REQUIRED READINGS: Text, Ch. 17, and all that follows here.

1. INTRODUCTION

Until this century, almost all astronomy involved the study of the planets, moons, comets, the Sun, and stars. It's easy to see why. When you look at the sky at night, all you see is a few planets, and occasionally some wondrous intruder, such as comet Hale-Bopp that passed our way last Spring -- and stars.

In Boulder, where the city lights make the sky bright at night, you can see a few dozen of the brightest stars that define the constellations. But at a dark place on a moonless night, you can see a few thousand stars. Now, with telescopes like Hubble, we can see literally billions of stars. We now know that many of these stars are rather similar to the Sun. But some are blue giant stars hundreds of thousands times more luminous than the Sun; some are equally luminous red giant stars, hundreds of times bigger than the Sun. The sky is also filled with stars too faint to see with the naked eye. Some are red dwarf stars, not much bigger than Jupiter and hundreds of times less luminous than the Sun; and some are white dwarf stars, as massive as the Sun but no bigger than the Earth.

Moreover, we know how these stars work. Beginning early in this century with the pioneering work of Lord Kelvin, Sir James Jeans, and Sir Arthur S. Eddington, and continuing after World War II with the work of nuclear physicists such as George Gamow and Hans Bethe, astrophysicists have developed a theory that explains how normal stars burn, why they are as bright as observed, how long they will live, and how they die. For all intents and purposes, normal stars are a done deal. The basic questions are answered. Nowadays, most astronomers have moved on to other subjects. But, as you will see, there are plenty of things left to learn about stars: there are many abnormal stars that we do not understand, and there are still some big unanswered questions, especially about the brightest and the faintest stars.

The story of how astronomers learned the secrets of the lives of stars is fascinating -- one of the triumphs of modern science. It is also a very good example of how science progresses: from observation and technology; to classification, or taxonomy; to theory. So here goes:

Like much of Western Civilization, it began in Greece, some 2100 years ago. Once upon a time, on the beautiful island of Rhodes, there lived a man named Hipparcos ...

2. STELLAR POSITIONS

Stars have three properties that are visible to the naked eye: position, brightness, and color. These properties are extrinsic, i.e., they depend not only on the properties of the actual star, but also on external conditions, such as the location of the observer. To understand stars, we must find ways to infer their intrinsic properties from these extrinsic properties. Intrinsic properties are properties that describe the star itself, independently of the observer, such as radius, luminosity, surface temperature, composition, and mass.

Let's start with position: the location of a star in the sky. Position is clearly an extrinsic property. If the Earth were located a few hundred light years away from its present position, all the constellations would be different. The positions of the brightest stars relative to each other would be totally different.

In fact, we can see the positions of the nearby stars change. The motion is very slight; the constellations look almost the same today as they did thousands of years ago. But with very accurate measurements, we can see the nearby stars move relative to the more distant stars by fractions of an arcsecond (denoted ") over timescales as short as a year. These apparent motions are the result of three actual motions: (1) the Earth's orbit around the Sun; (2) actual motion of the nearby stars through space; and (3) the actual motion of the Solar system through space.

The most important and useful kind of proper motion of nearby stars is that due the Earth's motion around the Sun. It causes the apparent positions of nearby stars to wobble back and forth annually [in circles if the star is toward the ecliptic poles (poles of the Earth's orbit), in straight lines if the star is in the ecliptic (equator of the Earth's orbit)]. The closer the star, the greater the angle of wobble, called parallax. (Actually, half the total wobble angle over a period of one year.) By measuring the parallax of a star, we can infer its distance, as described in your text, pp. 358-359.

Parallax angles are very small because the radius of the Earth's orbit is very small (< 10^-5 times) compared to the distance to the nearest star. In fact, the nearest bright star, a -Cen, has a parallax of 0.753". Such an angle is no greater than the size of the blurred stellar image seen through a telescope under conditions of very good seeing. Astronomers can actually locate the position of a star more accurately by measuring the average position of the center of the blur. That will give positions of the star that are accurate to about 0.01"; but that is about the limiting accuracy that can be achieved with ground-based telescopes.

A little trigonometry, explained in your text, gives the simple formula d = 1/p, where d stands for the distance of the star and p stands for the parallax angle of the star (measured in arcseconds). This formula works if the distance is measured in units called parsecs (pc). One parsec is equal to 3.24 light years, or 3.08 x 10¹⁶ meters. Therefore, the nearest star a -Cen, with p = 0.753", has a distance d = 1/(0.753) = 1.33 parsecs, or (1.33 x 3.24) = 4.3 light years from the solar system.

Since astronomers can measure parallax with an accuracy of about 0.01" from the ground, they can infer the distances of stars (with decreasing accuracy) out to about 100 pc. Table 5 in Appendix A4 of your text lists the parallaxes of the 30 known stars that are less than 4 pc distant from the Sun.

