| Interpreting the Emitted Light:
The Effects of Distance on Brightness.
Let us first dispose of a simple and unimportant geometrical effect by noting that the apparent brightness of a lamp depends on its distance. A single candle in your hand seems to outshine every star in the night sky, but only because of its proximity to you. This fading with distance is, like gravitation, described by an inverse-square law. (See the figure on page 523 of your text.) But such differences tell us nothing about the intrinsic nature and internal workings of luminous objects, the physical properties that determine how much energy flows out of them. Since we are interested in the inner workings, let us assume hereafter that these purely geometrical effects can be corrected for and ignored. (Imagine, for instance, placing your candle and the stars `side-by-side,' all at some common distance from you. You would quickly realize the complete insignificance of the candle.)
Thermal Radiators (Perfect Radiators, or Blackbodies).Having disposed of one unimportant geometrical effect, we can ask what it is that determines the nature of the light - in particular, the balance of colours, and the total quantity -- that we see when we look at some object. There is yet another complication to deal with: the distinction between reflected light, such as the blue colour you might see when you look at your jacket, and the emitted light, that given off by the object itself. In everyday life, we recognize an easy and automatic distinction between objects which we see by their emitted radiation (fires, stars, lamps, etc) and those we see by reflected light (everything else): the former are hot. But the second class of objects, although not heated to incandescence, are generally not absolutely stone-cold: perhaps they too emit light of some kind? If so, of what kind? If reflected light is only of superficial importance -- telling us, if you like, about the `paint job,' but nothing very profound about the inner workings of the body under scrutiny -- we need to find some way to dismiss it as irrelevant. How will we accomplish this? Physicists speak of, and can study, idealized objects called 'blackbodies,' a term that your text wisely tries to avoid (it gets a mention on pages 161-162). The reason this expression is confusing is that, to astronomers, the sun is a blackbody, or at least a close approximation to it; yet it is clearly not black in appearance! Where does this apparently ridiculous term come from? Why do we use such a misleading expression? The reason is this. To avoid confusion with reflected light, we simply focus our attention on objects which absorb any light which happens to fall on them. Such objects are usually called ``black.'' For instance, your black shoes look that way because most of the light falling on them is absorbed. But if you had infrared-sensitive eyes, you would see that the shoes are actually radiating or `glowing' at infrared wavelengths! Likewise, the sun is a pretty good approximation to a blackbody, because light falling on it plunges into its gaseous outer parts and is not reflected. The sun shines because it emits radiation from deep within it, not because it reflects light from some other source! So a study of any light we receive from the sun really does tell us about the nature of the sun itself, rather than describe some other object simply seen in reflection off the sun's surface. Let's try to use a less confusing term than 'blackbody.' In these notes, I will generally try to refer to such an object as a thermal radiator, or a ``thermal body'', an object which is emitting radiation of some sort because it has some non-zero temperature . (Since such idealised objects reflect no light whatsoever, we might also call them perfect radiators.) We now seek to understand the fundamental relationship between light and temperature. Why do hot bodies glow? What happens if they are made even hotter?
