Questions Arising from the HR Diagram:
A Point-Form Summary.
This section of the course notes, and the associated PowerPoint presentation, makes the following critical points:
in the HR diagram, we see stars of many kinds, but this does not mean that they are equally common. There are many faint stars low on the red part of the main sequence, and many white dwarfs; but there are relatively few very luminous stars, like the giants. Unfortunately, they are the conspicuous ones, so they catch our attention. We have to be very careful to pick an unbiassed sample if we want to know what an 'average' star is like
the sun itself lies in the middle of the range of stellar properties, but is not average. A truly average star is one of the much fainter ones
the HR diagram has a redundancy in it. For example, there are some stars of spectral type G which are like the sun -- that is, of middling luminosity. But there are some very luminous giant stars which are also said to be of spectral type G. These classifications mean that the spectral features are broadly similar, but the stars are clearly not the same objects! Fortunately there are some subtle differences in their spectra that allow us to distinguish the ultra-bright giants from the main sequence stars in other ways
this fine-tuning is important because we rely on the spectra of stars to derive their distances if they are too far away for a parallax measurement. For such stars, we do as follows:
get a spectrum for the target star
among the sample of nearby stars (those close enough to have their distances measured directly by parallax), find a star whose spectrum is exactly like that of the target, including the subtleties that allow us to distinguish giants from main sequence stars
then merely intercompare the brightnesses of the star of known distance and the target star to deduce how far away the latter must be in order to look as faint as it does
you can see that the spectrum plays the important role of allowing the target star to signal to us exactly which kind of star it is
in this way we can derive distances to any star which is bright enough to allow us to get a spectrum for it
the appearance of the main sequence is suggestive: it leads one right away to wonder if stars start out hot and get cooler and dimmer over time; or if instead they burn more vigorously with time, getting hotter and bluer. In short, do stars evolve along the main sequence? And what is the role of the red giants and white dwarfs? To resolve these questions, we need to know the masses of the stars
Associated Readings from the Text.Please look at: Chapter 16, especially pages 532-540.Selection Effects: The Demographics of Stellar Populations.In the original HR diagram, we saw about a dozen red giants but only one white dwarf. Is this because the former are more common than the latter? No! It turns out that white dwarfs are very common indeed. But we have an obvious problem: the red giants are bright and can be quite conspicuous even if they are far away. By contrast, the white dwarfs are faint and need to be practically on our doorstep if we are to discover or study them at all. They are easily missed! I used the following analogy in class. Imagine parachuting into the plains of Africa and trying to make quick notes on the relative numbers and diverse nature of the life forms there. You might notice a few elephants, a giraffe, perhaps some lions, and a lot of intermediate-sized things like small antelope. Unless you actually crawled around and took real pains, you would not notice that there are millions of bugs there as well -- and without a good microscope you would never notice the uncountable numbers of bacteria. If you wanted a truly unbiassed sample, you would have to cordon off a large area (say, one square kilometer of the Kenyan plains) and do an exhaustive study of every living creature in the sample, being especially careful not to overlook the large numbers of tiny creatures. So too with the stars: we have to take a representative region of space, some huge volume, and see what kinds of stars it contains. How might you accomplish this? Suppose I had not drawn your attention to this problem, but had merely asked you to find out something about `an average star.' You probably would have thought it sufficient simply to find out everything you can about the hundred brightest stars you see in the night sky. You could, for instance, determine the colours for all those stars, and work out an average colour, or go to the telescope on Ellis Hall and get a spectrum for each one, to see what a `typical' spectrum is like. But, as we have noted, that is not a truly representative sample. Indeed, if you restrict yourselves to these conspicuous stars, no white dwarfs or truly small stars show up at all in this sample, despite their great numbers. Most of the stars in your sample are big, bright objects; and indeed by these standards the sun itself is the faintest and smallest of all! To do this correctly, you need to imagine studying an `ice-cream scoop' of large volume surrounding the sun and including many hundreds of stars -- stars which were not chosen simply because of their prominence. What you find then is shown schematically in the figure on page 533. The most common stars of all are the so-called `red dwarfs', the faint red objects at the bottom end of the main sequence. There are quite literally millions of those for every one of the bright, hot O stars at the top end of the main sequence. Giants and supergiants are likewise very rare indeed -- one in a million or more. The white dwarfs are quite common, as I noted. In this more representative company, the sun is actually a fairly bright and big star, in fact considerably above average in both luminosity and size.The Sun as a Representative Star.In addition to its life-giving properties, the sun's importance stems from the fact that its proximity allows us to study it in intricate detail, and through it perhaps gain some deep insight into the internal structure of all stars. But if it has some truly exceptional or unique property, it may