Anyone who has spent a night looking at a dark sky full of stars will not find it hard to understand the fascination they have held for generations of human beings. But it is only in the last 100 years or so that we have known with any certainty that the stars are great spheres of hot, glowing gas like the Sun. For decades, astronomers have been seeking answers to a raft of questions about the nature of stars in our Galaxy; answers which will presumably also apply to the stars that make up the billions of other galaxies sprinkled throughout the Universe. We have compiled a great body of knowledge about stars--a census of different stellar population groups within the Galaxy. This chapter is devoted to surveying those data.
To go with a constellation's name, its brighter stars are designated by lowercase Greek letters, assigned approximately in descending order of brightness. When a letter is added to a constellation's Latin name, the case ending of the name changes to the possessive or genitive form, such as Alpha Orionis, which is the brightest star in the constellation of Orion. This scheme was devised by Johann Bayer for his 1603 map of the heavens, the Uranometria. Almost 900 of the brightest stars also have Greek, Latin, or Arabic names. The brightest star in the constellation Leo is Alpha Leonis, whose proper name is Regulus. Second brightest is Denebola, whose constellation designation is Beta Leonis. Table 13.4 lists the 30 brightest stars in the night sky by proper name and constellation designation.
Unfortunately, the Bayer scheme quickly runs out of Greek letters (the limit is 24 per constellation) for the 5400 naked-eye stars that dot the entire sky. John Flamsteed, first Astronomer Royal of England, cataloged the stars in 1725 and simply numbered them west to east across the constellation. For example, one of the bright stars marking the summer triangle, Deneb, is known in the Bayer system as Alpha Cygni and 50 Cygni in Flamsteed's system. By convention the Bayer designation is used in favor of the Flamsteed number where available.
Fainter stars may be identified by their number in any one of
several star catalogs. For example, HR 7925 is entry 7925 in
the Revised Harvard Photometry Catalog of 1908, which has numbers
for the brighter stars as well, such as HR 7924 for Deneb. Other
examples of catalog designations can be seen in Table 13.1 for
the nearest stars. There are still other systems for naming the
stars, and enough different designations may exist for a particular
star that even astronomers must exercise caution.
13.1.2. Locating Stars
Over time, astronomy has produced two coordinate systems for
organizing the sky and locating stars and other celestial objects.
Known as the horizon and equatorial coordinate systems, they
are analogous in principle to the grid lines of geographic longitude
and latitude. They differ from each other in that the grid lines
of the horizon system are fixed in the sky to the zenith and horizon,
whereas, the grid lines of the equatorial system are attached
to the poles of the celestial sphere and rotate daily with the
celestial sphere. In the horizon system, the coordinates are
called azimuth and altitude, while in the equatorial system, they
are known as right ascension and declination. The two systems
of coordinates can be related to each other by means of a star-based
timekeeping system known as sidereal time. (Further discussion
of the astronomical systems used to locate objects on the celestial
sphere can be found in Appendix 2; timekeeping is also covered
in Appendix 2).
13.1.3. Charting and Cataloging Stars
When astronomers began in the eighteenth and nineteenth centuries to study the great bulk of stars, it became vital to have star charts and catalogs of positions of stars. Otherwise it was impossible for them to know that in a debate they were discussing the same star. Although Bayer's charts were not the first, they were one of the first reasonable complete set of charts of the heavens. Later in 1725, John Flamsteed published a catalog containing the positions for about 3000 stars. Figure 13.1 shows illustrations from several of the early star charts which were not only scientific works but also works of art.
The first of the modern compilations and a most ambitious task was Friedrich Argelander's (18xx-18xx) publication between 1852 and 1862 of a catalog, known as the Bonner Durchmusterung, of over 300,000 stars located in the northern hemisphere skies. This catalog still forms the basis for newer catalogs used by astronomers. What makes Argelander's work such a spectacular achievement was that it was done without the aid of photography.
The first of the photographic catalogs was started in 1886 through the efforts of several observatories scattered worldwide, and the project took about a half a century to complete. The result was celestial charts containing about 100 million stars and a catalog of some six million of the brightest stars.
Today is the era of automated telescopes and electronic star
charts and catalogs. These star charts and catalogs are ones
that are stored on electronic media, such as magnetic tape or
disks, and that can be manipulated, sorted, and plotted by computers.