A Logical First Guess.The higher the temperature of a body, the more energy it contains (in the form of its constitutent particles jiggling about); thus temperature is a way of measuring total energy content. Remember now that photons carry energy, and that the shorter the wavelength the more energy the photon has. Even without any particular knowledge of physics, then, you might guess that hotter bodies, with more internal energy, would give off light which is more energetic (i.e. of shorter wavelength on average) than the light from cooler bodies. It could be the other way around, of course. For instance: When you heat up a body, perhaps it gives off light of exactly the same kind as before , just more of it. In that case, the colour of a body would not change at all as its temperature is raised. It could even be the case that hotter bodies give off proportionally more low-energy light for some reason I think that neither of these seems as logical than your first (correct) guess. Here is analogy which may make this clear. Imagine someone with an average income who spends it on necessary purchases: a book, a meal in a restaurant, a shirt. Now imagine a much richer person. This person has more money in total to spend, but in what way will he do so? Will he buy the same kinds of things as before, just many more of them? Does he buy even cheaper goods, in huge numbers? Or does he buy more expensive items, now that he can afford them: a really good meal in a fine restaurant, a silk shirt, ...? I think that the third of these is the most natural expectation (although of course some rich people become real misers and spend next to nothing). And so it is with radiating bodies: the hotter they are, the more energy they contain (the richer they are). As a consequence, they radiate their energy (spend their money) in larger lumps on average (more energetic photons/more expensive purchases), and the total amount emitted (money spent) is greater as well, since they have such a lot of available energy (money). If hot bodies behave like rich people, then, you might be tempted to predict that the hotter a body is, the more energetic the average photon it will emit, and the more light in total it will emit. These are both correct. One very familiar manifestation of this is provided by a fireplace poker, as follows: when you pick up the poker, it is at room temperature. It may, of course, look blue, or green, or red, if it is painted one of those colours, but we are paying no attention to reflected light here. It is in fact emitting some quantity of infrared radiation. if you stick it briefly into the fire, it will become warmer, and will start to feel quite hot to the touch. It is now emitting more infrared light in total, and that light is of a shorter wavelength than before (although your skin has no way of registering that). if you leave it in the fire for a time, it will start to glow red-hot, emitting visible light. if you use a bellows to get the fire really roaring, the poker can get white-hot (that is, it emits light with a range of colours rather like that of the sun itself). finally, if you could heat the poker to some extraordinarily high temperature, it would be hot enough to emit ultraviolet light, or even X-rays and gamma rays. (In practice, this would not be straightforward because the poker would first melt and then evaporate its constituent atoms into space! The X-rays and gamma rays would be emitted by the super-hot cloud of atoms rather than by an object which is recognizable as a poker any longer.) In what is to come, we will quantify these remarks, and see how they allow us to understand the stars. But the light from the stars tells us more than merely the temperature. As we will see, detailed study of stellar spectra also tells us, among other things: their compositions (the mix of elements they contain); the speeds with which they are moving. Then, when the speeds are looked at in the context of binary stars moving around each other in mutual orbits under the influence of gravity, we can also derive their masses. Once we know how much material in total a star contains, how hot it is, and what it is made of, we have enough information to address a lot of very deep questions about the very origins of stars, and how they evolve.
The Distribution of Radiated Energy: Shape, Total, and Peak.I have told you in qualitative terms that the temperature of a body determines the nature of the electromagnetic radiation which it emits. To make this more (astro)physically useful, we want - as any good scientist would - to make our remarks more quantitative, in three obvious ways. We first ask about the spectrum of light emitted by a body. For every blue photon emitted, how many red photons are emitted, and how many infrared photons? In other words, is there some characteristic form which adequately describes the shape of the spectrum? Next, we inquire about how much light in total is given off by a radiating body, and ask how that quantity depends upon the temperature. Finally, while recognizing that a radiating body emits photons of many different colours -- the sun, for instance, gives off ultraviolet light, blue light, red light, infrared, and so forth -- we ask what the ``average'' photon is like, and how that depends on temperature. To return briefly to our analogy of the rich person, you might imagine that someone with lots of money would spend money in lumps of all sizes, purchasing everything from cups of coffee to ocean-going yachts. Small items would probably still make up at least part of his budget -- it seems unlikely that he would buy expensive things only. By contrast, his less affluent friend would buy many small cheap items, but not be able to afford any big ones at all. someone with lots of money would spend more money in total; and the average purchase of the rich person would be more expensive than that of a poorer person. Let us see how well the analogy, and our expectations, hold up when we consider the varied behaviour of radiating objects with different temperatures.