As you will learn throughout this course, it's very important for astronomers to be able to measure the distances to more distant stars, galaxies, and other objects. They took a giant step in this direction with the launch of the Hipparcos satellite, built and operated from 1989 - 1993 by the European Space Agency. Above the blurring effect of the Earth's atmosphere, Hipparcos measured parallaxes with an accuracy of 0.001". With such accuracy, Hipparcos could measure the distances of stars out to about 1000 pc (or 1 kiloparsec, 1 kpc), or about 10 times better than could be done from the ground. The Hipparcos catalog contains some 120,000 stars.

Actually, you can think of the parallax method as a kind of binocular vision. You have depth perception because nearby objects in the image on the retina of your left eye are shifted a little to the right compared with the same objects in the image on the retina of your right eye. The part of your brain devoted to processing these images makes millions of calculations every second to combine these shifted images of objects and your mind that they actually come from nearby objects. You can think of two Hipparcos images of the sky taken 6 months apart as images from the left and right eyes. But in the case of the Hipparcos images, the "left eye" and the "right eye" are separated by the diameter of the Earth's orbit, or 2 AU.

Click here to see an image from Hipparcos of the part of the sky containing the Big Dipper (Ursa Major). It's actually a superposition of two images, one in blue and one in red, taken 6 months apart. The red image is taken when the Earth is to the right, so the nearby stars in the red Dipper are shifted to the left compared to those in the blue Dipper, while more distant stars are shifted less. In these images the shift (or parallax angle) is exaggerated by a large factor to make it easy to see. With red-blue ("3-d") glasses (put the red over your left eye), you can actually see the amazing depth of this field of stars. It may take a while for the 3-d effect to come in. (You might find it easier here.) Your brain must solve many thousands of equations to create this image in your mind. You can find a few more of these fantastic images in the Hipparcos home page.  (The Hubble Space Telescope can also measure parallaxes with comparable accuracy, but it has such a narrow field of view that it would take forever to compile a catalog with as many stars as the Hipparcos catalog.)

In addition to parallax, the positions of nearby stars move because the stars themselves are actually moving relative to each other. These motions, called proper motions, and can be distinguished easily from parallax motions because they do not oscillate with a one year period. They appear as a steady drift in a straight line that is combined with the oscillatory motion due to parallax to create a kind of loopy drift. If two stars have the same transverse (sideways) velocity, the nearer one will have a greater proper motion (i.e., will have a greater drift angle in a given time). The star with the greatest known proper motion is called Barnard's star (see Table 5, Appendix A-4). It has a proper motion of 10.31" per year, much greater than its parallax of 0.544", which implies that it is actually moving through space with a transverse velocity of about 90 km/s.

Even if all the nearby stars were stationary, they would have proper motions due to the Sun's actual motion. If the Sun were moving to the east, the nearby stars would appear to be moving toward the west (relative to the more distant stars). We can distinguish the apparent proper motions due to the Sun's actual motion from those due to the stars' actual motions because the Sun's motion will cause all the nearby stars to appear to be moving in the same direction, whereas those motions due to the stars' actual motions will have no such correlation.

3. STELLAR BRIGHTNESS

After position, the second obvious property of stars is their brightness. In about 150 BC, Hipparcos made the first known catalog of stars. He listed about 5000 stars and ranked them according to brightness. He gave the brightest stars (about 25 of them) the rank of 1st magnitude. Then he classified the rest 2nd through 5th magnitude, with 5th magnitude stars designating the faintest stars that he could see on a moonless night. Today, astronomers still use the magnitude system to define the brightness of stars, but they have refined the system to about 3 significant figures accuracy. Thus, the stars that Hipparcos classified as first magnitude now range in magnitude from about m = -1.47 for the brightest Star, Sirius, to a star such as Deneb (a-Cygni) with m = 1.26. The symbol m is called the apparent magnitude.

The magnitude scale is described in pp. 366-367 of your text. You don't need to memorize it. Just remember this: if the magnitude increases by adding 5, the actual brightness decreases by a factor of 100. That means, for example, that a star with m = 6.5 is 100 times fainter than a star with magnitude 1.5, and a star with magnitude 11.5 is 100 times fainter than a star with magnitude 6.5. So, a star with magnitude 11.5 is (100 x 100) = 10,000 times fainter than a star with magnitude 1.5.

The brightness (B), or magnitude, of a star is an extrinsic property, because it depends not only on the star's actual light output, or luminosity (L), but also on the distance of the star. Imagine two stars, Alpha and Beta, having equal luminosity. If Beta is the same distance from the Sun as Alpha, it will appear equally bright as Alpha. But if Beta is twice as far, it will be 1/4 as bright as Alpha. Three times as far, 1/9 as bright, and so on. This decrease in brightness with increasing distance (D) is called the inverse square law, which is expressed by the equation B = L/(4p D²). See if you can figure this one out: suppose that you have two identical stars, Alpha and Beta, with equal luminosity. Alpha has apparent magnitude m = 0. Beta is ten times more distant from the Sun than Alpha. What is the apparent magnitude of Beta?

