Although it has been known for over 200 years that some stars are gravitationally bound to each other in binary systems, multiple star systems, and clusters, evidence now suggests that the vast majority, if not all stars either are (or were) bound to other stars (or planetarylike bodies). Thus the fact that stars are members of a physical group of two or more stars is not the exceptional situation--it is more the rule. Obviously those processes that create stars and guide their evolution are ones which typically form several bodies at the same time, albeit that some of these bodies may be satellites rather than companion stars. And for many stars, these processes are ones involving the mutual gravitational interplay between bodies throughout the course of their lives. In Chapter 22, we will consider this evolutionary interplay and its meaning for binary systems.
14.1. Stellar Motions in the Solar Neighborhood
Most of the changes we see in the night sky result from the Earth's daily rotation on its axis and Earth's annual revolution about the Sun. But the stars are also moving as discovered in 1718 when Edmund Halley found that the positions of certain bright stars had altered appreciably, since the time of Hipparchus. Following Halley's discovery, the question arose as to whether stars move at random or there is some pattern to their motion. There is, as we shall see, a small, individual, random motion relative to their neighbors. And in turn this random motion is superimposed on a much larger, systematic motion in which all stars participate. This organized motion is the revolution of stars around the Galaxy's center. Let us consider the small random motion now and defer a discussion of the rotation of our Galaxy to Section 23.3. We will begin the discussion with a description of the Milky Way since it is such a prominent reference in any discussion of the nighttime sky.
If you pick a clear, moonless night, away from city lights, and look up, arching across the sky you will see a misty, irregular, beltlike cloud of light, the Milky Way. In temperate latitudes you will see different parts of the Milky Way inclined to the horizon at different angles, depending on the time of night and season of the year. About 30 percent of the Milky Way lies below our horizon; thus to see that portion you must be in the Southern Hemisphere. Visualizing how the stars are arranged to form the Galaxy and in turn how our Galaxy orientates relative to your horizon is much easier if you are familiar with the constellations (see star maps in Appendix 2). To begin the task of learning the constellations, one wants to locate first the summer triangle of bright stars--Deneb in Cygnus, Vega in Lyra, and Altair in Aquila--and the winter triangle of bright stars--Sirius in Canis Major, Procyon in Canis Minor, and Betelgeuse in Orion. Table 14.1 is a list of other prominent constellations that, if learned, make a reasonably good guide to the entire sky.
For midnorthern latitudes in the late summer months, the Milky Way (Figure 14.1) shines its richest an hour or so after sunset. Then the Northern Cross of Cygnus is directly overhead, along with the summer triangle. The bright star Arcturus is setting in the west, and the great W of the constellation Cassiopeia is rising in the northeast. The shimmering band of the Milky Way can be traced from Cassiopeia and Cepheus in the northeast through Cygnus and then down toward the southern horizon through the constellations Aquila, Ophiuchus, Sagittarius, and Scorpius. From Cassiopeia to Cygnus, the Milky Way is a single silvery band of varying width, but between Cygnus and Sagittarius it divides into two bright lanes flanking the dark band of interstellar clouds known as the Great Rift (Section 18.5). The center of our Galaxy lies beyond the great star cloud in Sagittarius (an area of the sky where great numbers of faint, distant stars give to the naked eye the impression of an almost continuous cloud of brightness; see Figure 14.2).
In winter we see the dimmer and more sparsely populated portion of the Milky Way. It starts with Canis Major in the southeast, passes through the winter triangle, and continues through Orion, Taurus, Auriga with its bright star Capella, and Perseus in the northwest. The direction between Orion and Auriga points away from the Galactic center (Figure 14.1). Notice that the brightness and width of the band of the Milky Way differ greatly from one section to another.