Shape: The Planck Law.If we want to understand a body like the sun, it is not enough simply to say that "it gives off light of all visible wavelengths", any more than you would be satisfied to know that your marks in university are going to range all the way from 60 to 95. You would be anxious to know details of how the marks are distributed. How many will be in the 90's? how many in the 60's? what average? and so on. So too in physics. To investigate this, we use a prism to spread the light of the sun out to create a spectrum, and then measure how much energy there is at each wavelength. We find in this way that the sun give off lots of yellow light, but less blue light (which is shorter in wavelength) and less red light (which is longer in wavelength). That is, the spectrum of the sun is peaked in the yellow part of the spectrum, right about where our eyes are most sensitive. (This, of course, is no accident. Our eyes have evolved to be especially responsive to the most abundant light around. It is also relevant that the Earth's atmosphere is transparent to light of this wavelength.) If we study the light from a body of a different temperature, we discover that its spectrum is once again peaked at some characteristic wavelength, although not the same as that for the sun. In fact, analysis reveals that the spectra of all thermal radiators are indistinguishable in shape, an important discovery which can be summarized in the simple physical law described by the figure on page 162 of your text. There is a mathematical function, called the Planck law, which tells you how much energy a body will radiate at each of the different wavelengths in the spectrum -- how much red light, how much blue light, and so on -- once you know the temperature, which is all you need. (The composition of the object, its shape, its total mass, and so forth are all irrelevant -- at least in the somewhat idealized case of the so-called perfect radiators. For such objects, the spectrum of light given off depends only on the temperature. ) The Planck Law, named after a German physicist named Max Planck, can be derived on theoretical grounds and written down as a mathematical function. The function itself is not simple, but its universal applicability is the important issue. To calculate the distribution of light emitted by one of the so-called thermal radiators, all you need to know is its temperature.
A Digression: Measuring Temperatures in Kelvins.Physicists express temperatures in Kelvins. Kelvins are the same size as degrees Celsius, but zero degrees Kelvin corresponds to absolute zero, the temperature at which all molecular and atomic motion (``jiggling'') ceases; nothing can be colder than that. By contrast, the temperature at which water freezes is taken to define the zero of the Celsius scale. Because it can get colder than that, as it often is during the Canadian winter, things can be at negative (below zero) Celsius temperatures. Absolute zero corresponds to a temperature of -273 Celsius, so to convert Celsius to Kelvin you merely add 273. In many astronomical situations, of course, temperatures are very high, and you can merely consider Kelvins to be roughly the same as degrees Centigrade (Celsius). For instance, the surface of the sun is about 5800 Kelvins, or about 5500 Celsius; for our purposes, we would call this `about six thousand degrees' and not worry about the small difference unless we were doing some detailed physical analysis. Deep inside the sun, the temperature is in the millions of degrees, so the differences are even less important. At lower temperatures, however, we have to be more careful. The human body temperature of about 37 Celsius corresponds to about 310 Kelvins, for instance, so if you were asked to work out the spectrum of electromagnetic radiation emitted by the human body, this is the temperature (310 K) that you would use in the formula for the Planck law.