If we know the distance of a star (say, as a result of measuring its parallax), we can infer its luminosity from its observed brightness and the inverse square law: L = 4p D²B. Thus, we can infer an intrinsic quantity, L, which describes a property of the star itself, from two extrinsic quantities, D, and B, which we can measure.

Besides for distance, another factor can cause the brightness of a star to decrease, more than the inverse square law would predict. That is absorption by interstellar dust. Suppose again two identical stars, Alpha and Beta, having the same luminosity and both at the same distance. But now suppose that Beta lies behind a dust cloud and Alpha doesn't. Beta will be fainter because the dust absorbs part some of its light. Thus a star's brightness (an extrinsic property) depends not only on its luminosity (an intrinsic property), but also on two other factors, its distance and whether or not a dust cloud happens to lie between the star and the Sun.

We can tell whether a star is dimmed as a result of obscuration by interstellar dust, because the dust makes the star appear redder than it actually is. That happens because the dust absorbs more blue light than red light. We see this effect on Earth, especially during a desert sunset. The Sun is very red when it sets because the dust in the Earth's atmosphere absorbs the blue sunlight and lets the red sunlight through.

To understand how stars work, astronomers want to measure intrinsic properties, such as luminosity (L). If we can measure a star's parallax (p), we can calculate its distance, D = 1/p. Then, if we measure its brightness (B), and if we know that the absorption due to dust is negligible, we can calculate its luminosity (L) from the inverse square law above. (In fact, there is very little interstellar dust within about 200 pc of the Sun, so for nearby stars we don't have to worry about this absorption.) So, we can infer the luminosity of thousands of nearby stars from this procedure.

There is one more term you need to know: Absolute Magnitude. Absolute magnitude (denoted capital M) is a measure of luminosity (L), while apparent magnitude (denoted m) is a measure of brightness (B). The important point to remember about the definition of absolute magnitude is this: the absolute magnitude is the apparent magnitude that a star would have if it were at a distance of 10 pc (with no absorption by interstellar dust). If the star is at a distance greater than 10 pc, its absolute magnitude is less than its apparent magnitude (see Appendix A3, Table 4). For example, the star Arcturus is at a distance D = 11 pc, and has apparent magnitude m = -0.06. The absolute magnitude of Arcturus is M = -0.3, only a small amount (-0.24) less than m. But now take a -Cen, with m = -0.01, almost equally bright as Arcturus. But since a-Cen (D = 1.3 pc) is much closer than Arcturus, its absolute magnitude (M = 4.4) is quite a bit greater than that of Arcturus (M = -0.3). That means that a-Cen is much (about 75 times) less luminous than Arcturus. In fact, a-Cen is just 1.6 times more luminous than the Sun (see Apendix A-4, Table 5).

Here are some simple mathematical formulas that define the magnitude scale:

4. SPECTRAL TYPE

The third obvious property of stars, visible to the naked eye, is its color. The color of a star is an extrinsic property because, as we have already noted, it may be reddened if the starlight passes through intervening dust. The related intrinsic property of the star is its true color, or, more precisely, its spectral type, which is determined by the temperature of its photosphere.

As a first approximation, a star radiates a blackbody spectrum. Therefore, according to Wein's Law (see text, Fig. 3.13), the true color of a star depends on its temperature: a hot star (say, 20,000 K) appears blue, while cool star (say, 3000 K) appears red (see text, Fig. 17.8). Thus, a nearby red star must be relatively cool, because we know that there is very little dust nearby to make the star appear redder than it actually is. A more distant star may appear red because it is cool. But it may be a hot blue star that only appears red because its blue light is absorbed by intervening dust. How can we tell?

It turns out that astronomers have a far more accurate method of measuring the temperature of a star's photosphere than by measuring colors. That is to measure the star's spectrum, which contains hundreds or thousands of spectral lines of different atoms and molecules. The presence or absence of certain spectral lines of an atom depends not only on the abundance of the atoms, but also on the temperature of the gas. Take the spectral lines of helium for example. Because it takes a lot of energy to excite the electrons in helium atoms, we can't see any lines due to neutral helium in the star's spectrum unless the gas in the photosphere has a temperature T > 15,000 K, and none due to ionized helium unless T > 25,000 K (see text, Fig. 17.10 and Table 17.2). Likewise, we can't see any spectral lines due to molecules unless T < 4500 K. By analyzing the ratios of the strengths of spectral lines from different atoms and molecules, we can determine the photospheric temperature quite accurately. So, for example, if we see a red star that has spectral lines due to helium, we know it must intrinsically hot and blue. Therefore, we know it must be behind a dust cloud.

