Stars and Their Distances: A Point-Form Summary. This section of the course notes, and the associated PowerPoint presentation, makes the following critical points: astronomers tried, literally for centuries, to detect heliocentric stellar parallax -- the effect by which the nearest stars should seem to shift about relative to the background of more distant stars as a result of the Earth's orbiting around the sun the problem was that stars were very remote. One can crudely estimate that the stars in the night sky appear to be about a trillion times fainter than the sun, which means (if the sun is a typical star) that the stars must be a million times farther away. This would explain the lack of readily observable parallax astronomers also picked the wrong targets to start with. They studied the stars that shine the brightest in the night sky, assuming that this was a consequence of the fact that they were the closest. This is not in general the case: the stars that appear so bright are often super-luminous stars that show up conspicuously even though they are very far away it was subsequently realized that the stars that appear to be moving quickly across the sky in their 'proper motions' are likely to be quite nearby. A remote star will scarcely seem to shift in position, even if moving quickly; a nearby star, even if moving rather slowly, will shift noticeably in position as time passes this realization led quickly to the first parallax measurements, in 1837, with many more to follow astronomers measure distances in parsecs , a contrived word which includes parts of the words parallax and the very small angle of a second of arc (which is about the size of the largest stellar parallax known, for the closest star of all). A shift of this small size means that stars will move almost imperceptibly between photographs taken six months apart in this course, I will tend to describe distances in light-years. This has the virtue of reminding us that we see the stars and galaxies as they were in the past there was one serendipitous discovery in the race to measure stellar parallaxes. Herschel had the bright idea of studying stars which appear to be closely side-by-side in the sky. He assumed that one of the stars was very far off, in the background, and that it would serve as a good reference point against which the other star would seem to jump back and forth as the Earth orbited the sun. Instead, he found that such pairs of stars orbited each other -- he had discovered binary stars! They are vital in our determinations of stellar masses, so this was fortunate indeed ground-based observations have yielded some tens of thousands of parallax measurements. Using satellites allows higher precision, and the HIPPARCOS satellite in particular allowed us to get parallaxes for over a million stars. There are, however, a hundred thousand times as many stars as this in our own galaxy, so the lesson is that parallax measurements work only for a tiny fraction of the stars - only the relatively nearby ones. (We will meet other tools for distance measurements later.) knowing the distances of many stars, we can now work out their absolute (or intrinsic ) brightnesses, free of any misleading effects caused by their varied distances thanks to having discovered binary stars, we can now now also work out masses of stars as well for those binaries whose distances we can determine (you need to know the distance to derive the mass) using eclipsing binaries, in which one star regularly passes in front of its partner, we can also determine the radii (sizes) of the stars, even if the two stars are merged hopelessly into a single point-like dot of light. There are other, more indirect ways of estimating sizes as well

Associated Readings from the Text.

Please look at: Chapter 16, pages 521-539. You can ignore the mathematical insights. Page 185, to see an image of the 'blurring' effects of the Earth's atmosphere

Stellar Variability.

In the previous section of the notes, I described the many things you can (or would like to) learn about the stars, pointing out that some of them required no knowledge of the distances at all. One thing I did not mention there in any detail is the fact tht some stars vary in brightness, for a variety of reasons. This is a topic I will return to later in several different contexts, but the existence of binary stars in particular turns out to of crucial importance in allowing us to develop a full understanding of the ways in which stars evolve -- so much so that we will devote an entire section of the notes to the nature of binary stars. Since binaries can be thought of as variable stars in a specialised sense of the word, and their discovery is itself closely related to the search for stellar parallax, I will digress for a moment here to define two broad classes of variable stars before resuming the narrative.

Extrinsic Variables: Binary Stars.