Electronic star charts and catalogs have advantages over printed
ones in that they can be readily updated at any time, can be communicated
to astronomers worldwide, and subsets of the data can be readily
made for particular projects. By having such catalogs of star
positions, additional stellar properties can be measured and immediately
incorporated into the catalog. Thus these catalogs become our
stellar census books.
13.1.4. Trigonometric Parallax and Finding Distance
Unless the distance of a star is known, it is impossible to determine most other stellar properties. The basic technique for determining distance is trigonometric parallax. You can understand how this works by looking first with one eye and then with the other at a pencil held at arm's length; as you do so, the pencil seems to shift first one way and then the other against a distant background, the apparent shift diminishing as the distance of the pencil from your eyes is increased. Applied to stars (Figure 13.2), the parallactic shift that results from the Earth's revolution around the Sun is very tiny. It was not until the period 1837 to 1839 that parallax for three fairly nearby stars, Alpha Centauri, 61 Cygni, and Vega, were first measured.
The tiny apparent displacements on photographs of a foreground star against background stars are measured at various times as the Earth orbits the Sun. Every 6 months, the parallactic shift is a maximum when Earth is at opposite ends of that diameter of its orbit which is perpendicular to the direction to the star. Even in a telescope of 15-m focal length the star closest to us, Alpha Centauri, has a parallactic displacement on a photographic plate of only 0.01 cm, so many photographs must be made over a period of years in order to measure such small shifts in position.
To calculate the distance of a star in astronomical units (AU), we insert the parallax angle p in the first equation of Figure 13.2 (derived by simple trigonometry) and solve for distance d. Because distance in astronomical units is an extremely large number that is difficult to remember, astronomers prefer two larger units of distance. One is the light year (ly), which is the distance that light travels in 1 year (the velocity of light times the number of seconds in 1 year). The second is a still larger unit known as the parsec (pc), the distance of a star whose parallax is 1 second of arc. This makes the parsec equal to 3.26 ly. The name comes from the first three letters in the words parallax and second.
Table 13.1 lists distances and other data on the known stars within 15 ly of the Solar System. As the distances in Table 13.1 suggest, the light year and the parsec are units characteristic of the separation between stars in the solar neighborhood.
The limit using current technology for measuring trigonometric
parallax is about 300 ly, or 100 pc. As many as 500,000 stars
may be within this range, and approximately 6000 of these have
had their distances determined. For stars whose parallax are
too small to be measured, astronomers fortunately have developed
other ways of estimating their distances.
13.1.5. Finding Distances of More Remote Stars
For two stars of the same intrinsic brightness, if one star is twice as distant as the other, its observed brightness compared to the other will appear reduced by a factor of 2E2 = 4 (inversely as the square of twice the distance); if it is three times as distant, the reduction is 3E2 = 9 times; if it is four times as distant, the reduction is 4E2 = 16 times; and so on. Hence, if we know the true brightness of the particular class of stars to which our subject star belongs, we can calculate from its observed apparent brightness a distance by the inverse-square law of light (Section 4.3). The trick is being able to recognize the subject star as a member of a class whose luminosity is known, so that we know the intrinsic brightness of the subject star. This is most often easier said than done.
Astronomers have used variations of the inverse-square technique to build step by step a set of overlapping distance scales. We begin with nearby stars, whose distances are known from trigonometric parallax, and move toward the more distant stars, applying the appropriate variation of the inverse-square law.
Before photography, the only way to determine a star's apparent brightness was by visual estimates. Ancient Greek astronomers were the first to use an apparent magnitude m as a measure of a star's brightness as it appeared in the sky. The scale they used for assigning a magnitude is the one governing the eye's behavior; namely, equal ratios of brightnesses correspond to equal differences of magnitude. Hipparchus and Ptolemy graded the apparent brightnesses of stars into six magnitude classes: the brightest stars were assigned to magnitude class 1, the next brightest to 2, and so on to 6 for the faintest stars. Later, William Herschel noted that a 1st-magnitude star was about 100 times brighter than a 6th-magnitude star.