For historical reasons, astronomers classify the temperatures of stars on a scale defined by spectral types, called O B A F G K M, ranging from the hottest (type O) to the coolest (type M) stars (see text, Fig. 17.10 and Table 17.2). These spectral types are further subdivided with a decimal system, ranging from 0 (hotter) to 9.5 (cooler). Thus, the coolest type O star (T = 23,000 K*) is called O 9.5, while the hottest type B star (T = 21,000 K*) is called B 0.

[*Experts will notice that I am being a bit loose in my definition of stellar temperatures. Because stars, especially the hotter ones, do not radiate exactly as blackbodies, the temperature of the gas in the photosphere is not exactly equal to the temperature defined by the color of the radiation. But this distinction is not critical for the argument here. When I quote a value of temperature here, I use "effective temperature", the temperature for which the Stefan-Boltzmann law is valid.]

Astronomers add a third Roman numeral to the spectral classification scheme to designate luminosity class, ranging from class I (for exceptionally luminous) to V (for relatively low luminosity). Thus, for example a K 0 I star means a relatively cool (T = 4000) K star that is exceptionally luminous for its spectral type -- a red giant. (Sometimes astronomers add yet more letters and numbers after the luminosity class, but that is getting to a level of detail that we'll overlook here.)

As described in your text, astronomers have been classifying stars according to spectral type for more than a century. We now have catalogs listing spectral types and magnitudes of several hundred thousands of stars.

5. STELLAR RADII

The luminosity radiated per unit of area of a stellar photosphere depends only on the photospheric temperature, i.e., on the spectral type. For example, we pointed out that each square centimeter of the Sun's photosphere radiates about 6500 Watts. Experiments on Earth show that the power per unit area radiated by a blackbody surface depends on the fourth power of the temperature. I.e., double the temperature and the power goes up by a factor 2⁴ = 16. Stellar photospheres radiate almost like a blackbody. So a hot star will radiate much more per unit surface area than a cool star.

But, given equal temperatures, a large star will radiate more luminosity than a small star, simply because it has more surface area. We can put these two facts together in the following formula for the stellar luminosity: L = 4p R²s T⁴, where R is the radius of the photosphere, s is a physical constant that can be measured from laboratory experiments, and T is the photospheric temperature. This equation is called the Stefan's Law (see text, p. 65 and 361). It is a powerful tool, because we can use it to infer a star's radius, a quantity that we cannot observe directly for most stars, from quantities that we can observe directly: the star's parallax (from which we can infer its distance), its brightness (from which, with distance, we can infer its luminosity L), and its spectral type (from which we can infer its photospheric temperature, T). Given L and T, and the known value of s, we can use Stefan's Law to solve for the star's radius R.

If a star is cool but very luminous, it must be very big. For example, the star Betelgeuse (spectral type M2 I -- see Appendix A3, Table 4) is cooler than the Sun but its luminosity is more than 10,000 times greater than the Sun's. That is possible because its photosphere extends to a huge radius -- about 400 times the Sun's radius. If the Sun were this big, its photosphere would extend beyond the Earth's orbit! We call such stars red giants.

This method of inferring the radii of stars based on physical laws discovered in laboratories on Earth depends on a giant principle, which we call The Universality of Physical Laws. This principle asserts that the laws of nature are the same everywhere, in Earth laboratory and thousands, even billions of light years away. Really, it's an assumption. We shouldn't take it for granted; we must constantly test it whenever we get a chance. Today, we are able to test the universality of the Stefan-Boltzmann law because we can measure the radius of nearby red giants such as Betelgeuse directly by stellar interferometry. We find that the radius measured by interferometry is, within the uncertainty of the measurement, the same as the radius inferred by the Stefan-Boltzmann law. So, at least in this case, the principle works.

Arthur S. Eddington understood all this in 1920. You can find his discussion of the radii of stars in The Internal Constitution of the Stars, which you should have read already while studying about the Sun.

Because they can't travel to stars, astronomers are always using the Principle of Universality of Physical Laws. For example, they use when they infer the temperatures and atomic abundances of stellar photospheres by comparing stellar spectra with laboratory spectra.

So far, nobody has found any deviation from the Principle of the Universality of Physical Laws. You might then ask: why not just take it for granted and forget about testing it. But that attitude goes against the whole philosophy by which science progresses.

There's a saying I like about the difference between religion and science: "Religion is based on faith; science is based on doubt." Most religions require that their followers take a leap of faith, accepting some tenets that cannot be proved by scientific experiments (and perhaps cannot be disproved either). Such acts of faith may greatly enhance one's religious experience.