In a binary system, two stars are in mutual orbit around their common center-of-mass. As noted above, the details will be presented later, so here I merely note that: for nearby systems, like Sirius, we can sometimes see both of the stars in the binary pair. As the years pass, we can actually monitor their slowly-changing positions and follow the complete orbit. for binary stars at more typical (i.e. larger!) distances, we cannot see two separate points of light at all. The two images are inextricably merged into one, regardless of the power of our telescope. This limitation is imposed partly because of the smearing of light by the Earth's atmosphere, and more fundamentally because of the wave nature of light. How then can we know of the presence of a binary star at all? There are a couple of ways: from time to time, if we are favourably located, one star may pass in front of the other so that the total light we receive is reduced. The system is then called an eclipsing binary. The eclipses cause a perceptible dimming of the spot of light. (You can understand why such a binary can logically be called a variable star -- the spot of light brightens and fades as time passes, even though neither of the stars in the binary undergoes any intrinsic change. Only our orientation with respect to the stars changes as the orbit progresses.) Remarkably enough, a detailed study of this behaviour allows us to determine things like the diameters of the stars, in real units like kilometers, even though we can not even see the separate points of light. as the stars orbit one another, each has a slowly changing velocity. At a particular instant, one of the stars may be approaching us, while the other is receding. Later each star will be on the other side of its orbit and moving the opposite direction in space. These varying motions show up as changes in the Doppler shifts of the absorption features in the stellar spectra, and such a system is called a spectroscopic binary. These binary stars are examples of extrinsic variables: the observed brightnesses, or the spectra, change in response to the presence of another star. That is, there is nothing intrinsic to a star itself which causes these changes. Indeed, in that sense the sun itself is such a variable star -- after all, from our point of view it goes out once a day! As it slips below the horizon, we experience an eclipse of a sort: the Earth itself cuts off the sunlight. But of course this tells us nothing about the structure of the sun.

Intrinsic Variables.

There are also stars which vary in brightness owing to some internal property, or rather because of changes in some internal property. For reasons to be encountered later in the course, such variables are astrophysically important in a number of different ways. For the moment, I merely note that there are: pulsating variables, stars which vary in brightness because they quite literally pulse in and out, rather like the beating of a giant heart; irregular variables, which display some of the same attributes but in a less periodic and indeed sometimes quite chaotic fashion; and exploding stars, such as novae and supernovae. Some of these episodes, such as the cataclysmic supernova explosions, are known to mark the deaths of the most massive stars. Obviously, one could monitor the changing brightnesses of stars to find and study variables of these different sorts even without a knowledge of the distances of any of the stars. As always, however, true physical insights only come after we know the distances precisely. That is still the pre-eminent objective in astronomy. How was it finally acccomplished?

A Few Lines on a Chart.

The sixth Dalai Lama, 1682-1705, wrote: "This girl was perhaps not born of a mother But blossomed in a peach tree: Her love fades Quicker than peach-flowers. Although I know her soft body I cannot sound out her heart; Yet we have but to make a few lines on a chart And the distance of the farthest stars In the sky can be measured. " The Dalai Lama, a fine poet, was no astronomer! Although it had been known since the time of Copernicus that the phenomenon of stellar parallax should be measureable if indeed the Earth was in orbit around the sun, as Galileo had subsequently shown to be the case, the search for parallax was unsuccessful for centuries. Indeed, the first stellar parallax was not measured until 1837, almost a century and a half after the death of the sixth Dalai Lama. Measuring the distances turned out to be harder than anyone had anticipated, despite enormous and continuous efforts!

First Thoughts.

This litany of failure did not discourage astronomers, who had known for a long time that parallaxes would be very hard to measure. This follows immediately from the observation that the stars are very much fainter than the sun, and therefore much more remote than it is -- possibly millions of times farther away. How did they know that? Remember that the apparent brightness of an object depends on the inverse square of its distance. For instance, a nearby luminous object will look one hundred times as bright as an identical object at ten times that distance (since 100 = 10 x 10). Given that, we can make some reasonable estimates. The brightest stars are about ten billion times fainter than the sun, a number which is hard to measure precisely but which can be crudely derived through various clever techniques. This implies that even the very nearest stars, if they are at all like the sun, are about one hundred thousand times as far away as the sun is. This kind of consideration immediately implies that any parallax effect of the sort shown on page 525 of the text will be very small indeed. Even the nearest stars will appear to show only tiny shifts when compared to the background of more remote stars. Astronomers knew the difficulty of the observations they were trying to make, and were not discouraged by a repeated lack of success. But many of them made a simple strategic error in their approach to the problem, an error which certainly slowed progress.

