "A", let us say. Another group of musicians were asked to stand in the ditch beside the train track. They had perfect pitch, and were able to tell what the note sounded like as the train approached. It was indeed raised in pitch (i.e. had a higher apparent frequency) as the train approached - it sounded like an A-sharp (A#).
A Modern Example.When you stand near a train crossing, and the train blows its horn as it approaches at high speed, two things can be noticed: The noise gets louder. This is not the Doppler effect. It is merely a consequence of the fact that the source is getting closer to you. The noise sounds higher in pitch than it would if the train were not moving. This is the Doppler effect. The higher pitch may be hard to judge, of course, if you have not got perfect pitch or don't know what it "should sound like." But just wait a moment! Once the train passes by, it is no longer approaching you, but is now instead receding from you. The pitch of the horn abruptly goes from high to low. This kind of change is readily detected, but please note again that you don't actually have to hear any change to use the Doppler effect. If you are sufficiently knowledgeable about the source of the sound -- that is, if you know what note the horn really produces -- then the unexpectedly high pitch that you hear already tells you about the train's state of motion. All you need to hear is one short blast; there is no need to monitor the sound over a period of time, or to hear it more than once. There are three things of which you should take special note: 1 The sound of the horn does not gradually change in pitch as the train approaches - indeed it does not change at all, except to get louder. It is a steady constant high pitch. Once the train has passed you and is moving away, you will hear a new steady lower pitch (which of course gets quieter as the train moves into the distance). 2 If you knew what the train's horn should sound like, you could use its actual sound to derive the speed of the train. The formula on page 166 of your text is for light, but an equivalent one applies to sound as well. For moderate speeds, the formula says essentially that the perentage change in frequency is the same as the train's speed, when this is expressed as a pecentage of the speed of sound. A train travelling at 70 miles per hour is moving at about 10% of the speed of sound, so its horn will sound about 10% too high in pitch as it approaches - a difference of almost a whole tone (from `doh' to `re' in the major scale). 3 Just to emphasise, please note especially that we don't need to listen to the train over a period of time to work out the speed: one brief sound is enough. Suppose the train is somewhere down the track approaching you, and sounds its horn briefly once. You determine that it sounds like an A-sharp (A#), say. If you already know, on independent grounds, that it would sound like an A if the train were at rest, then you can instantly deduce the train's speed towards you without having to hear the horn again.
The Question of Speed: the Perplexing Behaviour of Light.Let us return briefly to the analogy of the luggage carousel. I described what would happen if you were to travel with the wave, walking briskly alongside the conveyor belt. If you walk as fast as the belt itself moves, you can in fact keep pace with the wave pattern and stay right side-by-side with a bag of particular interest. If you walk at a slightly slower pace, the bags will still overtake you, but at a snail's pace from your point of view. Similar considerations apply to sound waves propagating through the atmosphere. It is possible to travel at and even beyond the speed of sound, so that the wave never overtakes you. If you run away from here at a speed greater than about 700 miles per hour, and I call after you, my shout - the disturbance in the air - will never catch up to you. Now, can we do something equivalent with light? This would be very interesting. In day to day life, photons absolutely whiz by us at phenomenally high speed; this makes their structure, if any, hard to study! But suppose we could travel very quickly, at 99.9999% the speed of light. Then we might hope to do the following: set up a source of light, like a searchlight, beaming out into space; hop into our speedy rocket ship and accelerate until we are moving at high speed right alongside the beam of light; and then examine the photons which pass by, which must logically seem to be overtaking us very slowly from our new point of view, just as we can walk alongside a particular bag on the conveyor belt. What would a photon look like, or an electromagnetic wave, if we could `travel with it' and seem to freeze its motion? The problem is that logic fails us here! Or, more correctly, our common-sense understanding of the way the universe works cannot be applied to situations involving such high speeds. Light simply does not work this way: that's what Einstein's Theory of Relativity was designed to explain! What will indeed happen is that the waves of light overtaking you