Astronomers cannot directly follow in one evening the motions of stars relative to the Sun, but they can detect their motions by comparing photographs of the same star field taken years apart. If we assumed that stars in the neighborhood of the Sun moved at about the same speed, then, like two small aircraft crossing our line of sight at different distances from us, the nearer stars should sweep across our line of sight faster than more distant ones. Thus the nearer stars readily reveal through their motion relative to the Sun that they are indeed nearby. From photographs taken over a span of years, astronomers measure minute angular changes in position for stars, which are expressed in arc seconds per year, and refer to these angular motions as a star's proper motion. In general, stars close to the Sun do have larger proper motions than more distant stars, which in the case of most distant stars are too small to even measure. However, it is not true that all the stars are moving at about the same speed as we assumed above. Actual star data reveals some sizable variations. For example, Barnard's star, with a distance of 6 ly, has the largest known proper motion, 10.3 arc seconds per year; while, the bright stars Sirius, Procyon, Altair, and Vega, which are at comparable distances to that of Barnard's star, display proper motions between 1.3 and 0.4 seconds of arc per year, figures closer to the average. Proper motions have now been measured for more than a quarter of a million stars.
The constellations appear to us very much the same as they did to people in ancient times, but eventually these familiar patterns of stars will change because of proper motion. This action of proper motion to change the relative positions of constellation stars is illustrated in Figure 14.3 for the Big Dipper, whose shape has changed drastically over the last 100,000 years and will change as dramatically over the next 100,000 years.
To convert a star's proper motion to speed at right angles to our line of sight, which is called the star's tangential velocity, one must multiple the star's proper motion by its distance (see caption to Figure 14.4). Thus the number of stars whose tangential velocity is known is limited by the number of stars for which we have distances, which is far fewer than those for which radial velocities have been measured. As discussed in Section 4.3, a star's line-of-sight motion, or radial velocity, is found from the Doppler shift of absorption lines in the star's spectrum. Radial velocities can be measured to an accuracy of a few kilometers per second or less, and there exist measurements for some 20,000 stars. Together, a star's tangential velocity and its radial velocity define its speed and direction, or its space velocity, relative to the Sun (Figure 14.4). Typical values for space velocities of stars in the Sun's neighborhood are around 30 km/s, with most under 60 km/s. For example, Barnard's star has a space velocity of 140 km/s, but those for Sirius, Procyon, Altair, and Vega are 30 km/s or less.
When it was realized that nearby stars are moving relative to the Sun, the question arose as to what the Sun's motion is relative to its neighbors as a group. To address this question assume for the moment that stars do not move and it is only the Sun that is moving. Think of this situation in relation to the familiar one of what you see when walking down a hallway crowded with people who are standing still. Relative to you people appear to be coming toward you from the direction in which you are walking, while behind you people appear to recede toward the direction from which you have just come. Our idealize case of motion of the Sun through fixed stars should be no different.
Now let us consider the real situation in which the stars are moving relative to the Sun which is analogous to the people in the hallway moving. If some people move toward you, some move away, and some remain stationary, then on the average you still have the impression that people are streaming toward and away from you as a consequence of your motion through them. The same will be true for the Sun's motion as long as the motion of nearby stars is basically random, which it appears to be. A nonrandom motion is analogous to all the people in the hallway moving in the same direction as you, for example, in which case you see no streaming effect.
Even without knowing how far away stars are, we can make some generalizations about their movements in the solar neighborhood from their proper motions and radial velocities. By analyzing proper motions statistically, we find that nearby stars generally appear to diverge outward from the point on the sky toward which we believe the Sun is headed. Simultaneously, on the opposite side of the sky, nearby stars appear to be converging toward the point from which the Sun is receding. This apparent motion, known as the solar motion (Figure 14.5), is taking the Sun toward a point on the celestial sphere that lies in Hercules within 10o of the bright star Vega. Radial velocities of the Sun's closest neighbors indicate that the Sun's motion relative to them is somewhat below the average space velocity for nearby stars, being about 16 km/s or 3.3 AU/y.
Overall, the motion of stars in the Sun's immediate neighborhood, distances within 500 ly or so, appears to be almost random in direction as seen from the Solar System; that is, as many stars are going in one direction as in any other direction. As we have stated, the typical speed for this random motion is about 30 km/s. We should also remind ourselves (see Section 13.4.4) that the typical star in the neighborhood of the Sun, which we are discussing, is a small, red-colored, main-sequence, M-type star.