Picking the Wrong Targets.

If you were told that the very closest stars might show a `parallactic shift' of the sort shown in the text, you would of course want to focus your attention on the best candidates. There is not much point in making repeated measurements of a very remote star for which the parallax will forever be immeasureably small. How would you choose your candidates? This is where astronomers got things wrong for a time. They made the perfectly reasonable assumption that the stars which look the brightest are the closest to us. This is certainly true in one special case: the sun looks fantastically bright because it is so near! But the assumption turns out to be wrong in general. For instance, Betelgeuse, one of the apparently brightest stars in the sky, is very far away, but looks bright simply because it actually is enormously luminous, an extremely bright beacon. Unfortunately, most of the really nearby stars are so faint that they can't even be seen with the naked eye! Astronomers wasted a lot of time, therefore, studying stars for which we now know parallaxes simply could not be measured. But the breakthrough came with the realization that the best stars to study are those that seem to move around relatively quickly in the ever-changing (but slowly changing) patterns in the sky. We went through this reasoning in the previous section of the notes. It depends on the fact that a truly remote star, even if it were travelling quite quickly, would only slowly change its apparent direction from our point of view, whereas a closer one, even if moving slowly, can change its position more quickly. For absolute clarity on this point, let me repeat the analogy I used in that earlier discussion. If you watch a distant airplane fly across the sky, it may take a few minutes to move across your whole field of view. But a fly can buzz past your nose and across your field of view in a matter of a few seconds, even though it is travelling much more slowly than the airplane. Its proximity makes all the difference. We also learned that centuries of study had shown that the stars do indeed move around in apparent position (the so-called `proper motions'). In the mid-1800's, astronomers focussed their studies on some of the most rapidly-moving stars and were able, in 1837, to measure the first heliocentric (sun-centered) parallaxes. The problem was solved at last, and the floodgates were open. Many more parallax measurements were to follow.

Units of Distance Measurement.

You should read through the discussions on pages 524-526 of the text. You will see there that astronomers use a unit of distance called a parsec. The rather oddly named unit comes from the measurements of the small parallactic angles. (See the figure on page 525, which is of course drawn very much out of scale! -- the star should be many hundreds of thousands of times as far from the Earth as the sun is.) A star which has a PAR allax of one SEC ond of arc is defined to be at a distance of one PARSEC. As it happens, a parsec is about 3.26 light years. Of course, the more remote stars have smaller parallaxes, in inverse proportion to their distances. Astronomers use parsecs all the time, but the text often uses light years instead. I approve of this unit for general discussion: it gives you a more easily grasped sense of the distances. (I remind you that one light year is the distance light travels in a year -- it is not a unit of time. But the use of the term 'light year' also reminds you that we are seeing the stars and the remote galaxies as they used to be, since it has taken time for the light to reach us.) One light year is about ten trillion kilometers, while a parsec is about three times that distance. Even the nearest stars are a few light years away. The immensity of these distances is underscored by the fact that there is no star within one parsec (3.26 light years) of the Sun -- or, equivalently, the largest measureable parallax (that of the closest known star, Proxima Centauri, which is 4.2 light years away) is less than one second of arc. In practice, this means that if you take a photograph of a field of stars, using your telescope, and then repeat the exercise six months later, even Proxima Centauri will have shifted in position by less than its own size on the image. (You can do better than that by using a telescope in space, since the images are not blurred by the atmosphere and can be much sharper and crisper. The tiny shifts are then easier to measure, but still very small.) Here's a good way of visualising this. Imagine asking a friend to 'move a bit to the left' for some reason. But suppose he is grumpy and uncooperative, and shifts only a centimeter or two, so that his head is still more-or-less above where his feet were to start with. If you are attentive, you will notice that he has moved, but it is not what you would call a dramatic effect! Given this stark reality, you can see that the figure shown on page 525 of the text is ridiculously optimistic. Rather than show the target star 'jumping' back and forth with respect to the background in the simulated images, it should have shown a shift comparable in size to the red dot of light itself -- or even smaller. That is why astronomers had such a hard time detecting and measuring the effects.