Total: Stefan's Law.When we look at the stars in the night sky, they do not all look equally bright. Why not? There are three possibilities. Perhaps the stars are all very much alike, but just happen to be at a variety of distances from us. Is this the case? Well, there is at least one obvious example where the effects of distance really matter -- the sun! Its close proximity makes it look extremely bright! Maybe, however, it is actually identical to all the other stars in existence, just closer to us, and just possibly all the stars are completely alike in every respect. That turns out not to be correct. When we determine the varied distances to many stars, and compensate for the geometrical effects (the inverse-square falloff of apparent brightness), we learn that the stars are not all alike. There are real differences in absolute brightnesses. Some stars are intrinsically bright, while others are faint. (The sun itself is fairly undistinguished, in the middle of the range.) Other factors, something to do with the properties of the stars themselves and not merely their location in space, must also be at play. The second possibility arises from a different geometrical consideration. Suppose you were hovering just above the surface of a star, measuring the total amount of light coming from a small patch of its surface (one square metre, say). If you now looked at a slightly larger patch of the star's surface, say a square patch two metres on a side (an area of four square meters), you would of course see more light in total. That is, the amount of light entering your eye, or your measuring instrument, depends on how much surface area is contributing to the outflowing radiation. ( Analogy: if your postage-stamp lawn and my hundred-acre estate -- at least, the one I'd like to own! -- both grow grass to a height of five centimeters, I will have more clippings to dispose of than you do after mowing. Each square metre of lawn pushes up the same amount of grass, but mine has more area in total. In like fashion, two stars of the same temperature will emit the same amount of light per unit area, but not necessarily the same amount of light in total: that will depend on their total surface areas.) Is this an important factor? Do stars actually differ much in size? Yes, they do, and rather dramatically at that. There are stars hundreds of times bigger in diameter than the sun, stars called `giants' and `supergiants'; and there are `dwarfs', small stars comparable in size to the Earth itself. (We will learn later how their sizes are determined). Even if all stars had the same temperature, therefore, they could be fantastically different in brightness. A supergiant star might have a hundred million times the surface area of a small white dwarf, for instance, and be proportionally brighter for this reason alone. But this is not yet the whole story. The total energy emitted by a thermal radiator also depends - as I am sure you were waiting to hear! - on the temperature. In fact, the energy emitted per unit area of a thermal radiator depends on the fourth power of the temperature (when we express the temperature in Kelvins, measured from absolute zero). In other words, if we consider two stars of identical size, so that their areas are the same, but with temperatures of 3000 and 6000 Kelvins, then the hotter one will emit sixteen times as much total energy as the cool star (since 2-to-the-fourth-power, or 2 x 2 x 2 x 2, is sixteen). Just like the rich man, then, the hotter stars, having a larger total energy content, radiate it in great abundance. This is Stefan's law, a law which applies to all thermal bodies, not just stars. The effects of these considerations are summarized in the following figure, one in which the scale along the bottom is in units which may be unfamiliar to you. It is sufficient to understand that the left end of the scale refers to ultraviolet (short-wavelength) light, while the right end refers to infrared (long-wavelength) light. As you can see, the hotter object delivers more light in total (as is indicated by the extreme height of the curve). Moreover, it delivers a higher proportion of short-wavelength light (as is shown by the leftward shift of the curve). The coloured bands labelled "B" and "V" represent the amount of light delivered in the Blue and Visual (roughly yellow) parts of the spectrum. The preponderance of blue light from the hotter body explains its blue appearance; the cooler object will look yellowish (or reddish, depending on the precise temperature).
The Importance of Size: A Specific Case, and a Warning.Consider the star Betelgeuse. It is relatively cool, with a surface temperature about half that of the sun. Given its lower temperature, Stefan's law suggests that it might appear only one-sixteenth as bright as the sun, -- and indeed, each square metre of the surface of Betelgeuse does emit radiation at this rather feeble rate when compared to a square metre of the sun's surface. But in fact Betelgeuse is intrinsically a very bright star, about ten thousand times as bright as the sun in its overall energy output. How can Betelgeuse be so luminous? The answer, of course, is that Betelgeuse is large, more than 100 times as big in diameter as the sun, and it has thousands of times more surface area than the sun does. To get some idea of its vast size, note that if you were to put Betelgeuse where the sun is, the Earth would be inside it. It is a giant star. Interestingly, we will learn later that the sun itself will swell up late in its life, and may get to be as large as Betelgeuse. The Earth then will be literally inside the outer parts of the sun. To repeat an important earlier warning: the size of a star does not necessarily mean that it is very massive. When we talk about the mass of an object, we mean the total amount of matter it contains - all the atoms added up. To understand this important point, think of a big, fluffy, lightly-rolled-up snowball. It contains a certain mass of snow. If you now pack and compress it in your hands, making it dense and icy, it is smaller in size but of exactly the same mass as before. Likewise, a star like Betelgeuse may contain no more atoms in total than the sun does: they may simply be more widely spread out. Indeed, the density of Betelgeuse is very low.