Half or more of all stars are in orbit around another star or stars. They are stars whose fates are permanently linked by gravity to that of their companion star or stars. Among the Sun's neighbors out to 15 ly at least half are in multiple systems (Table 13.1). In most of these multiple-star systems, there are just two stars, known as a binary star system, whose components may be separated by a large fraction of a light year, or they may be almost touching. In binaries, individual stars orbit in elliptical orbits around a common center of mass. The more massive component, which is not necessarily the brighter of the two stars, has the smaller orbit; the relative size of each star's orbit is inversely proportional to its mass, as shown in Figure 14.6. One can liken the relative motion of stars in a binary to the motion of a dumbbell with spheres of unequal mass on its ends. By imagining how the dumbbell would move if rotated around its center of balance, which is its center of mass, one can visualize the motion of the two stars in a binary.
Binary systems are classified according to the means by which we detect the fact that it is a binary. This in turn depends on the separation between the components, the distance of the system from Earth, and the orientation of the orbital plane to our line of sight. In practice, such considerations lead to three classes of binaries: visual, spectroscopic, and eclipsing binary systems. Some examples of each type of binary system are given in Table 14.2. Let's look at the details of these systems a little closer with particular regard to using them to determine the masses of the component stars. The problem of determining the masses of stars in a binary system has taken on renewed importance in recent years, as we will discuss in Chapters 20, 21, and 22.
Double-star systems that have separations large enough to see both companions are called visual binaries. In some visual systems, the secondary star's motion around the primary is obvious, while in others long spans of time are necessary to actually see one star orbit the other. The observed orbit is an ellipse, but it is the projection of the true elliptical orbit on the plane of the sky. To help you visualize the effect of projection of the true orbit, draw an ellipse on a piece of paper and incline the paper to your line of sight through various angles as shown in Figure 14.7.
If yearly measurements of the apparent separations and motions of the two stars around each other are made and if the system's distance from the Earth can be determined, then we can calculate the stars' individual orbits around their center of mass. These individual orbits have semimajor axes that vary in size from a few astronomical units to thousands of astronomical units, with periods of revolution from several years to many thousands of years. An example is shown in Figure 14.8 for the visual binary Xi Bootis.
Astrometric binaries are a subgroup of visual binaries and are distinguished from the main body of visual binaries by the fact that with today's technology the secondary star cannot be seen because it is much fainter than the primary. What we observe are tiny periodic wiggles in the visible star's proper motion across the celestial sphere; from these we infer that an invisible companion exists. The variation in proper motion arises from the visible star's orbital motion around the center of mass of the system which is moving more or less uniformly along a straight line relative to the Sun, as pictured in Figure 14.9. As the visible star (in white) orbits around the center of mass, which moves, let us say, upward toward the right-hand side of the page (its proper motion), it follows a wavy path across the sky. In a few astrometric binaries the invisible companion (or companions) seems to have a mass a few times that of Jupiter. As technology has improved over time, a number of binaries originally identified as astrometric binaries have been upgraded to visual binary because of the sighting of the secondary in the system.
The next category of binary systems are those whose components cannot be resolved visually because they are too close to each other. Nevertheless they may be shown to be binaries from periodic Doppler shifts of spectral lines in their composite spectrum (combined light from both stars). Such systems are called spectroscopic binaries, and they possess orbits for which the semimajor axis varies from a fraction of an astronomical unit to several tens of AUs. The corresponding orbital periods range from hours to several years. When absorption lines of both stars of such systems are visible in the composite spectrum, two sets of absorption lines shift periodically back and forth relative to each other. Such systems are called double-line spectroscopic binaries. The maximum shift occurs when the two stars are moving in a direction which is parallel to the line of sight, and there is no Doppler shift (the two sets of absorption lines merge) when the two stars are moving across the line of sight a quarter of a revolution later.
In most spectroscopic binary systems only the brighter star's spectrum is discernible on a spectrogram, and its absorption lines periodically shift back and forth. The secondary star is just too faint to have its light recorded. For every 100 of these so-called single-line spectroscopic binaries, there are only 15 or so of the double-line spectroscopic binaries mentioned above. For the single-line spectroscopic binary, astronomers can determine whether or not the system is a binary system by analyzing the velocity curve, that is, a plot of the change in radial velocity during the binary's period of revolution (Figure 14.10).