A Serendipitous Discovery.

As noted, stellar parallax was finally measured after attention was focussed onto the stars of highest proper motion (which are likely to be the closest stars). But on the way to this success, there was one interesting development which again exemplifies the role played by serendipitous discovery in astronomy. The story starts in the late 1700's with William Herschel. Now please remember that astronomers in those days had no photographic plates, no way of getting a permanent record of the positions of stars. In attempting to detect parallax effects, they would simply examine a field of stars by eye through a telescope and then, using fine micrometers and other instruments, measure how far apart the images seemed to be. They would reobserve the field some months later and repeat the measurements. The hope was that there would be perceptible changes in the position of one star with respect to its fellows, as shown in the figure on page 525 of your text. (I should emphasise again that the figure vastly exaggerates the size of the back-and-forth shift -- but Herschel and others did not know that. There were two problems: The first was that if you detected a change of position you had no idea whether it was real or not, since you had no permanent record (no photograph) of the first observation. Perhaps you merely wrote one number down incorrectly! So many repeated sets of observations, spread over many years, were required to confirm the reality of any detection. (But of course you would want to do this anyway, to confirm the long-sought goal of discovering a measureable stellar parallax. No scientist should be naive enough to base so important a conclusion on a single recorded observation!) The second problem was more difficult. Your bright target star might be seen more-or-less in isolation, with no other stars close to it. (The figure on page 525 exaggerates the probable richness of the field of background stars near a typical target star.) This meant that astronomers did not have many reference stars for comparison purposes, and few if any of them were seen to lie particularly close to the program star, a circumstance which limited the precision possible. Here is a poor analogy to make that point clearer. If you thought the wall of your house was slowly sagging, you might measure how big the gap was between it and (say) the nearby fence, looking for small differences as time passed. You would not try to make repeated measurements of how far the wall is from the CN tower in Toronto, or the flagpole in front of City Hall in Ottawa, and then look for changes! Small changes are more easily measured with respect to a much more local reference frame. For the second of these reasons, Herschel decided it would be most efficient to look at two stars that by chance seem to be close together from our point of view, on the assumption that one of them is quite nearby and the other very far away, in the remote background. Now suppose we watch this pair of stars for several years. As we orbit the sun, the nearer of the two stars will jump back and forth because of parallax, a behaviour which will show up very readily since there is a second point of light right beside it as an obvious background reference point. For this reason, Herschel expected to see one star `jump' back and forth relative to the other on six month intervals as the Earth swung back and forth around the sun in its yearly orbit. What he actually discovered is many of the pairs of stars that are in apparently close proximity are in orbit around each other! In other words, he was wrong in his assumption that one star was in the foreground and the other far away: instead, he had discovered binary stars! (In the figure, I have drawn the behaviour as if one star is at rest while the other revolves around it. In fact, of course, both stars are in motion around their common center of mass.) As we will see, binary stars are of critical importance in determining the masses of stars, so Herschel's discovery was one of the most important in all of astronomy. If not for the existence of binary stars, we would have very little understanding of astrophysics at all!

Increasing the Sample.

After the successes of 1837, parallax work was quickly extended to many hundreds of stars -- and more. In this way, we have built up quite a large sample of stars for which moderately precise distances are known. Of course, the closest ones are best-determined; the distances of the more remote stars are less accurately measureable. But please do not make the mistake of assuming that we can determine the distance to any star we like by making a parallax measurement! Parallax measurement of this sort is a tool which has a strictly limited range. In our Milky Way galaxy, for instance, we believe that there are about one hundred billion stars . Our ground-based parallax measurements allow us to get direct distance measurements for some tens of thousands of them, all relatively close to us. But we can improve on this with the use of telescopes borne by orbiting satellites. As noted above, the crisper images (free of atmospheric blurring) allow us to make more precise measurements, and to extend the range of our distance determinations. We have used a remarkable satellite called HIPPARCOS to measure the parallaxes for a million stars or more -- which sounds (and is!) wonderful until you realize that this is still only 0.001% of the total stellar population of the Milky Way. In short, our direct parallax measurements cover only a tiny fraction of the galaxy. Any estimates of the distances of more remote stars in the Milky Way, the size of the galaxy itself, and the distances to other galaxies, will need to be determined in some more complex way. (We will return to this question in due course.)