But scientists make the greatest advances when they can overturn some commonly held belief. For example, with his Theory of Relativity, Einstein revised Newton's laws of motion and overturned commonly held ideas about the absolute nature of time.

Some great scientists have proposed theories in which the laws of nature change over great distances or times. If anyone could devise a test to verify such a theory, he or she would certainly win the Nobel Prize. But so far, nobody has found any violation of the Principle of Universality of Physical Laws.

6. BINARY STARS

Slightly more than half of the stars in the sky belong to binary systems. For example, Sirius, the brightest star in the sky, has a faint blue companion, and a -Cen, the nearest star, has a fairly bright red companion. In these binary systems, the stars orbit each other, just as planets orbit the Sun and other stars.

Observations of binary stars are very important because they give us a chance to measure the stars' masses -- something we cannot do with single stars. The orbital motions of the stars are determined by a balance between centrifugal force and gravity. If we know the speed of the stars and the distance separating the stars, we can calculate the centrifugal force. But that must be equal to the attractive gravitational force, which is proportional to the product of the two masses divided by the separation distance. This principle of balance of forces yields Kepler's Third Law, which may be expressed in the following form (see text, p. 48):

where m₁ denotes the mass of the first star, m₂, the second star, a is the orbital separation (or, if the orbit is elliptical, the semimajor axis -- see text, p. 41), and P is the orbital period (the time to make one full orbit). K is a constant which has the value K = 1 if all quantities are expressed in solar system units: m₁ and m₂ in units of the Sun's mass, a in Astronomical Units (1 AU = the distance from Earth to Sun), and P in years.

(Actually, Kepler didn't really understand his Third Law -- he simply discovered that this kind of equation could account for the motion of the planets around the Sun as observed by Tycho Brahe. In 1687, 68 years after Kepler discovered his Third Law, Isaac Newton explained how it was a logical consequence of the balance of gravity and centrifugal force.)

Kepler's Third Law applies to any orbiting system: the Moon's orbit around the Earth; the Earth's (or any other planet's) orbit around the Sun; the orbit of planets around other stars; the orbits of two stars around each other; and even the orbit of the Sun around the center of the Milky Way. If we can observe P, the time it takes for one full orbit, and a, the orbital separation, we can calculate m₁ + m₂, the sum of the two masses. Try it! For example, take the Earth-Moon system, which has P = 0.075 years (slightly less than a month) and a = .0025 AU. If you plug these values into the above formula, you should find the sum of the mass of the Earth and Moon in units of the Sun's mass.

Note that Kepler's Third Law only tells the sum of the masses of the two orbiting objects. If one object is much heavier than the other, this sum will be almost equal to the mass of the heavier object. Thus, Kepler's Third Law applied to the Earth's orbit tells us the Sun's mass (plus a negligible contribution due to the Earth's mass), and applied to the Moon's orbit tells us the Earth's mass (plus a 1% contribution due to the Moon's mass).

For binary star systems, astronomers can directly measure the orbital separation a if the stars are far enough apart so that their images can be resolved separately. Such systems are called visual binaries (see text, Fig. 17.16). Typically, the stars in such systems are so widely separated that the orbital periods are decades or longer.

But in most binary star systems, the two stars are so close together (often only a few stellar diameters apart) that they cannot be resolved with ground-based telescopes. In those cases, astronomers cannot measure their orbital separation directly, but they can infer the separation from the orbital velocities and orbital period, which can be measured from the Doppler shifts of the absorption lines in the spectra of the stars. Such systems are called spectroscopic binaries.

If the two stars in a spectroscopic binary have comparable luminosities, astronomers can see absorption lines from both stars in the spectrum (e.g., text, Fig. 17.16). Such systems are called double-line spectroscopic binaries. The Doppler shifts of spectral lines of the two stars oscillate with opposite phase, and the heavier stars does not move as fast as the lighter star, as illustrated in this Binary star simulation. For such systems, astronomers can infer the mass ratio, m₁/m₂, from the ratio of Doppler shifts of the spectral lines from the two stars, as well as the sum of the two masses from Kepler's Third Law. Given both the ratio and the sum, they can infer the individual mass of each star.

If, however, one star is much more luminous than the other, astronomers may only be able to see absorption lines from the more luminous star in the spectrum. We call such systems single-line spectroscopic binaries; with these systems we may only be able to estimate the total mass of the system, not the masses of the individual stars.

There is a problem in inferring the masses of binary stars from Doppler shifts in their spectra. We often don't have any way of measuring the inclination of the orbit. The inclination angle, called i, is defined as the tilt of the orbital plane with respect to the Earth. If i = 90^o, we see the orbit edge-on; if i = 0^o, we see the orbit pole-on (see the Binary star simulation). Therefore, the Doppler velocity that we measure is less than the actual orbital velocity by sin i. It follows that the masses we infer from Kepler's Third Law will be less than the actual masses by the unknown factor sin i.