The ratio of apparent brightness of 1st-magnitude stars to 6th-magnitude stars has now been established to be 100 to 1, thereby quantifying the magnitude system. This ratio is the same for a five-magnitude difference anywhere on the scale, such as between 9th- and 14th-magnitude stars. (It is essential to remember that large negative numerical values of magnitude mean bright objects, whereas large positive values designate faint objects.) As shown in Table 13.2, this rule and a logarithmic equation can be used to calculate the brightness ratio corresponding to any magnitude difference. In addition, Table 13.3 illustrates the great range in apparent brightness covered by a relatively small range in apparent magnitude.
As an illustration of the use of the magnitude system, from Table 13.3 we find that the difference in visual apparent magnitude between Sirius and Venus (at its brightest) is -1.5 - (-4.4) = 3, which represents a brightness ratio of about 16 to 1; hence Venus can appear 16 times brighter than Sirius in the night sky. In fact, however, Sirius is intrinsically much brighter than Venus because Sirius is a large star and Venus is a tiny planet. The only reason Venus appears brighter is that it is very close and Sirius is far away. This leads to an important question as to whether the stars appear bright in our night skies because they are nearby or because they are intrinsically bright. The answer to this question requires what is termed a star's absolute magnitude.
To find the intrinsic brightness of a star, we must know its distance and apparent magnitude. Suppose that we calculate (using the inverse-square law of light) what the apparent magnitude of stars would be if they were all placed at the same distance. Comparing stars' brightnesses at the same distance is, of course, a comparison of their luminosities. The reference distance selected by astronomers is 32.6 ly or 10 pc. The magnitude a star would have if it were 32.6 ly from us is called the star's absolute magnitude. Why the distance 32.6 ly? Such a distance makes the range from brightest to faintest for absolute magnitude (roughly -7 to +17 for stars in our Galaxy) approximately the same as the range for apparent magnitude (roughly -1 to +17); a convenience that aids in memorization.
The difference between a star's apparent and absolute magnitude, that is m-M, is called its distance modulus, because it is related to the ratio of the star's distance to 32.6 ly. From its numerical value we can determine how many times brighter or fainter the object appears compared with its brightness at the reference distance of 32.6 ly.
To facilitate measuring apparent magnitudes of stars, astronomers have set up sequences of reference stars in various parts of the sky similar to the one shown in Figure 13.3. These are stars whose apparent magnitudes are known and against which the magnitudes of other stars can be determined.
As we discussed in Section 5.3, the color of radiation affects our perception of brightness. (At this point, you might wish to review the discussion on radiation detectors in Section 5.3.) Let us digress briefly to discuss the color of stellar radiation.
The numerical value of a star's magnitude depends on the spectral region that is used when we measure the magnitude. This is because the way in which radiant energy from stars is apportioned with wavelength is far from constant, as Figure 12.9 for the Sun shows. Because photographic emulsions and photoelectric devices do not respond equally to light from different color regions of the spectrum, they do not include the same wavelength range when used to determine the apparent magnitude of a star. One means of compensating for this effect is to use colored filters that transmit a narrow range of wavelengths. This technique ensures that the results obtained by using the same color filter with different radiation detectors are comparable. Some common filter designations are ultraviolet U, blue B, green or visual V (which approximates the human eye's color sensitivity), red R, near infrared I, and so forth. Hence the apparent magnitude should be qualified by the color region, such as "blue apparent magnitude" or "visual absolute magnitude," when making comparisons. One magnitude scheme, known as the UBV photometric system, covers the spectral range from approximately 3000 to 6000 A in three segments, as shown in Figure 13.4.
The difference between magnitudes measured in two different color regions of the spectrum is called a color index, or simply a color. A frequently used color index is the difference between blue and visual magnitudes, designated as B-V. A blue star, for example, has a brighter blue magnitude B than its visual magnitude V. Since brighter means algebraically smaller values on the magnitude scale, B-V is negative. The opposite is true for an orange or red star, where B-V is positive. The zero point of the B-V scale is arbitrarily assigned to a blue-white star whose spectrum is classified A0 V (Section 13.3).
Does the color index have any meaning? Yes indeed, because magnitudes
for different color regions of the same object differ because
the distribution of radiant energy varies with wavelength in accordance
with Planck's radiation law for blackbodies (Section 12.3). Therefore
the color index can be used to determine a star's temperature.