What Can We Derive?

Now that we know the distances to the stars, we can derive some important quantities, as follows: The Masses: Think again about how we work out the mass of the sun. We note that it takes the Earth one year to travel around it, in an orbit which is 150 million kilometers in radius. From this, we deduce how much mass the sun must have so that its gravitational force on the Earth is the right amount to keep us moving in this nearly circular orbit. Notice that to work this mass out we have to know the distance between the Earth and the Sun, since the gravitational force depends on the separation. Now suppose you could identify a planet (a small point of light) orbiting a nearby star, and discovered that it took exactly one year to go around. You might be tempted to conclude that the planet is in an orbit like that of the Earth and consequently that the star is of the same mass as the sun. Beware! This is not a safe conclusion! The period of the planet depends on the mass of the star and on how far the planet is from the sun. (If the sun was less massive, the Earth would move more slowly and take longer than a year to complete its orbit.) So unless you know how far the planet is from the star, the orbital period tells you nothing. How do you measure the true separation between the star and its planet? This is not difficult. First of all, you know how far apart they appear to be on the sky, measured as an angle. If the distance to the star can be determined by a parallax measurement, that angle can be converted to a true distance by the same reasoning you encountered in the first lab exercise. A simple application of Newton's laws then tells you the star's mass. But the important point is that we need to know the distances of the stars if we want to work out their masses in this way. In fact the stars have no planets that we can actually see (they would be too faint to detect), although their presence has been deduced in other ways for some stars. Consequently, we have to rely on binary stars to determine stellar masses. But the principle is exactly the same. We watch a pair of stars orbiting around their common center-of-mass to determine the orbital period, and (knowing the distance) derive the masses. Without binary stars, this would be an essentially impossible thing to work out! The Diameters (Sizes) of the Stars: In the first lab experiment of the winter term, you will discover (if you have not done it already) how one can derive the diameters of stars by measuring the duration of eclipses in an eclipsing binary. Such opportunities are rare because most stars are not in such systems. Fortunately, there are other ways of determining the size of a star, even one which is alone in space, provided we know its distance. We proceed as follows: From the spectral features or the colour of the star, we determine its surface temperature (for example, red stars are cool). This temperature is what determines how much light is emitted from every square meter of the surface (remember the laws of radiation). Meanwhile, we know how much light in total the star is emitting (because we know how bright it looks and how far away it is). These things together tell us what total area the star must have -- how many square meters in total -- in order to emit as much light as it does. This tells us its size. To make this sequence of thoughts come to life, let's think again about Betelgeuse. Parallax measurements reveal that Betelgeuse is quite far away. Given its large distance, how can it be one of the brightest stars in the sky? Obviously, it must be emitting a lot of light in total. But it is quite a red star -- you can even see that with your unaided eye! -- which means that it is cool. Consequently, every square meter of it emits only a relatively feeble amount of light. To produce as much light as it does in total, it must have a huge surface area. Conclusion: Betelgeuse is literally a giant star. (Of course these remarks can be made quite quantitative.) For the sun, this cumbersome technique is not necessary. We can see `how big the sun looks' -- that is, measure how big across it is in angular units -- and then deduce its real linear size through a knowledge of its distance. This is not in general possible for most other stars, which are so remote that we cannot see (`resolve') the faces: they merely appear as points of light. But for some of the very nearest really big stars, it is barely possible, using the very best telescopes, to see the stars as disks (rather like the way we see the sun as a circle of light) instead of just unresolved points. For these few stars, we can determine the sizes more directly. Previous chapter:Next chapter

0: 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:

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