Kirchhoff's Laws and Stellar Spectra. Kirchhoff's Three Laws: The Parable of the Piano Players. We have seen that the light of the sun can be spread out by a prism to reveal a spectrum which shows light at essentially all wavelengths from blue to red. We have also recognized that there is light at other, invisible, wavelengths, such as the infrared. Such continuous spectra are produced by thermal radiators -- hot, dense bodies -- like the sun, and the hot metal wires (filaments) inside incandescent lights. But now we consider a different kind of emission, starting with a homespun analogy. My three children and I are all taking piano lessons. My wife, in another room in the house, can tell which of us is at the keyboard by virtue of our varied levels of skill and by the music we play. But suppose the following: Imagine that my daughter Amy likes to play only the `A' notes on the piano -- high or low, but only an A (for Amy). (By the way, Amy does not do this! The analogy is a completely hypothetical one.) My wife would instantly recognize such a characteristic style. Imagine that I am more comprehensive, and play any and all of the keys, ranging up and down the whole keyboard (inclusively but perhaps not very musically). Again, this would be recognizable. Now ask yourself what my wife would be likely to hear if I was to sit on the piano bench with Amy on my knees, so that we could both reach the keyboard. She would probably expect to hear all of the notes (from me), and extra numbers of A's from Amy's participation, since she is generally so fond of playing them. My wife would be very perplexed to discover that, together, we produced sound consisting of everything but the A's! Remarkably, this is a very good analogy to how light behaves, as we shall see. Even more remarkably, there is an explanation which is both simple and compelling, one which depends on a clear understanding of the nature of the atom. [By the way, you can imagine a natural way in which Amy and I might produce the hypothetical effects described. Perhaps she is so protective of her favoured `A' keys that she jealously watches me, too busy to play them herself, and grabs my hands to prevent me every time I reach for an A. But it's hard to ascribe a motivation of 'jealousy' to atoms in their interaction with light, and we need to think of something else.]

The Parable in Light.

Kirchhoff's laws describe the spectra we get in three different circumstances, as follows: A hot, dense body emits light of all wavelengths (with the familiar Planck spectrum, depending on the temperature). This is analogous to me at the piano, playing every possible note. (Kirchhoff's First Law.) A hot, diffuse low-density gas emits light at only a number of discrete wavelengths. This is analogous to Amy at the piano, playing only her favoured 'A' keys. (Kirchhoff's Second Law.) [By the way, as you see in the figure, these emitted wavelengths show up in the spectrum as separate images of the slit of the spectrograph, so they are called emission lines. It is important to realize that if we had used a small circular hole instead to let the light in, we would have seen a set of images in the form of little circles of various colours. There is nothing particular significant about getting images in the form of lines. What you have to appreciate is the simple fact that the diffuse gas gives off light only at certain fixed wavelengths, while it emits nothing at all at many other wavelengths (=colours).] If we now let the light from a hot, dense body pass through a low-density gas which is cooler than the hot body, its spectrum is seen to be continuous except that it is missing light at certain wavelengths (the so-called absorption lines). Moreover, the absorption lines are at exactly the wavelengths which would have been produced by the diffuse gas on its own. This is analogous to me sitting at the piano with Amy on my knee between me and the keyboard. Suddenly the A's are missing instead of present in extra abundance! (Kirchhoff's Third Law.)

Spectral Signatures.

There is another very important thing to be said about Kirchoff's second law -- namely, it turns out that each element and molecule produces a distinctive set of emission lines. This is shown on page 160 of the text, for instance, and is the basis for certain kinds of forensic science. It is possible, therefore, to put a small amount of some substance into a flame and study the nature of the emitted light to determine the composition. (In fact, if you fling a handful of table salt into a fire, it flares up with a bright yellow colour because of the yellow emission lines in the spectrum of sodium - salt being sodium chloride, of course.) Naturally, physicists were keen to understand what makes the pattern of emission lines at all, and why it depends on the composition. Ultimately, it took the development of a whole new kind of physics called Quantum Mechanics before this could be fully understood. For many decades, people applied the effect , working purely phenomenologically -- that is, they use the spectra to deduce things about composition, with no understanding of the physical mechanisms at work.

The Spectrum of the Sun: The Solar Atmosphere.