tell us very little about the other stars! What we have just learned about stellar demographics is thus a bit of a worry. If the sun is not average, can we safely assume that any of its properties apply to stars in general? There is one point I would like to emphasize: The sun is in the middle of the range of properties. Notice, for instance, that there are stars 100 times bigger than the sun, and stars 100 times smaller; there are stars 3 times hotter, and stars 3 times cooler. Since there are many more faint stars than there are bright ones, the sun is not `average' in the mathematically strict sense of the word, but we nevertheless find reassurance in the fact that it is intermediate in its properties. To understand the important distinction between "average" and "middle of the range," think of the following helpful analogy. Suppose in your first year of university you take ten half-courses and wind up with marks of 50, 51, 51, 52, 53, 53, 55, 58, 75, and 100. You will see that your mark of 75 is halfway between your lowest (50) and your highest (100), but it is considerably better than your truly average mark! Or consider an NHL player who makes 4 million dollars a year while some people make essentially nothing. This does not mean that someone making 2 million dollars has an `average' salary. The sun, like your mark of 75, lies at mid-range: it is neither outstandingly bright nor outstandingly faint. As noted, though, it is considerably above average in size, mass, temperature, and so on. On the other hand, there are millions of stars just like it in our Milky Way galaxy, along with billions of even smaller ones, so in many respects it can be profitably studied as a reasonable representative of stars in general. In other words, we were not wasting our time examining the sun's structure, interior, and energy sources. What we learned earlier will have broad applicability.Telling Giants from Main Sequence Stars.Before we try to develop our understanding any further, I should remind you that stars of a particular spectral type can appear in more than one location in the HR diagram. Note for instance that there are some G stars of middling brightness on the main sequence -- the sun is one example -- but there are also some very bright G stars in the giant region. (Look at the data points near the top right of the figures on page 533, and consider a star like Polaris, which you see plotted there.) Thanks to their large surface areas, such stars are a thousand times as bright as the sun. Given this real physical difference, what does it mean to say that Polaris and the sun are both "G stars?" They are obviously not the same beast! To understand this, recall first how the spectral type is determined: I spread the light of one of these stars out to form a spectrum, and study the characteristic pattern of absorption lines. Calling them both G stars reflects the fact that the absorption line pattern is broadly the same, and (as you will recall) this depends fundamentally upon the fact that they have the same surface temperature. But now an obvious question arises. If I find that a particular star has a "G-type spectrum," is there any easy way of distinguishing whether it is a bright giant or a fainter main-sequence star? There is certainly one unimpeachable way: determine its distance with a direct parallax measurement. Once you know the distance, you can work out how bright the star is in absolute terms and categorize it. (If, for example, the parallax measurement shows that the star is very far away, but it still looks conspicuously bright in the night sky, it must be intrinsically extremely luminous, and thus a giant with a huge radiating area.) The problem with this approach is that measuring parallaxes is tough and time-consuming, and really only practical for the stars nearest us. We'd like to know as much as we can about any particular star away off in the distance. Might there not be another way of distinguishing giants from main-sequence stars? Could the spectrum itself contain some subtle clue which allows you to make the distinction? Or are the spectra of G giants always exactly like the spectra of G main sequence stars? The answer is `no,' the spectra are not exactly the same. Two stars which have the same temperature have pretty nearly identical spectra, so that they both get called type G (say). But there are subtle ways in which the spectra differ, and these small differences can be used to discriminate the big bright giants from the smaller main sequence stars. One such "luminosity indicator," for instance, is that the hydrogen absorption lines are sharper and narrower for the bright giants than for the lower-luminosity main sequence stars. The reason for this is that the giants are more `puffed up,' which means that the atoms are much more widely separated in the outer parts of the stellar atmosphere than they are in the smaller, more compact star. The atoms interact to a lesser degree in the giant star than they do in the smaller stars, and the absorption lines look subtly different. These distinctions allow us to categorize stars not just by spectral type (which depends on temperature) but also by luminosity class (which is dependent on the size and brightness of the star).Determining the Distances of Remote Stars by Spectroscopic Parallax.My discussion of the subtle differences between the spectra of the giants and the main sequence stars of similar temperature may strike you as an irrelevant digression, of less than Earth-shattering importance. But there is in fact a very important consequence of this kind of observation. In a technique known as spectroscopic parallax, the study of the spectral types and luminosity classes of stars allows us to derive distances for many millions of stars, far out beyond the limits to which we can successfully make direct parallax measurements. (As noted earlier, we can measure trigonometric parallaxes directly for only a small fraction of the billions of stars in the Milky Way galaxy.) The name "spectroscopic parallax" is actually a bit of a misnomer, because no parallax