The final type of binary system is eclipsing binaries, in which the eclipse of one star by the other is the key to identifying its binary nature. In such systems, an eclipse occurs because the stars are fairly close to each other and their orbits are seen more or less edgewise. Thus their periodic motion causes first one star and then the other to pass between its companion and us, temporarily cutting off all or part of the eclipsed star's light (Figure 14.11). Consequently there will be a decrease in the apparent brightness of the system each time an eclipse occurs.
When the two eclipsing stars are equal in brightness and size (Figure 14.11a), the minima in the light curve are V-shaped and of equal depth, whether the eclipses are partial or total. The more nearly total the eclipse is, the deeper are the minima. In Figure 14.11b, if most of the light comes from the smaller, brighter component, we see the deeper minimum when the larger, fainter star totally eclipses the smaller one. Halfway around the orbit the eclipse is annular and the minimum is much shallower when the brighter star passes in front of the fainter one. Both minima are flat-bottomed because it takes a period of time for the small star to cross the disk of the large one. In Figure 14.11c, the stars are tidally distorted into an ellipsoidal shape. The depths of the two minima differ, as in Figure 14.11b, because the two components are unequal in luminosity and size.
By analyzing the light curve's general shape and measuring how long the eclipses diminish the light and by how much, astronomers can derive a model of the system. If they also observe the eclipsing binary as a spectroscopic binary, they can combine data from the light and velocity curves to determine such properties as radii, masses, densities, temperatures of the stars, and the true orbital sizes.
Some gravitationally linked systems have more than two stars. For example, there may be a distant third star revolving around a close pair or possibly two close pairs (usually short-period spectroscopic binaries) revolving around a common center of mass as a long-period visual binary system. Other stellar systems have been shown to possess more than four stars. Take, for example, the second-magnitude star Castor in the constellation Gemini: With a small telescope one can easily pick out two visual components, which have a period of revolution of about 400 years. But when we observe the stars' spectra, we find that the primary is itself a spectroscopic binary, with a period of 9.2 days (Figure 14.12), and the secondary is also a spectroscopic binary, with a period of 2.9 days. A ninth magnitude star called Castor C, which is only 1.2 minutes of arc away from the visual pair and takes many thousands of years to orbit them, is an eclipsing binary, with a period of 0.8 day. Thus all three classes of binary are represented in Castor's sextuple system.
Observational evidence easily supports the contention that at least half of all stars are members of binary systems. We must qualify this remark by saying that astronomers are not able to sample at will stars across the Galaxy or in neighboring galaxies; our statement extends results found for nearby stars to a generality about all stars. Also, if the two components of a binary system are of approximately the same luminosity, their identification as a binary is fairly easy, but the greater the difference in luminosity, the more difficult binary identification becomes. As an illustration, the Sun does not appear to have a stellar companion, although we cannot entirely rule out the possibility that a companion star of very small mass (and consequently low luminosity) exists far beyond the orbit of Pluto.
In the case of the Sun, does the existence of a planetary system take the place of a stellar companion? This is a very difficult question to answer. Studies directed at this question have found that the relative percentages among samples of nearby stars like the Sun are typically 45:46:8:1 for single:double:triple:quadruple systems. When detailed study of the single systems is made, many are invariably found to have unseen companions, which may or may not be planetary systems, so that nearly 85 percent have some kind of companion. The suspicion is that probably all solarlike stars have one or several companions. When stars in other regions of the H-R diagram are studied, the binary or multiple nature of star systems again seems to be commonplace. From this conclusion it is possibly going too far to say that all stars everywhere in the Universe occur in some kind of companion relationship involving stellar or planetary objects. But certainly it is a point to ponder.