In Table 13.5, an observationally determined relationship is
given between the B-V color index and a star's surface temperature.
As discussed in the last chapter, the surface temperature is
an average over the depth of star's photosphere and its value
ranges from about 3000 to 50,000 K.
13.2.4. Bolometric Magnitudes and Stellar Luminosity
A star's luminosity is the total amount of energy in all wavelengths that it radiates per second from its entire surface. But the magnitudes we have considered so far measure the amount of radiation in a specific wavelength interval, such as the blue region. A more fundamental quantity, the bolometric magnitude, measures all of a star's radiation summed over all wavelengths received outside the Earth's atmosphere; that is, it is a measure of a star's luminosity. Unless this magnitude can be observed directly from an instrumented satellite, a theoretical correction using the star's surface temperature must be applied to the ground-observed visual magnitude to compensate for the radiation that does not pass through the Earth's atmosphere. The correction is largest where the peak of the energy curve lies outside the visual range, as it does for the hottest and coolest stars. In summary, a star's luminosity is derived by first measuring its distance and visual apparent magnitude, from which its visual absolute magnitude is calculated; a bolometric correction is added to the absolute magnitude to give the bolometric absolute magnitude, which can then be converted to the star's luminosity.
Values for luminosity are given in Table 13.4 for the 30 stars with the largest apparent brightnesses. As is evident from the table, most stars that appear bright are intrinsically bright and do not appear bright merely because they are nearby.
Techniques for the direct measurement of the radii of stars are of limited applicability, since direct measurement is difficult at best for even the largest stars and impossible for the vast majority of stars. Two such direct measurement techniques used for large stars, however, are optical or intensity interferometry, analogous to radio interferometry discussed in Section 5.4, and lunar occultation.
Assuming direct measurements are not possible, then if we know a star's temperature and luminosity we can calculate its radius from the Stefan-Boltzmann law in the following fashion. From the temperature we can determine the energy emitted per second from each square centimeter of a star's surface. The total amount of energy radiated per second by the entire star, which is its luminosity, depends not only on its temperature, but also on its surface area. We can thus calculate how large the star must be to have such a luminosity. The radii of stars are found to vary from one-hundredth of the Sun's radius (or about the size of the Earth) to about 1000 times that of the Sun, which is about 5 AU, or five times the radius of the Earth's orbit.
In the latter part of the last century and the first half of this one, it became clear that the spectra of stars could be classified into several broad groups according to which absorption lines are dominate. For example, there is a group in which the Balmer series of hydrogen form the dominate features. Such stars are known as A stars. In two other groups absorption lines of helium dominate. In one group known as B stars, it is the lines of neutral helium, while in the other group known as O stars, it is lines of singly ionized helium that are prominent. Other groups were identified on the basis of the absorption lines of singly ionized calcium, so prominent in the Sun's spectra, and lines of several singly ionized and neutral heavy elements. These groups are designated as F, G, K, and M stars. These seven groups--O,B,A,F,G,K,M--are known as spectral classes and they form the primary categories for the classification of stellar spectra.
The first large study of stellar spectra was a photographic survey of nearly a quarter of a million stars, which was begun in 1884 at the Harvard College Observatory. As that survey ended in the 1920s, when the relationship between atomic structure and emission of radiation was better understood, it was realized that the great diversity in spectral appearance (shown in Figure 13.5) resulted primarily from the stars' differing surface temperatures and not from differences in the abundance of elements in their photospheres. O stars are the hottest ones followed in succession by the B stars, A stars, F stars, G stars, K stars, and finally the coolest ones the M stars. Most stellar photospheres have a chemical composition very nearly the same as that of the Sun: an overwhelming amount of hydrogen, a little helium, and traces of other elements (Table 12.3).