Earlier, I described the sun as being a source of continuous radiation at all wavelengths. That is certainly how the solar spectrum appears when you see it as a rainbow, or when you shine light through a simple prism. But more detailed, careful analysis reveals that the sun is in fact a good example of Kirchhoff's Third Law. At certain specific wavelengths, there is not much light in the spectrum of the sun: it has absorption lines. The presence of these lines was first detected by a scientist named Fraunhofer, and they are still sometimes called Fraunhofer lines for that reason. This is shown in the following figure, from the original drawing by Fraunhofer: What causes the Fraunhofer lines? The situation must be as we described it in presenting Kirchhoff's Third Law. The interior parts of the sun are dense and hot, certainly, and this leads to the emission of light at all wavelengths. The light must then pass through some amount of low-density cooler gas, which gives rise to the absorption lines. But where is this diffuse gas? There are several possibilities. The Atmosphere of the Earth: Perhaps the diffuse gas is to be found in the Earth's atmosphere? The light which reaches our telescopes from the sun and stars certainly has to pass through the Earth's atmosphere, which is cooler than the stars and also of low density; thus the gases may absorb certain wavelengths. Indeed, this is part of the story. Some of the lines you see in the spectrum of the sun and stars are caused in this way -- but not all of them. We can tell which of the lines are formed in the Earth's atmosphere because: such lines are present in the spectrum of every star we look at from the ground, even though the spectra may differ in all other respects; such lines are less prominent when we work at observatories which are relatively high up on mountains, looking through less air; and these particular lines are not present in the spectra of stars studied from satellites completely above the Earth's atmosphere. Interplanetary or Interstellar Gas: Perhaps the space between the stars is filled with low density gas? Again, there is some truth to this: outer space is not a perfect vacuum. In the spectra of remote stars, we do see occasional absorption lines caused by interstellar gas. These effects are generally small, however. Another way to reach this conclusion is to examine the spectra of many stars. It is easy to find pairs of stars that seem to be side-by-side in space. Sometimes this is a coincidence, with one star very much in the foreground and one very remote; but there are also real binary stars, systems in which the two stars are in mutual orbit under the influence of each other's gravity. Whether we are studying a real binary or not is immaterial, however. The real point is that the spectra are often very different, which tells us that the absorption lines we detect are not caused by a big cloud of cool gas which lies between us and the pair. There must be something intrinsic to the stars which determines the nature of their spectra. The Solar and Stellar Atmospheres: The real answer is that the sun and stars are surrounded by thin gaseous atmospheres, and as the light comes flowing out from the dense, hot inner parts there is absorption at certain specific wavelengths, depending on the composition and physical state of the gas present. Do not think, by the way, of the stars as having a dense solid part and a thin detached atmosphere, as one visualises the rocky Earth and its thin atmosphere. The stars are gaseous throughout, thanks to their extreme temperatures which allow no solid structure; but they are nevertheless very dense in the inner parts (the centre of the Sun is about one hundred and fifty times as dense as water) and simply progressively less dense as one moves outwards towards the ill-defined surface. There is an additional informative way in which you can tell that the Sun's spectrum must be intrinsic to it, not the result of interplanetary gas. One can study the spectrum of light from the `left' and `right' sides of the sun, as seen from the Earth. We know, as did Galileo, that the sun is rotating because we can see sunspots move across it; thus we know which side of the sun is approaching us and which side is going away. The spectrum of light from the approaching side is not the same as that from the receding side: the absorption lines occur at slightly different wavelengths. The shifts in the positions of these spectral features is in the sense and of the size expected from so the absorption lines must be produced by gas sharing the rotational motion of the sun. It is impossible to understand how gas distributed between the planets could, by coincidence, give rise to spectral features which so perfectly accord with the known solar rotation.

The Composition of the Stars: Within Our Grasp?

To an extent, we can now study the stars rather as a forensic scientist might. From the presence of a certain pattern of absorption lines in the spectrum - say, that corresponding to the element calcium - we can presumably deduce that atoms of a certain type are present in the star. There are several important points to note here: This sort of analysis first demonstrated that the stars are made up of everyday elements found on Earth. Before this tool was available, one simply did not know that! The stars could, after all, be made of elements that simply do not exist elsewhere in nature. The absorption lines merely prove the presence of a particular element. To deduce the proportions (the actual composition) may require considerably more thought! This is a point I will return to in a moment. Finally, the spectra of different stars can be very different. (This is shown in greatly idealized form in a figure to be found on page 529 of your text. I will say more about this in a few paragraphs.) A star like Sirius, for instance, shows a very strong pattern of absorption lines attributable to hydrogen. Is this because it is particularly rich in hydrogen? The sun does not show strong hydrogen lines, but has a lot of lines which we recognize as due to the elements calcium and iron. Could this mean simply that the sun is rich in these elements, and perhaps deficient in hydrogen?

The Composition of the Sun: Helium.

Early study of the spectrum of the sun revealed lines of an element as yet undiscovered on Earth. It was consequently named helium , after helios , the Greek word for sun. Later, this element was discovered on Earth, and we now recognize that the sun contains all the elements known on Earth, and none that we cannot recognize. As noted in the section above, however, determining the proportions are not so easy. Indeed, it was not until the 1920's that the science of astrophysics was well enough developed to allow the interpretation of the spectrum in terms of the actual abundances. Look at the more detailed solar spectrum shown in the figure below. (The spectrum has been split into pieces, from the blue end at the top left to the red end at the lower right.) As you can see, there are many absorption lines present -- literally thousands, in fact, if one looks in fine enough detail. Many of the absorption lines seen are due to elements like iron, and there are some very prominent features due to calcium. The hydrogen lines are weak and the helium lines nearly undetectable. But what does this mean? Is it because the sun is rich in iron and calcium, poor in hydrogen and helium? In the early part of this century, that was the thinking of an astronomer named Henry Norris Russell, of Princeton University, who boldly tried to derive the sun's composition. He deduced that the sun was rather like the Earth in composition, which satisfied nearly everyone since it seemed so logical. (This came to be known as the Russell mixture.) After all, didn't the Earth form out of the same kind of material which we find in the sun? The only problem is that the conclusion is wrong!