will seem to be of lower energy (longer wavelength, redder colour, lower frequency), just as the luggage carousel analogy led us to expect , and as experiment confirms. But if you use a ruler and stopwatch to figure out how quickly the beam of light covers a measured distance as it passes you, you will discover to your amazement that it still seems to be travelling at the speed of light, 300 000 km/sec! You have not managed to freeze the light in its tracks, or slow it down one iota, despite your fantastically rapid pace. Experiment confirms this as well: it is not mere speculation or theory. How can this be? The reason is that determining speeds requires us, in effect, to use clocks and meter sticks to see how long it takes something to cross a measured distance. Einstein demonstrated that when we are in a state of motion, the rate at which clocks themselves run, and the apparent length of meter sticks, change in such a way that we always get `c' (300 000 km/sec) when we measure the speed of light. It is truly a universal constant, and will be measured as having the same value by every observer in every location, no matter his state of motion. One unexpected consequence of this is that two clocks moving with respect to each other run at different rates - and not just clocks with hands, but any physical phenomenon which records the lapse of time: the decay of radioactive atoms, the speed at which electrons orbit the nuclei of atoms, and so on. Once again, this has been experimentally demonstrated. But included in here are all the physical phenomena which are part of our day-to-day existence, so the conclusion is that we would age at a different pace than the rest of humanity if we were to travel at high speed with respect to the Earth. By high speed, of course, I mean to say at a significant fraction of the speed of light. (The effect is in fact always present, but only noticeable when things move near the speed of light.) We will return to this fascinating and perplexing topic later in the course, when we consider the prospect of interstellar travel - and the very structure of space and time.
Light and the Doppler Shift: An Everyday Example.As we have seen, we can use electromagnetic radiation to determine velocities by comparing observed to emitted frequencies. That is the principle behind the "radar gun" used by the police on the highway. They send out a radar signal (electromagnetic radiation) of a well-known frequency. It bounces off your car, with the reflected radiation picking up a bit of energy thanks to your motion (just as swinging a baseball bat sends the ball back with more energy than when it came in). The higher-energy, higher-frequency (i.e. "blueshifted") photons are picked up by the radar gun and compared to what was sent out. The result: a speeding ticket.
The Colours and Velocities of the Stars.Doppler thought that his effect might explain the colours of the stars. That is, he thought that the red-looking stars might be those that are moving away from us (so the light waves reaching us are of lower energy, and shifted towards the longer-wavelength or redder end of the spectrum - a "redshift"), while the bluer stars might be coming towards us (their light is "blueshifted"). It is easy to show that this cannot be the explanation: the stars in the galaxy move around at typical speeds of no more than a few hundred km/sec with respect to one another, and this is such a tiny fraction of the speed of light that the Doppler shifts are almost immeasureably small - certainly not enough to give perceptible changes in colour. (As noted earlier, the pronounced colour differences are actually due to differences in temperature.) But suppose the velocities of the stars were very much bigger, perhaps thousands or tens of thousands of kilometers a second, so that the Doppler shifts were quite significant. Even then, you would have trouble in using the different colours to deduce anything about the motions. To see why, imagine an orchestra consisting of countless numbers of instruments, all playing a great variety of notes and producing a welter of diffuse sound of some `average' pitch. The piccolos would sound high-pitched, the bassoons low-pitched, and so on. Now, let us imagine this huge orchestra on a railway flatcar somewhere in the distance. Suppose further that the flatcar starts to move rapidly towards you, carrying all the players together, still emitting their unchanged blast of music. You might think that this would result in a perceptibly higher pitch in the overall sound, thanks to the Doppler shift, but it is not quite so simple. It is true that the piccolos would be shifted to a higher pitch, and indeed their Doppler-shifted sound might be of such high pitch that it is above the threshold of human perception. (Of course, the orchestral players on the flatcar, moving with the instruments, would hear the notes at the original pitch.) The flutes might now sound like piccolos, and the cellos like violins. This would seem to imply that the average pitch would indeed rise, but a complication