Each of the seven spectral classes that form the spectral classification scheme is divided into 10 parts, known as spectral types, by assigning a number from 0 to 9 to the spectral class. The lower the number in the spectral type the hotter is the star's surface temperature. For example, A0 stars are hotter than A1, which in turn are hotter than A2, and so on. The spectral type is determined from a quantitative estimate of the strengths of certain absorption lines (some of which are labeled in Figure 13.5) in the star's spectrum. In this scheme, the Sun's spectral type is G2, 0.2 beyond G0 toward the next class, K. Table 13.5 describes the most distinguishing features of each spectral class, including the surface temperature and B-V color index for the hottest spectral type in each spectral class (that is, A0 in A, G0 in G, and so on). Often stars of spectral classes O, B, and A are referred to as early-type stars, whereas stars of classes G, K, and M are known as late-type stars. Early and late are also used to denote direction along the spectral sequence from O to M. For example, B0 stars are earlier stars than B5 stars, while B9 is a later spectral type than B5.
Spectral types are given in Table 13.4 for the 30 brightest stars, whereas those for the nearest known stars are given in Table 13.1. In these tables, the significance of the Roman numeral following the spectral type is discussed next. The point to remember here is that the spectral classification scheme is a grading of stars by their surface temperatures.
A second aspect to spectral classification was added to the Harvard scheme by astronomers at Yerkes Observatory in the early 1940s; it subclassified stars of similar surface temperatures into luminosity classes. The luminosity classification scheme is based on behavior more subtle than that characterizing temperature differences in stars' spectra. The luminosity classes are designated by Roman numerals:
In Figure 13.6, although the spectra of the A0 supergiant, the A0 giant, and the A0 main-sequence stars are much alike (and similarly for the B2 stars), differences are visible in the lines of the Balmer series. The differences in the Balmer lines are not so pronounced in cooler stars as they are in these hotter stars in the figure.
Since luminosity differences in stellar spectra are more subtle effects than temperature differences are, it is not always possible to determine a star's luminosity class with low-quality spectrograms even though its spectral type can be roughly estimated. Once a star's luminosity class is known, though, we can find its distance from its apparent and absolute magnitudes, that is, its distance modulus m-M.
The names of the luminosity classes are more than poetic. As we have already seen, two stars of the same spectral type (or temperature) must differ in radius if they are to have different luminosities. The more luminous the star, the larger will be its radius. Supergiants are the largest stars, the bright giants next largest, and so forth. The significance of the name main sequence is that approximately 90 percent of all stars are main-sequence stars.
In addition to stars that are brighter than main-sequence stars
but have the same spectral type, there are stars that are less
luminous than main-sequence stars. These are the subdwarfs and
the white dwarfs. We shall not have a great deal to say about
the subdwarfs, but we will about the white dwarfs, which are a
fascinating stellar species.
13.3.3. Chemical Composition Effects
As mentioned in Chapter 14 and above, the principal differences in stellar spectra are temperature effects and not chemical composition effects. However, that is not to say that astronomers have not found effects they believe to be the result of differences in chemical composition. Figure 13.7 illustrates what are some of the more pronounced spectral differences that are believed to be attributable to chemical composition differences. Most of the differences are in those elements heavier than hydrogen and helium, which are known to astronomers as the heavy elements.
Between 1911 and 1913, the Danish astronomer Ejnar Hertzsprung (1873-1967) and the American astronomer Henry Norris Russell (1877-1957) independently developed the diagram that is now called the Hertzsprung-Russell, or H-R, diagram. Plotting the spectral type (or, equivalently, the color index or surface temperature) for many stars on the horizontal axis against the absolute magnitude (or luminosity) on the vertical axis, they found that the resulting points were not scattered at random. Instead, the points lie in well-defined regions, as illustrated in Figure 13.8, which suggests a common relationship exists among the stars of each region.
The most conspicuous region of the H-R diagram is the sequence of stars running from extremely bright, hot stars in the upper left-hand corner to faint, cool stars in the lower right-hand corner. This sequence is called the main sequence, and it contains most of the stars that could be plotted on the diagram. The Sun, for example, is a G2 main-sequence star and lies in roughly the middle of the diagram among what are often referred to as the yellow dwarfs. Earlier we said that these stars formed one class in the luminosity classification scheme (luminosity class V). Clearly, main-sequence stars vary from extremely luminous O stars to very faint M dwarfs--a range of about a billion in luminosity. Nevertheless, because of their number and the common features of their internal structure, main-sequence stars are considered to belong to a single class.