14.3. Mass-Luminosity Relation
Masses of stars, as we will discuss later, are critically important data when it comes to the theory of the evolution of stars, and that this is so has been known since the 1930s. Thus there has been a major effort made in both the methodology and the actual measurement of stellar masses over the last forty or so years. The only means of determining the masses of stars is through their gravitational influence on other stars (or satellite bodies). We cannot directly obtain, however, the mass of single stars because their gravitational effects on stars close to them are insignificant. When stars are gravitationally bound together, as binaries are, their orbital motion about each other is a consequence of their mutual gravitational attraction. As a result, we can find the masses of these stars relative to the Sun's mass through Newton's modification of Kepler's third law (as discussed in the accompanying box). Determining the masses of component stars in binary systems is at least doable in principle for every known binary system, if not in actual practice. What prevents the determination in practice is knowing the distance of the system and the inclination of the orbital plane to the line of sight, among other things. However, enough masses have been determined for main sequence stars and some other luminosity classes that astronomers feel fairly confident that we have a reasonable grasp of this most important piece of stellar data. There are two points to remember here. Measurements show that the masses of main-sequence stars increase in the order M, K, G, F, A, B, to O, or in the same sense that their surface temperatures increase. The second point is that the observed range in mass for main-sequence stars is approximately 0.1M. (read as one-tenth solar mass) for a late M-type star to 100M. (read as one hundred solar masses) for an early O-type star.
When masses for main-sequence stars were measured and analyzed, an important correlation, the mass-luminosity relation, was found to exist between the mass and luminosity of these stars (Figure 14.13). The relationship has also been derived theoretically from the fundamental laws governing matter and radiation in stellar interiors (to be discussed in Chapter 17). One can see in Figure 14.13 that a main-sequence star's luminosity is approximately proportional to between the third and fourth power of its mass. More specifically from about 0.5M. up to 4M., stars comparable to the Sun, a star's luminosity is approximately proportional to its mass to the fourth power. But for stars either less massive or more massive than this range, luminosity depends on mass to more like the 3.3 power. A reasonable compromise is that the luminosity is proportional to the 3.8 power of the mass for the entire range of stellar masses. For example, a star of 4M., which is approximately a B8 main-sequence star, is almost 200 times more luminous than the Sun, while a 0.5M. star, which is a K8 main-sequence star, is only 0.07 times as bright as the Sun.
The correspondence between mass and luminosity in Figure 14.13 does not apply to stars that are not main-sequence stars. The very luminous supergiants, bright giants, and red giants lie somewhat above the curve, and the white dwarfs lie considerably below it. The fact that a correlation does exist for main-sequence stars suggests that all main-sequence stars have something in common about their internal structure. And the fact that there appears to be no simple correlation between mass and luminosity for any other luminosity class besides main sequence suggests that the internal structures of non-main-sequence luminosity classes is different in a very substantial way from that of main-sequence stars.
Another important point to remember is that because of the dependence of luminosity on a high power of the mass in the mass-luminosity law, the observed range of masses may be relatively modest for main-sequence stars, going from several hundredths of the Sun's mass to about 100 times the Sun's mass, but their luminosities vary over a much bigger range--from a few ten-thousandths of the Sun's brightness to almost 100,000 times the Sun's brightness.
Over the history of telescopic observations, astronomers as they searched the heavens found, in addition to binary and multiple-star systems, even larger aggregates of stars called star clusters. One of the primary reasons that the great eighteenth-century astronomer William Herschel constructed ever larger telescopes was to pursue his interest in star clusters. During the first few decades of this century it became clear that clusters had much to tell us about the organization of stars in our Galaxy and the evolution of the Galaxy itself. If we could imagine ourselves backing away from the disk of our Galaxy, we would reach a distance at which the aspects of the Galaxy that would catch our eye would be these clusters of stars that are so prominent a part of the Galactic landscape.
Today astronomers recognize three types of star clusters, open clusters, stellar associations, and globular clusters. Star clusters exist because their member stars are close enough to each other to be bound into a physical group by the stars' mutual gravitational attraction. Investigations over periods of years show that it is unlikely that any cluster achieved its present appearance starting from a multiple-star system and then gravitationally capturing first one nearby star followed by another and another until the cluster arrived at the large aggregate of stars we see today. But on the contrary, evidence suggests that clusters are born with their observed stellar membership, and if anything, they may loose stars over long periods of time rather than gain them. Consequently, we must keep in mind that the process for forming stars must form a large number of stars at approximately the same time and close enough to each other to be bound gravitationally in some cases for billions of years. Our evidence that clusters, specifically the globular clusters, are billions of years old will be covered in later chapters.