would arise if there were additional instruments producing notes of a pitch which is ordinarily below the threshold of perception. The motion of the orchestra would raised these notes into the range of frequency in which you could hear them, and the net effect would be to fill out the whole range of sound to something quite like what you heard before! You may object, rightly, that there are no orchestral parts written for instruments which play at frequencies below the threshold of perception - what would be the point? - but in the production of light, this is a real consideration. The hot body of a star produces invisible infrared light as well as visible, and if the star were made to approach us very rapidly these photons would look like visible light to us. The overall colour of the star would probably not be perceptibly changed, even for quite large velocities. This seems to imply that we can never work out the velocities of the stars! If the Doppler shift has essentially no effect on the colours we perceive, how can we tell whether it is influencing the spectrum of light at all? There are two possible ways out of this: If stars emitted only one single wavelength of light, rather than at all wavelengths, then we could compare the perceived colour of the star to that which is emitted from a reference star (perhaps the sun). Differences would tell us about different states of motion. Unfortunately, the stars are not so obliging: they are indeed thermal radiators, hot dense bodies which emit light of a vast range of colours, rather as the imagined orchestra emits a welter of sound. But suppose, on the other hand, that there were some identifiable feature in the spectrum of the star. To return to our analogy of the orchestra, suppose that amid the welter of diffuse sound there was one powerful trumpeter playing a long continous note of well-known pitch (an A, say), so loudly that it can be heard above the roar of everything else. Now, suppose you turn your attention to a second orchestra which you think is similar in all respects to the first one. If you can pick out this prominent trumpet sound, you can ask whether it is also a well-defined `A.' If it sounds like an A-sharp instead, you can deduce that the orchestra is rushing towards you at a few percent of the speed of sound. If the trumpet sounds flat, the orchestra is rushing away. Of course, this will only work if you completely understand the physics of the source. If the trumpeter in the second orchestra has decided to play something other than an A this time, you are out of luck, and your conclusions about the state of motion will be mistaken. You must know, on other grounds, what kind of light (or sound) is actually being emitted by the source. Are there any helpful features in the spectra of the stars, something equivalent to the note played by our orchestral trumpet player? Of course there are! We can study the absorption lines. Here is how it works: In the laboratory, we can study the pattern of emission lines in the spectrum produced by a glass tube full of hydrogen gas (and other elements) to determine their `rest' wavelengths. (They are like the notes played by our hypothetical trumpeter.) When we get the spectrum of a distant star, we may see aborsption lines in the same pattern -- the same 'fingerprint'. This confirms the presence of those elements in the star (and gives us interesting information on the star's temperature, among other things). Finally, if there is a modest shift of all the lines to longer wavelengths (since all photons or waves of light are subject to the Doppler effect), then we know that the distance between us and the star is increasing. Likewise, a shift to shorter wavelengths tells us that we and the remote star are drawing ever closer together. Moreover, we can quantify this finding and measure the actual speeds of the stars, in kilometers per second (or whatever units are convenient). And I remind you that a single spectrum is all we require. There is no need to observe the star again and again, or to look for any changes in its spectrum. The information is already there in the very first spectrum! Although one spectrum gives us the star's velocity, there may be interesting reasons to look for changes with other spectra taken at later times. For instance, in June we are moving one direction in our orbit of the Sun, but by December we are moving the other direction as we continue our near-circular orbit. A star which lies in the plane of our orbit will be seen to vary in apparent Doppler shift by about 30 km/sec up and down as the year progresses, thanks to the changing direction of our motion. This is one of the proofs that Similarly, we can deduce that binary stars move back and forth around each other in mutual orbits: see the figure on page 531 of your text. Studies of binaries allow us to determine the masses of the stars, the key to understanding stellar evolution. 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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(Wednesday, 21 April, 2021.)