However, we see some spectroscopic binaries so nearly edge-on that they actually eclipse each other (see text, Fig. 17.16). We call such systems eclipsing binaries. With eclipsing binaries, we know that sin i @ 1. This fact greatly reduces the uncertainty in inferring the masses.

The most famous eclipsing binary is the star Algol, which the ancients called the "Demon star" because its brightness dips periodically as the cool red star in the binary system eclipses the hotter blue star. There's a very good scientific story about Algol that we'll return to later in this course.

With eclipsing binaries, we not only diminish the uncertainty of the stellar masses resulting from the uncertainty in sin i, we can also infer the radius of each star from the way the light varies during the eclipse (Fig. 17.16). By comparing the radius inferred this way from the radius as inferred from the Stefan-Boltzmann Law, we can check the accuracy of these methods and test the validity of the Principle of the Universality of Physical Laws.

7. CLASSIFYING STARS

As described above, astronomers can now measure the distances of single stars out to about 1000 pc, their brightnesses, and hence can infer their luminosities (L). From their spectra, astronomers can infer the spectral types, or photospheric temperatures (T) of stars. Then, using the Stefan-Boltzmann Law they can infer their radii (R) from their temperatures and luminosities. If the stars are binaries, as many are, astronomers can also infer their masses (M) from observations of their orbits. These quantities -- L, T, R, and M, are intrinsic quantities. With many years of painstaking work, astronomers measure them for many thousands of stars. How then, do they make sense of all these measurements?

This problem is the same one faced by Charles Darwin when he arrived on the Galapagos Islands and found all sorts of strange birds and lizards, and by any botanist who begins to try to make sense out of the plants in a tropical forest. The answer is to begin organizing and classifying the information, a technique called taxonomy.

The first giant breakthrough in classifying stars was achieved independently by two astronomers, Einar Hertzsprung in Denmark, and Henry Norris Russell at Princeton University, around 1913. They plotted the locations of stars on a graph with the horizontal coordinate being spectral type (equivalent to temperature) and the vertical coordinate being absolute magnitude (equivalent to luminosity). The result, called the Hertzsprung-Russell diagram, or H-R diagram, is illustrated in Figs. 17.12 and 17.13 of your text (and also here and here). As you will learn in the next chapter, it was the first major clue to understanding how stars work.

Above is an H-R diagram of 8784 fairly nearby stars with distances measured by the Hipparcos satellite. In this diagram the spectral type (or photospheric temperature) is represented by the ratio of star brightness as measured in two different filters, the B filter that is sensitive to blue light and the V filter that is sensitive to yellow light. Here's a little table to translate these filter ratios to spectral type and effective photospheric temperature:

Notice that the sample of nearby stars that Hipparcos measured contains no stars with spectral type earlier than A0, relatively few red giant stars, and very few white dwarfs. Most of the nearby stars are cooler than the Sun (spectral type G2V).

The most striking thing about the H-R diagram is that the locations of most (about 90%) of the stars on this diagram are clustered in a relatively thin curved band that stretches from the upper left to the lower right. This band is called the main sequence. Main sequence stars have spectral types ending with the label V, indicating relatively low luminosity. For example, when we see from Table 17-5 that Sirius has spectral type AIV, we can conclude that it is a main sequence star. The stars at the upper left of the main sequence are hot (35,000 K) type O5 hot blue giant stars with luminosities more than 10⁵ times that of the Sun, while those at the lower right are cool (3000 K) type M5 red dwarf stars with luminosities less than 10^-4 times that of the Sun. The range in photospheric temperature is slightly more than a factor of 10, but the net range in luminosity is enormous -- more than 10⁹ (1 billion)!

Since the Stefan-Boltzmann Law says that the radius of a star can be determined from its temperature and luminosity, we can draw dashed curves on the H-R diagram corresponding to a given radius, as illustrated in Fig. 17.12 and 17.13. We see that the radii of the hottest blue giants are slightly more than ten times the Sun's radius, while the radii of the faintest red dwarfs are about 0.1 times the Sun's radius.

There seems to be a rule that most stars should belong to the main sequence, but about 10% of the stars are exceptions. So, classifying stars with the H-R diagram gives us two big questions: why do most stars obey such a rule; and why are some 10% of the stars exceptions to the rule? These simple questions led astronomers to an understanding of how stars live and die, as we shall learn in Ch. 20.

Before we leave the H-R diagram, we should notice that most of the exceptional stars fall into two loose bunches. In the upper right of the diagram are the red giant stars; with radii ranging up to more than 100 times the Sun's radius; they are the biggest stars on the diagram. On the lower left are the white dwarf stars; with radii less than 0.1 times the Sun's radius, they are the smallest stars on the diagram.