Whereas typical separations between the stars in the solar neighborhood that are not members of a star cluster (such stars are known as field stars) are about 5 to 6 ly, the typical separation for members of an open cluster is probably 2 to 3 ly, and for a globular cluster it is a few tenths of a light year. Another way of characterizing the separation between stars is to state it in terms of star size: Using the solar radius as a typical star size, we find that field stars are separated by several tens of millions of solar radii, the stars of an open cluster by a few tens of millions, and the stars of a globular cluster by a few million. Even though cluster members are not much closer to each other compared with field stars, their gravitational attraction is capable of holding a cluster together for up to tens of billions of years in the case of globular clusters and up to a few billion years for some open clusters.
A few clusters have proper names, such as the Hyades, the Pleiades, and Praesepe, in addition to various catalog designations, but most clusters are known only by a number in some catalog. For example, many of the entries in the Messier catalog (see Appendix 4) are clusters, the Pleiades being M45 and Praesepe being M44.
Astronomers have identified some 1100 open clusters close to the band of the Milky Way that runs around the sky. These open clusters contain anywhere from a score of stars to many hundreds of stars distributed over volumes that are several tens of light years across. Open clusters have a somewhat loose appearance and only approximate a spherical shape. Information on some typical open clusters is given in Table 14.3, and Figure 14.14 shows two pictures of open clusters.
Several of the open clusters are easily visible to the naked eye. Two of the best known are both in the constellation of Taurus--the Hyades and the Pleiades (shown in the frontispiece to this chapter). To the eye, the Hyades is a V-shaped group of stars marking the face of the bull. With telescopes, astronomers have found several hundred stars in the Hyades cluster which are spread over a 15 ly diameter at a distance of 140 ly. The Pleiades contains about the same number of stars and is approximately the same size as the Hyades. But because it is almost three times farther away than the Hyades cluster is, the Pleiades, looking like a tiny dipper of six stars in the shoulder of the bull, occupies a much smaller area on the sky than does the Hyades.
In the 1930s, open clusters were instrumental in demonstrating that there was matter lying between the stars--the interstellar medium. Astronomers now know that interstellar matter, which is composed of gas and a fine dust, hides from our view most of the open clusters in the Galaxy. Undoubtedly there are many more than 1100 open clusters; estimates suggest that as many as 18,000 open clusters are buried in the Galaxy's spiral arms and disk.
For open clusters, the stellar composition varies from cluster to cluster. There are some whose members are predominantly or almost exclusively main-sequence stars; the brightest in these clusters are brilliant blue, main-sequence, giant, or supergiant stars. In contrast with these open clusters containing bright blue stars are those whose membership does not include any bright blue stars of spectral classes O, B, or A. For such open clusters, their brightest stars are red giants and supergiants of spectral classes K and M. What we have just described seems to be opposite ends of a reasonably continuous gradation of stellar composition in open clusters. That is, there are also open clusters with a mixed stellar composition, containing some bright blue stars and some bright red stars. In nearby open clusters, faint yellow and red dwarfs of the main-sequence luminosity class are also observable. And some even fainter white dwarf stars have been found in those nearby open clusters which contain bright red stars. We shall have more to say about the physical significance of the stellar composition of open clusters in Chapter 19.
A Russian astronomer, V. A. Ambartsumian (b. 1908) was the first to note that far from the Solar System there exist a considerable number of loose stellar groupings in the spiral arms that lie in the Galactic disk. These sparsely populated associations contain either highly luminous O and B stars (and no bright red stars), in which case they are called OB associations, or T Tauri stars (a type of variable star), in which case they are known as T associations. Conspicuously presence between the stars in an association is an interstellar gas composed primarily of hydrogen and a very fine dust. Associations contain up to 100 or so stars and are up to several hundred light years in diameter. Estimates place the number of associations in our Galaxy at around a hundred or so. Three representative OB associations are listed in Table 14.3. To the naked eye, one of them appears to be the middle star in the sword of Orion, but it is actually an association of O stars known as the Trapezium.
Because an OB association contains hot O and B stars, these hot stars ionize the hydrogen gas surrounding the association and produce the highly luminous gaseous nebula shown in Plate 15. This is a common Galactic environment in which many associations are found. The ionizing is done by ultraviolet photons rather than visible ones. This is because the resident energy in ultraviolet photons is sufficient to ionize hydrogen atoms while that in visible photons is not. Also important is the fact that the luminosity of a 40,000o O star is some one hundred thousand times greater than that of a 6000o star like the Sun. And of that greater luminosity for the O star, about 96 percent of the energy is in the ultraviolet region of the spectrum below 3500 A, while only 7 percent of the Sun's luminosity is in the ultraviolet region.