Peak: The Wien Displacement Law.The final important feature of the Planck law is that hotter bodies give off light which is peaked at shorter wavelengths than cooler bodies, consistent with our earlier expectation that a more energetic (hotter) body would give off more energetic photons on average. An everyday example is a poker: when you put it in the fire, it warms up and glows a dull red; if you use a bellows and get the fire really roaring, you can make it glow white-hot. This law is given numerically at the bottom of page 162 in the text, and you can put in a temperature to see what it implies. Suppose for example we ask about something at "room temperature" - about 300 Kelvins (27 Centigrade). The equation implies that such a body should radiate energy at a wavelength of about 10 micrometers, which is in the infrared. And indeed, everything around us is doing exactly that - our bodies, the tables and chairs, the very air in the room, and so on. If we had infrared-sensitive eyes, we would see this glow. Thus the poker is 'glowing' in the infrared part of the spectrum even before we put it into the fire. Let us consider a couple of applications of the infrared glow of objects at everyday temperatures: In the winter, the walls and the very structure of our houses are cooler than in the summer since they are surrounded by cold air. As a result, they give off less infrared (heat) radiation. That is partly why it feels cooler at home in the winter, even if the thermostat is set to a quite comfortable level. The air is warmed by the furnace, but your body notices the lack of radiant heat (infrared energy) from the surrounding walls. (Your comfort is also affected by the dryness of the air in the winter, by the way.) The military uses "sniperscopes" to detect enemy soldiers. These are infrared-sensitive devices which can pick up the radiation from warm bodies, truck motors, and so on, so that troop movements can be monitored even at night. Some of the same technology is used in medicine - in the detection of cancers, for example, since they are characteristically of a different temperature than the surrounding tissue and radiate infrared differently. At the other end of the spectrum, you can show that it would require a substance with a temperature of about one million Kelvins to give off X-rays. Indeed, we find such material in great clouds of hot gas between and surrounding the galaxies in remote parts of the universe.
When Temperature Isn't a Factor.There are several contexts in which temperature has nothing to do with colour. When you see someone wearing a blue shirt, you do not immediately conclude that the shirt is hotter than the surface of the sun! As we have already noted, this is simply an example of something being seen by reflected light. The sunlight falling on it is white, but the reflected light has a different colour balance because certain wavelengths and colours are absorbed by dyes and pigments. Similarly, the daytime looks blue, but is not at a temperature of tens of thousands of degrees! Its blue colour, however, is not attributable to any pigments or to a 'paint job'; instead, it looks blue for the reasons explained on page 305 of your text. The blue wavelengths of sunlight are preferentially scattered into your eyes by the molecules in the atmosphere, in a phenomenon known as Rayleigh scattering, while the redder photons pass through unaffected. Sometimes, however, you see truly radiant objects which have different colours. Think, for example, of traffic lights. When you are at an intersection, are the red "stop" lights very much cooler than the green "go" lights? Do traffic engineers use light bulbs which heat up their filaments to different temperatures to create the desired colours? The answer is no; the colours look different to your eyes simply because there are green or red filters in front of ordinary light bulbs. These filters selectively absorb some wavelengths and let others pass through. The colour in this case does not tell you anything about the intrinsic nature of the light source. What you are looking at is not a "perfect radiator" since there has been a modification imposed by an external influence (the filters). Finally, there are special kinds of bulbs like neon lights which look very colourful (neon is brilliantly red, for instance; mercury lamps look blue) even though they are not seen through filters. Nor are they merely reflecting light from some other source: these lamps are truly radiant - that is, we are seeing light which they give off from within. Are they, then, at vastly different temperatures? The short answer is no: they are not thermal radiators at all. As it happens, a deep understanding of what is going on is very important to astronomy since this kind of physics provides us the tools for determining the compositions of the stars. We will need to explore how these lamps function, and how we can extend this understanding to the stars. 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, 20 January, 2021.)