We know the masses of many stars in binary systems, and for those stars we can organize the data in another way, called the mass-luminosity relation, which is plotted in Fig. 17.18b of your text and reproduced below. As the figure shows, the luminosity of main-sequence stars is closely correlated with their masses. The straight line in that diagram can be represented roughly by the equation L/L_Sun = (M/M_Sun)^3.5. This equation implies that the luminosity of a main sequence star is extremely sensitive to its mass. For example, a star with mass 10 times that of the Sun will be roughly 10^3.5 = 3,000 times as luminous as the Sun. The most luminous known main sequence stars have masses about 50 times that of the Sun and luminosities about 10⁶ times that of the Sun, while the least luminous ones have masses about 1/10th that of the Sun and luminosities about 10^-3.5 times that of the Sun.

The figure above is identical to Figure 17.18 of your text, except that I have added a few white dwarf stars and red giant stars. You can see that white dwarf stars have masses ranging from about 0.7 - 1.2 times that of the Sun, but luminosities ranging from 10^-3 to 10^-2 times that of the Sun. Red giant stars, on the other hand, have masses ranging from a few solar masses to perhaps 20 solar masses, and luminosities ranging from 100 to 10⁵ solar luminosity units.

The main point of this figure is that main sequence stars, even though their luminosities span a range of more than 10⁹ (one billion!), all seem to belong to the same family -- they obey the same rule. But red giants and white dwarfs march to a different drummer.

There is yet another way to organize the observations of binary stars, and that turns out to provide the best clue of all, as we shall see in Ch. 20. That is to plot the locations of the stars in a mass-radius diagram, as illustrated in Fig. 17.18 of your text. You can see that the main sequence stars again cluster about a narrow band, such that the radius of a star is nearly proportional to its mass [approximately, R/R_Sun = (M/M_Sun)^0.8 ]. Now, since the volume of a star is proportional to R³, the average density (mass divided by volume) is approximately proportional to M^(1-3x0.8) = M^-1.4. This argument shows: the main sequence blue giant stars are much less dense than the Sun.

The red giant stars are even bigger than the blue giant stars, but no more massive, so they have the lowest average densities. The white dwarf stars, on the other hand, are small and heavy. The average density of a white dwarf star is about 1 ton per cubic centimeter!

8. STAR CLUSTERS:

As we shall see, stars are not born in isolation; they are born in clusters containing many thousands of stars. Star clusters are among the most beautiful things you can see in the sky with a small (or large) telescope. There's a very nice picture gallery of several star clusters from the Anglo-Australian Telescope here, and a few images of star clusters from the Hubble Space Telescope here.

Besides for the fact that they are beautiful, star clusters are very important to astronomers for two reasons: (1) they are cosmic archives of stellar evolution; and (2) we can infer their distances without measuring their parallaxes.

As you will see from these pictures, star clusters can be divided roughly into two very distinct types: open clusters and globular clusters. (There are intermediate types, but they are relatively rare.) The table below lists the main distinguishing characteristics.

With your naked eye you can easily see a nearby open cluster, the Pleiades. It contains several blue giants (spectral types B7 - A0) having luminosities hundreds of times that of the Sun. Below is a color-magnitude diagram of the Pleiades. A color-magnitude diagram is like an H-R diagram, except that the vertical axis is labeled by apparent magnitude (a measure of brightness) rather than absolute magnitude (a measure of luminosity). To convert this scale to absolute magnitude, subtract 5.6. (the "distance modulus" of the Pleiades). Figure 17.19 of your text also shows an image and an H-R diagram of the Pleiades.

With the telescopes at the Sommers-Bausch Observatory, you can see a very different type of star cluster, called a globular cluster. The figure below is an H-R diagram for the globular cluster M3. [Click here to see an image of M3.] Figure 17.20 of your text shows an image and H-R diagram for another globular cluster, Omega Centauri.

Note that M3 has no luminous blue main sequence stars. In fact, the only main sequence stars (labeled MS) are solar type or redder. M3 has many red giant stars, labeled RGB, and stars that lie on a horizontal strip, labeled HB, called horizontal branch stars. The HB stars have roughly the same temperature as the Sun and so have spectral types in the range A - G, but they are almost 100 times as luminous as the Sun. Their luminosity class is II or III - e.g., a star with spectral type F0III would be a horizontal branch star.

Until recently, white dwarf stars were too faint to see at the typical distances of globular clusters (5000 - 15000 parsecs). But the Hubble Space Telescope changed that. For example, here is a description of HST observations of white dwarf stars in the globular cluster M4.