The individual stars in an association are separating rapidly from each other because the association has too little mass in its member stars to bind them permanently. Hence associations are highly unstable, with a maximum life expectancy of only a few million years before they completely disperse into the mainstream of stars in the disk of the Galaxy.
The largest and most densely concentrated of all stellar groups are globular clusters. A photograph of the magnificent globular cluster that is just visible to the naked eye in the constellation Hercules appears in Figure 14.15. Astronomers have discovered more than a hundred globular clusters surrounding the center of our Galaxy, somewhat like an immense spherical frame of reference within which the Galaxy rotates. Other clusters undoubtedly are hidden from us by dust in or near the Galactic plane. The largest globular cluster may have over a half million stars packed into a diameter of 100 ly or so. The nearest clusters are about 7000 to 8000 ly from the Sun, whereas the farthest are over 100,000 ly.
Photographs of the stars in globular clusters look crowded, but the individual stars, although more closely spaced than are those in the solar neighborhood (up to a thousand times, or several stars per cubic light year), are sufficiently far apart to escape collision with each other. They move in extremely elliptical orbits from one side through the center to the other side of the cluster. In some of the nearer globular clusters, binaries have even been found, suggesting that the closer proximity of stars in them is not sufficient to disrupt binaries. In fact, the common occurrence of binaries among nearby field stars may also extend to clusters, although this point is still in question. If the Earth orbited a star in a globular cluster, the nearest stars would be light months from us instead of light years, and although there would be many more bright stars in the night sky than we see around the Sun, the night sky would still not be close to being as bright as the daytime sky.
Red giants dominate photographs of globular clusters because they are more luminous, even though they may exist in fewer numbers, than main-sequence stars. RR Lyrae variable stars are also commonly found in globular clusters and are useful for determining the distance of the cluster. Because they vary in brightness in periods of less than 1 day, the RR Lyrae stars are readily identifiable. Also, all the RR Lyrae stars are of very nearly the same visual absolute magnitude (= +0.5). Thus they are good candidates on which to apply the inverse-square law of light so that we can determine their parent cluster's distance. For clusters too distant to resolve into individual stars, astronomers can use (with precautions) angular diameters and integrated light to estimate distance. Totally unexpected and as yet unexplained has been the discovery of a few faint but extremely hot blue stars in several clusters. Along with the detection of X-rays coming from several globular clusters, the hot blue stars indicate that there is still a great deal to be known about such stellar groupings.
The H-R diagrams in Figures 13.6, 13.7, and 13.8 are for field stars or the brightest stars of the night sky or the nearby stars. Other than their common location (they are relatively near the Sun), the stars of these diagrams need not have any significant relationship with each other. However, the stars in a cluster have been with each other since birth, and they should be more comparable than stars selected at random. Therefore, it is important to consider H-R diagrams for clusters. As the story of the evolution of stars unfolded in the 1950s and 1960s, the H-R diagrams of clusters played a vital part in unlocking the secret of how stars evolve.
Figure 14.16 is an H-R diagram for two nearby open clusters: the Hyades, the nearer one, and the Pleiades, some three times farther away. Both clusters have well-delineated main sequences but not the substantial number of red giants one sees in the H-R diagram for bright stars. The Pleiades cluster has no red giants, and the Hyades has four. When astronomers compare the H-R diagrams for a number of open clusters (Figure 21.3), a gradation is found from those with a well-developed upper main sequence and no red giants to those with no massive blue stars on the upper main sequence and a large number of red giants.
A dramatic contrast to H-R diagrams for open clusters are those for globular clusters, which are fairly similar to each other but distinctly different from those of open clusters. One such H-R diagram is shown in Figure 14.17 for the globular cluster known as M3 in the constellation Canes Venatici. In it one sees that there are no massive blue stars on the upper end of the main sequence. The striking feature is the extremely well developed red giant branch and the existence of a horizontal branch, which is not present in H-R diagrams for open clusters.