measurement is directly involved. The name arises for historical reasons: Once you know the distance to a star, you can readily work out what parallax the star would have if only you could measure it, and distances used to be quoted in terms of the parallax, whether measured or inferred. To understand how the technique works, consider the simplest imaginable situation. Suppose you are studying an extremely faint star, one which is obviously very remote -- much too far away for you to hope to measure its parallax directly. But suppose you discover that its spectrum is identical to that of the sun, in every minute respect. It would then seem fairly logical to assume that the star itself is identical to the sun in all important respects, including its intrinsic brightness. We could then calculate how far away it is by comparing its apparent brightness to that of the sun. In spectroscopic parallax, we do essentially this, as follows: We start with a catalogue of many thousand stars whose distances are known from direct parallax measurements. In this catalogue are stars of a great variety of spectral types and luminosity classes. We now turn our attention to a star so far away that its parallax cannot be measured directly, but bright enough that you can collect enough light to get a decent spectrum. Suppose that the spectrum tells you that this star is, say, of K spectral type (a moderately cool star). As noted, this is only part of the story. Is the star a bright K giant; or is it instead a much fainter K main sequence star? What local standard star do we compare it to? As described above, the subtle differences in the star's spectrum tell you whether the star is a giant or a main sequence star. Let us suppose, for instance, that it is deduced to be a giant. You now know which of the more local stars is its "twin", and simply intercompare the brightnesses of these two to determine how much farther away the new target is. Note that the spectroscopy plays the critical role of allowing you to intercompare identical stars -- the remote targets and the more local stars of known distance which act as standards. If this sort of distinction were not possible, we would be in serious trouble, at least for stars which are isolated in space. (We will see later that determining the distances to clusters of stars is somewhat easier.) To return to our example, if K giants were spectroscopically identical to K main sequence stars, you would have no way of knowing whether a particular K star was a moderately close main sequence object just barely beyond the reach of parallax, or else a giant 10,000 times brighter but at one hundred times the distance!Do the Stars Change As Time Passes?The HR diagram relates the brightnesses and temperatures of stars in an interesting way, and allows us to identify the big, cool giants and the small, hot white dwarfs in addition to the main sequence stars. But science explores relationships; origins; evolution; and cause and effect. What questions arise as you look at the HR diagram? You would first of all like to know whether the observable properties of stars change over the aeons. Regardless of what the physics is, it is obvious that stars have limited "fuel supplies." They are emitting energy, but that must be a finite resource, so eventually they will `run out.' What will happen then? One possibility is that stars will evolve (change their properties) in a way which explains the HR diagram. Suppose you were to study a collection of people -- say, the population of Kingston -- and to plot their masses (weights) against their heights. It is easy to visualize the graph which would result. At the lower left, we would have data points representing babies of small size and weight; at the upper right would be the larger grown-ups, tall and heavy. Of course there would be some scatter about the general relationship, because some people are short but of heavy build while others are tall and slender, but the trend would be well established. What the figure would not show, but what is well-known to you all, is that this is a plot within which individual points change position as time passes. If I were to make another plot of the same sort five years from now, the point representing my eight-year-old daughter would not be in the same location as it is now. (She will be taller and heavier.) In general, individuals start at the "lower left" -- small size and weight -- and progress to the upper right. The positions of individual points on the plot depend on time. Is this true for the stars as well? Do their properties change in measurable ways as time passes? And can this explain the distribution of points in the HR diagram? A couple of plausible-sounding explanations for the HR diagram spring to mind: Perhaps the stars start out faint and cool, and gradually heat up and get brighter as the nuclear reactions within them (or whatever the energy source is) really take hold. Perhaps, therefore, the stars move up the main sequence from lower right (cool, faint) to upper left (hot, bright) as the aeons pass. Perhaps, instead, the stars start out fairly hot and bright, and gradually cool down as the fuel supply dwindles, rather like a hot coal dying away. Perhaps, therefore, they move down the main sequence from the upper left to the lower right; or Perhaps they do something more complex -- after all, we have yet to explain how the red giants and white dwarfs fit in. On the other hand, you might speculate that the stars don't change much over time. We will see, for instance, that the sun has not changed much in its appearance over the last 4-5 billion years (a conclusion we reach from a study of the fossil record). But what tell-tale piece of information would allow us to judge the correctness (or not) of these varied speculations? The critical information turns out to be the masses of the stars, which we derive from the binary stars. That is the subject of the next section of the notes. Previous chapter:Next chapter0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts: Part 1:Part 2:Part 3: |
(Wednesday, 22 April, 2026.)