Distances of star clusters: Most star clusters in the Milky Way are at distances too great to measure by means of parallax, even with the Hipparcos satellite. But still, we can infer the distance of a star cluster because we know what to expect for the luminosity of its main sequence stars. The argument goes as follows: when we plot the stars of a given cluster on a color-magnitude diagram, such as the one shown above for the Pleiades, we can recognize a familiar pattern - the main sequence. If we know that a star in the Pleiades (or any other cluster) is a main sequence star (luminosity class V) and we know its spectral type, we can say what its luminosity should be. (For example, table 17.5 shows that the Sun has spectral type G2V. Therefore, we expect that every G2V star has roughly the same luminosity as the Sun. Likewise, we expect that all type A1V stars will have roughly the same luminosity as Sirius: L = 23.5 Solar units.) But if we know its luminosity (L), and we can measure its brightness (B), we can calculate its distance (D) from the inverse square law, which can be written in the form D = (L/4pB)^1/2.

This method of inferring distances from the spectral type and brightness of a star depends on two assumptions: (1) all main sequence stars of a given spectral type have the same luminosity; and (2) the star really is a main sequence star. Assumption (1) is another example of the principle of universality of physical laws. It says that the underlying physical rule that makes most stars fit on the main sequence when plotted on the H-R diagram should be true in every group of stars, everywhere in the universe. [In fact, the "rule of the main sequence" is not an exact rule, as you can see from the fact that the star a -Cen has spectral type G2V - the same as the Sun - but luminosity L = 1.56 Suns. Evidently, there is a margin of error in the main sequence relationship, as indicated by the fact that main sequence stars have some scatter above and below the curve defined by the gray bands in Figures 17.12 and 17.13.] Assumption (2) can be verified by measuring the magnitudes and spectral types of many stars in the cluster and plotting them in a color-magnitude diagram. Then one can see which stars in the cluster belong to the main sequence.

Thus, the fact that most stars belong to the main sequence (i.e., have a well-defined relationship between spectral type and luminosity) becomes a powerful tool for measuring distances to stars that are so distant that their parallax cannot be detected, even by the Hipparcos satellite.

There's a wonderful logic to this technique that might appear circular but is not. Astronomers first measured the distances (D) of nearby stars by parallax methods and their brightness (B). Then they used the inverse square law to infer their luminosities (D & B Þ L), and discovered that most stars had a relationship between spectral type and luminosity called the main sequence (Spectral Type Þ L). Assuming that the relationship is universal, astronomers can use it to infer the luminosities of more distant stars from their spectra, and then use the inverse square law to infer their distances from their luminosity and brightness (L & B Þ D). It's almost like pulling yourself up by your own bootstraps!

[Actually, since some 90% of all stars belong to the main sequence, you could simply measure the brightness and spectral type of any star in the sky and estimate its distance by assuming that it has the luminosity of a main-sequence star with that spectral type. That would give a fairly good estimate of distance 90% of the time. But 10% of the time, you might be way off. By making color-magnitude diagrams of stars in clusters, we have a way of ensuring that a star is a main sequence star.]

9. SUMMARY:

Astronomers have found ways to measure four intrinsic properties of stars: their luminosities (L), photospheric temperatures (T), radii (R), and masses (M). They can classify thousands of stars by plotting these stars on diagrams characterized by any pair of these properties: the H-R diagram (L,T); the mass-luminosity diagram (L,M); and the mass-radius diagram (R,M). When they do, they find that about 90% of all stars cluster in thin bands on each of these three diagrams. We call these stars main sequence stars. They range from about 0.1 to 100 times the Sun's mass and have an enormous range of luminosities, from 10^-3 to 10⁶ times the Sun's. Only one quantity - M - is sufficient to determine all the other quantities (L, R, T) of a main sequence star. But some stars do not obey these relationships: they include faint white dwarf stars, about as massive as the Sun but not much larger than Earth; and very luminous red giant stars, hundreds of times larger than the Sun.

There are two distinct types of star clusters: dense clusters called globular clusters containing many red giant stars and white dwarf stars but no luminous blue stars; and open clusters containing luminous blue main sequence stars and relatively few red giants and white dwarfs.

By organizing the observations in these ways, astronomers raised the following questions:

• Why do 90% of all stars fit into the tight band called the main sequence?
• Why do the intrinsic properties of main sequence stars obey the relationships seen on the various diagrams?
• Why are red giants and white dwarf stars different from main sequence stars?
• Why are the types of stars found in globular clusters so different from those in open clusters?

As we shall see in Chapter 20, the quest to answer these questions led astronomers to a deep understanding of how stars burn and how they die. They are the right questions, and once astronomers knew enough to ask them, they were more than halfway to the answers. But, as you can see from the above, it wasn't even possible for astronomers to ask these questions until they had done a great deal of painstaking work in gathering and organizing their data. That process, called taxonomy, is basic to every kind of science.

Last modified February 5, 1998
All rights reserved by Richard McCray