The second most prominent region in the H-R diagram is the region labeled as red giants. They are luminous stars in spectral classes F, G, K, and M lying above the main sequence in a region that angles up toward the upper right-hand corner. Despite their being in the same luminosity class (III), the red giants vary by at least a factor of 100 in luminosity. On average, they are 100 times more luminous than the Sun, and they vary in surface temperature from 3000 to 7000 K.
Stars of luminosity classes I and II are called red supergiants if they lie on the cool side of the diagram in spectral classes G, K, and M and blue supergiants if they are early-type stars of classes O and B. The red supergiants and the blue supergiants can be hundreds of thousands of times more luminous than our Sun.
The last region of importance, which spans spectral classes B,
A, and F, contains faint stars lying below the main sequence;
these are called white dwarfs. (When we refer to a star as being
on or off the main sequence, we refer to its position in the H-R
diagram and not to its actual position in space.) White dwarfs
are typically a few thousandths of the luminosity of the Sun,
even though their outer layers are hotter than those of the Sun.
13.4.3. Radius and Mean Density on the H-R Diagram
Stars of similar spectral type or surface temperature can be vastly different in size (recall that luminosity is proportional to the radius squared and the fourth power of the surface temperature). You can see in Tables 13.1 and 13.4 how dwarfs, giants, and supergiants differ in size. These tables show (as you would expect) that because of its larger surface, a large star radiates more energy than does a small star of the same spectral type (temperature). Figure 13.9 is an H-R diagram for the bright stars listed in Table 13.4. On it are plotted lines along which all stars have the same radius. As is evident, the radii of stars increase from the lower left-hand corner toward the upper right-hand corner of the diagram.
Although there is no unique position on the H-R diagram for stars
of a given mass, except for main-sequence stars (as we shall discuss
in Section 14.3), the more massive stars are, very roughly, in
the upper part of the diagram. The range between the least and
the most massive stars is about 10,000, whereas radii vary by
almost 100,000. For typical white dwarf stars, whose mass and
radius are about 1 and 0.01 in units of the Sun's values, the
mean density is about 106 g/cm3. The mass and radius of typical
red giants are about 1 and 100 in solar units, so that their mean
density is about 10-6 g/cm3. Thus the mean density of the stars
roughly decreases in the same direction as the radius increases.
13.4.4. Bright Stars Versus Nearby Stars
The H-R diagram for the brightest stars in the sky in Figure 13.9 is striking in that the majority of the stars are not part of the main sequence. Of the 30 brightest stars, roughly 70 percent are stars above the main sequence, whereas only 30 percent are main-sequence stars. This again emphasizes that stars appear bright in the night sky because they are intrinsically bright and not because they are nearby.
In the H-R diagram for stars within 15 ly (Figure 13.10) from Table 13.1, it is striking that no giants or supergiants appear. The Sun outranks all but three stars in size and luminosity. This sample also contains three white dwarfs, but the majority are red dwarfs from the lower end of the main sequence.
For the nearby stars in Table 13.1 there are approximately 50 stars in a sphere of 15-ly radius, or about 0.004 star per cubic light year (one star in almost 300 ly3). Figure 13.11 displays the number of main-sequence stars from Table 13.1 by spectral type and approximate mass. As is evident, there are many more faint, small-mass red dwarfs than anything else. For every
Obviously, the physical process that forms stars (to be discussed in Chapter 20) appears to favor the formation of faint, small-mass stars.
Astronomers have identified several tens of thousands of stars that vary in brightness and have cataloged them into approximately 30 types. To analyze the variability, astronomers usually obtain both photometric measurements (such as UBV) of the light variations and spectrograms for velocity variations from Doppler shifts. From the photometric observations they can plot the observed changes in apparent magnitude over a specific interval of time, which is called a light curve; a plot of the Doppler line shift over time is known as a velocity curve (Figure 13.12).
Many of the variable stars belong to the group of pulsating variables, which we believe owe their variability to a regular expansion and contraction (Table 13.6). For example, most supergiant stars are varying in brightness to some extent because of pulsation. Within a range of surface temperature from about 7000 to 3500 K, there are no known supergiants that are not variables.
A simple theory for stellar pulsation predicts that the period
of oscillation is inversely proportional to the square root of
a star's mean density. Hence a star whose mean density is four
times greater than another will pulsate with half the period.