Spectral Classification.

Kirchhoff's laws were known by the late 1800's, although the understanding was meagre. But a new technological breakthrough allowed a profitable marriage between the science of astronomy and that of spectroscopy. For the first time, astronomers could take photographs of the sky. By putting a photographic plate behind a prism, they could do even more: they could take pictures which recorded the spectra of dozens of stars at the same time. By pointing the telescope in different directions, astronomers were able to collect images of the spectra of literally hundreds of thousands of stars - although no one knew how to interpret them! In science, when a new discovery is made -- something like the discovery of a new species, say -- an early stage in its pursuit is the creation of a catalogue of various subclasses. This organizational chore expedites subsequent analysis by providing clearly defined subsamples for later examination once the basic physics is understood. In this spirit, an astronomer named Annie Jump Cannon (and some others before and alongside her) examined and classified a lot of spectra, in a very simple way. In essence, Ms Cannon merely looked at them, one after another, and assigned them to classes of her own devising, using a simple alphabetical naming scheme. According to this scheme, for instance, `A' stars had the very strongest hydrogen lines; `B' stars had hydrogen lines plus some helium lines as well; and so on. The point of doing this was to provide a big database which might eventually be put to use once our astronomical insights had improved. The breakthrough came with the development of atomic theory and the new quantum mechanics: suddenly it became possible to understand why the absorption lines looked the way they did. The greatest contribution came from a Harvard astronomer named Cecilia Payne, whose Ph.D. thesis established the new science of Stellar Atmospheres. Building on the work of others, she was able to show: that the stars are almost all the same composition, containing about 3/4 hydrogen, about 1/4 helium, and mere traces (a few percent at most) of other species; and that the differences in spectral line strengths are almost all attributable to temperature differences. Moreover, she showed that Annie Cannon's classifications were not in a physically meaningful order (for instance, A stars are of middling temperature, B stars are hotter, G stars are cooler, O stars are the hottest of all, etc.). Annie Cannon's whole classification scheme was re-ordered in light of this new insight, and various redundant categories were eliminated, so that now we have stars that run from hottest to coolest in the order OBAFGKM (RNS). (The stars called R,N,S don't always appear in the list, by the way. They are not cooler than the M stars, but instead differ in some other subtle ways.) Astronomers all know this sequence by heart, and when I hear of an `O' star I instantly think of a hot star. But there is a famous mnemonic to help you remember the sequence: O h B e A F ine G irl (or G uy), K iss M e ( R ight N ow - S mack!). The surface temperatures range from about 30,000 Kelvins for the O stars down to about 3000 Kelvins for the M stars. The sun, a G star, has a surface temperature of about 6000 Kelvins. Cecilia Payne, and those who learned from her work, were now able to study the spectrum of a star and work out both its temperature (from the presence of tell-tale spectral lines) and its composition (from the details of the strengths of those lines). This was perhaps the most important breakthrough in stellar astrophysics in the first half of this century. Look again at the idealized stellar spectra on page 529 of your text, or in the figure below. It is interesting to note just how different they appear; yet the fact remains that the compositions are very similar.

More From Stellar Spectra.

It is worth repeating, yet again, exactly how rich a bounty we get from the detailed analysis of stellar spectra. For example: tells us the speed with which the star is approaching us or receding; if we find indications of a hot star (helium absorption lines, say) and a cool star (molecular absorption lines) in a single spectrum, the conclusion must be that the light we are studying must be from a binary pair, with the light inextricably mixed; the widths of the absorption lines can tell us about the rotation rate of a star, as explained on page 167 of the text (although they should have been talking there about absorption lines instead of emission lines); and other details of the absorption lines can tell us about the strength of the star's magnetic field; whether it has a particularly turbulent atmosphere; whether there are active regions (flares) on its surface; whether the star has a fairly dense atmosphere or if it is distended and diffuse; and a variety of other things.

An Important Warning.

When we use the spectrum to deduce the composition of a star, we are really studying only the outermost parts (which is where the absorption lines are formed, in the diffuse atmosphere). We will learn later that stars undergo nuclear reactions deep inside, a process which can profoundly change the composition in the core. But that processed material mostly stays buried deep inside, unobservable - at least until the star enters its death throes, which we will learn about in due course. Consequently, for most stars the outer parts still consist of the mix of elements out of which the star was made when it formed, and (with occasional exceptions) it is this original mixture that we mean when we talk about the composition of a star. 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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