Thus stars possessing the lowest mean density have the longest
periods, and these stars are the largest and brightest ones--the
red supergiants. This dependence of period on mean density is
borne out by observational data for Cepheids.
13.5.2. Cepheid Variables
The classical Cepheids form an important subdivision of the pulsating variables. Named after the first star of their kind discovered, Delta Cephei, whose slow variation in brightness was detected some 200 years ago, Cepheids have the following characteristics: They are supergiants of great brilliance and spectral types late F to early K, their brightness varies periodically over about one magnitude during an interval of 2 to 40 days, and their spectra show Doppler shifts synchronized with the periodic changes in brightness (Figure 13.12).
The classical Cepheids are the most luminous of all Cepheids. Sparsely sprinkled within the Galaxy's disk, they are a very small segment of the population of the spiral arms (Chapter 25). Another group of Cepheids, somewhat less luminous than classical Cepheids with periods generally between 10 and 30 days, is the W Virginis stars (named after their prototype). They are found in the population of stars in the halo portion of our Galaxy and in several globular star clusters (Section 14.4). Still another type of pulsating variable present by the thousands in the Galaxy's central and halo regions and in globular clusters constitutes a third class of Cepheids. They are known as cluster variables or, more often, as RR Lyrae variables (also named for their prototype). They are bluish-white giant stars fainter than the other groups of Cepheids but up to a hundred times brighter than the Sun. Their periods of brightness variation average about a half day.
The more luminous the Cepheid, the longer is the period of variation. This extremely important correlation between period and intrinsic brightness is known as the period-luminosity relation. It is represented graphically by a plot of the median absolute magnitude against period (Figure 13.13). (The median magnitude is halfway between maximum and minimum magnitudes.) Because there are no Cepheids close enough to have their parallax measured, the period-luminosity relation can be used to determine distances for individual Cepheids assuming that absolute magnitudes are known for various types of Cepheids. (This is another application of the inverse-square law of light.) Inasmuch as a Cepheid is relatively easy to recognize because of its brightness variation, a knowledge of its distance becomes an important tool. From a Cepheid's distance, the distances of other astronomical objects that are associated with it can be estimated.
A number of red giants are observed to have cyclic brightness changes of several magnitudes with a period of about a year. Known as long-period variables, these stars form another large group of pulsating variables. Variations in the spectra and velocity curves of long-period variables are complex, with a poorly understood cause that is apparently not identical with what makes Cepheids pulsate.
Hundreds of other variable stars covering a wide range of luminosities
and colors go through such irregular and often baffling changes
that the causes of their behavior remain obscure. More will be
said about the reasons for a star's varying brightness in Chapter
21, but the important points to remember here are that a small
percentage of all stars vary in brightness and that their variability
attracts attention to them and to their location in the Galaxy.
13.5.4. Explosive Variables
Not all stars whose brightness varies at one time or other do
so because they are pulsating. Another class of variable stars
is made up of those which undergo sudden and usually rapid changes
in brightness. Some stars do so only once as far as we know,
but others have a recurrent pattern of sudden outbursts in brightness.
The reason for the sudden change has been traced to explosive
ejections of matter; consequently, this class is called explosive
variables. Included in this class are such objects as novae and
supernovae, which will be discussed in Chapter 16.
13.5.5. Pulsating Variable Stars on the H-R Diagram
Although the pulsating stars that we considered, the Cepheids and the red variables, are intrinsically bright stars, there are less luminous stars that also appear to pulsate. If in the H-R diagram of Figure 13.14 we mark the boundaries of regions in which the Cepheids and red variable stars are located, we find that Cepheids of all types occupy a strip of the diagram that astronomers called the Cepheid instability strip. It lies between the upper end of the main sequence, containing bright blue stars, and the region where red giants and supergiants are located. However, long-period and irregular variable stars are red giants and supergiants, and they lie to the cool side of the Cepheid instability strip. Why this might be the case will be discussed in Section 22.1.
Having covered the general features and characteristics of individual stars, we move on in the next three chapters to discuss first the motions of stars and groups of stars (binary star systems and clusters of stars), second the outer layers of stars, and third the interstellar medium which surrounds stars and groups of stars in the disk of the Galaxy.